A mixed relaxed singular maximum principle for linear SDEs with random coefficients

TL;DR

This paper develops a new maximum principle for singular stochastic control of linear SDEs with random coefficients, providing existence and optimality conditions applicable to complex financial models.

Contribution

It introduces a mixed relaxed-singular maximum principle for linear SDEs with random coefficients, extending control theory to more realistic stochastic systems.

Findings

Established existence of optimal relaxed controls.

Derived necessary conditions for optimality.

Applied results to an investment and consumption problem.

Abstract

We study singular stochastic control of a two dimensional stochastic differential equation, where the first component is linear with random and unbounded coefficients. We derive existence of an optimal relaxed control and necessary conditions for optimality in the form of a mixed relaxed-singular maximum principle in a global form. A motivating example is given in the form of an optimal investment and consumption problem with transaction costs, where we consider a portfolio with a continuum of bonds and where the portfolio weights are modeled as measure-valued processes on the set of times to maturity.