Adjoint-based optimal control of jump-diffusion processes

TL;DR

The paper develops an adjoint-based optimization framework for controlling jump-diffusion stochastic dynamics directly at the microscopic level, avoiding the curse of dimensionality inherent in PDE-based methods. It uses a discretize-then-optimize approach with Monte Carlo sampling to derive a fully discrete optimality system and a reduced gradient for a control of the form $u({\boldsymbol z},t)= {\boldsymbol \mu}(t)^T {\boldsymbol \boldsymbol \Phi}({\boldsymbol z})$. Numerical experiments on centering, stabilization, trajectory tracking, and interacting particles validate the method and demonstrate its parallelizability and memory-saving variants. This approach offers a robust, scalable path for calibrating and optimally controlling jump-diffusion processes in physics, finance, and related fields, with potential extensions to more complex jump structures and learning-based strategies.

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.

Figures (11)

• Figure 3.1: Different discretization of the time interval $[0,T]$ and exemplary trajectory. Gray: Deterministic splitting in $N_t^u$ intervals; Black dashed: Stochastic splitting of the time interval into $N_j$ jump times, different for every realization and particle. (a) Visualization of stochastic and deterministic parts of the time discretization. (b) Visualization of an exemplary SDE trajectory of a single particle in a single realization together with the corresponding discretization.
• Figure 5.1: A radial basis function $\varphi$ and its derivative $\varphi_x$.
• Figure 5.2: UML flowchart of the optimization procedure \ref{['algo:OptAlgo']}.
• Figure 6.1: Results of numerical experiments trying to center the particles starting from a normal distribution. (a) Initial (gray) and final (black) configuration; (b) Mean and standard deviation in phase space; (c) Mean and standard deviation in position and velocity over time.
• Figure 6.2: Results of numerical experiments trying to center the particles starting from a uniform distribution. (a) Initial and final configuration; (b) Evolution of mean and variance; (c) Control ${\boldsymbol \mu}(t)$ at final time $t=T$.

Theorems & Definitions (5)

• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 4.1