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Probability and Computational Finance Seminar
Daniel Lacker
Brown University
Title: From convex risk measures to large deviations and concentration of measure

Abstract: The convex duality between relative entropy and the entropic risk measure (a.k.a. cumulant generating functional) underlies several arguments in large deviations (especially the weak convergence approach of Dupuis-Ellis) and concentration inequalities (particularly transport inequalities and tensorization). In fact, essentially only the basic convex duality relations and the chain rule for relative entropy are needed to derive Sanov's theorem as well as various tensorization properties of concentration inequalities. We use the rich duality theory for convex risk measures along with a suitable substitute for the chain rule to derive a vast generalization of Sanov's theorem in which the entropic risk measure appearing in the Laplace principle is replaced by an arbitrary convex risk measure. Some of the many applications include non-exponential large deviations for i.i.d. samples, uniform large deviation principles, and asymptotics for variational problems involving optimal transport costs and constrained hedging problems. More fundamentally, certain risk measure inequalities are shown to encode tail behavior of random variables in analogy with Chernoff bounds, and these inequalities behave well with respect to both duality and tensorization.

Date: Monday, February 13, 2017
Time: 4:30 pm
Location: Wean Hall 8220
Submitted by:  Steve Shreve