Linear-quadratic stochastic delayed control and deep learning resolution

TL;DR

This paper studies stochastic control problems with delayed controls, characterizes solutions via Riccati PDEs, establishes conditions for existence and uniqueness, and employs deep learning to analyze delay effects on portfolio optimization.

Contribution

It introduces a new Riccati PDE-based characterization for delayed stochastic control problems and applies deep learning to portfolio allocation with execution delay.

Findings

Established existence and uniqueness conditions based on horizon and delay parameters.

Developed a deep learning scheme to analyze delay effects on portfolio optimization.

Illustrated the impact of delay on Markowitz portfolio allocation through numerical experiments.

Abstract

We consider a class of stochastic control problems with a delayed control, both in drift and diffusion, of the type dX t = $\alpha$ t--d (bdt + $\sigma$dW t). We provide a new characterization of the solution in terms of a set of Riccati partial differential equations. Existence and uniqueness are obtained under a sufficient condition expressed directly as a relation between the horizon T and the quantity d(b/$\sigma$) 2. Furthermore, a deep learning scheme is designed and used to illustrate the effect of delay on the Markowitz portfolio allocation problem with execution delay.