arXiv subjects
Publications and source records attributed to Hannah Geiss.
We study the quantitative approximation of certain stochastic integrals, where we use discrete time approximations under initial enlargement of filtration. It turns out that the approximation rate is in general the same as in the case of no additional information, however, the asymptotic constant improves.
This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a L{\'e}vy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D 1,2 by S. Geiss and Zhou.
We study the rate of convergence w.r.t.~a Wasserstein type distance for random walk approximations of mean field BSDEs. Our method does not use the particle method but instead a freezing technique. We extend results by Briand, Ch. Geiss, S. Geiss, and Labart [Bernoulli, 27(2) 2021] about the rate of convergence of a Donsker-type theorem for BSDEs from the classical setting to the mean field setting. In this connection the mean field setting leads to new phenomena and requires new techniques that should be of independent interest: The H\"older continuous terminal condition causes a singularity in time of the generator when seen as a generator in the non-mean field setting. To handle this singularity we introduce a concept of modified H\"older continuity by which we are able to achieve, up to a logarithmic term, the same polynomial approximation rates as in the classical non-mean field setting (in fact, already when approximating the Brownian motion itself a logarithmic term is necessary). Moreover, the exploited freezing technique of the mean field terms yields to the problem to handle the quantitative behavior of several different generators. Using BMO-techniques we obtain the rate of convergence for the integrated gradient process in the scale of Lorentz (type) spaces of exponential type.