Finite difference schemes for linear stochastic integro-differential equations

Abstract

We study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump-diffusion processes. We show that the rate is of order one in space and order one-half in time.

On finite difference schemes for partial integro-differential equations of Lévy type

“…We introduce an approximation that treats the integro-differential operator as a second order operator on the whole unit ball. Our approximation is similar to the one that we introduced in [3], [5]. However, in these works the results and their proofs rely on the nondegeneracy of the second order differential operator.…”

Numerical Solution of Singular Stochastic Integral Equations of Abel’s Type Using Operational Matrix Method

“…The study on various deterministic integral equations of Volterra type, Fredholm type, Volterra-Fredholm type, integro-differential equations, etc., can be seen in [12,17,9,19,20,21,31,4,3]. Also, various problems modelled using stochastic integral equations and stochastic integro-differential equations can be found in [18,22,6,16].…”

Stability and error analysis of an implicit Milstein finite difference scheme for a two-dimensional Zakai SPDE

“…A finite difference scheme for a filtered jump-diffusion process resulting in a stochastic integrodifferential equation is studied in [DL16], where convergence of order 1 in space and 1/2 in time, in L 2 and L ∞ in space, is proven for an Euler time stepping scheme.…”

“…The theoretical results in this paper are an extension from those in [GR12] to the multi-dimensional case. They are more specific than those in [DL16] in that we analyse only the case of constant coefficient local SPDEs. In contrast to [BL12,BLS13], we consider only finite-dimensional Brownian motions, as is relevant in our applications.…”