The composite Milstein methods for the numerical solution of Itô stochastic differential equations
Three composite Milstein methods are proposed for approximating strong solutions of the Itô stochastic differential equation (SDE) \[ dy(t)= f(t, y(t))\,dt+ \sum^d_{j=1} g_j(t, y(t))\,dW_j(t), \] where the \(W_j(t)\) are standard Wiener processes. Starting with the case where \(d=1\), the three methods are constructed by combining a semi-implicit Milstein method and an implicit Milstein method. For a linear test equation, the three methods are shown to converge to the exact solution and then mean-square stability, \(T\)-stability, and \(T(A)\)-stability are examined. Summaries of numerical results for a linear SDE and two nonlinear SDEs are provided which suggest that the proposed methods have better accuracy than a known composite Euler method. Then for the case where \(d=2\) similarly favorable stability properties are reported for a linear test equation and for a Duffing-Van der Pol oscillator driven by white noise.
scientific article; zbMATH DE number 5863134