Riemannian-like structures on the set of probability measures: a comparison between Euclidean and discrete spaces

The purpose of this thesis is to present in detail two theories, not deductible from each other, but which obtain very similar results, giving a Riemannian-like structure to the set of probability measures. Both theories lead to a correspondence between heat equation and gradient flow of entropy. Chapter 1 contains a quick but self-contained treatment of the theory of optimal transport, including Kantorovich’s duality, Brenier’s theorem, the Wasserstein spaces P_p and the characterization of their geodesics. Chapter 2 focuses on P_2(R^n). Using the continuity equation, we describe its geodesics and absolutely continuous curves, and determine a “tangent” velocity to them; this enables the definition of (sub)differential of a functional. We study convexity and differentiability of the “internal energy” and “potential energy” functionals. We give a definition of gradient flows, and exploit its equivalence with the purely metric EVI formulation to show some of its basic properties. We conclude characterizing the gradient flows of the entropy by the condition that the densities solve the heat equation. In Chapter 3, we move to a finite space endowed with an irreducible Markov kernel. Guided by an explicit study of the two-point space, we define a family of distances between probabilities via a discrete analogue of “Benamou-Brenier’s formula” from Chapter 2, and characterize their finiteness. The AC curves are described as in the continuous case. Appropriate subsets of probability measures turn out to be Riemannian manifolds, on which we can reproduce results like the Eulerian description of geodesics, the differentiability of potential energy, the identification of heat flow with gradient flow of the entropy

Fast and Stable Credit Gamma of CVA

Credit Valuation Adjustment is a balance sheet item which is nowadays subject to active risk management by specialized traders. However, one of the most important risk factors, which is the vector of default intensities of the counterparty, affects in a non-differentiable way the most general Monte Carlo estimator of the adjustment, through simulation of default times. Thus the computation of first and second order (pure and mixed) sensitivities involving these inputs cannot rely on direct path-wise differentiation, while any approach involving finite differences shows very high statistical noise. We present ad hoc analytical estimators which overcome these issues while offering very low runtime overheads over the baseline computation of the price adjustment. We also discuss the conversion of the so-obtained sensitivities to model parameters (e.g. default intensities) into sensitivities to market quotes (e.g. Credit Default Swap spreads).Comment: 22 pages, 6 figure