Deep Learning for Solving Initial Path Optimization of Mean-Field Systems With Memory

Abstract

We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean-Vlasov (mean-field) stochastic differential equation with delay delta > 0, driven by a Brownian motion and a pure jump Poisson random measure. The problem is to find the optimal initial values for the system in this period [-delta,0] before the system starts at t=0. Because of the delay in the dynamics, the system will after startup be influenced by these initial investment values.

It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By using this equivalence we can find implicit expression for the optimal investment. We use machine learning algorithms to solve explicitly some examples.