Rüdiger Frey

Rüdiger Frey, Thorsten Schmidt, Ling Xu

This paper studies Galerkin approximations applied to the Zakai equation of stochastic filtering. The basic idea of this approach is to project the infinite-dimensional Zakai equation onto some finite-dimensional subspace generated by smooth basis functions; this leads to a finite-dimensional system of stochastic differential equations that can be solved numerically. The contribution of the paper is twofold. On the theoretical side, existing convergence results are extended to filtering models with observations of point-process or mixed type. On the applied side, various issues related to the numerical implementation of the method are considered; in particular, we propose to work with a subspace that is constructed from a basis of Hermite polynomials. The paper closes with a numerical case study.

Rüdiger Frey, Verena Köck

In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep neural network algorithms for solving linear and semilinear parabolic partial integro-differential equations with boundary conditions in high dimension. To show the viability of our approach we discuss several case studies from insurance and finance.