Invariant measures of the stochastic theta method for stochastic differential equations with super-linearly growing coefficients
For researchers in numerical SDEs, this is an incremental extension of known results to a broader class of methods.
The paper extends existing results on numerical invariant measures for SDEs with super-linearly growing coefficients from specific methods (backward Euler-Maruyama) to the more general stochastic theta method, proving existence, uniqueness, and convergence of the numerical invariant measure.
The stochastic theta method is proposed to approximate invariant measures of stochastic differential equations (SDEs), both of whose drift and diffusion coefficients may grow super-linearly. For the numerical solution generated by the stochastic theta method, we show the existence and uniqueness of the numerical invariant measure first. Then, we prove that the numerical invariant measure is convergent to the exact invariant measure of the underlying SDE. We also provide some numerical simulations to illustrate our theoretical results. This work could be regarded as an extension of the results in [Y. Jiang et al, Numer. Algorithms 83(4)(2020), pp. 1531-1553] to the case of super-linearly growing diffusion coefficient. As the backward Euler-Maruyama (EM) method is a special case of the stochastic theta method, the results derived in this work could also be regarded as a generalization of the results for the backward EM method in [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the stochastic theta method.