Wong-Zakai approximations of SDEs and SPDEs with jump noise
Final Report Abstract
The goal of this project was to study convergence of Wong–Zakai approximations of Lévy-driven stochastic (partial) differential equations. The idea behind the setting is quite natural. If one formally replaces the cádlàg Lévy process by its continuous piece-wise smooth approximations, one obtains a random ordinary or partial differential equation that can be solved in the classical sense. The question is what happens in the limit. It is well-known that stochastic equations driven by approximations of the Brownian motion converge to the stochastic equations that have to be understood in the Stratonovich sense. In the jump case, one obtains the so-called Marcus or canonical equations. It should be taken into account that in certain cases existence and uniqueness of solutions of canonical stochastic partial differential equations is not automatic and has to be established. The results obtained in this project are subdivided into four topics. First we studied the Wong–Zakai approximations of SDEs with Lévy noise and established first order weak convergence of the Wong–Zakai approximation scheme. These results refine general convergence results by Kunita and complement similar results by Protter, Talay et al. for Lévy-driven Itô SDEs. It is interesting to emphasise that Marcus SDEs demand existence of certain exponential moments of the Lévy measure opposite to conventional moments in the Itô case. This seemingly restrictive condition appears due to non-linear nature of jumps dynamics in the Marcus setting. Second, the Wong–Zakai approximations of an advection-diffusion equation with boundary Lévy noise were studied. This problem is motivated by works devoted to the contaminant transport in water flows. The Lévy noise serves here as an idealization of random contaminant source. We determined the closed form solutions of the advection-diffusion equation and proved convergence of the Wong–Zakai approximations in the non-standard M1-Skorokhod topology. One of the technical difficulties here was to determine an appropriate space (a fractional Sobolev space) on which the stochastic solution is well defined. The most interesting and challenging topic was to study stochastic transport equation of Marcus type. Here we followed the approach by Kunita and Fujiwara & Kunita and made use of canonical SDEs based on cádlàg semimartingales with a spatial parameter. This approach allows to solve a stochastic transport equation with Lévy noise by the gradient term by the method of stochastic characteristics that are given by a canonical SDE. The crux of the argument is the new generalized Itô formula for canonical Lévy driven SDEs. Once a solution to a transport equation is written down in terms of the stochastic characteristics, the convergence of Wong–Zakai approximations is straightforward (at least in the sense of finite dimensional distributions). Finally, we studied the Lévy-driven second order equation of Marcus type with noise acting an the gradient term. The solution is obtained here with the help of the integration theory 10 w.r.t. Poisson random measures in Banach spaces developed by Brze´zniak and Hausenblas. As in the case of advection-diffusion equations, it turned out to be convenient to work in certain fractional Sobolev spaces.
Publications
- Advection-diffusion equation on a half-line with boundary Levy noise. Discrete and Continuous Dynamical Systems - B, 24(2):637– 655, 2019
L.-S. Hartmann and I. Pavlyukevich
(See online at https://dx.doi.org/10.3934/dcdsb.2018200) - First order convergence of weak Wong–Zakai approximations of Levy-driven Marcus SDEs. Theory of Stochastic Processes
T. Kosenkova, A. Kulik, and I. Pavlyukevich
