Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems

Tao Hao,Juan Li

doi:10.1155/2014/262713

Abstract

We get a new type of controlled backward stochastic differential equations (BSDEs), namely, the BSDEs, coupled with value function. We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. We get the related dynamic programming principle (DPP) with the help of the stochastic backward semigroup which was introduced by Peng in 1997. By making use of a new, more direct approach, we prove that our nonlocal Hamilton-Jacobi-Bellman (HJB) equation has a unique viscosity solution in the space of continuous functions of at most polynomial growth. These results generalize the corresponding conclusions given by Buckdahn et al. (2009) in the case without control.

Highlights

In the recent years, many authors have studied models of large stochastic particle systems with mean-field interaction
There have been a lot of works published on the dynamic programming approach, which gives with the help of dynamic programming principle (DPP) a stochastic interpretation to the associated partial differential equations (PDEs); we refer, for instance, to Buckdahn and Li [22], Peng [23, 24], and Yong and Zhou [25]
− Zst,x;vdBs, s ∈ [t, t + δ], Ỹtt+,xδ;v = η, where Xt,x;v is the unique solution of SDE (22) and W is the value function given by the backward stochastic differential equations (BSDEs) coupled with value function (26) in Theorem 9

Summary

Introduction

Many authors (see [1,2,3,4,5,6]) have studied models of large stochastic particle systems with mean-field interaction. Ŵ (t, x) := esssupYtt,x;v, (t, x) ∈ [0, T] × Rn. the fact that the control v ∈ V0,T is frozen in (4) has as a consequence that the value function Ŵ is a viscosity solution of the following classical PDE (there are many references, such as [22,23,24]):. Which can be regarded as an equation with the solution (Yt,x, Zt,x, W) in some sense Inspired by this idea, we change to study the following BSDE coupled with value function:.

Mean-Field SDEs and Mean-Field BSDEs

BSDEs Coupled with Value Function

The Dynamic Programming Principle

Associated Nonlocal Hamilton-Jacobi-Bellman Equations