Invariant measures of the Milstein method for stochastic differential equations with commutative noise
Provides theoretical guarantees for the Milstein method's long-term behavior in approximating invariant measures, relevant for computational stochastic dynamics.
The paper proves that the Milstein method for SDEs with commutative noise has an exponential convergence rate to its unique invariant measure, and the numerical invariant measure converges to the true one with rate 1.
In this paper, the Milstein method is used to approximate invariant measures of stochastic differential equations with commutative noise. The decay rate of the transition probability kernel generated by the Milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable. The convergence rate of the numerical invariant measure to the underlying one is shown to be a one. Numerical simulations are presented to demonstrate the theoretical results.