BSDE Representation and Randomized Dynamic Programming Principle for Stochastic Control Problems of Infinite-Dimensional Jump-Diffusions

TL;DR

This paper develops a probabilistic representation for stochastic control problems in infinite-dimensional spaces with jumps, using backward stochastic differential equations and randomization, extending existing results and linking to Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a novel probabilistic representation for infinite-dimensional jump-diffusion control problems via backward SDEs and randomization, broadening applicability and theoretical understanding.

Findings

Probabilistic representation formula for control problem value function

Extension of non-linear Feynman-Kac formula to infinite dimensions

Value function as viscosity solution of HJB equation in Hilbert space

Abstract

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.