## The Annals of Probability

### Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE

#### 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
Revised: November 2013
First available in Project Euclid: 3 June 2015

https://projecteuclid.org/euclid.aop/1433341321

Digital Object Identifier
doi:10.1214/14-AOP920

Mathematical Reviews number (MathSciNet)
MR3353816

Zentralblatt MATH identifier
1333.60150

#### 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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