The Euler scheme for Feller processes

TL;DR

This paper analyzes the Euler scheme for simulating Feller processes with jumps, providing explicit formulas for increment characteristic functions and convergence conditions, enabling efficient simulation of complex jump processes.

Contribution

It introduces a general convergence condition for the Euler scheme applied to Feller processes with jumps and derives explicit formulas for the characteristic functions of the increments.

Findings

Explicit formulas for increments' characteristic functions in terms of the symbol.

Convergence conditions for the Euler scheme in this setting.

The scheme can be used for efficient simulation of jump processes.

Abstract

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of L\'evy processes and thus the Euler scheme can be used for simulation by applying standard techniques from L\'evy processes.