High order local linearization methods: an approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise
New explicit numerical methods called high order local linear (HOLL) methods are introduced for the purpose of strong approximation of the solution of the \(d\)-dimensional stochastic differential equation with additive noise \[ dX(t)= f(t,X(t))\,dt+ \sum^m_{j=1} g_j(t)\,dW^j_t,\quad t\in [t_0,T],\quad X(t_0)= x_0, \] where \(W_t\) is an \(m\)-dimensional standard Wiener process. Theorems are proved that establish the global order of convergence and the A-stability of the methods. The methods are shown, for linear equations with stationary solutions, to produce a random attractor that converges to the actual one as the stepsize approaches zero, a property dubbed random A-stability. Two types of HOLL methods, one employing Taylor schemes and the other employing Runge-Kutta schemes, are presented and then shown in examples to have faster convergence with less computation cost than some comparable known methods.
scientific article; zbMATH DE number 5799390
scientific article