Stochastic B-Series and Order Conditions for Exponential Integrators

Abstract

We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order conditions for stochastic exponential integrators. The resulting general order theory covers both Itô and Stratonovich integration.

B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

“…in [4,16]. This has been extended to exponential integrators in [2] and more recently in [31]. Consider a semi-linear non-autonomous SDE of the form…”

“…We will here provide the details on how the result presented in Example 2.7 for the elementary differential looks like more concrete for the SDE introduced in Example 2.1. The partitioned SDE in Example 2.1 is of the form dX (1) = g (1,1) 0 (X (1) , X (2) )dt + g (1,2) 0 (X (1) , X (2) )dt + g (1,1) 1 (X (1) , X (2) )dW 1 (t) dX (2) = g (2,1) 0 (X (1) , X (2) )dt with…”