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-expectation of the sum a terminal and running cost is constrained. We derive the explicit form of the measure change and characterize the optimal drift and compensator adjustments under the optimal perturbed measure. We apply our results to a risk management setting where a trader seeks to ask the question: what dynamics induces a perturbation of the value-at-risk (VaR) or conditional VaR of the base process?