A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations (Q1378460)

This paper concerns the strong order of convergence of stochastic Runge-Kutta methods for approximating the solution of a Stratonovich stochastic differential equation of the form \[ dy=f(t,y) dt+g(t,y)dW \] where \(f\) is a vector function, \(g\) is a matrix function, and \(W\) is a vector process whose components are independent Wiener processes. It is proved that if Stratonovich integrals of order no greater than \(p\) are used in an \(s\)-stage Runge-Kutta method, then that method can have order no greater than 1 when \(p=1\) and no greater than \(\min \{{p+1 \over 2}, {s-1 \over 2}\}\) when \(p\geq 2\), \(s\geq 3\).