Composition of Stochastic B–series with Applications to Implicit Taylor Methods

preprint numerics no. 1/2010 norwegian university of science and technology trondheim, norway Abstract. In this article, we construct a representation formula for stochastic B–series evaluated in a B–series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to Itô SDEs. 1. Introduction. Taylor methods have for a long time been a common choice for solving stochastic differential equations (SDEs), and with increased use of automatic differentiation techniques, their popularity will hardly subside. The Taylor expansions of the exact solutions of SDEs, from which the Taylor methods are derived, can take one out of two forms: either as Wagner–Platen series [9, 10] or as B–series [6, 1, 2, 12, 14], the relation between the two series has been demonstrated in [7]. In the present paper, we focus on B–series. In short, the exact solution X(t) evaluated after one step, starting at (t 0 , x 0) can be expressed in terms of a B–series: