Risk Minimization for Insurance Products via                                  F-Doubly Stochastic Markov Chains

Francesca Biagini; Andreas Groll; Jan Widenmann

doi:10.3390/risks4030023

Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains

Biagini, Francesca;Groll, Andreas;Widenmann, Jan 2016-07-07 00:00:00 risks Article Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains 1,2, 3 4 Francesca Biagini *, Andreas Groll and Jan Widenmann Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, Germany Department of Mathematics, University of Oslo, Boks 1072 Blindern, Norway Department of Statistics, University of Munich, Akademiestr.1, 80799 Munich, Germany; andreas.groll@stat.uni-muenchen.de BMW Financial Services, BMW Bank GmbH, 80787 Munich, Germany; jan-maximilian.widenmann@bmw.de * Correspondence: francesca.biagini@math.lmu.de; Tel.: +49-89-2180-4492 Academic Editor: Mogens Steffensen Received: 21 March 2016; Accepted: 27 June 2016; Published: 7 July 2016 Abstract: We study risk-minimization for a large class of insurance contracts. Given that the individual progress in time of visiting an insurance policy’s states follows an F-doubly stochastic Markov chain, we describe different state-dependent types of insurance beneﬁts. These cover single payments at maturity, annuity-type payments and payments at the time of a transition. Based on the intensity of the F-doubly stochastic Markov chain, we provide the Galtchouk-Kunita-Watanabe decomposition for a general insurance contract and specify risk-minimizing strategies in a Brownian ﬁnancial market setting. The results are further illustrated explicitly within an afﬁne structure for the intensity. Keywords: insurance liabilities; doubly stochastic Markov chains; risk minimization MSC: 60J27; 62P05; 91G99 JEL: C02 1. Introduction The management of an insurance portfolio’s risk is one of the core challenges in actuarial science. While the classic form of risk mitigation is based on reinsurance contracts, in some cases it is also possible to hedge claim payments by appropriately trading in different assets. This particularly applies if the assets are correlated to the insurance contract’s beneﬁts or their (conditional) probability of occurrence. Practical examples in this direction are unit-linked life insurance products, where beneﬁts depend on the performance of the assets, or unemployment insurance products, where the occurrence of a claim payment may depend to some extend on economic and ﬁnancial conditions of the markets. Moreover, there is an ongoing discussion about the introduction of so called longevity bonds which would establish the possibility for life insurance companies and pension funds to hedge parts of their longevity risk, see [1,2] or [3]. Due to their unsystematic risk part, most insurance claims are not hedgeable completely through a self-ﬁnancing trading strategy which particularly means that a hybrid market, consisting among others of ﬁnancial and insurance markets, is incomplete. A reasonable method for optimally choosing an investment strategy is then important to cover at least parts of the risk. In the present paper we choose the risk-minimization approach and determine hedging strategies in the sense of this criterion for insurance contracts in a very general setting. This quadratic hedging approach bases on the results in [4] for European type payments and to [5] for payment Risks 2016, 4, 23; doi:10.3390/risks4030023 www.mdpi.com/journal/risks Risks 2016, 4, 23 2 of 26 processes. In most cases, the risk-minimizing strategies can be derived from the well known Galtchouk-Kunita-Watanabe (GKW-) decomposition, see [6] or [7]. Similar to the works of [5,8] or [9–11], we describe an insured person’s progress of sojourning different states of an insurance policy as a right continuous stochastic process with ﬁnite state space K = f1, ..., Ng, 1 being a.s. the initial state. More speciﬁcally, we adopt the class of F-doubly stochastic Markov chains as introduced in [12], see Appendix A and the comments therein. This family of processes has several properties which make them very suitable for applications in credit risk and insurance market modeling. Being a sub-class of F-conditional Markov chains, they extend the classic notion of Markov chains by including a reference ﬁltration F which in our case represents additional market information. In this way we are able to take in consideration the inﬂuence of external risk factors and economic and ﬁnancial conditions on transition probabilities of an insured person’s progress. In particular, F-doubly stochastic Markov chains behave like time inhomogeneous Markov chains, if we know all the information concerning the underlying risk factors. This corresponds to the intuition that the transition probabilities would be completely speciﬁed, if we would dispose of full knowledge on the underlying economic and ﬁnancial situation. Another important feature is that, if we specify the information as given by the ﬁltration X X G := F _ F, where F is the natural ﬁltration of the F-doubly stochastic Markov chain X, then we have that predictable representation theorems and the so-called hypothesis (H) , or immersion property, hold. These properties play a fundamental role in order to compute the optimal strategy for insurance contracts according to the risk-minimization method. Furthermore, F-doubly stochastic Markov chains may admit matrix-valued stochastic intensity processes. This allows to investigate more ﬂexible models compared to the results e.g., in Møller [5] where a (classical) Markov chain with deterministic intensity matrix function is considered. One further advantage is that F-doubly stochastic Markov chains with intensity are fully characterized by some martingale properties, which can be used for the estimation of the underlying intensity processes, see Biagini et al. [13]. Well known examples of F-doubly stochastic Markov chains are reduced form or intensity based models in the case that hypothesis (H) is satisﬁed. Here, the state space consists of two states with the second state being absorbing such that there can only occur one transition in time. There exist many works on quadratic hedging for these models particularly in the context of credit risk or life insurance theory, see e.g., [1,2,14–19] or [20]. In particular, the present paper extends these works to a multi-state framework where several subsequent transitions, driven by F-adapted stochastic intensity processes, are considered. This general setting allows to investigate a larger class of insurance contracts, e.g., income protection insurance contracts with the states “healthy”, “sick” and “deceased”, and to include the inﬂuence of market conditions and external risk factors on the insured person’s progress. Given an F-doubly stochastic Markov chain, we propose a general insurance contract, deﬁned by three different types of insurance beneﬁts: state-dependent payments at maturity, state-dependent annuity-type payments, and (transition-dependent) payments at the time of a transition from one state to another. This deﬁnition covers a large set of currently adopted insurance policies. In particular, we illustrate the deﬁnitions for pure endowment, term insurance, general life annuity and payment protection insurance contracts. Similar to the results in [21], who applied F-doubly stochastic Markov chains in the context of hedging rating-sensitive ﬁnancial claims, we obtain the GKW-decomposition for the payment process of general insurance contracts with respect to a particular F-martingale. In this context, we generalize and complement the proofs in [21] in order to adapt the results for the risk-minimization approach which is not investigated there. Given that the reference information F is generated by an N-dimensional Brownian motion W, we then introduce a ﬁnancial market model, driven by W. In this setting we infer risk-minimizing For the deﬁnition and further comments on hypothesis (H), see Proposition A.4 and the text below. Risks 2016, 4, 23 3 of 26 hedging strategies for insurance contracts with deterministic payment structure with respect to the assets on the ﬁnancial market. Similarly to the work in [2] we then assume a general afﬁne structure for the intensity of the underlying F-doubly stochastic Markov chain and obtain explicit formulas for the strategies and their residual risk processes. We apply these results in the speciﬁc example of an income protection insurance, where we assume that the intensities follow a (multi-dimensional) Ornstein-Uhlenbeck process. We discuss the resulting expected cumulative payment, which may be considered as a fair premium in the interpretation of [22,23], as a function of the time horizon, the payment amounts and the underlying interest rates. The paper is organized as follows. In Section 2 we introduce the notion of general insurance contracts and discuss several examples. In Section 3 we prove our main results for the risk minimization of this kind of contracts in full generality. The risk minimizing strategies are then further illustrated within a general afﬁne speciﬁcation for the intensities and in a numerical example in an Ornstein-Uhlenbeck framework. We conclude the paper with Appendixs A and B, where we summarize important results and concepts of risk-minimization and F-doubly stochastic Markov chains for the reader ’s convenience. 2. General Insurance Contracts We now introduce the notion of general insurance contracts and provide some well known examples of actuarial practice. In the same notation as in Appendix A, let (W,G ,G,P) be a ﬁltered probability space with G = F _ F for some F-doubly stochastic Markov chain X with state spaceK = f1, ..., Ng. We assume P(X = 1) = 1. The following deﬁnition of general insurance contracts is based on the deﬁnitions for payment processes on rating sensitive claims as given e.g., in [21] or [20]. The deﬁnition also covers the concepts of insurance contracts as given in [5] or [9]. Deﬁnition 1. A general insurance contract is given by the quadruple (X; A; Y; Z), where X = (X ) t2[0,T] 1 N is an F-doubly stochastic Markov chain, A = ( A , ..., A ) is an F-adapted, N-dimensional process of t t t2[0,T] 1 N ﬁnite variation, Y = (Y , ..., Y ) is an F -measurable, N-dimensional random vector, and Z = (Z ) T t2[0,T] h i j,k with Z = Z is an F-adapted, N N-dimensional process with zeros on the diagonal. j,k2K The different elements of a general insurance product’s quadruple are interpreted as follows. The process X is the insured person’s progress in time of sojourning in the states j 2 K, considered by the insurance policy. The N-dimensional process A characterizes the cumulative state-dependent payment streams which are continuously paid up to maturity. For example, one can take A = C P , t 2 [0, T] t t t 1 N | with C = (C , ...C ) representing the cumulative state-dependent claim payments (e.g., annuities) t t 1 N and P = (P , ..., P ) the cumulative state-dependent insurance premiums up to maturity. Both t t processes, P and C are then taken to be F-adapted, càdlàg and increasing. The vector Y characterizes state-dependent “extra” claim payments at maturity T and the process Z the “immediate” claim payments at the transition times from one state to another. For every general insurance contract (X; A; Y; Z) the cumulative payment process D = (D ) t2[0,T] is given by R R | | D : = Y H 1 + H dA + (Z H ) dH t T s s s s s ft=Tg [0,t] ]0,t] R R (1) j j j j,k jk N N = Y H 1 + H d A + Z dN å å s s s s j=1 ft=Tg k=1 T [0,t] ]0,t] k6=j j jk with H , j 2 K, as deﬁned in Equation (A7) and counting processes N from Equation (A8). Note that t t D is of ﬁnite variation. We now provide some well known examples of insurance contracts. Example 1. A pure endowment is an insurance contract which guarantees to the insured person some ﬁxed payment if she is alive at maturity. For the sake of simplicity, we only consider the payment to be equal to 1. Risks 2016, 4, 23 4 of 26 We set K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. A pure endowment | | contract is then given as the quadruple (X; 0; (1, 0) ; 0) or (X;P; (0, 1) ; 0) if premium payments are considered, respectively. Example 2. A term insurance is an insurance contract which guarantees the heirs of an insured person some ﬁxed payment at the time of decease. For the sake of simplicity, we only consider the payment to be equal to 1. Again, we set K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. Then a term insurance contract is given as the quadruple (X; 0; 0; Z) or (X;P; 0; Z) if premium payments are considered, 0 1 respectively, with Z := . 0 0 Example 3. A general annuity as deﬁned in [2] is an insurance contract which guarantees the insured person an F-progressively measurable, non-negative continuous rate payment (c ) as long as she is alive. The state t2[0,T] space is again K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. Then a general | | R R annuity contract is given as the quadruple (X; c ds, 0 ; 0; 0) or (X; c ds, 0 P; 0; 0) s s ]0,t] ]0,t] t2[0,T] t2[0,T] if premium payments are considered, respectively. Example 4. A payment protection insurance (PPI) is an insurance contract which is usually offered as an add-on product to some payment obligations, e.g., a loan. In the case of an insured event, the insurance company takes over the respective instalments of the payment obligation for the insured person or her heirs. Generally, the insured events are “disability”, “unemployment” and “decease”. Hence, the state space for PPI products is given as K = f1, 2, 3, 4g with “2” being the state “disabled”, “3” the state “unemployed”, “4” the absorbing state “deceased” and “1” the state where no insured event is present. 2 3 4 | Then a PPI contract is given as the quadruple (X; (0, C , C , C ) ; (0, Y, Y, Y) ; 0) or t t t t2[0,T] 2 3 4 | (X; (0, C , C , C ) P; (0, Y, Y, Y) ; 0) if premium payments are considered, respectively. t t t t2[0,T] As the underlying payment obligation usually stipulates ﬁxed instalments c , ..., c at some given payment 1 K i i dates 0 < T < ... < T = T, the processes C , i = 2, 3, 4, are generally given as C = c 1 . 1 K j t t j=1 fT tg Moreover, some payment obligations also contain a so-called balloon rate B at the end of the contract, which has to be paid on top of the usual instalment. If there exists a balloon rate and it is insured, then we set Y = B, if there exists no balloon rate or it is not insured, then we set Y = 0. Remark 1. (1) The extra claim payment could also be included in the continuous claim payments. For the reader’s convenience, however, we explicitly separate continuous and extra claim payments. (2) The main concepts of premium payment are - a single premium P, where the complete price for the insurance contract is paid at its beginning. In this case, the vector P would be given as P = P1 , P1 , ..., P1 . ft0g ft0g ft0g t2[0,T] - periodically paid premiums. Here, the insurance price is paid according to periodically paid premiums p at a priori speciﬁed dates 0 = T < T < ... < T T. Moreover, some insurance policies 0 L i 1 consider premium freedoms which allow the insured person to intermit premium payments while sojourning (some) insured states. In this case, we have for each vector entry P , i 2 f1, ..., Ng, of P å p 1 j fT tg i j=1 P = depending on whether state i is guarantees premium freedom or not. Risks 2016, 4, 23 5 of 26 3. Risk-Minimization for General Insurance Contracts Aim of this paper is to provide the risk minimizing strategy for a general insurance contract by applying the approach presented in Appendix B. Risk minimization provides hedging strategies which perfectly replicate the claim. Since the market is incomplete, these strategies may not be self-ﬁnancing and a readjustment (or cost) is needed to achieve perfect replication. According to this method we choose then the optimal strategy, i.e., the strategy with minimal cost. For this sake, we ﬁrst consider a general setting and then focus on a deterministic payment structure and an underlying market which is driven by some N-dimensional Brownian motion. These results are then further speciﬁed within a general afﬁne setting for the different entries of the matrix-valued intensity. 