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

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