Optimal simulation schemes for Lévy driven stochastic differential equations

Abstract

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic splitting argument. The resulting error bound involves separate contributions of the compound Poisson approximation and of the discretization scheme for the Brownian part, and allows, on one hand, to balance the two contributions in order to minimize the computational time, and on the other hand, to study the optimal design of the approximating compound Poisson process. For driving processes whose Lévy measure explodes near zero in a regularly varying way, this procedure allows to construct discretization schemes with arbitrary order of convergence.

Weak Euler Approximation for Itô Diffusion and Jump Processes

“…For discrete-time approximations of Itô processes with jumps, Mikulevičius and Platen showed first weak-order convergence in the case that the coefficient functions possess fourthorder continuous differentiability [28]. Similar results have been presented for Lévy-driven stochastic differential equations [17,18,20,21,36,37].…”

“…Let ψ, ϕ k be functions defined by (19) and (20), respectively. If u n → u uniformly on compact sets, then…”

An Euler–Poisson scheme for Lévy driven stochastic differential equations

“…The first part (i) follows substituting the upper bounds E[( * )] and E[( * * )] into (18). For the second part (ii) we consider H i := G σ Y t j , 0 ≤ j ≤ i and reproduce the above derivations up to (18). By the definition of ∆ Y t i , we have…”

“…A more sensible approach is to substitute the small jumps by a Gaussian correction as performed in Dereich [8], but this method has its limitations as discussed in Asmussen and Rosiński [2]. A novel approach described in Kohatsu-Higa et al [18] is to approximate the small jumps with an extra compound Poisson process matching a given number of moments of the original driving process provided these moments exist. Convergence rates for weak errors are derived under further assumptions on the smoothness of the function f .…”

Markov chain approximations for transition densities of Lévy processes

“…In discrete time, Markov chains have been used to approximate the much larger class of Feller processes and [4] proves convergence in law of such an approximation in the Skorokhod space of càdlàg paths, but does not discuss rates of convergence; [32] has a finite state-space path approximation and applies this to option pricing together with a discussion of the rates of convergence for the prices. With respect to Lévy process driven SDEs, [21] (resp. [34]) approximates solutions Y thereto using a combination of a compound Poisson process and a high order scheme for the Brownian component (resp.…”