3.1. Martingale Decomposition for Payment Processes of General Insurance Claims We consider the payment process D in Equation (1). Let S denote the market’s discounting factor, which will be further speciﬁed in Section 3.2. If djDj < ¥, we get by Equation (B1) that the 0 u ]0,T] b b discounted cumulative payment stream D = (D ) is given as t2[0,T] | R R Y H | | 1 1 T | D = 1 + H dA + (Z H ) dH t 0 0 s s 0 s s s ft=Tg [0,t] ]0,t] S S S s s T ! (2) j R R Y H j j j,k jk N 1 N 1 = 1 + H d A + Z dN å å 0 0 s s 0 s s j=1 ft=Tg k=1 [0,t] ]0,t] S S S s s k6=j We further assume the underlying F-doubly stochastic Markov chain X to admit an intensity j,k Y Y Y = [y ] , as introduced in Deﬁnition A.5. j,k2K t2[0,T] Assumption 2. For every general insurance contract (X; A; Y; Z), let 2 3 4 5 E < ¥ , j 2 K (3) " # sup E d A < ¥ , j 2 K (4) [0,s] S s2[0,T] u " # Z Z E d A jy jdu < ¥ , j 2 K (5) v X ,j ]0,T] [0,u] 2 3 Z j,k 4 5 E y (u)du < ¥ , j, k 2 K, j 6= k (6) j,k ]0,T] 2 3 Z j,k 4 5 E y (u)du < ¥ , j, k 2 K, j 6= k (7) j,k ]0,T] S where H , t 2 [0, T] is deﬁned in Equation (A7). Note that Equations (3), (4), (6) and (7) ensure that the discounted payment stream D generated by the general insurance contract (X; A; Y; Z) is square integrable. We remark that the following Lemma is given similarly in [21] (Theorem 16.38) under the assumption the local martingale M, deﬁned in Equation (A9), is square integrable and the processes A and Z are bounded. Here, we generalize their proof to the case where A and Z satisfy the conditions of Assumption 2. Risks 2016, 4, 23 6 of 26 1 N For notational convenience, we introduce the process G = (G ) = G , ..., G with t2[0,T] t t j | j,k G := Z Y Y Y = Z y (t) , j 2 K, t 2 [0, T] (8) t å j,k t t t j,j k=1 k6=j Lemma 3. Let (X; A; Y; Z) be a general insurance contract, satisfying Assumption 2, then " # h i j j,k R R Y H j j jk N 1 N T u b b E D D G = E + H d A + dN G t t å å t T 0 0 u u 0 u j=1 k=1 ]t,T] ]t,T] S S S u u k6=j " # j j,k R R Y p (t,T) i,j Z N i N 1 N (9) = H E + p (t, u)d A + p (t, u)y (u)du F å å å t 0 0 i,j u 0 i,j j,k i=1 t j=1 ]t,T] k=1 ]t,T] S S S u u k6=j R R P(t,T)Y P(t,u) P(t,u) = E + dA + G du F H u u t t 0 0 0 ]t,T] ]t,T] S S S u u where the conditional transition probability process P = P(s, t) = p (s, t) , 0 s t T is deﬁned i,j i,j2K in Deﬁnition A.1. Proof. We proove the theorem by investigating the different conditional expectations separately. First note that because Y is taken to be F -measurable and S to be F-adapted, by Equation (A3) we get for every j 2 K " # " # " # j j N h i j N Y H Y Y T i i E G = E H E H jG G = E H p (t, T) G t t t t å t å t i,j 0 0 0 S S S T T i=1 T i=1 " # N j = H E p (t, T) F å t i,j i=1 T where G := F _F . t T Next, by Equation (A10) of Theorem A.7, we get for j, k 2 K, j 6= k that Z j,k Z j,k Z j,k Z Z Z u j,k u j,k u j dN = d M + H y (u)du u u u j,k 0 0 0 ]t,T] S ]t,T] S ]t,T] S u u u jk Note that because of Equations (A15) and (6), the integral-process with respect to M is a square integrable G-martingale. Hence, for every j, k 2 K, j 6= k, we have " # " # Z jk Z jk Z Z jk j u u E dN G = E H y (u)du G u t u j,k t 0 0 ]t,T] S ]t,T] S u u " # Z jk u j = E H y (u) G du u t j,k ]t,T] S " " # # Z jk = E E H y (u) G G du t t u j,k ]t,T] S " ! # Z jk = E y (u) H p (t, u) G du j,k å i,j t ]t,T] S i=1 " # jk = H E p (t, u)y (u) F du å i,j t t j,k ]t,T] S i=1 " # Z jk i u = H E p (t, u)y (u)du F å t i,j j,k ]t,T] S i=1 Risks 2016, 4, 23 7 of 26 by the conditional version of Fubini’s theorem, the deﬁnition of F-doubly stochastic Markov chains and hypothesis (H). j j Finally, for every j 2 K and for ﬁxed t 2 [0, T] we deﬁne A := d A , u 2 [t, T]. By u 0 v ]t,u] Proposition A.9 we get Z Z j j j j E H d A jG = E H d A jG t t u u u u ]t,T] ]t,T] j j j j j j e e e = E A H A H A d H jG T T t t u ]t,T] j j j j e e = E A H A d H jG = I I u t 1 2 T T ]t,T] with j j j j e e I := E[ A H jG ], I := E A d H jG 1 t 2 u t T T ]t,T] Since A is F -measurable, it follows by the hypothesis (H) that T T h h i i K h i j j j e e I = E A E H jG jG = H E A p (t, v)jF t t t 1 å t i,j T T T i=1 Again by the conditional version of Fubini’s theorem, hypothesis (H) and with the Kolmogorov forward Equation (A6) it follows that Z Z Z j j j j j e e e I = E A d H G = E A d M + A y (u)du G 2 u t u X ,j t u u u u ]t,T] ]t,T] ]t,T] " # " # Z Z K K j j k k e e = E A H y (u)du G = E A H y (u) G du t t u å u k,j u å u k,j ]t,T] ]t,T] k=1 k=1 " # h i e e = E A E H j G y (u) G du å t k,j t u u ]t,T] k=1 " ! # K K = H E A p (t, u)y (u) F du å å i,k k,j t t u ]t,T] i=1 k=1 " ! # K K = H E A p (t, u)y (u) du F å å i,k k,j t t u ]t,T] i=1 k=1 = H E A d p (t, u) F i,j t å t u ]t,T] i=1 Hence, by integration by parts and since p(t,) is continuous, we get e e I I = H E A p (t, T) A d p (t, u)jF 1 2 å T i,j i,j t t u ]t,T] i=1 e e = H E A p (t, t) + p (t, u)d A jF å t i,j i,j u t ]t,T] i=1 = H E p (t, u)d A jF å i,j u t ]t,T] i=1 This completes the proof. Risks 2016, 4, 23 8 of 26 Now we are ready to provide the Galtchouk-Kunita-Watanabe decomposition for payment processes of general insurance contracts. Theorem 4. Let (X, A, Y, Z) be a general insurance contract, satisfying Assumption 2, with discounted payment process D, deﬁned in Equation (2). Then the GKW-decomposition of the square-integrable discounted h i D D D b b b b value process U = (U ) with U = E D jG is given as T t t t2[0,T] t Z Z D D | b b U = U + aaa dm + bbb dM (10) u u u t 0 u ]0,t] ]0,t] where M is given by Equation (A9), m = (m ) is a square-integrable F-martingale, given by t2[0,T] " # Z Z P(0, T)Y P(0, u) P(0, u) m := E + dA + G du F (11) t u u t 0 0 0 S [0,T] S ]0,T] S u u and aaa, bbb are G-predictable R -valued processes deﬁned by F(t, T) + Z H aaa = L = Q (0, t)H , bbb = (12) t t t 1 N with H = ( H , ..., H ), t 2 [0, T], deﬁned by Equation (A7), Q(0, t), t 2 [0, T], deﬁned in Equation (A11), t t F(t, T), t 2 [0, T], deﬁned by " # Z Z P(t, T)Y P(t, u) P(t, u) F(t, T) := S E + dA + G du F (13) u u t 0 0 0 S ]t,T] S ]t,T] S u u b b and U = E[D ] = m H . T 0 0 0 Proof. The statement and the proof of this theorem can be found in [21] (Theorem 16.62). The authors there, however, prove Decomposition 10 only for t 2 [0, T). Because the integrals on the r.h.s. are not all continuous, it is a priori not clear if the decomposition also holds for U . Here we refer to [24] (Theorem 4.2.3) for an extension of the proof of [21] to the case t = T. 3.2. Risk Minimization for General Insurance Contracts with Deterministic Payment Structure In this section we focus on a more speciﬁc setting, where we specify the underlying ﬁnancial market and derive risk-minimizing hedging strategies for insurance contracts with deterministic payment structures. We start by specifying the underlying market. First of all, we assume the reference ﬁltration F = F to be the augmented ﬁltration, generated by some N-dimensional Brownian motion W. For computational reasons, particularly in the afﬁne setting of the next section, we set the dimension N of the Brownian motion equal to the number of states under consideration. 0 d Consider then a ﬁnancial market consisting of (d + 1) traded assets S = (S , ..., S ) , assumed t t t2[0,T] 1 d d b b b to be F-adapted, non-negative stochastic processes. Let S = (S , ..., S ) denote the R -valued t t t2[0,T] 1 d 0 i i 0 stochastic process of the primary assets S , ..., S , discounted with the asset S , i.e., S = S /S , t t t 0 0 i = 1, ..., d. Here S is taken to be continuous with S > 0 for all t 2 [0, T] and shall generally represent the value of a self-ﬁnancing portfolio on the primary assets. In the sequel, we assume S to be a local (F,P)-martingale, which particularly implies that the market model is arbitrage-free. Remark 2. The requirement that S is a local martingale may appear restrictive. However it is always satisﬁed if we choose S to be the numéraire portfolio deﬁned in [25], since we assume the underlying ﬁnancial market to contain only continuous asset price processes. Risks 2016, 4, 23 9 of 26 We could also start with a general situation where the discounted asset price processes are given by semimartingales. In this case one has to assume some technical conditions to guarantee the existence of the optimal strategy, see [26,27]. Here we prefer to avoid technical complications since our aim is to compute explicitly the risk-minimizing strategy when it exists. By the representation theorem with respect to Brownian motion it follows that there exists a N dN measurable map ssse : [0, T] R ! R , such that b b S = S + sss(s, S )dW t s s ]0,t] se Assumption 5. We assume that ss(t, S ) is a.s. left-invertible, i.e., that for almost every (w, t) 2 W [0, T] there exists an F -adapted N d-valued matrix GGG (w) such that GGG ssse(t, S ) = I . This particularly t t t N implies N d. From now on we focus on discount factors and insurance contracts with deterministic payment structure. Assumption 6. (1) Y is a deterministic vector in R . (2) The payment A = (A ) is of the form A = nnn(s)ds for some bounded deterministic function t t t2[0,T] nn : [0, T] ! R . NN (3) Z : [0, T] ! R is a bounded deterministic matrix-valued function. (4) S : [0, T] ! R is a deterministic continuous function. j,k (5) For every j, k 2 K, j 6= k, C := sup E y < ¥. u2[0,T] Assumption 6 particularly implies that the integrability conditions of Assumption 2 hold. Note also that the insurance contracts, given in Examples 1, 2 and 4 all satisfy (1), (2) and (3) of Assumption 6. The assumption on S being deterministic is applied very frequently in the literature, 0 rt e.g if P is assumed to be some risk-neutral probability measure and S = e for some constant r > 0. Remark 3. Here we assume constant interest rates for the sake of simplicity, since the focus of this paper is primarily to evaluate the role of a multi-state progression of the insured person on the risk-minimizing strategy. The following computations can be easily extended to the case of stochastic interest rates if S is assumed to be independent of X. In more general models, the investigation of dependency structures will become inevitable. This goes beyond the scope of the paper and is left to further research. Due to the representation theorem with respect to Brownian motion, for every u 2 [0, T] and i,j every i, j 2 K, there exists some xxx (u,) 2 L (W) such that i,j E p (0, u)jF = E p (0, u) + 1 (s)xxx (u, s)dW (14) t s i,j i,j ]0,u] ]0,t] Similarly, because of Assumption 6 (5), for every u 2 [0, T] and every i, j, k 2 K, j 6= k, there exists i,j,k 2 some qqq (u,) 2 L (W) such that h i h i j,k j,k i,j,k E p (0, u)y jF = E p (0, u)y + 1 (s)qq (u, s)dW (15) i,j u t i,j u s ]0,u] ]0,t] Risks 2016, 4, 23 10 of 26 Theorem 7. Given Assumptions 5 and 6, the unique risk-minimizing hedging strategy xxx = (xxx ) , t t2[0,T] characterized in Theorem B.5 for a general insurance claim (X; A; Y; Z), satisfying Assumption 6, is given as Z Z Z j,k N N N Y 1 Z j u i T i,j i,j i,j,k xxx = L xxx (T, t) + xxx (u, t)n du + qqq (u, t)du GGG (16) å å u å t t t 0 0 0 ]0,t] S [t,T] S [t,T] S u u i=1 j=1 T k=1 k6=j 0 D b b b x = U D xxx S (17) t t t t t D D b b where G is the left-inverse of the volatility matrix es(t, S ) and U = (U ) is the discounted value t t t t2[0,T] process of the cumulative payment process D. Proof. Because of Assumption 6, the i-th component m of the martingale m in Equation (11) is given as N h i h i i T m = E p (0, T)jF + E p (0, u)jF n du t t t å i,j i,j u 0 0 S [0,T] S j=1 T Z j,k N h i u j,k + E p (0, u)y jF du å i,j u ]0,T] S k=1 k6=j j Z Z j,k h i h i h i N N Y 1 Z j j,k = E p (0, T) + E p (0, u) n du + E p (0, u)y du å i,j i,j u å i,j u 0 0 0 S [0,T] S ]0,T] S u u j=1 T k=1 k6=j Z Z Z T i,j i,j x x + xx (T, s)dW + 1 (s)xx (u, s)dW n du s s u å ]0,u] 0 0 S ]0,t] [0,T] S ]0,t] j=1 T Z j,k Z i,j,k + 1 (s)qq (u, s)dW du å s ]0,u] ]0,T] S ]0,t] k=1 k6=j By Assumption 6, Fubini’s theorem and the Itô isometry it then follows for every i, j 2 K that " # Z Z Z Z i,j i,j 2 2 E 1 (s)kxxx (u, s)kn duds K E kxxx (u, s)k ds du ]0,u] u ]0,T] ]0,T] S ]0,T] ]0,T] " # Z Z i,j = K E xx (u, s)dW du ]0,T] ]0,T] h i = K E E p (0, u)jF E p (0, u) du 1 i,j i,j ]0,T] K T < ¥ for some constant K > 0. Due to Assumption 6 (5), we similarly have for every i, j, k 2 K, j 6= k that 2 3 Z Z j,k Z Z i,j,k 2 i,j,k 2 4 5 E 1 (s)kqqq (u, s)k duds K E kqqq (u, s)k ds du ]0,u] 2 ]0,T] ]0,T] S ]0,T] ]0,T] " # Z Z 2 i,j,k = K E qqq (u, s)dW du ]0,T] ]0,T] h i h i j,k = K E E p (0, u)y jF E p (0, u) du i,j u T i,j ]0,T] h i j,k 2 2 K E (y ) du ]0,T] K CT < ¥ 2 Risks 2016, 4, 23 11 of 26 for some constant K > 0. Therefore, we can apply the stochastic version of Fubini’s theorem, see [28], and obtain Z Z j,k N N h i Y 1 Z j j,k i T m = E p (0, T) + E p (0, u) n du + E p (0, u)y du t å i,j i,j u å i,j u 0 0 0 S [0,T] S ]0,T] S u u j=1 T k=1 k6=j Z j Z Z j,k N N 1 Z i,j i,j u T i,j,k x x q + xx (T, s) + xx (u, s)n du + qq (u, s)du dW å u å 0 0 0 S S S ]0,t] [s,T] [s,T] u u j=1 T k=1 k6=j This ﬁnally implies Z Z j,k N N Y 1 Z i T i,j i,j i,j,k dm = xxx (T, t) + xxx (u, t)n du + qqq (u, t)du GGG dS t å u å t t 0 0 0 S [t,T] S [t,T] S u u j=1 T k=1 k6=j The result then follows immediately with Theorem A.7 and the results of Theorem B.5. 3.3. Risk Minimization for General Insurance Contracts with Deterministic Payment Structure under an Afﬁne Speciﬁcation for the Intensities In the same setting as in Section 3.2, we now specify the risk-minimizing hedging strategies, computed in Theorem 7 within a general afﬁne setting for the intensities. In addition to Assumption 2 we also consider Assumption 8. (1) For every 0 t u T and every j, k 2 K, j 6= k, we assume the entries p (t, u) of the transition j,k matrix P(t, u) are of the form j,k y dv p (t, u) = 1 e (18) j,k j,k where y are the respective entries of the intensity matrix YY. j,k (2) For every u 2 [0, T] and every j, k 2 K, j 6= k, y is of the form j,k j,k | j,k y = (b ) mmm + c (19) j,k N j,k N where b 2 R , c 2 R and mmm = (mmm ) an R -valued afﬁne process as speciﬁed e.g., in [29] t t2[0,T] (Section 3 and Appendix A) or [30]. Here, mmm is a Markov process with respect to F , given as the strong solution to the SDE dmmm = ddd(t,mmm )dt + sss(t,mmm )dW (20) t t t where for t 2 [0, T], x 2 R and i, j 2 f1, ..., Ng 0 1 dd(t, x) = d (t) + (d (t)) x (21) | 0 1 [sss(t, x)sss(t, x) ] = [V (t)] + (V (t)) x (22) i,j i,j i,j 0 1 0 1 N NN NN NNN with coefﬁcient functions d , d , V and V , taking values in R ,R ,R and R , respectively. (3) The process mmm is such that C := sup E[kmm k ] < ¥ (23) u2[0,T] Risks 2016, 4, 23 12 of 26 This particularly implies that for every j, k 2 K, j 6= k, we have j,k sup E[(y ) ] < ¥ (24) u2[0,T] With these assumptions and under some technical conditions, presented in [30], we obtain for every 0 t u T and every i, j 2 K with i 6= j that i,j u i,j i,j y dv a (t)+(bbb (t)) mmm v u u t t (25) E p (t, u)jF = E[1 e jF ] = 1 e i,j t t u i,j N N y dv E [ p (t, u)jF ] = E[1 p (t, u)jF ] = 1 E[1 e jF ] å å i,i t i,j t t j=1 j=1 j6=i j6=i (26) i,j i,j N a (t)+(bbb (t)) mmm u u = 2 N + e j=1 j6=i Similarly, we obtain for every 0 t u T and every i, j, k 2 K with i 6= j, j 6= k h i h i h i h i R R u i,j u i,j j,k j,k j,k j,k y du y du u u t t E p (t, u)y jF = E (1 e )y jF = E y jF E e y jF i,j u t u t u t u t j,k e | j,k ea (t)+(bbb (t)) mmm | u e (27) u t = e (ea (t) + (bbb (t)) mmm ) u t i,j i,j | j,k j,k a (t)+(bbb (t)) mmm | u u b e (ba (t) + (bbb (t)) mmm ) u t h i h i u j,l j,k j,k j,k N y du E p (t, u)y jF = (2 N)E y jF + E e y jF j,j u t u t u t l=1 l6=j j,k e j,k ea (t)+(bbb (t)) mmm | u e (28) u t = (2 N)e (ea (t) + (bbb (t)) mmm ) u t j,l j,l j,k | j,k N a (t)+(bbb (t)) mmm | u u + e (ba (t) + (bbb (t)) mmm ) l=1 u t l6=j For every 0 t u T and every combination of i, j, k, l 2 K, considered in Equations (25)–(28), i,j i,j the functions a , bbb solve the ODEs i,j dbb 1 i,j 1 i,j i,j i,j u | | 1 b b b (t) = b d (t) bb (t) (bb (t)) V (t)bb (t) (29) u u u dt 2 i,j da 1 i,j 0 | i,j i,j | 0 i,j (t) = c d (t) bbb (t) (bbb (t)) V (t)bbb (t) (30) u u u dt 2 i,j i,j with terminal conditions a (u) = 0 and bbb (u) = 0, whereas the functions ea , bbb solve the ODEs u u u u dbb 1 u 1 | | 1 e e e (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (31) u u u dt 2 dea 1 0 | | 0 e e e (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (32) u u u dt 2 with terminal conditions ea (u) = 0 and bbb (u) = 0. k,l k,l i,j k,l k,l i,j e b e e b b b e b b The functions a , bb , a and bb , k, l 2 K, corresponding to a and bb or to a and bb for u u u u u u u u i, j 2 K as considered in Equations (25)–(28), then solve the ODEs k,l dbb k,l k,l u | | 1 e e e b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (33) u u u dt k,l dea k,l k,l u 0 | | 0 e e e b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (34) u u u dt Risks 2016, 4, 23 13 of 26 k,l k,l k,l k,l with terminal conditions ea (u) = c , bbb (u) = b and k,l dbbb k,l k,l u 1 | i,j | 1 b b (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (35) u u u dt k,l dba k,l k,l u 0 i,j | | 0 b b b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (36) u u u dt k,l k,l k,l k,l with terminal conditions ba (u) = c , bbb (u) = b . Note that with these speciﬁcations, we obtain by Equation (20) that for every 0 t < u T and every i, j 2 K, i 6= j R R R t i,j u i,j t i,j i,j i,j y dv y dv y dv a (t)+(bbb (t)) mmm v v v u u 0 t 0 E p (0, u)jF = 1 e E e jF = 1 e e t t i,j i,j i,j i,j i,j i,j | | b m b m i,j a (0)+(bb (0)) mm y dv a (s)+(bb (s)) mm | u u v u u 0 0 s =1 e e e (sss(s,mmm )) bbb (s)dW s u and N R N R u i,j t i,j i,j i,j y dv y dv a (t)+(bbb (t)) mmm v v u u t 0 t E [ p (0, u)jF ] = 1 E[1 e jF ] = 2 N + e e i,i t t å å j=1 j=1 j6=i j6=i t R i,j i,j s i,j i,j i,j | | i,j a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | u u v u u 0 s =2 N + e + e e (sss(s,mmm )) bbb (s)dW s u j=1 j6=i Moreover, as every martingale with respect to the Brownian ﬁltration F is continuous, we have for 0 u t T that E p (0, u)jF = p (0, u) = lim E p (0, u)jF t w i,j i,j i,j w%u i,j i,j i,j i,j i,j | | b m b m i,j a (0)+(bb (0)) mm y dv a (s)+(bb (s)) mm | u u v u u 0 0 s s m b = lim 1 e e e (ss(s,mm )) bb (s)dW s u w%u i,j i,j s i,j i,j i,j | | a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | i,j u u v u u 0 0 s s m b =1 e e e (ss(s,mm )) bb (s)dW s u and E [ p (0, u)jF ] = p (0, u) = lim E [ p (0, u)jF ] t w i,i i,i i,i w%u N R i,j i,j s i,j i,j i,j | | a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | i,j u u v u u 0 0 s =2 N + e + e e (sss(s,mmm )) bbb (s)dW s u j=1 j6=i Hence, for arbitrary u, t 2 [0, T], we obtain i,j i,j E p (0, u)jF = c (u) + J (s, u)1 (s)dW (37) i,j t s [0,u] 1 1 i i E [ p (0, u)jF ] = c (u) + J (s, u)1 (s)dW (38) t s i,i [0,u] 2 2 0 Risks 2016, 4, 23 14 of 26 where for u, s 2 [0, T] and every i, j 2 K, i 6= j i,j i,j i,j a (0)+(bbb (0)) mmm u u 0 c (u) := 1 e i,j i,j i,j i,j b m i,j y dv a (s)+(bb (s)) mm | v u u 0 s J (s, u) := e e (sss(s,mmm )) bbb (s) 1 s u i,j i,j i a (0)+(bbb (0)) mmm u u c (u) := 2 N + e j=1 j6=i s i,j i,j i,j i y dv a (s)+(bbb (s)) mmm | i,j v u u s J (s, u) := e e (sss(s,mmm )) bbb (s) 2 s u j=1 j6=i Similarly we get for 0 t < u T and every i, j, k 2 K with i 6= j, j 6= k that h i h i R R i,j i,j t u j,k j,k j,k y dv y dv v v 0 t E p (0, u)y jF = E y jF e E e y jF u t u t u t i,j i,j i,j i,j | j,k | j,k e j,k j,k e b m b m a (t)+(bb (t)) mm | y dv a (t)+(bb (t)) mm | u e v u u b u t 0 t b m b m = e (ea (t) + (bb (t)) mm ) e e (ba (t) + (bb (t)) mm ) u u u t u t i,j i,j j,k j,k e | | j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | e u u b u 0 0 = e (ea (0) + (bbb (0)) mmm ) e (ba (0) + (bbb (0)) mmm ) u u u 0 u 0 | j,k j,k e j,k ea (s)+(bbb (s)) mmm | | u e e e u s + e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) s u s u u s j,k i,j i,j | j,k j,k y dv a (s)+(bbb (s)) mmm | | i,j v u u b 0 s b b m s m b + e e (a (s) + (bb (s)) mm )(ss(s,mm )) bb (s) u s s u s j,k i,j i,j | j,k y dv a (s)+(bbb (s)) mmm | v u u b 0 s + e e (sss(s,mmm )) bbb (s) dW s u and h i h i N R h R i t j,l u j,l j,k j,k j,k y dv y dv v v 0 t E p (0, u)y jF = (2 N)E y jF + e E e y jF j,j u t u t å u t l=1 l6=j j,l j,l e | j,k | j,k j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | u e u b u 0 u 0 = (2 N)e (ea (0) + (bbb (0)) mmm ) + e (ba (0) + (bbb (0)) mmm ) u u u 0 u 0 l=1 l6=j | j,k j,k j,k ea (s)+(bbb (s)) mmm | | u e e e u s s m e b m b b + (2 N) e (ss(s,mm )) (a (s) + (bb (s)) mm )bb (s) + bb (s) s u s u u s j,l j,l j,l | j,k j,k y dv a (s)+(bbb (s)) mmm | | j,l v u b 0 u s + e e (ba (s) + (bbb (s)) mmm )(sss(s,mmm )) bbb (s) å u u s s u l=1 l6=j s j,l j,l j,l | j,k y dv a (s)+(bbb (s)) mmm | v u b 0 u s + e e (sss(s,mmm )) bbb (s) dW s u Note that by Jensen’s inequality and Assumption 8 (4), we get for every 0 t < u T and every i, j, k 2 K with j 6= k that j,k j,k j,k 2 2 2 2 E[E[ p (0, u)y j F ] ] E[E[ p (0, u) (y ) j F ]] E[(y ) ] C i,j u t i,j u t u Finally, with the same limit-arguments as above, we obtain for arbitrary u, t 2 [0, T] and every i, j, k 2 K with i 6= j, j 6= k that h i j,k i,j,k E p (0, u)y jF = c (u) + J (s, u)1 (s)dW (39) i,j u t 3 s 3 [0,u] h i j,k j,k E p (0, u)y jF = c (u) + J (s, u)1 (s)dW (40) j,j u t 4 s [0,u] 0 Risks 2016, 4, 23 15 of 26 where for u, s 2 [0, T], i, j, k 2 K with i 6= j, j 6= k i,j i,j e | j,k | j,k j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | u u e b u 0 u 0 c (u) := e (ea (0) + (bbb (0)) mmm ) e (ba (0) + (bbb (0)) mmm ) 3 u u u 0 u 0 | j,k j,k i,j,k e j,k e b m a (s)+(bb (s)) mm | | u e e e u s J (u, s) := e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) 3 s u s u u s j,k i,j i,j | j,k j,k j,k y dv a (s)+(bbb (s)) mmm | | i,j v u b b 0 u s s m b m b b + e e (ss(s,mm )) (ba (s) + (bb (s)) mm )bb (s) + bb (s) s u s u u e | j,k j,k ea (0)+(bbb (0)) mmm | u e u 0 c (u) := (2 N)e (ea (0) + (bbb (0)) mmm ) 4 u u 0 j,l j,l | j,k j,k a (0)+(bbb (0)) mmm | u 0 b + e (ba (0) + (bbb (0)) mmm ) å u u 0 l=1 l6=j | j,k j,k j,k e j,k e b m a (s)+(bb (s)) mm | | u e e e u s J (u, s) := (2 N)e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) 4 s u s u u N R s j,l j,l j,l | j,k j,k j,k y dv a (s)+(bbb (s)) mmm | | j,l v u b b 0 u s s m b b m b b + e e (ss(s,mm )) (a (s) + (bb (s)) mm )bb (s) + bb (s) å u s u s u u l=1 l6=j We can now apply these results to compute explicitly the risk minimizing strategy as given in Theorem 4 and more speciﬁcally by Theorem 7 in the Brownian setting in consideration. From i,j i,j,k Equations (37)–(40) it follows immediately that the processes x (u,) and q (u,), i, j, k 2 K, j 6= k, of Equations (14) and (15) are given as i,j i,j i,i i x (u, t) = J (u, t) , x (u, t) = J (u, t) 1 2 i,j,k j,k i,j,k j,j,k q (u, t) = J (u, t) , q (u, t) = J (u, t) 3 4 Moreover, with Equations (25)–(28) the i-th component F (t, T), i 2 K, of F(t, T) in Equation (13) is given as h i 0 0 S (t) S (t) i i N j F (t, T) = Y E p (t, T)jF + å Y E p (t, T)jF 0 i,i t 0 i,j t j=1 S (T) S (T) j6=i h i R R 0 0 T S (t) T S (t) i j + E p (t, u)jF n (u)du + å E p (t, u)jF n (u)du 0 t 0 t i,i i,j t j=1 t S (u) S (u) j6=i h i T S (t) i,k i,k + Z (u)E p (t, u)y (u)jF du å t 0 i,i k=1 t S (u) k6=i h i T S (t) N N j,k j,k + Z (u)E p (t, u)y (u)jF du å å t 0 i,j j=1 k=1 t S (u) k6=j j6=i 0 0 i,j i,j i,j i,j S (t) S (t) i N a (t,T)+bbb (t,T)mmm N j a (t,T)+bbb (t,T)mmm t t = Y 2 N + e + Y (1 e ) å å 0 0 j=1 j=1 S (T) S (T) j6=i j6=i R 0 i,j i,j T S (t) N a (t,u)+bbb (t,u)mmm i + 2 N + e n (u)du j=1 t S (u) (41) j6=i 0 i,j i,j T S (t) N a (u)+bbb (u)mmm j t t + å 1 e n (u)du j=1 t S (u) j6=i i,k T S (t) e N i,k ea(t,u)+bbb(t,u)mmm i,k t b m + å Z (u) (2 N)e (ea (t, u) + bb (t, u) mm ) k=1 t S (u) t k6=i i,l i,k i,l N a (t,u)+bbb (t,u)mmm i,k b b m + å e (a (t, u) + bb (t, u) mm ) du l=1 l6=i R 0 j,k T S (t) N N j,k ea(t,u)+bbb(t,u)mmm j,k + Z (u) e (ea (t, u) + bbb (t, u) mmm ) å å j=1 k=1 t t S (u) k6=j j6=i i,j j,k i,j a (t,u)+bbb (t,u)mmm j,k e (ba (t, u) + bbb (t, u) mmm ) du and can hence be expressed explicitly in terms of mmm. Risks 2016, 4, 23 16 of 26 3.4. Application: The Expected Cumulative Payment in an Ornstein-Uhlenbeck Framework In this section, we illustrate in a speciﬁc example how the expected (discounted) cumulative payment E[D ] from Lemma 3 can be calculated by using the explicit expression of the components F (t, T) from the previous section and the connection established through Equation (9). In the following, we regard a speciﬁc insurance product, namely an income protection insurance, and for simplicity we assume that the corresponding process X of the insured person can only take the three states K = f1 = healthy, 2 = sick/unﬁt for work, 3 = deathg, where state 3 is absorbing. The corresponding Version May 27, 2016 submitted to Risks 18 of 28 transitions are illustrated in Figure 1. 1 2 3 Figure 1. Possible transitions for an insured person’s process in an income protection insurance with Figure the three 1. states Possible K = transitions f1 = healthy, for an 2 insur = sick/unﬁt ed person’s for work, process 3 = indeath an income g withpr absorbing otection insurance state 3. with the three states K = {1 = healthy, 2 = sick/unﬁt for work, 3= death} with absorbing state 3. j,k j,k 3 j,k j,k Hence, for b 2 R , c 2 R from Equation (19) we set b = 0 0 0 and c = 0 for (j, k) 2 / I := f(1, 2), (2, 1), (1, 3), (2, 3)g. In the following, we assume a speciﬁc form for the Markov processμμμ from (19), namely a simple, In the following, we assume a speciﬁc form for the Markov process mmm from Equation (19), namely 3-dimensional Ornstein-Uhlenbeck (OU) process with corresponding SDE a simple, 3-dimensional Ornstein-Uhlenbeck (OU) process with corresponding SDE μ μ σ μ ξ dμμ = aμμ dt + σσdW , μμ = ξξ ∈ R . t t 0 dmmm = ammm dt + sssdW , mmm = xxx 2 R t t 0 Note that for the OU process it is well-known that condition (23) for the expectation is fulﬁlled. A Note that for the OU process it is well-known that Condition (23) for the expectation is fulﬁlled. major drawback, though, of choosing this process for the intensity is the undesirable feature that A major drawback, though, of choosing this process for the intensity is the undesirable feature that it it can become negative with positive probability. However, [31] provide calibrated parameters for can become negative with positive probability. However, [31] provide calibrated parameters for which which the probability of negative values for μμμ turns out to be negligible. For this reason, we choose the probability of negative values for mmm turns out to be negligible. For this reason, we choose similar similar parameter values and set parameter values and set   0 0.0003 0 0 1   | 0.0003 0 0 | a = (0.07, 0.11, 0.09) ,σσσ = 0 0.0007 0 and ξξξ = (0.1, 0.1, 0.1) .   B C | | s x a = (0.07, 0.11, 0.09) ,ss = 0 0.0007 0 and xx = (0.1, 0.1, 0.1) @ A 0 0 0.0005 0 0 0.0005 Hence, equations (21) and (22) simplify and yield to Hence, Equations (21) and (22) simplify and yield to δδ(t, x) = (d ) x 1 | ddd(t, x) = (d ) x | 0 [σσσ(t, x)σσσ(t, x) ] = [V ] , i,j i,j | 0 [sss(t, x)sss(t, x) ] = [V ] i,j i,j with     with 0 a 0 0 1 0 σσσ 0 0 1     1 0 a 0 0 sss 2 0 0 d = and V = 11 σ .  0 a 0   0 σσ 0  B C B C 1 0 d = 0 a 0 and V = 0 sss 2 0 @ A @ A 0 0 a 0 022 σσσ 0 0 a 0 0 sss As a consequence of these assumptions, the ODEs (29)-(36) substantially simplify and can be explicitly As a consequence of these assumptions, the ODEs Equations (29)–(36) substantially simplify and solved, see e.g. [32]. For example, for every 0 ≤ t ≤ u ≤ T and every choice of (i, j) ∈ I , the ODEs can be explicitly solved, see e.g., [32]. For example, for every 0 t u T and every choice of (29) and (30) now are given by (i, j) 2 I , the ODEs Equations (29) and (30) now are given by i,j dββ i,j 1 i,j i,j (t) = b − d ββ (t) , dbbb i,j 1 i,j dt (t) = b d bbb (t) dit,j dα 1 i,j i,j | 0 i,j i,j (t) = c − (βββ (t)) V βββ (t) , da 1 u u i,j i,j | 0 i,j dt 2 (t) = c (bbb (t)) V bbb (t) u u dt 2 i,j i,j i,j i,j (k) with terminal conditions α (u) = 0 and βββ (u) = 0. For the k-th components of βββ (t), i.e. β (t), u u u u one obtains the explicit solutions i,j i,j (k) 1 β (t) = 1− exp(d (u− t) , k ∈ {1, 2, 3} , kk kk Risks 2016, 4, 23 17 of 26 i,j i,j i,j i,j (k) with terminal conditions a (u) = 0 and bbb (u) = 0. For the k-th components of bbb (t), i.e., b (t), u u u u one obtains the explicit solutions i,j i,j (k) 1 b (t) = 1 exp(d (u t) , k 2 f1, 2, 3g kk kk and i,j 3 2 2 1 s (b ) 2 i,j i,j k k 1 a (t) = c (u t) + (u t) 1 exp(d (u t) u å kk 1 1 (d ) d k=1 kk kk + 1 exp(2d (u t) kk 2d kk Similarly, the ODEs of the remaining Equations (31)–(36) can be derived. j,k j,k Next, we need to specify all remaining parameters, i.e., b , c for (j, k) 2 I , the components A, rt Y, Z of the quadruple (X; A; Y; Z) and the discounting factor S . For simplicity, we set S = e with a 0 0 constant interest rate r and choose 0 1 0 1 0 1 3 1 2 B C B C B C 1,2 2,1 1,3 2,3 1,2 2,1 1,3 2,3 b = b = , b = , b = , c = c = 1 , c = 0.1 , c = 0.2 @3A @1A @2A 3 1 2 j,k We have chosen the components of b to be equal in order to emphasize the general dependence j,k of the process y in Equation (19) on mm and not on the speciﬁc linear combination. t t Furthermore, similar to Example 2, we set 0 1 0 0 z B C Z := 0 0 z , Y 0 , and A = C P , t 2 [0, T] @ A t t t 0 0 0 with C and P representing the cumulative state-dependent claim payments (e.g., annuities) and t t insurance premiums up to maturity, respectively, and z the “immediate” claim payment if the insured person dies. More precisely, we assume monthly equal insurance premiums and claim payments equal to 1, if the insured person is in the states 1 (healthy) or 2 (sick/unﬁt for work), respectively, and which are paid at the end of each month in a proportional way. Let the payment dates 0 < T < T < . . . denote 1 2 the ﬁnal day of each month, i.e., T = , then we set 0 1 1/Dt B C nnn(t) = 1/Dt @ A with Dt = 1/12 and A = nnn(s)ds. Clearly, with these speciﬁcations Assumption 6 is fulﬁlled. Finally, assuming that the insured persons process X starts in the state 1 (healthy) at t = 0, one obtains H = (1, 0, 0). Based on the explicit result for F (0, T) from Equation (41), now we are able to calculate the expected (discounted) cumulative payment E[D ] from Lemma 3 in t = 0, depending on the claim payment amount z, which is payed when the insured person dies, and on the constant interest rate r. The corresponding integrals involved in Equation (41) are approximated using the integrate function in the statistical software program R ([33]). Figure 2 illustrates the expected cumulative payment E[D ] as a function of the claim payment amount z, a time horizon of one year (T = 1) and three different values of the constant interest rate Risks 2016, 4, 23 18 of 26 r. In Figure 3, the expected cumulative payment E[D ] are displayed against the time horizon T, for three different values of the claim payment amount z and for a constant interest rate r = 0.1. r = 0.05 r = 0.1 r = 0.2 0 20 40 60 80 100 claim payment amount z Figure 2. Expected (discounted) cumulative payments E[D ] as a function of the claim payment amount z and for three different constant interest rate values r for a time horizon of one year, i.e., T = 1. z = 1 z = 10 z = 20 5 10 15 20 time horizon T (in years) Figure 3. Expected (discounted) cumulative payments E[D ] as a function of the time horizon T (in years) and three different claim payment amounts z and for a constant interest rate r = 0.1. 