Open Access
July 2015 Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE
Idris Kharroubi, Huyên Pham
Ann. Probab. 43(4): 1823-1865 (July 2015). DOI: 10.1214/14-AOP920


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.


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Idris Kharroubi. Huyên Pham. "Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE." Ann. Probab. 43 (4) 1823 - 1865, July 2015.


Received: 1 December 2012; Revised: 1 November 2013; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1333.60150
MathSciNet: MR3353816
Digital Object Identifier: 10.1214/14-AOP920

Primary: 35K55 , 60H10 , 60H30 , 93E20

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

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 4 • July 2015
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