Convergence of Stochastic Runge-kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

preprint numerics no. 4/2007 norwegian university of science and technology trondheim, norway Abstract. In the last years, implicit SRK methods have been developed both for strong and weak approximation. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes like simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for Itô-and Stratonovich-SDEs and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts. 1. Introduction. In many applications, e.g., in epidemiology and financial mathematics , taking stochastic effects into account when modeling dynamical systems often leads to stochastic differential equations (SDEs). Therefore, in recent years, the development of numerical methods for the approximation of SDEs has become a field of increasing interest, see e.g [16, 22] and references therein. Many stochastic schemes fall into the class of stochastic Runge-Kutta (SRK) methods. SRK methods have been studied both for strong approximation [1, 10, 11, 16], where one is interested in obtaining good pathwise solutions, and for weak approximation [7, 9, 16, 19, 21, 30], which focuses on the expectation of functionals of solutions. For solving SDEs which are stiff, implicit SRK methods have to be considered, which also has been done both for strong [4, 11, 12] and weak [8, 12, 17] approximation. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes like simple iteration, modified Newton iteration or full Newton iteration. For the numerical solution of ODEs such iterative schemes have been studied by the means of B-series and rooted trees [13, 14] and it was shown that certain growth functions defined on trees give a precise description of the development of the iterations. The aim of the present paper is to do a similar analysis in the SDE case. In particular, it will be shown that the growth functions describing the iterative schemes are the same in the ODE and the SDE case. Let (Ω, A, P) be a probability space. We denote by (X(t)) t∈I the stochastic process which is the solution of either a d-dimensional Itô or Stratonovich SDE defined by