4. Conclusions In this paper, we consider pricing and hedging of general insurance contracts by means of risk minimization. We model the individual progress in time of visiting an insurance policy’s states by using F-doubly stochastic Markov chains. In this way we are able to consider a multi-state setting to describe different types of insurance beneﬁts and to include the inﬂuence of market conditions and external risk factors on the evolution of the insured person among the policy’s states as well as on the insurance beneﬁts, when they are linked to some ﬁnancial performance. We explicitly provide the risk-minimizing strategy for an insurance contract in a Brownian ﬁnancial market setting and E[D ] E[D ] −50 0 50 100 150 200 0 20 40 60 80 100 Risks 2016, 4, 23 19 of 26 specify it within an afﬁne structure for the intensity. The results are illustrated by a numerical example, which shows how this technical setting can actually be easily implemented. Acknowledgments: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No [228087]. Author Contributions: All authors have contributed equally to all aspects of this paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. Appendix A. F-Doubly Stochastic Markov Chains In this section we introduce brieﬂy to some basic properties of F-doubly stochastic Markov chains, which we are going to use in the sequel. Main references are [12,21]. On a probability space (W,G ,P), let X = (X ) be a right-continuous stochastic process with t2[0,T] X X state space K := f1, ..., Ng. We denote by F the ﬁltration generated by X, i.e., F = s(X(u) : u t) for all t 2 [0, T], and consider the ﬁltration G to be the enlargement of F through some reference X X ﬁltration F, i.e., we assumeG = F _F for all t 2 [0, T]. Further, we setG = F _F , t 2 [0, T] and t t t t t assume that all ﬁltrations satisfy the usual conditions of completeness and right-continuity, see [28]. Deﬁnition A.1. A process X = (X ) is called an F-doubly stochastic Markov chain with state space K, t2[0,T] if there exists a family of stochastic matrices P(s, t) = [ p (s, t)] , 0 s t T j,k j,k2K such that (1) the matrix P(s, t) is F -measurable, and P(s, .) is progressively measurable, (2) for every j, k 2 K we have 1 P(X = k j G ) = 1 p (s, t) (A1) t s fX =jg fX =jg j,k s s The process P is called the conditional transition probability process of X. By Deﬁnition A.1 we can see that the class of F-doubly stochastic Markov chains contains Markov chains, compound Poisson processes with integer-valued jumps, Cox processes as in [34] and processes of rating migration as in [35]. The adjective “double” refers to the fact that there are two sources of uncertainty in their deﬁnition. We remark that an F-doubly stochastic Markov chain is a different object than a doubly stochastic Markov chain which is a Markov chain with a doubly stochastic transition matrix. Furthermore, in [12] it is shown that F-doubly stochastic Markov chains are a subclass of F-conditional G = F _ F Markov chains. In particular, F-doubly stochastic Markov chains behave like time inhomogeneous Markov chains conditioned on F , i.e., if we know all the information concerning the underlying risk factors. Deﬁnition A.2. We say that a state N 2 K is an absorbing state, if p (s, t) = 0 for all 0 s < t T and N,j all j 2 K with j 6= N. Proposition A.3. Let X be an F-doubly stochastic Markov chain with transition matrices P(s, t), then for every 0 s < t < u T we have P(s, u) = P(s, t)P(t, u) a.s. (A2) Risks 2016, 4, 23 20 of 26 Proposition A.4. If X is an F-doubly stochastic Markov chain, then for every bounded,F -measurable random variable Y and for each t 2 [0, T], we have E[YjG ] = E[YjF ] (A3) t t Property Equation (A3) is well-known in the context of survival analysis and credit risk as hypothesis (H) or immersion property. According to Proposition A.4, F-martingales remain martingales with respect to the enlarged ﬁltration G. If we think of a martingale as a process describing a fair game, this property means that the additional information contained in G does not change the valuation of processes which are considered fair by taking in account only the information F. Another property, which makes the class of F-doubly stochastic Markov chains interesting for applications is that they may admit matrix-valued stochastic intensity processes in the following sense. Deﬁnition A.5. An F-doubly stochastic Markov chain X with state space K is said to have an intensity, if h i j,k there exists an F-adapted matrix-valued stochastic process Y Y Y = (Y Y Y ) with Y Y Y = y such that t t t2[0,T] j,k2K (1) Y Y Y is integrable, i.e., j,j jy jds < ¥ (A4) ]0,T] j2K (2) Y Y Y satisﬁes the following conditions: j,k j,j j,k (A5) y 0 8j, k 2 K, j 6= k, y = å y 8j 2 K, t 2 [0, T] k6=j t t t P(v, t) I = Y Y Y(u)P(u, t)du 8v t (Kolmogorov backward equation) ]v,t] (A6) P(v, t) I = P(v, u)Y Y Y(u)du 8v t (Kolmogorov forward equation) ]v,t] A process Y Y Y, satisfying the above conditions, is called an intensity of the F-doubly stochastic Markov chain X. Theorem A.6. Let (Y Y Y ) be an F-adapted N N matrix-valued stochastic process, satisfying the t2[0,T] Conditions (A4) and (A5) of Deﬁnition A.5. Then there exists an F-doubly stochastic Markov chain X with intensity (Y Y Y ) . t2[0,T] For j 2 K, let H := 1 , t 2 [0, T] (A7) t fX =jg 1 N | be the indicator function for X, being in state j at time t and denote by H = ( H , ..., H ) the t t jk jk corresponding N-variate vector. Moreover, for j, k 2 K, j 6= k, let N = (N ) with t t2[0,T] jk j j k k N := H d H = H 4 H (A8) t u u u u ]0,t] 0<ut deﬁne the counting processes of the jumps of X from state j to k up to time t, t 2 [0, T]. The following theorem provides a martingale characterization of F-doubly stochastic Markov chains and is the core connection of the theory of F-doubly stochastic Markov chains and the counting process theory, underlying for example several estimation schemes for intensity processes, see [13]. Risks 2016, 4, 23 21 of 26 Theorem A.7. Let X = (X ) be a stochastic process with state space K and Y Y Y = (Y Y Y ) be t t t2[0,T] t2[0,T] a matrix-valued process, satisfying Equations (A4) and (A5) of Deﬁnition A.5. The following conditions are equivalent: (i) X is an F-doubly stochastic Markov chain. (ii) The process M = (M ) with t2[0,T] M := H YY H du (A9) t t u u ]0,t] is a G-local martingale. jk jk (iii) For j, k 2 K, j 6= k, the processes M = ( M ) with t2[0,T] jk jk j j,k M := N H y du (A10) u u t t ]0,t] are G-local martingales. (iv) The process L = (L ) with t2[0,T] L := Q(0, t) H t t is a G-local martingale. Here Q(0, t) is a unique solution to the random integral equation dQ(0, t) = YY Q(0, t)dt, Q(0, 0) = I (A11) Note that then L = H + Q (0, u)dM , t 2 [0, T] (A12) t 0 u ]0,t] Remark A.1. (1) For every t 2 [0, T], the matrix Q(0, t) is the unique inverse matrix of P(0, t). More generally, for 0 s t T, we denote by Q(s, t) the unique inverse matrix of P(s, t). The existence and further properties of the family Q(s, t) is given in [12]. It follows immediately from Equation (A2) that for every 0 s < t < u T, we have P(t, u) = Q(s, t)P(s, u) (A13) jk (2) As the processes M, L and M , j, k 2 K, j 6= k, are G-adapted, they are also G-local martingales. Corollary A.8. For every j, k 2 K, j 6= k, and for every t 2 [0, T] we have jk jk [ M ] = N (A14) jk h M i = H y (u)du (A15) u j,k ]0,t] j j k,j N k Moreover, with M = H y H du, j 2 K, t 2 [0, T], we have u u t t k=1 ]0,t] j k j jk j 2 N [ M ] = (D H ) = (N + N ) t å å 0<st s k=1 t t k6=j R R (A16) j N k h M i = å H y (u)du H y (u)du t u u k,j j,j k=1 ]0,t] ]0,t] k6=j Risks 2016, 4, 23 22 of 26 jk Proof. Equalities (A14) and (A15) follow directly by the deﬁnition of M in Equation (A10). k,j Moreover, we observe that å H y du is a continuous ﬁnite variation process. It k2K u u ]0,t] t2[0,T] follows that j k j jk j 2 [ M ] = (D H ) = N + N t å s å t t 0<st k=1 k6=j j j j,k as (D H ) counts the jumps of X into and out of the state j up to time t. As H y du s u u 0<st ]0,t] t2[0,T] jk is the compensator of N it follows that Z Z k,j j j,k j k h M i = H y du + H y du t u u u å u ]0,t] ]0,t] k=1 k6=j Z Z k,j j j,j = H y du H y du å u u u u ]0,t] ]0,t] k=1 k6=j where the last equality follows from Equation (A5). This ends the proof. Proposition A.9. Let X be an F-doubly stochastic Markov chain with intensity and jump times t := 0 and t := infft < t T : X 6= X g (A17) k k1 t t k1 Then every jump time t , k 1, avoids F-stopping times, i.e., P(t = $) = 0 for every F-stopping time k k $, provided that t < ¥ a.s.. The following proposition is the crucial result in order to compute the risk-minimizing strategies for general insurance claims which we provide in Section 3. Proposition A.10. Let X be an F-doubly stochastic Markov chain. Then the local martingale M, given in Equation (A9), is orthogonal to every F-local martingale N, in the sense that for each i 2 K, the product M N is a G-local martingale. i i Proof. First note that M is a ﬁnite variation local martingale. Its sequence (te ) of jump times with k0 te := 0 and i i i i e e t := infft > t j M 6= M g , k 1 t t k k1 is a subsequence of the jump times (t ) of X, as given by Equation (A17). As the jump times of j j0 the càdlàg local martingale N are F-stopping times, the processes M and N have almost surely no common jumps due to Proposition A.9. This implies that for all t 2 [0, T] we have i i [ M , N] = M N + D M DN = 0 t s 0 0 å s 0<st and ends the proof. Remark A.2. It is easily seen that hazard-rate models, as applied frequently in the context of credit risk or life insurance, are particular examples of F-doubly stochastic Markov chains, provided they satisfy hypothesis (H). A thorough description of this relation is given in [24]. Risks 2016, 4, 23 23 of 26 Appendix B. Risk-Minimization for Payment Processes The following survey of risk-minimization for payment processes is borrowed to some extend from [1], as well as [22]. As in the foregoing sections, we provide the results with respect to a general numéraire process S such that one could also consider e.g., the P-numéraire portfolio as discount factor, see [25]. The results base on the proofs, given in [4] for the case of European type contingent claims and in [5,36,37] (Chapter 4) for the case of payment processes. In the market model, deﬁned in Section 3.2, we would like to ﬁnd a hedging strategy for a b b G-adapted, square integrable payment process D = (D ) , representing the cumulative discounted t2[0,T] payments up to time t, t 2 [0, T]. Note that if an undiscounted cumulative payment stream D = (D ) is a stochastic process of t2[0,T] ﬁnite variation and we have djDj < ¥ with jDj denoting the absolute variation process of D, [0,T] then D is given as D = dD (B1) t u [0,t] S D D Deﬁnition B.1. If djDj < ¥, then the value process U = (U ) of a payment process D is 0 u t t2[0,T] [0,T] deﬁned as h i D 0 0 U := S E D G = S E dD G (B2) T t u t t t t [0,T] S Since the market is not necessarily complete, it is in general not possible to ﬁnd a self-ﬁnancing hedging strategy that perfectly replicates the discounted cumulative payment process D. In this context, the idea of risk-minimization is to relax the self-ﬁnancing assumption, allowing for a wider class of admissible strategies, and to ﬁnd an optimal hedging strategy with “minimal risk” within this class of strategies that perfectly replicate D. For the local martingale S, we denote ( ) | | 2 1 d x x x x L (S) := xx = (x , ..., x ) xx is G-predictable, E xx d[S] xx < ¥ (B3) t s s t2[0,T] [0,T] It is well known that for every xxx 2 L (S), the process xxx dS is a square [0,t] t2[0,T] integrable martingale. In the following we now explain how to ﬁnd the risk-minimizing strategy and explain in what sense this strategy is optimal. We begin with some deﬁnitions. 2 0 2 0 Deﬁnition B.2. An L -strategy is a pair jjj = (xxx , x ), such that xxx 2 L (S) and x is a real-valued G-adapted process, such that the discounted portfolio value process | 0 b b V = xx S + x , t 2 [0, T] t t is right-continuous and square integrable. 2 j For an L -strategy jjj the discounted (cumulative) cost process C is deﬁned as j j b b b b C := V xxx d S + D , t 2 [0, T] s t t t s ]0,t] describing the accumulated costs of the trading strategy jjj during [0, t], including the payments D . Note that V should therefore be interpreted as the discounted value of the portfolio jjj held at time t t t Risks 2016, 4, 23 24 of 26 b b after the payments D have been made. In particular, V is the discounted value of the portfolio upon settlement of all liabilities, and a natural condition is then to restrict to 0-admissible strategies, satisfying V = 0 P-a.s. The risk process of jj is given by the conditional expected value of the squared future costs j j j b b R = E[(C C ) jG ], t 2 [0, T] (B4) t t and is taken as a measure of the hedger ’s remaining risk. We would like to ﬁnd a trading strategy that minimizes the risk in the following sense. 2 0 Deﬁnition B.3. An L -strategy jjj = (xxx , x ) is called risk-minimizing for the discounted payment process D, je j 2 0 e e b b if for any L -strategy jjj = (xxx, x ) such that V = V = 0 P-a.s., we have T T j je R R P-a.s., t 2 [0, T] t t i.e., j minimizes pointwise the risk process introduced in Equation (B4). The key to ﬁnding the strategy with minimal risk in our setting is the so-called Galtchouk-Kunita-Watana decomposition. b b Deﬁnition B.4. Given a square integrable martingale U 2 M and the local martingale S, the b b Galtchouk-Kunita-Watanabe decomposition for U with respect to S is given as U | U b b b U = U + (JJJ ) dS + L , t 2 [0, T] (B5) t s 0 s t ]0,t] U 2 D where JJJ 2 L (S) and L is a square integrable martingale null at 0 which is strongly orthogonal to the space 2 | 2 b b b I (S) of all integral processes y y y dS with y y y 2 L (S). [0,t] t2[0,T] 2 2 It is well known that the set I (S) is a closed stable subset of M , the set of all square integrable martingales, zero at 0. Due to Jensen’s inequality and the fact that D is square-integrable, the discounted value process D U U = is a square-integrable martingale and may be decomposed according to Equation (B5). Theorem B.5. For every (discounted) square integrable payment stream D, there exists a unique 0-admissible 2 0 risk-minimizing L -strategy jjj = (xxx, x ), given by xxx := xxx t t 0 D D | b b b x := U D (xx ) S t t t t t with discounted portfolio value process j b b b b b b V = E[D jG ] D = E[D jG ] + xx dS + L D t t 0 s t T T s t ]0,t] discounted optimal cost process b b j j D D b b C = E[D jG ] + L = C + L T 0 t t t 0 and minimal risk process j b b D D 2 R = E[(L L ) jG ] t T t Risks 2016, 4, 23 25 of 26 D D D t 2 [0, T], where xxx and L are given by Equation (B5) for the square integrable martingale U . Proof. See [26] for the single payoff case or [5,36] for the extension to the case of payment streams. Note that the approach, described above, relies heavily on the fact that the discounted asset prices are local martingales under the measure P. 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Risk Management

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ISSN: 2227-9091
DOI: 10.3390/risks4030023
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Abstract

