Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

TL;DR

This paper proves the uniqueness of viscosity solutions for fully nonlinear stochastic Hamilton-Jacobi equations arising in optimal control problems with random coefficients, ensuring well-posedness of the value function.

Contribution

It establishes the uniqueness of viscosity solutions for stochastic HJ equations with random coefficients under standard Lipschitz conditions, advancing the theory of stochastic control.

Findings

Value function is the unique viscosity solution.

Uniqueness holds under Lipschitz continuity assumptions.

Results ensure well-posedness of stochastic control problems.

Abstract

This paper is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.