Abstract
In this paper, we discuss the numerical approximation of random periodic solutions of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to -infty along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler–Maruyama scheme and modified Milstein scheme. Subsequently, we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of sqrt{Delta t} in the mean square sense in Euler–Maruyama method and Delta t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.
Highlights
Periodic solution has been a central concept in the theory of dynamical systems since Poincare’s pioneering work [18,19,20,21]
The definition of random periodic solutions and their existence for semi-flows generated by non-autonomous stochastic differential equations (SDEs) and SPDEs with additive noise were given in [5,6]
We study stochastic differential equations, which possess random periodic solutions, and approximate them by Euler–Maruyama and Milstein schemes
Summary
Periodic solution has been a central concept in the theory of dynamical systems since Poincare’s pioneering work [18,19,20,21]. There have been a number of recent works such as [3] on random attractors of the stochastic TJ model in climate dynamics; [2] on stochastic lattice systems; [4] on stochastic resonance; [7] for SDEs with multiplicative linear noise; and [25] on bifurcations of stochastic reaction diffusion equations All these results are theoretical on the existence of random periodic paths. We will show that when k → ∞, the pull-back Xr−kτ (ξ) has a limit Xr∗ in L2(Ω) and Xr∗ is the random periodic solution of SDE (1.3) It satisfies the infinite horizon stochastic integral equation (IHSIE). We will consider the convergence of transition probabilities generated by (1.3) and its numerical scheme along the integral multiples of period to the periodic measure and discretised periodic measure, respectively, and error estimate of the two periodic measures in the weak topology
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