Adjoint-based optimal control of jump-diffusion processes

TL;DR

This paper introduces an adjoint-based optimization method for jump-diffusion stochastic processes at the microscopic level, enabling efficient control and calibration with reduced computational costs and parallelization capabilities.

Contribution

It develops a novel microscopic optimization approach using adjoint processes and Monte Carlo methods, avoiding the curse of dimensionality and enhancing computational efficiency.

Findings

Effective optimization of jump-diffusion processes demonstrated

Parallelizable method reduces computational time

Validation through extensive numerical experiments

Abstract

Stochastic differential equations (SDEs) using jump-diffusion processes describe many natural phenomena at the microscopic level. Since they are commonly used to model economic and financial evolutions, the calibration and optimal control of such processes are of interest to many communities and have been the subject of extensive research. In this work, we develop an optimization method working at the microscopic level. This allows us also to reduce computational time since we can parallelize the calculations and do not encounter the so-called curse of dimensionality that occurs when lifting the problem to its macroscopic counterpart using partial differential equations (PDEs). Using a discretize-then-optimize approach, we derive an adjoint process and an optimality system in the Lagrange framework. Then, we apply Monte Carlo methods to solve all the arising equations. We validate our optimization strategy by extensive numerical experiments. We also successfully test a optimization procedure that avoids storing the information of the forward equation.