Stochastic Recursive Optimal Control Problem with Time Delay and Applications

TL;DR

This paper develops a framework for stochastic recursive optimal control problems with time delay, deriving generalized HJB equations and maximum principles, and applies these to financial optimization problems with explicit solutions.

Contribution

It introduces a generalized HJB equation and maximum principle for stochastic control with delays, and demonstrates their application in finance.

Findings

Derived a generalized HJB equation for delayed stochastic control.

Established a maximum principle involving forward-backward SDDEs.

Showed explicit solutions in financial portfolio optimization.

Abstract

This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.