risks Article Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains 1,2, 3 4 Francesca Biagini *, Andreas Groll and Jan Widenmann Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, Germany Department of Mathematics, University of Oslo, Boks 1072 Blindern, Norway Department of Statistics, University of Munich, Akademiestr.1, 80799 Munich, Germany; andreas.groll@stat.uni-muenchen.de BMW Financial Services, BMW Bank GmbH, 80787 Munich, Germany; jan-maximilian.widenmann@bmw.de * Correspondence: francesca.biagini@math.lmu.de; Tel.: +49-89-2180-4492 Academic Editor: Mogens Steffensen Received: 21 March 2016; Accepted: 27 June 2016; Published: 7 July 2016 Abstract: We study risk-minimization for a large class of insurance contracts. Given that the individual progress in time of visiting an insurance policy’s states follows an F-doubly stochastic Markov chain, we describe different state-dependent types of insurance beneﬁts. These cover single payments at maturity, annuity-type payments and payments at the time of a transition. Based on the intensity of the F-doubly stochastic Markov chain, we provide the Galtchouk-Kunita-Watanabe decomposition for a general insurance contract and specify risk-minimizing strategies in a Brownian ﬁnancial market setting. The results are further illustrated explicitly within an afﬁne structure for the intensity. Keywords: insurance liabilities; doubly stochastic Markov chains; risk minimization MSC: 60J27; 62P05; 91G99 JEL: C02 1. Introduction The management of an insurance portfolio’s risk is one of the core challenges in actuarial science. While the classic form of risk mitigation is based on reinsurance contracts, in some cases it is also possible to hedge claim payments by appropriately trading in different assets. This particularly applies if the assets are correlated to the insurance contract’s beneﬁts or their (conditional) probability of occurrence. Practical examples in this direction are unit-linked life insurance products, where beneﬁts depend on the performance of the assets, or unemployment insurance products, where the occurrence of a claim payment may depend to some extend on economic and ﬁnancial conditions of the markets. Moreover, there is an ongoing discussion about the introduction of so called longevity bonds which would establish the possibility for life insurance companies and pension funds to hedge parts of their longevity risk, see [1,2] or [3]. Due to their unsystematic risk part, most insurance claims are not hedgeable completely through a self-ﬁnancing trading strategy which particularly means that a hybrid market, consisting among others of ﬁnancial and insurance markets, is incomplete. A reasonable method for optimally choosing an investment strategy is then important to cover at least parts of the risk. In the present paper we choose the risk-minimization approach and determine hedging strategies in the sense of this criterion for insurance contracts in a very general setting. This quadratic hedging approach bases on the results in [4] for European type payments and to [5] for payment Risks 2016, 4, 23; doi:10.3390/risks4030023 www.mdpi.com/journal/risks Risks 2016, 4, 23 2 of 26 processes. In most cases, the risk-minimizing strategies can be derived from the well known Galtchouk-Kunita-Watanabe (GKW-) decomposition, see [6] or [7]. Similar to the works of [5,8] or [9–11], we describe an insured person’s progress of sojourning different states of an insurance policy as a right continuous stochastic process with ﬁnite state space K = f1, ..., Ng, 1 being a.s. the initial state. More speciﬁcally, we adopt the class of F-doubly stochastic Markov chains as introduced in [12], see Appendix A and the comments therein. This family of processes has several properties which make them very suitable for applications in credit risk and insurance market modeling. Being a sub-class of F-conditional Markov chains, they extend the classic notion of Markov chains by including a reference ﬁltration F which in our case represents additional market information. In this way we are able to take in consideration the inﬂuence of external risk factors and economic and ﬁnancial conditions on transition probabilities of an insured person’s progress. In particular, F-doubly stochastic Markov chains behave like time inhomogeneous Markov chains, if we know all the information concerning the underlying risk factors. This corresponds to the intuition that the transition probabilities would be completely speciﬁed, if we would dispose of full knowledge on the underlying economic and ﬁnancial situation. Another important feature is that, if we specify the information as given by the ﬁltration X X G := F _ F, where F is the natural ﬁltration of the F-doubly stochastic Markov chain X, then we have that predictable representation theorems and the so-called hypothesis (H) , or immersion property, hold. These properties play a fundamental role in order to compute the optimal strategy for insurance contracts according to the risk-minimization method. Furthermore, F-doubly stochastic Markov chains may admit matrix-valued stochastic intensity processes. This allows to investigate more ﬂexible models compared to the results e.g., in Møller [5] where a (classical) Markov chain with deterministic intensity matrix function is considered. One further advantage is that F-doubly stochastic Markov chains with intensity are fully characterized by some martingale properties, which can be used for the estimation of the underlying intensity processes, see Biagini et al. [13]. Well known examples of F-doubly stochastic Markov chains are reduced form or intensity based models in the case that hypothesis (H) is satisﬁed. Here, the state space consists of two states with the second state being absorbing such that there can only occur one transition in time. There exist many works on quadratic hedging for these models particularly in the context of credit risk or life insurance theory, see e.g., [1,2,14–19] or [20]. In particular, the present paper extends these works to a multi-state framework where several subsequent transitions, driven by F-adapted stochastic intensity processes, are considered. This general setting allows to investigate a larger class of insurance contracts, e.g., income protection insurance contracts with the states “healthy”, “sick” and “deceased”, and to include the inﬂuence of market conditions and external risk factors on the insured person’s progress. Given an F-doubly stochastic Markov chain, we propose a general insurance contract, deﬁned by three different types of insurance beneﬁts: state-dependent payments at maturity, state-dependent annuity-type payments, and (transition-dependent) payments at the time of a transition from one state to another. This deﬁnition covers a large set of currently adopted insurance policies. In particular, we illustrate the deﬁnitions for pure endowment, term insurance, general life annuity and payment protection insurance contracts. Similar to the results in [21], who applied F-doubly stochastic Markov chains in the context of hedging rating-sensitive ﬁnancial claims, we obtain the GKW-decomposition for the payment process of general insurance contracts with respect to a particular F-martingale. In this context, we generalize and complement the proofs in [21] in order to adapt the results for the risk-minimization approach which is not investigated there. Given that the reference information F is generated by an N-dimensional Brownian motion W, we then introduce a ﬁnancial market model, driven by W. In this setting we infer risk-minimizing For the deﬁnition and further comments on hypothesis (H), see Proposition A.4 and the text below. Risks 2016, 4, 23 3 of 26 hedging strategies for insurance contracts with deterministic payment structure with respect to the assets on the ﬁnancial market. Similarly to the work in [2] we then assume a general afﬁne structure for the intensity of the underlying F-doubly stochastic Markov chain and obtain explicit formulas for the strategies and their residual risk processes. We apply these results in the speciﬁc example of an income protection insurance, where we assume that the intensities follow a (multi-dimensional) Ornstein-Uhlenbeck process. We discuss the resulting expected cumulative payment, which may be considered as a fair premium in the interpretation of [22,23], as a function of the time horizon, the payment amounts and the underlying interest rates. The paper is organized as follows. In Section 2 we introduce the notion of general insurance contracts and discuss several examples. In Section 3 we prove our main results for the risk minimization of this kind of contracts in full generality. The risk minimizing strategies are then further illustrated within a general afﬁne speciﬁcation for the intensities and in a numerical example in an Ornstein-Uhlenbeck framework. We conclude the paper with Appendixs A and B, where we summarize important results and concepts of risk-minimization and F-doubly stochastic Markov chains for the reader ’s convenience. 2. General Insurance Contracts We now introduce the notion of general insurance contracts and provide some well known examples of actuarial practice. In the same notation as in Appendix A, let (W,G ,G,P) be a ﬁltered probability space with G = F _ F for some F-doubly stochastic Markov chain X with state spaceK = f1, ..., Ng. We assume P(X = 1) = 1. The following deﬁnition of general insurance contracts is based on the deﬁnitions for payment processes on rating sensitive claims as given e.g., in [21] or [20]. The deﬁnition also covers the concepts of insurance contracts as given in [5] or [9]. Deﬁnition 1. A general insurance contract is given by the quadruple (X; A; Y; Z), where X = (X ) t2[0,T] 1 N is an F-doubly stochastic Markov chain, A = ( A , ..., A ) is an F-adapted, N-dimensional process of t t t2[0,T] 1 N ﬁnite variation, Y = (Y , ..., Y ) is an F -measurable, N-dimensional random vector, and Z = (Z ) T t2[0,T] h i j,k with Z = Z is an F-adapted, N N-dimensional process with zeros on the diagonal. j,k2K The different elements of a general insurance product’s quadruple are interpreted as follows. The process X is the insured person’s progress in time of sojourning in the states j 2 K, considered by the insurance policy. The N-dimensional process A characterizes the cumulative state-dependent payment streams which are continuously paid up to maturity. For example, one can take A = C P , t 2 [0, T] t t t 1 N | with C = (C , ...C ) representing the cumulative state-dependent claim payments (e.g., annuities) t t 1 N and P = (P , ..., P ) the cumulative state-dependent insurance premiums up to maturity. Both t t processes, P and C are then taken to be F-adapted, càdlàg and increasing. The vector Y characterizes state-dependent “extra” claim payments at maturity T and the process Z the “immediate” claim payments at the transition times from one state to another. For every general insurance contract (X; A; Y; Z) the cumulative payment process D = (D ) t2[0,T] is given by R R | | D : = Y H 1 + H dA + (Z H ) dH t T s s s s s ft=Tg [0,t] ]0,t] R R (1) j j j j,k jk N N = Y H 1 + H d A + Z dN å å s s s s j=1 ft=Tg k=1 T [0,t] ]0,t] k6=j j jk with H , j 2 K, as deﬁned in Equation (A7) and counting processes N from Equation (A8). Note that t t D is of ﬁnite variation. We now provide some well known examples of insurance contracts. Example 1. A pure endowment is an insurance contract which guarantees to the insured person some ﬁxed payment if she is alive at maturity. For the sake of simplicity, we only consider the payment to be equal to 1. Risks 2016, 4, 23 4 of 26 We set K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. A pure endowment | | contract is then given as the quadruple (X; 0; (1, 0) ; 0) or (X;P; (0, 1) ; 0) if premium payments are considered, respectively. Example 2. A term insurance is an insurance contract which guarantees the heirs of an insured person some ﬁxed payment at the time of decease. For the sake of simplicity, we only consider the payment to be equal to 1. Again, we set K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. Then a term insurance contract is given as the quadruple (X; 0; 0; Z) or (X;P; 0; Z) if premium payments are considered, 0 1 respectively, with Z := . 0 0 Example 3. A general annuity as deﬁned in [2] is an insurance contract which guarantees the insured person an F-progressively measurable, non-negative continuous rate payment (c ) as long as she is alive. The state t2[0,T] space is again K = f1, 2g with 1 being the state “alive” and 2 the absorbing state “deceased”. Then a general | | R R annuity contract is given as the quadruple (X; c ds, 0 ; 0; 0) or (X; c ds, 0 P; 0; 0) s s ]0,t] ]0,t] t2[0,T] t2[0,T] if premium payments are considered, respectively. Example 4. A payment protection insurance (PPI) is an insurance contract which is usually offered as an add-on product to some payment obligations, e.g., a loan. In the case of an insured event, the insurance company takes over the respective instalments of the payment obligation for the insured person or her heirs. Generally, the insured events are “disability”, “unemployment” and “decease”. Hence, the state space for PPI products is given as K = f1, 2, 3, 4g with “2” being the state “disabled”, “3” the state “unemployed”, “4” the absorbing state “deceased” and “1” the state where no insured event is present. 2 3 4 | Then a PPI contract is given as the quadruple (X; (0, C , C , C ) ; (0, Y, Y, Y) ; 0) or t t t t2[0,T] 2 3 4 | (X; (0, C , C , C ) P; (0, Y, Y, Y) ; 0) if premium payments are considered, respectively. t t t t2[0,T] As the underlying payment obligation usually stipulates ﬁxed instalments c , ..., c at some given payment 1 K i i dates 0 < T < ... < T = T, the processes C , i = 2, 3, 4, are generally given as C = c 1 . 1 K j t t j=1 fT tg Moreover, some payment obligations also contain a so-called balloon rate B at the end of the contract, which has to be paid on top of the usual instalment. If there exists a balloon rate and it is insured, then we set Y = B, if there exists no balloon rate or it is not insured, then we set Y = 0. Remark 1. (1) The extra claim payment could also be included in the continuous claim payments. For the reader’s convenience, however, we explicitly separate continuous and extra claim payments. (2) The main concepts of premium payment are - a single premium P, where the complete price for the insurance contract is paid at its beginning. In this case, the vector P would be given as P = P1 , P1 , ..., P1 . ft0g ft0g ft0g t2[0,T] - periodically paid premiums. Here, the insurance price is paid according to periodically paid premiums p at a priori speciﬁed dates 0 = T < T < ... < T T. Moreover, some insurance policies 0 L i 1 consider premium freedoms which allow the insured person to intermit premium payments while sojourning (some) insured states. In this case, we have for each vector entry P , i 2 f1, ..., Ng, of P å p 1 j fT tg i j=1 P = depending on whether state i is guarantees premium freedom or not. Risks 2016, 4, 23 5 of 26 3. Risk-Minimization for General Insurance Contracts Aim of this paper is to provide the risk minimizing strategy for a general insurance contract by applying the approach presented in Appendix B. Risk minimization provides hedging strategies which perfectly replicate the claim. Since the market is incomplete, these strategies may not be self-ﬁnancing and a readjustment (or cost) is needed to achieve perfect replication. According to this method we choose then the optimal strategy, i.e., the strategy with minimal cost. For this sake, we ﬁrst consider a general setting and then focus on a deterministic payment structure and an underlying market which is driven by some N-dimensional Brownian motion. These results are then further speciﬁed within a general afﬁne setting for the different entries of the matrix-valued intensity. 