Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

Abstract

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

“…al. [7,8], Rößler [12,13,14] and Tocino and Vigo-Aguiar [19]. However, due to the knowledge of the author, all proposed second order SRK methods suffer from an inefficiency if they are applied to SDEs with a multi-dimensional Wiener process.…”

“…On the other hand, the number of random variables that have to be simulated for each step is only 2m − 1. The paper is organized as follows: Firstly, in Sections 2-4 we briefly review the main results of the rooted tree analysis for weak approximation [12,15]. In Section 5, the new class of SRK methods is introduced and order conditions are calculated by the rooted tree analysis given in Section 2-4.…”

“…We consider a very general class of stochastic Runge-Kutta methods which has been introduced in [12]: Let M be a finite set of multi-indices with κ = |M| elements and let θ ν (h), ν ∈ M, be some suitable random variables. For the weak approximation of the solution (X t ) t∈I of SDE (1.1), a general class of s-stage stochastic Runge-Kutta methods is given by Y 0 = x 0 and…”

Families of efficient second order Runge–Kutta methods for the weak approximation of Itô stochastic differential equations

“…By the application of the multi-colored rooted tree analysis [8], order conditions for the coefficients of the SRK method (3) can be easily determined. As a result of this, the following Theorem 2.1 due to Rößler [10] gives order conditions for the SRK method (3) up to order two.…”

“…where T S(∆) denotes a set of trees, ρ(t) the order of the tree t, F (t) the elementary differential connected with the tree t and lec t a coefficient depending only on t and the numerical method (see [8,9,10] for details). Let lec = (lec t ) t∈T S(∆) be the vector of these coefficients.…”