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Stochastic viscosity solutions for stochastic integral-partial differential equations and singular stochastic control

Jinbiao Wu

Abstract

In this article, we mainly study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs). We investigate a new class of generalized backward doubly stochastic differential equations (GBDSDEs) driven by two independent Brownian motions and an independent Poisson random measure, which involves an integral with respect to a càdlàg increasing process. We first derive existence and uniqueness of the solution of GBDSDEs with general jumps. We then introduce the definition of stochastic viscosity solutions of SIPDEs and give a probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions. Finally, we establish stochastic maximum principles for the optimal control of a stochastic system modelled by a GBDSDE with general jumps.

Stochastic viscosity solutions for stochastic integral-partial differential equations and singular stochastic control

Abstract

In this article, we mainly study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs). We investigate a new class of generalized backward doubly stochastic differential equations (GBDSDEs) driven by two independent Brownian motions and an independent Poisson random measure, which involves an integral with respect to a càdlàg increasing process. We first derive existence and uniqueness of the solution of GBDSDEs with general jumps. We then introduce the definition of stochastic viscosity solutions of SIPDEs and give a probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions. Finally, we establish stochastic maximum principles for the optimal control of a stochastic system modelled by a GBDSDE with general jumps.

Paper Structure

This paper contains 8 sections, 21 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GBDSDEs with general jumps
  4. Viscosity solutions to IPDEs with nonlinear Neumann boundary conditions
  5. Stochastic viscosity solutions to SIPDEs with nonlinear Neumann boundary conditions
  6. Singular stochastic optimal control
  7. Sufficient Maximum Principle
  8. Necessary Maximum Principle

Key Result

Proposition 1

let $\alpha\in \mathcal{S}^2(\mathbb{F},[0,T])$, $\beta\in L^2(\mathbb{F},[0, T])$, $\gamma\in L^2(\mathbb{F},[0,T]; \mathbb{R}^m)$, $\delta\in L^2(\mathbb{F},[0,T]; \mathbb{R}^d)$, $\theta\in L^2(\mathbb{F},[0,T]; \nu)$, $\lambda\in L^2(A,\mathbb{F},[0,T])$ be such that then where $\Delta A(s)=A(s)-A(s-)$ and

Theorems & Definitions (31)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Definition 1
  • Theorem \oldthetheorem
  • Remark 3
  • Theorem \oldthetheorem
  • Remark 4
  • Lemma 1
  • ...and 21 more