Order conditions of stochastic Runge--Kutta methods by B-series (Q2706380)

Convergence of stochastic Runge-Kutta (SRK) methods for approximating the solution of a vector Stratonovich stochastic differential equation of the form NEWLINE\[NEWLINEdy= g_0(y) dt+ \sum^d_{j=1} g_j(y) dW_j,\quad y(t_0)= y_0,NEWLINE\]NEWLINE where the \(W_j(t)\), \(j= 1,\dots, d\), are independent Wiener processes, is studied. Theorems are proved that establish the relationship between local order of convergence and global order of convergence of SRK methods. The development of SRK methods by employing B-series is explained and then practiced to obtain 5 stage SRK method with strong global order 1.5.