3.1. Martingale Decomposition for Payment Processes of General Insurance Claims We consider the payment process D in Equation (1). Let S denote the market’s discounting factor, which will be further speciﬁed in Section 3.2. If djDj < ¥, we get by Equation (B1) that the 0 u ]0,T] b b discounted cumulative payment stream D = (D ) is given as t2[0,T] | R R Y H | | 1 1 T | D = 1 + H dA + (Z H ) dH t 0 0 s s 0 s s s ft=Tg [0,t] ]0,t] S S S s s T ! (2) j R R Y H j j j,k jk N 1 N 1 = 1 + H d A + Z dN å å 0 0 s s 0 s s j=1 ft=Tg k=1 [0,t] ]0,t] S S S s s k6=j We further assume the underlying F-doubly stochastic Markov chain X to admit an intensity j,k Y Y Y = [y ] , as introduced in Deﬁnition A.5. j,k2K t2[0,T] Assumption 2. For every general insurance contract (X; A; Y; Z), let 2 3 4 5 E < ¥ , j 2 K (3) " # sup E d A < ¥ , j 2 K (4) [0,s] S s2[0,T] u " # Z Z E d A jy jdu < ¥ , j 2 K (5) v X ,j ]0,T] [0,u] 2 3 Z j,k 4 5 E y (u)du < ¥ , j, k 2 K, j 6= k (6) j,k ]0,T] 2 3 Z j,k 4 5 E y (u)du < ¥ , j, k 2 K, j 6= k (7) j,k ]0,T] S where H , t 2 [0, T] is deﬁned in Equation (A7). Note that Equations (3), (4), (6) and (7) ensure that the discounted payment stream D generated by the general insurance contract (X; A; Y; Z) is square integrable. We remark that the following Lemma is given similarly in [21] (Theorem 16.38) under the assumption the local martingale M, deﬁned in Equation (A9), is square integrable and the processes A and Z are bounded. Here, we generalize their proof to the case where A and Z satisfy the conditions of Assumption 2. Risks 2016, 4, 23 6 of 26 1 N For notational convenience, we introduce the process G = (G ) = G , ..., G with t2[0,T] t t j | j,k G := Z Y Y Y = Z y (t) , j 2 K, t 2 [0, T] (8) t å j,k t t t j,j k=1 k6=j Lemma 3. Let (X; A; Y; Z) be a general insurance contract, satisfying Assumption 2, then " # h i j j,k R R Y H j j jk N 1 N T u b b E D D G = E + H d A + dN G t t å å t T 0 0 u u 0 u j=1 k=1 ]t,T] ]t,T] S S S u u k6=j " # j j,k R R Y p (t,T) i,j Z N i N 1 N (9) = H E + p (t, u)d A + p (t, u)y (u)du F å å å t 0 0 i,j u 0 i,j j,k i=1 t j=1 ]t,T] k=1 ]t,T] S S S u u k6=j R R P(t,T)Y P(t,u) P(t,u) = E + dA + G du F H u u t t 0 0 0 ]t,T] ]t,T] S S S u u where the conditional transition probability process P = P(s, t) = p (s, t) , 0 s t T is deﬁned i,j i,j2K in Deﬁnition A.1. Proof. We proove the theorem by investigating the different conditional expectations separately. First note that because Y is taken to be F -measurable and S to be F-adapted, by Equation (A3) we get for every j 2 K " # " # " # j j N h i j N Y H Y Y T i i E G = E H E H jG G = E H p (t, T) G t t t t å t å t i,j 0 0 0 S S S T T i=1 T i=1 " # N j = H E p (t, T) F å t i,j i=1 T where G := F _F . t T Next, by Equation (A10) of Theorem A.7, we get for j, k 2 K, j 6= k that Z j,k Z j,k Z j,k Z Z Z u j,k u j,k u j dN = d M + H y (u)du u u u j,k 0 0 0 ]t,T] S ]t,T] S ]t,T] S u u u jk Note that because of Equations (A15) and (6), the integral-process with respect to M is a square integrable G-martingale. Hence, for every j, k 2 K, j 6= k, we have " # " # Z jk Z jk Z Z jk j u u E dN G = E H y (u)du G u t u j,k t 0 0 ]t,T] S ]t,T] S u u " # Z jk u j = E H y (u) G du u t j,k ]t,T] S " " # # Z jk = E E H y (u) G G du t t u j,k ]t,T] S " ! # Z jk = E y (u) H p (t, u) G du j,k å i,j t ]t,T] S i=1 " # jk = H E p (t, u)y (u) F du å i,j t t j,k ]t,T] S i=1 " # Z jk i u = H E p (t, u)y (u)du F å t i,j j,k ]t,T] S i=1 Risks 2016, 4, 23 7 of 26 by the conditional version of Fubini’s theorem, the deﬁnition of F-doubly stochastic Markov chains and hypothesis (H). j j Finally, for every j 2 K and for ﬁxed t 2 [0, T] we deﬁne A := d A , u 2 [t, T]. By u 0 v ]t,u] Proposition A.9 we get Z Z j j j j E H d A jG = E H d A jG t t u u u u ]t,T] ]t,T] j j j j j j e e e = E A H A H A d H jG T T t t u ]t,T] j j j j e e = E A H A d H jG = I I u t 1 2 T T ]t,T] with j j j j e e I := E[ A H jG ], I := E A d H jG 1 t 2 u t T T ]t,T] Since A is F -measurable, it follows by the hypothesis (H) that T T h h i i K h i j j j e e I = E A E H jG jG = H E A p (t, v)jF t t t 1 å t i,j T T T i=1 Again by the conditional version of Fubini’s theorem, hypothesis (H) and with the Kolmogorov forward Equation (A6) it follows that Z Z Z j j j j j e e e I = E A d H G = E A d M + A y (u)du G 2 u t u X ,j t u u u u ]t,T] ]t,T] ]t,T] " # " # Z Z K K j j k k e e = E A H y (u)du G = E A H y (u) G du t t u å u k,j u å u k,j ]t,T] ]t,T] k=1 k=1 " # h i e e = E A E H j G y (u) G du å t k,j t u u ]t,T] k=1 " ! # K K = H E A p (t, u)y (u) F du å å i,k k,j t t u ]t,T] i=1 k=1 " ! # K K = H E A p (t, u)y (u) du F å å i,k k,j t t u ]t,T] i=1 k=1 = H E A d p (t, u) F i,j t å t u ]t,T] i=1 Hence, by integration by parts and since p(t,) is continuous, we get e e I I = H E A p (t, T) A d p (t, u)jF 1 2 å T i,j i,j t t u ]t,T] i=1 e e = H E A p (t, t) + p (t, u)d A jF å t i,j i,j u t ]t,T] i=1 = H E p (t, u)d A jF å i,j u t ]t,T] i=1 This completes the proof. Risks 2016, 4, 23 8 of 26 Now we are ready to provide the Galtchouk-Kunita-Watanabe decomposition for payment processes of general insurance contracts. Theorem 4. Let (X, A, Y, Z) be a general insurance contract, satisfying Assumption 2, with discounted payment process D, deﬁned in Equation (2). Then the GKW-decomposition of the square-integrable discounted h i D D D b b b b value process U = (U ) with U = E D jG is given as T t t t2[0,T] t Z Z D D | b b U = U + aaa dm + bbb dM (10) u u u t 0 u ]0,t] ]0,t] where M is given by Equation (A9), m = (m ) is a square-integrable F-martingale, given by t2[0,T] " # Z Z P(0, T)Y P(0, u) P(0, u) m := E + dA + G du F (11) t u u t 0 0 0 S [0,T] S ]0,T] S u u and aaa, bbb are G-predictable R -valued processes deﬁned by F(t, T) + Z H aaa = L = Q (0, t)H , bbb = (12) t t t 1 N with H = ( H , ..., H ), t 2 [0, T], deﬁned by Equation (A7), Q(0, t), t 2 [0, T], deﬁned in Equation (A11), t t F(t, T), t 2 [0, T], deﬁned by " # Z Z P(t, T)Y P(t, u) P(t, u) F(t, T) := S E + dA + G du F (13) u u t 0 0 0 S ]t,T] S ]t,T] S u u b b and U = E[D ] = m H . T 0 0 0 Proof. The statement and the proof of this theorem can be found in [21] (Theorem 16.62). The authors there, however, prove Decomposition 10 only for t 2 [0, T). Because the integrals on the r.h.s. are not all continuous, it is a priori not clear if the decomposition also holds for U . Here we refer to [24] (Theorem 4.2.3) for an extension of the proof of [21] to the case t = T. 3.2. Risk Minimization for General Insurance Contracts with Deterministic Payment Structure In this section we focus on a more speciﬁc setting, where we specify the underlying ﬁnancial market and derive risk-minimizing hedging strategies for insurance contracts with deterministic payment structures. We start by specifying the underlying market. First of all, we assume the reference ﬁltration F = F to be the augmented ﬁltration, generated by some N-dimensional Brownian motion W. For computational reasons, particularly in the afﬁne setting of the next section, we set the dimension N of the Brownian motion equal to the number of states under consideration. 0 d Consider then a ﬁnancial market consisting of (d + 1) traded assets S = (S , ..., S ) , assumed t t t2[0,T] 1 d d b b b to be F-adapted, non-negative stochastic processes. Let S = (S , ..., S ) denote the R -valued t t t2[0,T] 1 d 0 i i 0 stochastic process of the primary assets S , ..., S , discounted with the asset S , i.e., S = S /S , t t t 0 0 i = 1, ..., d. Here S is taken to be continuous with S > 0 for all t 2 [0, T] and shall generally represent the value of a self-ﬁnancing portfolio on the primary assets. In the sequel, we assume S to be a local (F,P)-martingale, which particularly implies that the market model is arbitrage-free. Remark 2. The requirement that S is a local martingale may appear restrictive. However it is always satisﬁed if we choose S to be the numéraire portfolio deﬁned in [25], since we assume the underlying ﬁnancial market to contain only continuous asset price processes. Risks 2016, 4, 23 9 of 26 We could also start with a general situation where the discounted asset price processes are given by semimartingales. In this case one has to assume some technical conditions to guarantee the existence of the optimal strategy, see [26,27]. Here we prefer to avoid technical complications since our aim is to compute explicitly the risk-minimizing strategy when it exists. By the representation theorem with respect to Brownian motion it follows that there exists a N dN measurable map ssse : [0, T] R ! R , such that b b S = S + sss(s, S )dW t s s ]0,t] se Assumption 5. We assume that ss(t, S ) is a.s. left-invertible, i.e., that for almost every (w, t) 2 W [0, T] there exists an F -adapted N d-valued matrix GGG (w) such that GGG ssse(t, S ) = I . This particularly t t t N implies N d. From now on we focus on discount factors and insurance contracts with deterministic payment structure. Assumption 6. (1) Y is a deterministic vector in R . (2) The payment A = (A ) is of the form A = nnn(s)ds for some bounded deterministic function t t t2[0,T] nn : [0, T] ! R . NN (3) Z : [0, T] ! R is a bounded deterministic matrix-valued function. (4) S : [0, T] ! R is a deterministic continuous function. j,k (5) For every j, k 2 K, j 6= k, C := sup E y < ¥. u2[0,T] Assumption 6 particularly implies that the integrability conditions of Assumption 2 hold. Note also that the insurance contracts, given in Examples 1, 2 and 4 all satisfy (1), (2) and (3) of Assumption 6. The assumption on S being deterministic is applied very frequently in the literature, 0 rt e.g if P is assumed to be some risk-neutral probability measure and S = e for some constant r > 0. Remark 3. Here we assume constant interest rates for the sake of simplicity, since the focus of this paper is primarily to evaluate the role of a multi-state progression of the insured person on the risk-minimizing strategy. The following computations can be easily extended to the case of stochastic interest rates if S is assumed to be independent of X. In more general models, the investigation of dependency structures will become inevitable. This goes beyond the scope of the paper and is left to further research. Due to the representation theorem with respect to Brownian motion, for every u 2 [0, T] and i,j every i, j 2 K, there exists some xxx (u,) 2 L (W) such that i,j E p (0, u)jF = E p (0, u) + 1 (s)xxx (u, s)dW (14) t s i,j i,j ]0,u] ]0,t] Similarly, because of Assumption 6 (5), for every u 2 [0, T] and every i, j, k 2 K, j 6= k, there exists i,j,k 2 some qqq (u,) 2 L (W) such that h i h i j,k j,k i,j,k E p (0, u)y jF = E p (0, u)y + 1 (s)qq (u, s)dW (15) i,j u t i,j u s ]0,u] ]0,t] Risks 2016, 4, 23 10 of 26 Theorem 7. Given Assumptions 5 and 6, the unique risk-minimizing hedging strategy xxx = (xxx ) , t t2[0,T] characterized in Theorem B.5 for a general insurance claim (X; A; Y; Z), satisfying Assumption 6, is given as Z Z Z j,k N N N Y 1 Z j u i T i,j i,j i,j,k xxx = L xxx (T, t) + xxx (u, t)n du + qqq (u, t)du GGG (16) å å u å t t t 0 0 0 ]0,t] S [t,T] S [t,T] S u u i=1 j=1 T k=1 k6=j 0 D b b b x = U D xxx S (17) t t t t t D D b b where G is the left-inverse of the volatility matrix es(t, S ) and U = (U ) is the discounted value t t t t2[0,T] process of the cumulative payment process D. Proof. Because of Assumption 6, the i-th component m of the martingale m in Equation (11) is given as N h i h i i T m = E p (0, T)jF + E p (0, u)jF n du t t t å i,j i,j u 0 0 S [0,T] S j=1 T Z j,k N h i u j,k + E p (0, u)y jF du å i,j u ]0,T] S k=1 k6=j j Z Z j,k h i h i h i N N Y 1 Z j j,k = E p (0, T) + E p (0, u) n du + E p (0, u)y du å i,j i,j u å i,j u 0 0 0 S [0,T] S ]0,T] S u u j=1 T k=1 k6=j Z Z Z T i,j i,j x x + xx (T, s)dW + 1 (s)xx (u, s)dW n du s s u å ]0,u] 0 0 S ]0,t] [0,T] S ]0,t] j=1 T Z j,k Z i,j,k + 1 (s)qq (u, s)dW du å s ]0,u] ]0,T] S ]0,t] k=1 k6=j By Assumption 6, Fubini’s theorem and the Itô isometry it then follows for every i, j 2 K that " # Z Z Z Z i,j i,j 2 2 E 1 (s)kxxx (u, s)kn duds K E kxxx (u, s)k ds du ]0,u] u ]0,T] ]0,T] S ]0,T] ]0,T] " # Z Z i,j = K E xx (u, s)dW du ]0,T] ]0,T] h i = K E E p (0, u)jF E p (0, u) du 1 i,j i,j ]0,T] K T < ¥ for some constant K > 0. Due to Assumption 6 (5), we similarly have for every i, j, k 2 K, j 6= k that 2 3 Z Z j,k Z Z i,j,k 2 i,j,k 2 4 5 E 1 (s)kqqq (u, s)k duds K E kqqq (u, s)k ds du ]0,u] 2 ]0,T] ]0,T] S ]0,T] ]0,T] " # Z Z 2 i,j,k = K E qqq (u, s)dW du ]0,T] ]0,T] h i h i j,k = K E E p (0, u)y jF E p (0, u) du i,j u T i,j ]0,T] h i j,k 2 2 K E (y ) du ]0,T] K CT < ¥ 2 Risks 2016, 4, 23 11 of 26 for some constant K > 0. Therefore, we can apply the stochastic version of Fubini’s theorem, see [28], and obtain Z Z j,k N N h i Y 1 Z j j,k i T m = E p (0, T) + E p (0, u) n du + E p (0, u)y du t å i,j i,j u å i,j u 0 0 0 S [0,T] S ]0,T] S u u j=1 T k=1 k6=j Z j Z Z j,k N N 1 Z i,j i,j u T i,j,k x x q + xx (T, s) + xx (u, s)n du + qq (u, s)du dW å u å 0 0 0 S S S ]0,t] [s,T] [s,T] u u j=1 T k=1 k6=j This ﬁnally implies Z Z j,k N N Y 1 Z i T i,j i,j i,j,k dm = xxx (T, t) + xxx (u, t)n du + qqq (u, t)du GGG dS t å u å t t 0 0 0 S [t,T] S [t,T] S u u j=1 T k=1 k6=j The result then follows immediately with Theorem A.7 and the results of Theorem B.5. 3.3. Risk Minimization for General Insurance Contracts with Deterministic Payment Structure under an Afﬁne Speciﬁcation for the Intensities In the same setting as in Section 3.2, we now specify the risk-minimizing hedging strategies, computed in Theorem 7 within a general afﬁne setting for the intensities. In addition to Assumption 2 we also consider Assumption 8. (1) For every 0 t u T and every j, k 2 K, j 6= k, we assume the entries p (t, u) of the transition j,k matrix P(t, u) are of the form j,k y dv p (t, u) = 1 e (18) j,k j,k where y are the respective entries of the intensity matrix YY. j,k (2) For every u 2 [0, T] and every j, k 2 K, j 6= k, y is of the form j,k j,k | j,k y = (b ) mmm + c (19) j,k N j,k N where b 2 R , c 2 R and mmm = (mmm ) an R -valued afﬁne process as speciﬁed e.g., in [29] t t2[0,T] (Section 3 and Appendix A) or [30]. Here, mmm is a Markov process with respect to F , given as the strong solution to the SDE dmmm = ddd(t,mmm )dt + sss(t,mmm )dW (20) t t t where for t 2 [0, T], x 2 R and i, j 2 f1, ..., Ng 0 1 dd(t, x) = d (t) + (d (t)) x (21) | 0 1 [sss(t, x)sss(t, x) ] = [V (t)] + (V (t)) x (22) i,j i,j i,j 0 1 0 1 N NN NN NNN with coefﬁcient functions d , d , V and V , taking values in R ,R ,R and R , respectively. (3) The process mmm is such that C := sup E[kmm k ] < ¥ (23) u2[0,T] Risks 2016, 4, 23 12 of 26 This particularly implies that for every j, k 2 K, j 6= k, we have j,k sup E[(y ) ] < ¥ (24) u2[0,T] With these assumptions and under some technical conditions, presented in [30], we obtain for every 0 t u T and every i, j 2 K with i 6= j that i,j u i,j i,j y dv a (t)+(bbb (t)) mmm v u u t t (25) E p (t, u)jF = E[1 e jF ] = 1 e i,j t t u i,j N N y dv E [ p (t, u)jF ] = E[1 p (t, u)jF ] = 1 E[1 e jF ] å å i,i t i,j t t j=1 j=1 j6=i j6=i (26) i,j i,j N a (t)+(bbb (t)) mmm u u = 2 N + e j=1 j6=i Similarly, we obtain for every 0 t u T and every i, j, k 2 K with i 6= j, j 6= k h i h i h i h i R R u i,j u i,j j,k j,k j,k j,k y du y du u u t t E p (t, u)y jF = E (1 e )y jF = E y jF E e y jF i,j u t u t u t u t j,k e | j,k ea (t)+(bbb (t)) mmm | u e (27) u t = e (ea (t) + (bbb (t)) mmm ) u t i,j i,j | j,k j,k a (t)+(bbb (t)) mmm | u u b e (ba (t) + (bbb (t)) mmm ) u t h i h i u j,l j,k j,k j,k N y du E p (t, u)y jF = (2 N)E y jF + E e y jF j,j u t u t u t l=1 l6=j j,k e j,k ea (t)+(bbb (t)) mmm | u e (28) u t = (2 N)e (ea (t) + (bbb (t)) mmm ) u t j,l j,l j,k | j,k N a (t)+(bbb (t)) mmm | u u + e (ba (t) + (bbb (t)) mmm ) l=1 u t l6=j For every 0 t u T and every combination of i, j, k, l 2 K, considered in Equations (25)–(28), i,j i,j the functions a , bbb solve the ODEs i,j dbb 1 i,j 1 i,j i,j i,j u | | 1 b b b (t) = b d (t) bb (t) (bb (t)) V (t)bb (t) (29) u u u dt 2 i,j da 1 i,j 0 | i,j i,j | 0 i,j (t) = c d (t) bbb (t) (bbb (t)) V (t)bbb (t) (30) u u u dt 2 i,j i,j with terminal conditions a (u) = 0 and bbb (u) = 0, whereas the functions ea , bbb solve the ODEs u u u u dbb 1 u 1 | | 1 e e e (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (31) u u u dt 2 dea 1 0 | | 0 e e e (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (32) u u u dt 2 with terminal conditions ea (u) = 0 and bbb (u) = 0. k,l k,l i,j k,l k,l i,j e b e e b b b e b b The functions a , bb , a and bb , k, l 2 K, corresponding to a and bb or to a and bb for u u u u u u u u i, j 2 K as considered in Equations (25)–(28), then solve the ODEs k,l dbb k,l k,l u | | 1 e e e b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (33) u u u dt k,l dea k,l k,l u 0 | | 0 e e e