07 Oct

Damiano Brigo, “Intrinsic stochastic differential equations as jets: theory and applications”

Monday October 10 2016
16:00
Scuola Normale Superiore
Aula Mancini

Damiano Brigo
Imperial College, London

Abstract

We quickly introduce Stochastic Differential Equations (SDEs) and their two main calculi: Ito and Stratonovich. Briefly recalling the definition of jets, we show how Ito SDEs on manifolds may be defined intuitively as 2-jets of curves driven by Brownian motion and show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can lead to intuitive and intrinsic representations of Ito SDEs, presenting several plots and numerical examples. We give a new geometric interpretation of the Ito-Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows. We interpret classic quantities and operators in stochastic analysis geometrically. We hint at applications of the jet representation to i) dimensionality reduction by projection of infinite dimensional stochastic partial differential equations (SPDEs) onto finite dimensional submanifolds for the filtering problem in signal processing, and ii) consistency between dynamics of interest rate factors and parametric form of term structures in mathematical finance. We explain that the mainstream choice of Stratonovich calculus for stochastic differential geometry is not optimal when combining geometry and probability, using the mean square optimality of projection on submanifolds as a fundamental
application.