Optimal control of predictive mean-field equations and applications to finance

TL;DR

This paper develops maximum principles for controlling coupled stochastic differential equations with predictive mean-field terms, applying these to optimal portfolio and consumption problems in finance involving insider influence and recursive utility.

Contribution

It introduces a novel maximum principle framework for predictive mean-field SDEs and applies it to complex financial optimization problems.

Findings

Derived necessary and sufficient maximum principles for predictive mean-field control systems.

Applied the theoretical results to insider-influenced asset pricing and recursive utility optimization.

Demonstrated the effectiveness of the approach in complex financial models.

Abstract

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$. The driver of the BSDE at time $t$ may depend not just upon the unknown processes $Y(t), Z(t), K(t,\cdot)$, but also on the predicted future value $Y(t+\delta)$, defined by the conditional expectation $A(t):= E[Y(t+\delta) | \mathcal{F}_t]$. \\ We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems:\\ (i) Optimal portfolio in a financial market with an \emph{insider influenced asset price process.} \\ (ii) Optimal consumption rate from a cash flow modeled as a geometric It\^ o-L\' evy SDE, with respect to \emph{predictive recursive utility}.