b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (34) u u u dt Risks 2016, 4, 23 13 of 26 k,l k,l k,l k,l with terminal conditions ea (u) = c , bbb (u) = b and k,l dbbb k,l k,l u 1 | i,j | 1 b b (t) = d (t) bbb (t) (bbb (t)) V (t)bbb (t) (35) u u u dt k,l dba k,l k,l u 0 i,j | | 0 b b b b b (t) = d (t) bb (t) (bb (t)) V (t)bb (t) (36) u u u dt k,l k,l k,l k,l with terminal conditions ba (u) = c , bbb (u) = b . Note that with these speciﬁcations, we obtain by Equation (20) that for every 0 t < u T and every i, j 2 K, i 6= j R R R t i,j u i,j t i,j i,j i,j y dv y dv y dv a (t)+(bbb (t)) mmm v v v u u 0 t 0 E p (0, u)jF = 1 e E e jF = 1 e e t t i,j i,j i,j i,j i,j i,j | | b m b m i,j a (0)+(bb (0)) mm y dv a (s)+(bb (s)) mm | u u v u u 0 0 s =1 e e e (sss(s,mmm )) bbb (s)dW s u and N R N R u i,j t i,j i,j i,j y dv y dv a (t)+(bbb (t)) mmm v v u u t 0 t E [ p (0, u)jF ] = 1 E[1 e jF ] = 2 N + e e i,i t t å å j=1 j=1 j6=i j6=i t R i,j i,j s i,j i,j i,j | | i,j a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | u u v u u 0 s =2 N + e + e e (sss(s,mmm )) bbb (s)dW s u j=1 j6=i Moreover, as every martingale with respect to the Brownian ﬁltration F is continuous, we have for 0 u t T that E p (0, u)jF = p (0, u) = lim E p (0, u)jF t w i,j i,j i,j w%u i,j i,j i,j i,j i,j | | b m b m i,j a (0)+(bb (0)) mm y dv a (s)+(bb (s)) mm | u u v u u 0 0 s s m b = lim 1 e e e (ss(s,mm )) bb (s)dW s u w%u i,j i,j s i,j i,j i,j | | a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | i,j u u v u u 0 0 s s m b =1 e e e (ss(s,mm )) bb (s)dW s u and E [ p (0, u)jF ] = p (0, u) = lim E [ p (0, u)jF ] t w i,i i,i i,i w%u N R i,j i,j s i,j i,j i,j | | a (0)+(bbb (0)) mmm y dv a (s)+(bbb (s)) mmm | i,j u u v u u 0 0 s =2 N + e + e e (sss(s,mmm )) bbb (s)dW s u j=1 j6=i Hence, for arbitrary u, t 2 [0, T], we obtain i,j i,j E p (0, u)jF = c (u) + J (s, u)1 (s)dW (37) i,j t s [0,u] 1 1 i i E [ p (0, u)jF ] = c (u) + J (s, u)1 (s)dW (38) t s i,i [0,u] 2 2 0 Risks 2016, 4, 23 14 of 26 where for u, s 2 [0, T] and every i, j 2 K, i 6= j i,j i,j i,j a (0)+(bbb (0)) mmm u u 0 c (u) := 1 e i,j i,j i,j i,j b m i,j y dv a (s)+(bb (s)) mm | v u u 0 s J (s, u) := e e (sss(s,mmm )) bbb (s) 1 s u i,j i,j i a (0)+(bbb (0)) mmm u u c (u) := 2 N + e j=1 j6=i s i,j i,j i,j i y dv a (s)+(bbb (s)) mmm | i,j v u u s J (s, u) := e e (sss(s,mmm )) bbb (s) 2 s u j=1 j6=i Similarly we get for 0 t < u T and every i, j, k 2 K with i 6= j, j 6= k that h i h i R R i,j i,j t u j,k j,k j,k y dv y dv v v 0 t E p (0, u)y jF = E y jF e E e y jF u t u t u t i,j i,j i,j i,j | j,k | j,k e j,k j,k e b m b m a (t)+(bb (t)) mm | y dv a (t)+(bb (t)) mm | u e v u u b u t 0 t b m b m = e (ea (t) + (bb (t)) mm ) e e (ba (t) + (bb (t)) mm ) u u u t u t i,j i,j j,k j,k e | | j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | e u u b u 0 0 = e (ea (0) + (bbb (0)) mmm ) e (ba (0) + (bbb (0)) mmm ) u u u 0 u 0 | j,k j,k e j,k ea (s)+(bbb (s)) mmm | | u e e e u s + e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) s u s u u s j,k i,j i,j | j,k j,k y dv a (s)+(bbb (s)) mmm | | i,j v u u b 0 s b b m s m b + e e (a (s) + (bb (s)) mm )(ss(s,mm )) bb (s) u s s u s j,k i,j i,j | j,k y dv a (s)+(bbb (s)) mmm | v u u b 0 s + e e (sss(s,mmm )) bbb (s) dW s u and h i h i N R h R i t j,l u j,l j,k j,k j,k y dv y dv v v 0 t E p (0, u)y jF = (2 N)E y jF + e E e y jF j,j u t u t å u t l=1 l6=j j,l j,l e | j,k | j,k j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | u e u b u 0 u 0 = (2 N)e (ea (0) + (bbb (0)) mmm ) + e (ba (0) + (bbb (0)) mmm ) u u u 0 u 0 l=1 l6=j | j,k j,k j,k ea (s)+(bbb (s)) mmm | | u e e e u s s m e b m b b + (2 N) e (ss(s,mm )) (a (s) + (bb (s)) mm )bb (s) + bb (s) s u s u u s j,l j,l j,l | j,k j,k y dv a (s)+(bbb (s)) mmm | | j,l v u b 0 u s + e e (ba (s) + (bbb (s)) mmm )(sss(s,mmm )) bbb (s) å u u s s u l=1 l6=j s j,l j,l j,l | j,k y dv a (s)+(bbb (s)) mmm | v u b 0 u s + e e (sss(s,mmm )) bbb (s) dW s u Note that by Jensen’s inequality and Assumption 8 (4), we get for every 0 t < u T and every i, j, k 2 K with j 6= k that j,k j,k j,k 2 2 2 2 E[E[ p (0, u)y j F ] ] E[E[ p (0, u) (y ) j F ]] E[(y ) ] C i,j u t i,j u t u Finally, with the same limit-arguments as above, we obtain for arbitrary u, t 2 [0, T] and every i, j, k 2 K with i 6= j, j 6= k that h i j,k i,j,k E p (0, u)y jF = c (u) + J (s, u)1 (s)dW (39) i,j u t 3 s 3 [0,u] h i j,k j,k E p (0, u)y jF = c (u) + J (s, u)1 (s)dW (40) j,j u t 4 s [0,u] 0 Risks 2016, 4, 23 15 of 26 where for u, s 2 [0, T], i, j, k 2 K with i 6= j, j 6= k i,j i,j e | j,k | j,k j,k j,k ea (0)+(bbb (0)) mmm | a (0)+(bbb (0)) mmm | u u e b u 0 u 0 c (u) := e (ea (0) + (bbb (0)) mmm ) e (ba (0) + (bbb (0)) mmm ) 3 u u u 0 u 0 | j,k j,k i,j,k e j,k e b m a (s)+(bb (s)) mm | | u e e e u s J (u, s) := e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) 3 s u s u u s j,k i,j i,j | j,k j,k j,k y dv a (s)+(bbb (s)) mmm | | i,j v u b b 0 u s s m b m b b + e e (ss(s,mm )) (ba (s) + (bb (s)) mm )bb (s) + bb (s) s u s u u e | j,k j,k ea (0)+(bbb (0)) mmm | u e u 0 c (u) := (2 N)e (ea (0) + (bbb (0)) mmm ) 4 u u 0 j,l j,l | j,k j,k a (0)+(bbb (0)) mmm | u 0 b + e (ba (0) + (bbb (0)) mmm ) å u u 0 l=1 l6=j | j,k j,k j,k e j,k e b m a (s)+(bb (s)) mm | | u e e e u s J (u, s) := (2 N)e (sss(s,mmm )) (ea (s) + (bbb (s)) mmm )bbb (s) + bbb (s) 4 s u s u u N R s j,l j,l j,l | j,k j,k j,k y dv a (s)+(bbb (s)) mmm | | j,l v u b b 0 u s s m b b m b b + e e (ss(s,mm )) (a (s) + (bb (s)) mm )bb (s) + bb (s) å u s u s u u l=1 l6=j We can now apply these results to compute explicitly the risk minimizing strategy as given in Theorem 4 and more speciﬁcally by Theorem 7 in the Brownian setting in consideration. From i,j i,j,k Equations (37)–(40) it follows immediately that the processes x (u,) and q (u,), i, j, k 2 K, j 6= k, of Equations (14) and (15) are given as i,j i,j i,i i x (u, t) = J (u, t) , x (u, t) = J (u, t) 1 2 i,j,k j,k i,j,k j,j,k q (u, t) = J (u, t) , q (u, t) = J (u, t) 3 4 Moreover, with Equations (25)–(28) the i-th component F (t, T), i 2 K, of F(t, T) in Equation (13) is given as h i 0 0 S (t) S (t) i i N j F (t, T) = Y E p (t, T)jF + å Y E p (t, T)jF 0 i,i t 0 i,j t j=1 S (T) S (T) j6=i h i R R 0 0 T S (t) T S (t) i j + E p (t, u)jF n (u)du + å E p (t, u)jF n (u)du 0 t 0 t i,i i,j t j=1 t S (u) S (u) j6=i h i T S (t) i,k i,k + Z (u)E p (t, u)y (u)jF du å t 0 i,i k=1 t S (u) k6=i h i T S (t) N N j,k j,k + Z (u)E p (t, u)y (u)jF du å å t 0 i,j j=1 k=1 t S (u) k6=j j6=i 0 0 i,j i,j i,j i,j S (t) S (t) i N a (t,T)+bbb (t,T)mmm N j a (t,T)+bbb (t,T)mmm t t = Y 2 N + e + Y (1 e ) å å 0 0 j=1 j=1 S (T) S (T) j6=i j6=i R 0 i,j i,j T S (t) N a (t,u)+bbb (t,u)mmm i + 2 N + e n (u)du j=1 t S (u) (41) j6=i 0 i,j i,j T S (t) N a (u)+bbb (u)mmm j t t + å 1 e n (u)du j=1 t S (u) j6=i i,k T S (t) e N i,k ea(t,u)+bbb(t,u)mmm i,k t b m + å Z (u) (2 N)e (ea (t, u) + bb (t, u) mm ) k=1 t S (u) t k6=i i,l i,k i,l N a (t,u)+bbb (t,u)mmm i,k b b m + å e (a (t, u) + bb (t, u) mm ) du l=1 l6=i R 0 j,k T S (t) N N j,k ea(t,u)+bbb(t,u)mmm j,k + Z (u) e (ea (t, u) + bbb (t, u) mmm ) å å j=1 k=1 t t S (u) k6=j j6=i i,j j,k i,j a (t,u)+bbb (t,u)mmm j,k e (ba (t, u) + bbb (t, u) mmm ) du and can hence be expressed explicitly in terms of mmm. Risks 2016, 4, 23 16 of 26 3.4. Application: The Expected Cumulative Payment in an Ornstein-Uhlenbeck Framework In this section, we illustrate in a speciﬁc example how the expected (discounted) cumulative payment E[D ] from Lemma 3 can be calculated by using the explicit expression of the components F (t, T) from the previous section and the connection established through Equation (9). In the following, we regard a speciﬁc insurance product, namely an income protection insurance, and for simplicity we assume that the corresponding process X of the insured person can only take the three states K = f1 = healthy, 2 = sick/unﬁt for work, 3 = deathg, where state 3 is absorbing. The corresponding Version May 27, 2016 submitted to Risks 18 of 28 transitions are illustrated in Figure 1. 1 2 3 Figure 1. Possible transitions for an insured person’s process in an income protection insurance with Figure the three 1. states Possible K = transitions f1 = healthy, for an 2 insur = sick/unﬁt ed person’s for work, process 3 = indeath an income g withpr absorbing otection insurance state 3. with the three states K = {1 = healthy, 2 = sick/unﬁt for work, 3= death} with absorbing state 3. j,k j,k 3 j,k j,k Hence, for b 2 R , c 2 R from Equation (19) we set b = 0 0 0 and c = 0 for (j, k) 2 / I := f(1, 2), (2, 1), (1, 3), (2, 3)g. In the following, we assume a speciﬁc form for the Markov processμμμ from (19), namely a simple, In the following, we assume a speciﬁc form for the Markov process mmm from Equation (19), namely 3-dimensional Ornstein-Uhlenbeck (OU) process with corresponding SDE a simple, 3-dimensional Ornstein-Uhlenbeck (OU) process with corresponding SDE μ μ σ μ ξ dμμ = aμμ dt + σσdW , μμ = ξξ ∈ R . t t 0 dmmm = ammm dt + sssdW , mmm = xxx 2 R t t 0 Note that for the OU process it is well-known that condition (23) for the expectation is fulﬁlled. A Note that for the OU process it is well-known that Condition (23) for the expectation is fulﬁlled. major drawback, though, of choosing this process for the intensity is the undesirable feature that A major drawback, though, of choosing this process for the intensity is the undesirable feature that it it can become negative with positive probability. However, [31] provide calibrated parameters for can become negative with positive probability. However, [31] provide calibrated parameters for which which the probability of negative values for μμμ turns out to be negligible. For this reason, we choose the probability of negative values for mmm turns out to be negligible. For this reason, we choose similar similar parameter values and set parameter values and set   0 0.0003 0 0 1   | 0.0003 0 0 | a = (0.07, 0.11, 0.09) ,σσσ = 0 0.0007 0 and ξξξ = (0.1, 0.1, 0.1) .   B C | | s x a = (0.07, 0.11, 0.09) ,ss = 0 0.0007 0 and xx = (0.1, 0.1, 0.1) @ A 0 0 0.0005 0 0 0.0005 Hence, equations (21) and (22) simplify and yield to Hence, Equations (21) and (22) simplify and yield to δδ(t, x) = (d ) x 1 | ddd(t, x) = (d ) x | 0 [σσσ(t, x)σσσ(t, x) ] = [V ] , i,j i,j | 0 [sss(t, x)sss(t, x) ] = [V ] i,j i,j with     with 0 a 0 0 1 0 σσσ 0 0 1     1 0 a 0 0 sss 2 0 0 d = and V = 11 σ .  0 a 0   0 σσ 0  B C B C 1 0 d = 0 a 0 and V = 0 sss 2 0 @ A @ A 0 0 a 0 022 σσσ 0 0 a 0 0 sss As a consequence of these assumptions, the ODEs (29)-(36) substantially simplify and can be explicitly As a consequence of these assumptions, the ODEs Equations (29)–(36) substantially simplify and solved, see e.g. [32]. For example, for every 0 ≤ t ≤ u ≤ T and every choice of (i, j) ∈ I , the ODEs can be explicitly solved, see e.g., [32]. For example, for every 0 t u T and every choice of (29) and (30) now are given by (i, j) 2 I , the ODEs Equations (29) and (30) now are given by i,j dββ i,j 1 i,j i,j (t) = b − d ββ (t) , dbbb i,j 1 i,j dt (t) = b d bbb (t) dit,j dα 1 i,j i,j | 0 i,j i,j (t) = c − (βββ (t)) V βββ (t) , da 1 u u i,j i,j | 0 i,j dt 2 (t) = c (bbb (t)) V bbb (t) u u dt 2 i,j i,j i,j i,j (k) with terminal conditions α (u) = 0 and βββ (u) = 0. For the k-th components of βββ (t), i.e. β (t), u u u u one obtains the explicit solutions i,j i,j (k) 1 β (t) = 1− exp(d (u− t) , k ∈ {1, 2, 3} , kk kk Risks 2016, 4, 23 17 of 26 i,j i,j i,j i,j (k) with terminal conditions a (u) = 0 and bbb (u) = 0. For the k-th components of bbb (t), i.e., b (t), u u u u one obtains the explicit solutions i,j i,j (k) 1 b (t) = 1 exp(d (u t) , k 2 f1, 2, 3g kk kk and i,j 3 2 2 1 s (b ) 2 i,j i,j k k 1 a (t) = c (u t) + (u t) 1 exp(d (u t) u å kk 1 1 (d ) d k=1 kk kk + 1 exp(2d (u t) kk 2d kk Similarly, the ODEs of the remaining Equations (31)–(36) can be derived. j,k j,k Next, we need to specify all remaining parameters, i.e., b , c for (j, k) 2 I , the components A, rt Y, Z of the quadruple (X; A; Y; Z) and the discounting factor S . For simplicity, we set S = e with a 0 0 constant interest rate r and choose 0 1 0 1 0 1 3 1 2 B C B C B C 1,2 2,1 1,3 2,3 1,2 2,1 1,3 2,3 b = b = , b = , b = , c = c = 1 , c = 0.1 , c = 0.2 @3A @1A @2A 3 1 2 j,k We have chosen the components of b to be equal in order to emphasize the general dependence j,k of the process y in Equation (19) on mm and not on the speciﬁc linear combination. t t Furthermore, similar to Example 2, we set 0 1 0 0 z B C Z := 0 0 z , Y 0 , and A = C P , t 2 [0, T] @ A t t t 0 0 0 with C and P representing the cumulative state-dependent claim payments (e.g., annuities) and t t insurance premiums up to maturity, respectively, and z the “immediate” claim payment if the insured person dies. More precisely, we assume monthly equal insurance premiums and claim payments equal to 1, if the insured person is in the states 1 (healthy) or 2 (sick/unﬁt for work), respectively, and which are paid at the end of each month in a proportional way. Let the payment dates 0 < T < T < . . . denote 1 2 the ﬁnal day of each month, i.e., T = , then we set 0 1 1/Dt B C nnn(t) = 1/Dt @ A with Dt = 1/12 and A = nnn(s)ds. Clearly, with these speciﬁcations Assumption 6 is fulﬁlled. Finally, assuming that the insured persons process X starts in the state 1 (healthy) at t = 0, one obtains H = (1, 0, 0). Based on the explicit result for F (0, T) from Equation (41), now we are able to calculate the expected (discounted) cumulative payment E[D ] from Lemma 3 in t = 0, depending on the claim payment amount z, which is payed when the insured person dies, and on the constant interest rate r. The corresponding integrals involved in Equation (41) are approximated using the integrate function in the statistical software program R ([33]). Figure 2 illustrates the expected cumulative payment E[D ] as a function of the claim payment amount z, a time horizon of one year (T = 1) and three different values of the constant interest rate Risks 2016, 4, 23 18 of 26 r. In Figure 3, the expected cumulative payment E[D ] are displayed against the time horizon T, for three different values of the claim payment amount z and for a constant interest rate r = 0.1. r = 0.05 r = 0.1 r = 0.2 0 20 40 60 80 100 claim payment amount z Figure 2. Expected (discounted) cumulative payments E[D ] as a function of the claim payment amount z and for three different constant interest rate values r for a time horizon of one year, i.e., T = 1. z = 1 z = 10 z = 20 5 10 15 20 time horizon T (in years) Figure 3. Expected (discounted) cumulative payments E[D ] as a function of the time horizon T (in years) and three different claim payment amounts z and for a constant interest rate r = 0.1. 