Minimal Kullback-Leibler Divergence for Constrained L\'evy-It\^o Processes

Abstract

Given an n-dimensional stochastic process X driven by P-Brownian motions and Poisson random measures, we seek the probability measure Q, with minimal relative entropy to P, such that the Q-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterise the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting -- including an algorithm to simulate under the optimal measure -- where an agent seeks to answer the question: what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?