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References (38)
-
Mark H.A.Davis, G. Burstein (1992)
A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative ControlStochastics and Stochastics Reports, 40
-
J. Diehl, P. Friz, Harald Oberhauser (2014)
Regularity Theory for Rough Partial Differential Equations and Parabolic Comparison Revisited -
(1980)
Functional Analysis, 6th edn -
M. Haugh, L. Kogan (2001)
Pricing American Options: A Duality ApproachDerivatives eJournal
-
P Friz, N Victoir (2010)
Multidimensional Stochastic Processes as Rough Paths. Theory and Applications -
R. Buckdahn, Jin Ma (2007)
Pathwise Stochastic Control Problems and Stochastic HJB EquationsSIAM J. Control. Optim., 45
-
D. Nualart, É. Pardoux (1988)
Stochastic calculus with anticipating integrandsProbability Theory and Related Fields, 78
-
M Bardi, I Capuzzo (1997)
Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations -
David Brown, James Smith, Peng Sun (2010)
Information Relaxations and Duality in Stochastic Dynamic ProgramsFound. Trends Optim., 5
-
Mark Davis (1993)
A deterministic approach to optimal stopping with application to a prophet inequality -
MB Haugh, L Kogan (2004)
Pricing American options: a duality approachOper. Res., 52
-
P. Lions, P. Souganidis (2000)
Fully nonlinear stochastic PDE with semilinear stochastic dependenceComptes Rendus De L Academie Des Sciences Serie I-mathematique, 331
-
DB Brown, JE Smith, P Sun (2010)
Information relaxations and duality in stochastic dynamic programsOper. Res., 58
-
Terry Lyons, Z. Qian (2003)
System Control and Rough Paths -
H. Sussmann (1978)
On the Gap Between Deterministic and Stochastic Ordinary Differential EquationsAnnals of Probability, 6
-
Nicolas Perkowski, David Promel (2013)
Pathwise stochastic integrals for model free financeBernoulli, 22
-
W. Fleming, H. Soner, H. Soner, Div Mathematics, Florence Fleming, Serpil Soner (1992)
Controlled Markov processes and viscosity solutions -
T. Duncan, B. Pasik-Duncan (2013)
Linear-Quadratic Fractional Gaussian ControlSIAM J. Control. Optim., 51
-
Terry Lyons (1998)
Di erential equations driven by rough signals -
L. Mazliak, I. Nourdin (2006)
OPTIMAL CONTROL FOR ROUGH DIFFERENTIAL EQUATIONSStochastics and Dynamics, 08
-
N. Krylov (2000)
On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficientsProbability Theory and Related Fields, 117
-
D. Ocone, É. Pardoux (1989)
A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equationsAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 25
-
J. Yong, X. Zhou (1999)
Stochastic Controls: Hamiltonian Systems and HJB Equations -
R. Wets (1975)
On the Relation between Stochastic and Deterministic Optimization -
P. Friz, Nicolas Victoir (2010)
Multidimensional Stochastic Processes as Rough Paths: Variation and Hölder spaces on free groups -
L. Rogers (2002)
Monte Carlo valuation of American optionsMathematical Finance, 12
-
D. Crisan, J. Diehl, P. Friz, Harald Oberhauser (2012)
Robust filtering: Correlated noise and multidimensional observationAnnals of Applied Probability, 23
-
L. Rogers (2007)
Pathwise Stochastic Optimal ControlSIAM J. Control. Optim., 46
-
Terry Lyons, M. Caruana, Thierry Lévy, École Saint-Flour (2007)
Differential equations driven by rough paths -
(2011)
A (rough) pathwise approach to a class of nonlinear SPDEs Annales de l’Institut Henri Poincaré / Analyse non linéaire -
P. Friz, Martin Hairer (2014)
A Course on Rough Paths: With an Introduction to Regularity Structures -
L. Coutin, P. Friz, Nicolas Victoir (2007)
Good rough path sequences and applications to anticipating stochastic calculusAnnals of Probability, 35
-
M. Gubinelli (2003)
Controlling rough pathsJournal of Functional Analysis, 216
-
G. Barles, E. Jakobsen (2002)
On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equationsMathematical Modelling and Numerical Analysis, 36
-
V. Borkar (2005)
Controlled diffusion processesProbability Surveys, 2
-
H. Doss (1977)
Liens entre equations di erentielles stochastiques et ordinaires -
P. Lions, P. Souganidis (1998)
Fully nonlinear stochastic partial differential equations: non-smooth equations and applicationsComptes Rendus De L Academie Des Sciences Serie I-mathematique, 327
-
K Yosida (1980)
Functional Analysis
- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Springer Science+Business Media New York
- Subject
- Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
- ISSN
- 0095-4616
- eISSN
- 1432-0606
- DOI
- 10.1007/s00245-016-9333-9
- Publisher site
- See Article on Publisher Site
Abstract
We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of Lyons. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically involve anticipating stochastic analysis. We make the link to old work of Davis and Burstein (Stoch Stoch Rep 40:203–256, 1992) and then prove a continuous-time generalization of Roger’s duality formula [SIAM J Control Optim 46:1116–1132, 2007]. The generic case of controlled volatility is seen to give trivial duality bounds, and explains the focus in Burstein–Davis’ (and this) work on controlled drift. Our study of controlled rough differential equations also relates to work of Mazliak and Nourdin (Stoch Dyn 08:23, 2008).
Journal
Applied Mathematics and Optimization – Springer Journals
Published: Feb 11, 2016