4. Conclusions In this paper, we consider pricing and hedging of general insurance contracts by means of risk minimization. We model the individual progress in time of visiting an insurance policy’s states by using F-doubly stochastic Markov chains. In this way we are able to consider a multi-state setting to describe different types of insurance beneﬁts and to include the inﬂuence of market conditions and external risk factors on the evolution of the insured person among the policy’s states as well as on the insurance beneﬁts, when they are linked to some ﬁnancial performance. We explicitly provide the risk-minimizing strategy for an insurance contract in a Brownian ﬁnancial market setting and E[D ] E[D ] −50 0 50 100 150 200 0 20 40 60 80 100 Risks 2016, 4, 23 19 of 26 specify it within an afﬁne structure for the intensity. The results are illustrated by a numerical example, which shows how this technical setting can actually be easily implemented. Acknowledgments: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No [228087]. Author Contributions: All authors have contributed equally to all aspects of this paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. Appendix A. F-Doubly Stochastic Markov Chains In this section we introduce brieﬂy to some basic properties of F-doubly stochastic Markov chains, which we are going to use in the sequel. Main references are [12,21]. On a probability space (W,G ,P), let X = (X ) be a right-continuous stochastic process with t2[0,T] X X state space K := f1, ..., Ng. We denote by F the ﬁltration generated by X, i.e., F = s(X(u) : u t) for all t 2 [0, T], and consider the ﬁltration G to be the enlargement of F through some reference X X ﬁltration F, i.e., we assumeG = F _F for all t 2 [0, T]. Further, we setG = F _F , t 2 [0, T] and t t t t t assume that all ﬁltrations satisfy the usual conditions of completeness and right-continuity, see [28]. Deﬁnition A.1. A process X = (X ) is called an F-doubly stochastic Markov chain with state space K, t2[0,T] if there exists a family of stochastic matrices P(s, t) = [ p (s, t)] , 0 s t T j,k j,k2K such that (1) the matrix P(s, t) is F -measurable, and P(s, .) is progressively measurable, (2) for every j, k 2 K we have 1 P(X = k j G ) = 1 p (s, t) (A1) t s fX =jg fX =jg j,k s s The process P is called the conditional transition probability process of X. By Deﬁnition A.1 we can see that the class of F-doubly stochastic Markov chains contains Markov chains, compound Poisson processes with integer-valued jumps, Cox processes as in [34] and processes of rating migration as in [35]. The adjective “double” refers to the fact that there are two sources of uncertainty in their deﬁnition. We remark that an F-doubly stochastic Markov chain is a different object than a doubly stochastic Markov chain which is a Markov chain with a doubly stochastic transition matrix. Furthermore, in [12] it is shown that F-doubly stochastic Markov chains are a subclass of F-conditional G = F _ F Markov chains. In particular, F-doubly stochastic Markov chains behave like time inhomogeneous Markov chains conditioned on F , i.e., if we know all the information concerning the underlying risk factors. Deﬁnition A.2. We say that a state N 2 K is an absorbing state, if p (s, t) = 0 for all 0 s < t T and N,j all j 2 K with j 6= N. Proposition A.3. Let X be an F-doubly stochastic Markov chain with transition matrices P(s, t), then for every 0 s < t < u T we have P(s, u) = P(s, t)P(t, u) a.s. (A2) Risks 2016, 4, 23 20 of 26 Proposition A.4. If X is an F-doubly stochastic Markov chain, then for every bounded,F -measurable random variable Y and for each t 2 [0, T], we have E[YjG ] = E[YjF ] (A3) t t Property Equation (A3) is well-known in the context of survival analysis and credit risk as hypothesis (H) or immersion property. According to Proposition A.4, F-martingales remain martingales with respect to the enlarged ﬁltration G. If we think of a martingale as a process describing a fair game, this property means that the additional information contained in G does not change the valuation of processes which are considered fair by taking in account only the information F. Another property, which makes the class of F-doubly stochastic Markov chains interesting for applications is that they may admit matrix-valued stochastic intensity processes in the following sense. Deﬁnition A.5. An F-doubly stochastic Markov chain X with state space K is said to have an intensity, if h i j,k there exists an F-adapted matrix-valued stochastic process Y Y Y = (Y Y Y ) with Y Y Y = y such that t t t2[0,T] j,k2K (1) Y Y Y is integrable, i.e., j,j jy jds < ¥ (A4) ]0,T] j2K (2) Y Y Y satisﬁes the following conditions: j,k j,j j,k (A5) y 0 8j, k 2 K, j 6= k, y = å y 8j 2 K, t 2 [0, T] k6=j t t t P(v, t) I = Y Y Y(u)P(u, t)du 8v t (Kolmogorov backward equation) ]v,t] (A6) P(v, t) I = P(v, u)Y Y Y(u)du 8v t (Kolmogorov forward equation) ]v,t] A process Y Y Y, satisfying the above conditions, is called an intensity of the F-doubly stochastic Markov chain X. Theorem A.6. Let (Y Y Y ) be an F-adapted N N matrix-valued stochastic process, satisfying the t2[0,T] Conditions (A4) and (A5) of Deﬁnition A.5. Then there exists an F-doubly stochastic Markov chain X with intensity (Y Y Y ) . t2[0,T] For j 2 K, let H := 1 , t 2 [0, T] (A7) t fX =jg 1 N | be the indicator function for X, being in state j at time t and denote by H = ( H , ..., H ) the t t jk jk corresponding N-variate vector. Moreover, for j, k 2 K, j 6= k, let N = (N ) with t t2[0,T] jk j j k k N := H d H = H 4 H (A8) t u u u u ]0,t] 0<ut deﬁne the counting processes of the jumps of X from state j to k up to time t, t 2 [0, T]. The following theorem provides a martingale characterization of F-doubly stochastic Markov chains and is the core connection of the theory of F-doubly stochastic Markov chains and the counting process theory, underlying for example several estimation schemes for intensity processes, see [13]. Risks 2016, 4, 23 21 of 26 Theorem A.7. Let X = (X ) be a stochastic process with state space K and Y Y Y = (Y Y Y ) be t t t2[0,T] t2[0,T] a matrix-valued process, satisfying Equations (A4) and (A5) of Deﬁnition A.5. The following conditions are equivalent: (i) X is an F-doubly stochastic Markov chain. (ii) The process M = (M ) with t2[0,T] M := H YY H du (A9) t t u u ]0,t] is a G-local martingale. jk jk (iii) For j, k 2 K, j 6= k, the processes M = ( M ) with t2[0,T] jk jk j j,k M := N H y du (A10) u u t t ]0,t] are G-local martingales. (iv) The process L = (L ) with t2[0,T] L := Q(0, t) H t t is a G-local martingale. Here Q(0, t) is a unique solution to the random integral equation dQ(0, t) = YY Q(0, t)dt, Q(0, 0) = I (A11) Note that then L = H + Q (0, u)dM , t 2 [0, T] (A12) t 0 u ]0,t] Remark A.1. (1) For every t 2 [0, T], the matrix Q(0, t) is the unique inverse matrix of P(0, t). More generally, for 0 s t T, we denote by Q(s, t) the unique inverse matrix of P(s, t). The existence and further properties of the family Q(s, t) is given in [12]. It follows immediately from Equation (A2) that for every 0 s < t < u T, we have P(t, u) = Q(s, t)P(s, u) (A13) jk (2) As the processes M, L and M , j, k 2 K, j 6= k, are G-adapted, they are also G-local martingales. Corollary A.8. For every j, k 2 K, j 6= k, and for every t 2 [0, T] we have jk jk [ M ] = N (A14) jk h M i = H y (u)du (A15) u j,k ]0,t] j j k,j N k Moreover, with M = H y H du, j 2 K, t 2 [0, T], we have u u t t k=1 ]0,t] j k j jk j 2 N [ M ] = (D H ) = (N + N ) t å å 0<st s k=1 t t k6=j R R (A16) j N k h M i = å H y (u)du H y (u)du t u u k,j j,j k=1 ]0,t] ]0,t] k6=j Risks 2016, 4, 23 22 of 26 jk Proof. Equalities (A14) and (A15) follow directly by the deﬁnition of M in Equation (A10). k,j Moreover, we observe that å H y du is a continuous ﬁnite variation process. It k2K u u ]0,t] t2[0,T] follows that j k j jk j 2 [ M ] = (D H ) = N + N t å s å t t 0<st k=1 k6=j j j j,k as (D H ) counts the jumps of X into and out of the state j up to time t. As H y du s u u 0<st ]0,t] t2[0,T] jk is the compensator of N it follows that Z Z k,j j j,k j k h M i = H y du + H y du t u u u å u ]0,t] ]0,t] k=1 k6=j Z Z k,j j j,j = H y du H y du å u u u u ]0,t] ]0,t] k=1 k6=j where the last equality follows from Equation (A5). This ends the proof. Proposition A.9. Let X be an F-doubly stochastic Markov chain with intensity and jump times t := 0 and t := infft < t T : X 6= X g (A17) k k1 t t k1 Then every jump time t , k 1, avoids F-stopping times, i.e., P(t = $) = 0 for every F-stopping time k k $, provided that t < ¥ a.s.. The following proposition is the crucial result in order to compute the risk-minimizing strategies for general insurance claims which we provide in Section 3. Proposition A.10. Let X be an F-doubly stochastic Markov chain. Then the local martingale M, given in Equation (A9), is orthogonal to every F-local martingale N, in the sense that for each i 2 K, the product M N is a G-local martingale. i i Proof. First note that M is a ﬁnite variation local martingale. Its sequence (te ) of jump times with k0 te := 0 and i i i i e e t := infft > t j M 6= M g , k 1 t t k k1 is a subsequence of the jump times (t ) of X, as given by Equation (A17). As the jump times of j j0 the càdlàg local martingale N are F-stopping times, the processes M and N have almost surely no common jumps due to Proposition A.9. This implies that for all t 2 [0, T] we have i i [ M , N] = M N + D M DN = 0 t s 0 0 å s 0<st and ends the proof. Remark A.2. It is easily seen that hazard-rate models, as applied frequently in the context of credit risk or life insurance, are particular examples of F-doubly stochastic Markov chains, provided they satisfy hypothesis (H). A thorough description of this relation is given in [24]. Risks 2016, 4, 23 23 of 26 Appendix B. Risk-Minimization for Payment Processes The following survey of risk-minimization for payment processes is borrowed to some extend from [1], as well as [22]. As in the foregoing sections, we provide the results with respect to a general numéraire process S such that one could also consider e.g., the P-numéraire portfolio as discount factor, see [25]. The results base on the proofs, given in [4] for the case of European type contingent claims and in [5,36,37] (Chapter 4) for the case of payment processes. In the market model, deﬁned in Section 3.2, we would like to ﬁnd a hedging strategy for a b b G-adapted, square integrable payment process D = (D ) , representing the cumulative discounted t2[0,T] payments up to time t, t 2 [0, T]. Note that if an undiscounted cumulative payment stream D = (D ) is a stochastic process of t2[0,T] ﬁnite variation and we have djDj < ¥ with jDj denoting the absolute variation process of D, [0,T] then D is given as D = dD (B1) t u [0,t] S D D Deﬁnition B.1. If djDj < ¥, then the value process U = (U ) of a payment process D is 0 u t t2[0,T] [0,T] deﬁned as h i D 0 0 U := S E D G = S E dD G (B2) T t u t t t t [0,T] S Since the market is not necessarily complete, it is in general not possible to ﬁnd a self-ﬁnancing hedging strategy that perfectly replicates the discounted cumulative payment process D. In this context, the idea of risk-minimization is to relax the self-ﬁnancing assumption, allowing for a wider class of admissible strategies, and to ﬁnd an optimal hedging strategy with “minimal risk” within this class of strategies that perfectly replicate D. For the local martingale S, we denote ( ) | | 2 1 d x x x x L (S) := xx = (x , ..., x ) xx is G-predictable, E xx d[S] xx < ¥ (B3) t s s t2[0,T] [0,T] It is well known that for every xxx 2 L (S), the process xxx dS is a square [0,t] t2[0,T] integrable martingale. In the following we now explain how to ﬁnd the risk-minimizing strategy and explain in what sense this strategy is optimal. We begin with some deﬁnitions. 2 0 2 0 Deﬁnition B.2. An L -strategy is a pair jjj = (xxx , x ), such that xxx 2 L (S) and x is a real-valued G-adapted process, such that the discounted portfolio value process | 0 b b V = xx S + x , t 2 [0, T] t t is right-continuous and square integrable. 2 j For an L -strategy jjj the discounted (cumulative) cost process C is deﬁned as j j b b b b C := V xxx d S + D , t 2 [0, T] s t t t s ]0,t] describing the accumulated costs of the trading strategy jjj during [0, t], including the payments D . Note that V should therefore be interpreted as the discounted value of the portfolio jjj held at time t t t Risks 2016, 4, 23 24 of 26 b b after the payments D have been made. In particular, V is the discounted value of the portfolio upon settlement of all liabilities, and a natural condition is then to restrict to 0-admissible strategies, satisfying V = 0 P-a.s. The risk process of jj is given by the conditional expected value of the squared future costs j j j b b R = E[(C C ) jG ], t 2 [0, T] (B4) t t and is taken as a measure of the hedger ’s remaining risk. We would like to ﬁnd a trading strategy that minimizes the risk in the following sense. 2 0 Deﬁnition B.3. An L -strategy jjj = (xxx , x ) is called risk-minimizing for the discounted payment process D, je j 2 0 e e b b if for any L -strategy jjj = (xxx, x ) such that V = V = 0 P-a.s., we have T T j je R R P-a.s., t 2 [0, T] t t i.e., j minimizes pointwise the risk process introduced in Equation (B4). The key to ﬁnding the strategy with minimal risk in our setting is the so-called Galtchouk-Kunita-Watana decomposition. b b Deﬁnition B.4. Given a square integrable martingale U 2 M and the local martingale S, the b b Galtchouk-Kunita-Watanabe decomposition for U with respect to S is given as U | U b b b U = U + (JJJ ) dS + L , t 2 [0, T] (B5) t s 0 s t ]0,t] U 2 D where JJJ 2 L (S) and L is a square integrable martingale null at 0 which is strongly orthogonal to the space 2 | 2 b b b I (S) of all integral processes y y y dS with y y y 2 L (S). [0,t] t2[0,T] 2 2 It is well known that the set I (S) is a closed stable subset of M , the set of all square integrable martingales, zero at 0. Due to Jensen’s inequality and the fact that D is square-integrable, the discounted value process D U U = is a square-integrable martingale and may be decomposed according to Equation (B5). Theorem B.5. For every (discounted) square integrable payment stream D, there exists a unique 0-admissible 2 0 risk-minimizing L -strategy jjj = (xxx, x ), given by xxx := xxx t t 0 D D | b b b x := U D (xx ) S t t t t t with discounted portfolio value process j b b b b b b V = E[D jG ] D = E[D jG ] + xx dS + L D t t 0 s t T T s t ]0,t] discounted optimal cost process b b j j D D b b C = E[D jG ] + L = C + L T 0 t t t 0 and minimal risk process j b b D D 2 R = E[(L L ) jG ] t T t Risks 2016, 4, 23 25 of 26 D D D t 2 [0, T], where xxx and L are given by Equation (B5) for the square integrable martingale U . Proof. See [26] for the single payoff case or [5,36] for the extension to the case of payment streams. Note that the approach, described above, relies heavily on the fact that the discounted asset prices are local martingales under the measure P. In a more general setting, when the vector of discounted asset is a semimartingale under P, one has to apply the local risk-minimization technique, see [36] or [37] (Chapter 4). For more information on (local) risk-minimization and other quadratic hedging approaches we refer to the survey paper of [26]. References 1. Biagini, F.; Schreiber, I. Risk minimization for life insurance liabilities. SIAM J. Financ. Math. 2013, 4, 243–264. 2. Biagini, F.; Rheinländer, T.; Widenmann, J. Hedging mortality claims with longevity bonds. ASTIN Bull. 2013, 43, 123–157. 3. Blake, D.; Cairns, A.J.G.; Dowd, K. The birth of the life market. Asia-Pac. J. Risk Insur. 2008, 3, 6–36. 4. Föllmer, H.; Sondermann, D. Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics; Mas-Colell, A., Hildenbrand, W., Eds.; North Holland: Haarlem, The Netherlands, 1986; pp. 205–223. 5. Møller, T. Risk-Minimizing Hedging Strategies for Insurance Payment Processes. 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Mortality Risk via afﬁne stochastic intensities: Calibration and empirical relevance. Belg. Actuar. Bull. 2008, 8, 5–16. 32. Filipovic, ´ D. Term-Structure Models; Springer: Berlin, Germany, 2009. 33. Team, R.C. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing: Vienna, Austria, 2015. 34. Lando, D. On Cox processes and credit risk securities. Rev. Deriv. Res. 1998, 2, 99–120. 35. Bielecki, T.; Rutkowski, M. Credit Risk: Modelling, Valuation and Hedging, 2nd ed.; Springer-Finance, Springer: Berlin, Germany, 2004. 36. Schweizer, M. Local Risk-Minimization for Multidimensional Assets and Payment Streams. Banach Center Publications: Warszawa, Poland, 2008; Volume 83, pp. 213–229. 37. Barbarin, J. Valuation, Hedging and the Risk Management of Insurance Contracts. Ph.D. Thesis, Université Catholique de Louvain, Louvain, Belgium, 2008. c 2016 by the authors; licensee MDPI, Basel, Switzerland. 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Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains

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Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains

Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains

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