Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models

TL;DR

This paper develops a framework for optimal control of stochastic delay differential equations, characterizing the value function via viscosity solutions and applying it to path-dependent financial and economic models.

Contribution

It introduces a novel infinite-dimensional approach and proves regularity results for the value function in stochastic delay control problems.

Findings

Value function characterized as unique viscosity solution

Established partial regularity of the value function

Applied framework to financial and economic models

Abstract

In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,\alpha}$-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).