Research seminar by Prof. Silvana Pesenti (Assistant Professor, Dept of Statistics, University of Toronto, Canada)

Title: Model modification and combination using Kullback-Leibler divergence and barycentre

Abstract: In many settings, a modeler may have trained a model on data but that trained model is not consistent with their beliefs - e.g., the average time the process spends below some barrier is 10% from the estimated model, but their belief about the future evolution is that it should be 20%. Hence, they would like to modify this model in a minimal manner that respects their beliefs. Here we study such questions. More generally, 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 the 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. In a second part we look at model combinations, in particular how to combine different modellers / experts views using a barycentre approach. We discuss several examples stemming from finance and insurance.