Anticipating random periodic solutions—I. SDEs with multiplicative linear noise

Abstract

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward-backward infinite horizon stochastic integral equations (IHSIEs), using the "substitution theorem" of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward-backward IHRIE in C(R, L 2 loc (Ω)) using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener-Sobolev spaces in C([0, T ], L 2 (Ω)) and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in C(R, L 2 (Ω)). Thus we obtain the existence of a random periodic solution and a periodic measure.

On Random Periodic Solution to a Neutral Stochastic Functional Differential Equation

“…But the existence of the periodic solutions of the stochastic differential systems is more complex than the deterministic case. Recently, there are some works studying existence random periodic solutions for SDEs, such as Feng et al [29][30][31]. In [30], Feng et al study the existence of random periodic solutions to semilinear stochastic differential equations.…”

“…Recently, there are some works studying existence random periodic solutions for SDEs, such as Feng et al [29][30][31]. In [30], Feng et al study the existence of random periodic solutions to semilinear stochastic differential equations. In [31], an ergodic theory of random dynamical systems has been built under the stationary regime, in which stationary solutions and stationary measures, which are "equivalent," are fundamental objects.…”

Numerical approximation of random periodic solutions of stochastic differential equations

“…Let X be a separable Banach space, (Ω, F, P, (θ t ) t∈R ) be a metric dynamical system. Consider a stochastic periodic semi-flow u : ∆ × Ω × X → X of period τ , which satisfies the semiflow relation u(t, r, ω) = u(t, s, ω) • u(s, r, ω), (1.1) and the periodic property u(t + τ, s + τ, ω) = u(t, s, θ τ ω), (1.2) for all r ≤ s ≤ t. SDEs and SPDEs with time-dependent coefficients which are periodic in time generate periodic semiflows satisfying (1.1) and (1.2) ([5]- [7]).…”

“…We will show that when k → ∞, the pull-back X −kτ r (ξ) has a limit X * r in L 2 (Ω) and X * r is the random periodic solution of SDE (1. We separate the linear term AX from the nonlinear term in (1.3) to enable us to represent the random periodic solution by IHSIE ( [5], [7]). This is helpful to formulate the scheme for SPDEs for which random periodic solutions were considered in [6].…”

“…Numerical analysis for random periodic solutions was not considered in previous work. The infinite horizon stochastic integral equation (IHSIE) method can deal with anticipated cases ([5]- [7]). But it is still not clear how to numerically approximate two-sided IHSIE and anticipating random periodic solutions.…”