The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching

Abstract

The existence and uniqueness of the numerical invariant measure of the backward Euler-Maruyama method for stochastic differential equations with Markovian switching is yielded, and it is revealed that the numerical invariant measure converges to the underlying invariant measure in the Wasserstein metric. Under the polynomial growth condition of drift term the convergence rate is estimated. The global Lipschitz condition on the drift coefficients required by Bao et al., 2016 and Yuan et al., 2005 is released. Several examples and numerical experiments are given to verify our theory.

Continuous stage stochastic Runge–Kutta methods

“…However, most SDEs arising in practice are nonlinear, and cannot be solved explicitly. There has been tremendous interests in developing effective and reliable numerical methods for SDEs during the last few decades, for example see [4][5][6][7][8][9][10][11][12][13][14]. Runge-Kutta (RK) methods with continuous stage were firstly presented by Butcher in 1970s [15], and they have been investigated and discussed by several authors recently because of the great advantages in conserving symplecticity [16], preserving energy [17] and so on.…”

The stochastic θ method for stationary distribution of stochastic differential equations with Markovian switching

“…Although the classical Euler-Maruyama method is easy to use in computation and implementation, the numerical solutions of SDEs with super-linear coefficients may diverge to infinity in finite time. To tackle this drawback, Li used the backward Euler-Maruyama method to prove the numerical invariant measure under one-sided Lipschitz condition for those SDEs with Markovian switching, see [10].…”

Invariant measures of the Milstein method for stochastic differential equations with commutative noise

“…When the Markovian switching is combined with SDEs, authors in [11] and [2] studied the Euler-Maruyama method. In a more recent paper [8], the authors investigated the backward Euler-Maruyama method.…”