# On numerical approximation to stationary distribution of diffusion process

Suppose a vector-valued diffusion process X satisfies the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t) dW_t,$$ in which $W$ is a Brownian motion and $b,\sigma$ are such that strong existence and uniqueness of solution to the SDE hold. Assume also that $X$ has a stationary distribution. Sufficient condition for the existence of a stationary distribution can be found in, e.g. Khasminskii, Stochastic Stability of Differential Equations. In a recent book by Mao and Yuan, Stochastic differential equations with Markovian switching, a Euler Mariamman method is proposed to numerically approximate the stationary distribution of $X$. But there is no mentioning on the efficiency of the algorithm, say, the convergence rate. What will be the convergence rate? And are there any other methods in the literature available? Many thanks for your comments and suggestions.

For details on why the algorithm works, Mao has a paper on Euler methods for SDEs with Markovian Switching. Here he shows that the EM method converges with order 1/2 in the $L^2$ sense. The key from the analysis is that the Higham, Mao, Stuart proof for SDEs pretty much carries over, so your intuition about Euler methods in SDEs with Markovian switching can pretty much follow your Euler method intuitions from SDEs. Again, this error is just the pathwise error and not the steady state error, but this estimate is required for the convergence to the steady state.