Clark representation for the local times of Gaussian integrators

Andrey Dorogovtsev, Institute of Mathematics, National Academy of Sciences of Ukraine. Part of the Probability, Stochastic Modelling and Financial Mathematics seminar.

Speaker: Andrey Dorogovtsev

Title: Clark representation for the local times of Gaussian integrators

Abstract: Gaussian integrators are the processes which admit integration of deterministic square integrable functions. Among it there are such processes as Wiener process, Brownian bridge, fractional Brownian motion for certain value of the Hurst parameter and so on. Gaussian integrators can be obtained from the Wiener process with the help of second quantization. This allows to develop anticipating stochastic calculus for them. Also in most cases they have Berman's local nondeterminism property, which leads to the existence of the local time. In the talk we discuss the analog of the Clark representation for these local times. Instead of Ito stochastic integral we use Skorokhod integral with respect to integrator. It occurs that there exist a lot of representations (which is a difference with Wiener case).  We give one similar to Wiener case and discuss the representation with the minimal norm. The talk is based on the joint work with Olga Izyumtseva, Georgii Riabov and Naoufel Salhi.