A white noise approach to optimal insider control of systems with delay

TL;DR

This paper develops a white noise approach using Hida-Malliavin calculus to analyze optimal insider control in stochastic delay systems, with applications to population harvesting and financial markets, revealing market viability issues with inside information.

Contribution

It introduces a white noise method with maximum principles for insider control in delay systems, extending analysis to financial markets with delay effects.

Findings

Optimal control characterized via maximum principles.

Market viability is compromised with inside information, even with delays.

Applications to population harvesting and insider trading models.

Abstract

We use a white noise approach to study the problem of optimal inside control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system.These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the optimal insider portfolio in a financial market where the risky asset price is given by a stochastic delay equation. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable. Our results show that with this inside information the market is not viable even if there is delay in the equations.