Multivalued Stochastic Delay Differential Equations and Related Stochastic Control Problems

TL;DR

This paper investigates the existence, uniqueness, and control of solutions to multivalued stochastic delay differential equations with constraints, establishing a dynamic programming principle and viscosity solutions for related control problems.

Contribution

It introduces a framework for multivalued stochastic delay equations with constraints and derives the dynamic programming principle and viscosity solutions for associated control problems.

Findings

Proved existence and uniqueness of solutions for multivalued stochastic delay equations.

Established the dynamic programming principle for the control problems.

Showed the value function as a viscosity solution to a Hamilton-Jacobi-Bellman equation.

Abstract

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type): \[ \left\{\begin{array} [c]{r} dX(t)+\partial\varphi\left(X(t)\right) dt\ni b\left(t,X(t),Y(t),Z(t)\right) dt+\sigma\left(t,X(t),Y(t),Z(t)\right)dW(t), \medskip\\ t\in(s,T],\medskip\\ \multicolumn{1}{l}{X(t)=\xi\left(t-s\right) ,\;t\in\left[ s-\delta,s\right] .} \end{array} \right. \] Specify that in this case the coefficients at time $t$ depends also on previous values of $X\left(t\right) $ through $Y(t)$ and $Z(t)$. Also $X$ is constrained with the help of a bounded variation feedback law $K$ to stay in the convex set $\bar{\mathrm{Dom}\left(\varphi\right)}$. Afterwards we consider optimal control problems where the state $X$ is a solution of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.