Kolmogorov equations for stochastic Volterra processes with singular kernels
A talk in the Bielefeld Stochastic Afternoon series by
Ioannis Gasteratos
| Abstract: | We associate backward and forward Kolmogorov equations to a
class of fully nonlinear Stochastic Volterra Equations (SVEs) with
convolution kernels K that are singular at the origin. Working on a
carefully chosen Hilbert space H1, we rigorously establish a link
between solutions of SVEs and Markovian mild solutions of a Stochastic
Partial Differential Equation (SPDE) of transport-type. Using this
Markovian lift, we obtain novel Itˆo formulae for functionals of mild
solutions and, as a byproduct, show that their laws solve
corresponding Fokker–Planck equations. Finally, we introduce a
natural notion of “singular” directional derivatives along K and
prove that (conditional) expectations of SVE solutions can be
expressed in terms of the unique solution to a backward Kolmogorov
equation on H1. Our analysis relies on stochastic calculus in Hilbert
spaces, the reproducing kernel property of the state space H1, as well
as crucial invariance and smoothing properties that are specific to
the SPDEs of interest. In the special case of singular power-law
kernels, our conditions guarantee well-posedness of the backward
equation either for all values of the Hurst parameter H, when the
noise is additive, or for all H > 1/4 when the noise is
multiplicative. Time permitting, we shall discuss applications to
mathematical finance as well as a few open problems. Based on joint
work with Alexandre Pannier (Université Paris Cité). Within the CRC this talk is associated to the project(s): A5 |