Mathematical Finance, Stochastic Analysis and Machine Learning Seminar by William Hammersley: Rearranged Stochastic Heat Equation

Speaker:

William Hammersley, 3IA, Université Côte D'Azur

Title:

Rearranged Stochastic Heat Equation

Abstract:

The ideas presented in this talk are motivated by the desire to regularise by noise certain partial differential equations written over the space of probability measures on $\mathbb{R}$ with finite second moment. Considering the quantile representation of random variables, the problem is translated to function space, where we introduce a stochastic process expected to aid in our regularisation objective. Valued in the set of symmetric non-increasing square integrable functions on the unit circle, we hope this process will allow us to gain by randomising over the space of quantile functions rather than over the space of $L^2$ functions. This process obtained through explicit schemes composing the stochastic heat equation and the rearrangement operator. After recalling some key results from the theory of rearrangements, I will outline how one may characterize the limit process, which exhibits desirable smoothing properties for the associated semigroup flow. This is a joint work with François Delarue.

Mathematical Finance, Stochastic Analysis, and Machine Learning