Numerical simulation of nonlinear dynamical systems driven by commutative noise

Abstract

The local linearization (LL) approach has become an effective technique for the numerical integration of ordinary, random and stochastic differential equations. One of the reasons for this success is that the LL method achieves a convenient trade-off between numerical stability and computational cost. Besides, the LL method reproduces well the dynamics of nonlinear equations for which other classical methods fail. However, in the stochastic case, most of the reported works has been focused in Stochastic Differential Equations (SDE) driven by additive noise. This limits the applicability of the LL method since there is a number of interesting dynamics observed in equations with multiplicative noise. On the other hand, recent results show that commutative noise SDEs can be transformed into a random differential equation (RDE) by means of a random diffeomorfism (conjugacy). This paper takes advantages of such conjugacy property and the LL approach for defining a LL scheme for SDEs driven by commutative noise. The performance of the proposed method is illustrated by means of numerical simulations.