Access the full text.
Sign up today, get DeepDyve free for 14 days.
(2014)
Viability and martingale measures in jump diffusion markets under partial information
N. Karoui, M. Quenez (1995)
Dynamic Programming and Pricing of Contingent Claims in an Incomplete MarketSiam Journal on Control and Optimization, 33
David Kreps (1981)
Arbitrage and equilibrium in economies with infinitely many commoditiesJournal of Mathematical Economics, 8
A. Schied, H. Föllmer, Stefan Weber (2009)
Robust Preferences and Robust Portfolio ChoiceHandbook of Numerical Analysis, 15
B Øksendal, A Sulem (2007)
Applied Stochastic Control of Jump Diffusions
M Royer (2006)
Backward stochastic differential equations with jumps and related non-linear expectationsStoch. Process. Appl., 116
N. Karoui, S. Peng, M. Quenez (1997)
Backward Stochastic Differential Equations in FinanceMathematical Finance, 7
H Föllmer, A Schie, S Weber (2009)
Mathematical Modelling and Numerical Methods in Finance
A. Gushchin (2011)
Dual Characterization of the Value Function in the Robust Utility Maximization ProblemTheory of Probability and Its Applications, 55
Giuliana Bordigoni, A. Matoussi, M. Schweizer (2007)
A Stochastic Control Approach to a Robust Utility Maximization Problem
D. Kramkov, W. Schachermayer (2003)
Necessary and sufficient conditions in the problem of optimal investment in incomplete marketsAnnals of Applied Probability, 13
Shanjian Tang, Xunjing Li (1994)
Necessary Conditions for Optimal Control of Stochastic Systems with Random JumpsSiam Journal on Control and Optimization, 32
(2013)
BSDEswith jumps , optimization and applications to dynamic riskmeasures
B. Øksendal, A. Sulem (2014)
Risk minimization in financial markets modeled by Itô-Lévy processesAfrika Matematika, 26
M-C Quenez (2004)
Seminar on Stochastic Analysis
M. Quenez (2004)
Optimal Portfolio in a Multiple-Priors Model
(1970)
Convex Analysis
G Bordigoni, A Matoussi, M Schweizer (2007)
Stochastic Analysis and Applications
M. Jeanblanc, A. Matoussi, Armand Ngoupeyou (2012)
Robust utility maximization in a discontinuous filtration
Pascal Maenhout (2004)
Robust Portfolio Rules and Asset PricingReview of Financial Studies, 17
B. Øksendal, A. Sulem (2012)
Forward–Backward Stochastic Differential Games and Stochastic Control under Model UncertaintyJournal of Optimization Theory and Applications, 161
T. Lim, M. Quenez (2010)
Exponential utility maximization and indifference price in an incomplete market with defaults
(2000)
Local martingales, arbitrage, and viability
M. Royer (2004)
Backward Stochastic Ddifferential Equations with Jumps and Related Non Linear ExpectationsDerivatives eJournal
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth $$X^*(T) : = X_{\varphi ^*}(T)$$ X ∗ ( T ) : = X φ ∗ ( T ) of the problem to maximize the expected U-utility of the terminal wealth $$X_{\varphi }(T)$$ X φ ( T ) generated by admissible portfolios $$\varphi (t); 0 \le t \le T$$ φ ( t ) ; 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario $$\frac{dQ^*}{dP}$$ d Q ∗ d P of the dual problem to minimize the expected V-value of $$\frac{dQ}{dP}$$ d Q d P over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all $$t \in [0,T]$$ t ∈ [ 0 , T ] . We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process; $$0 \le t \le T$$ 0 ≤ t ≤ T . In the terminal time case $$t=T$$ t = T we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio $$\varphi ^*$$ φ ∗ and the optimal measure $$Q^*$$ Q ∗ . We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.
Applied Mathematics and Optimization – Springer Journals
Published: Jan 21, 2016
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.