The Annals of Probability

Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE

Idris Kharroubi and Huyên Pham

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Abstract

We aim to provide a Feynman–Kac type representation for Hamilton–Jacobi–Bellman equation, in terms of forward backward stochastic differential equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton–Jacobi–Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework, and importantly we do not make any ellipticity condition. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.

Article information

Source
Ann. Probab., Volume 43, Number 4 (2015), 1823-1865.

Dates
Received: December 2012
Revised: November 2013
First available in Project Euclid: 3 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1433341321

Digital Object Identifier
doi:10.1214/14-AOP920

Mathematical Reviews number (MathSciNet)
MR3353816

Zentralblatt MATH identifier
1333.60150

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K55: Nonlinear parabolic equations 93E20: Optimal stochastic control

Keywords
BSDE with jumps constrained BSDE regime-switching jump-diffusion Hamilton–Jacobi–Bellman equation nonlinear Integral PDE viscosity solutions inf-convolution semiconcave approximation

Citation

Kharroubi, Idris; Pham, Huyên. Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE. Ann. Probab. 43 (2015), no. 4, 1823--1865. doi:10.1214/14-AOP920. https://projecteuclid.org/euclid.aop/1433341321


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