Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis
This work provides new numerical methods for solving stochastic differential equations, with stability analysis that helps determine step size restrictions for reliable approximation.
The authors derived coefficient families for diagonally drift-implicit stochastic Runge-Kutta methods of weak order one and two for Itô SDEs, and analyzed their asymptotic and mean-square stability, comparing stability domains to those of the test equation.
The class of stochastic Runge-Kutta methods for stochastic differential equations due to Rößler is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.