[Paper Review] Random periodic solutions and ergodicity for stochastic differential equations

Kenneth Uda, Huaizhong Zhao|arXiv (Cornell University)|Nov 13, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance25 references3 citations

TL;DR

This paper establishes sufficient conditions for the existence of stable random periodic solutions and ergodicity in stochastic differential equations on R^d using Lyapunov functions, the two-point generator, strong Feller property, and weak convergence. The key contribution is proving the convergence of the Krylov–Bogolyubov scheme for periodic measures under integrability conditions on the drift coefficient, ensuring long-term statistical stability in random periodic regimes.

ABSTRACT

In this paper, we establish some sufficient conditions for the existence of stable random periodic solutions of stochastic differential equations and ergodicity in the random periodic regime. The techniques involve the existence of Lyapunov type function, using two-point generator of the stochastic flow map, strong Feller argument and weak convergence.

Motivation & Objective

• To establish sufficient conditions for the existence of stable random periodic solutions in stochastic differential equations on R^d.
• To investigate ergodicity in the random periodic regime, extending classical ergodic theory to non-autonomous, randomly perturbed systems.
• To develop a rigorous mathematical framework for random periodic solutions, addressing the lack of formal definition and tools in prior work.
• To prove convergence of the Krylov–Bogolyubov scheme for periodic measures under integrability and decay conditions on the drift coefficient.
• To extend the theory of invariant measures to periodic measures in the context of random dynamical systems with time-periodic noise structure.

Proposed method

• Utilizes a Lyapunov-type function to control the growth of solutions and ensure stability in the random periodic regime.
• Employs the two-point generator of the stochastic flow map to analyze the infinitesimal evolution of transition probabilities.
• Applies the strong Feller property to ensure regularity of transition probabilities and smooth dependence on initial conditions.
• Uses weak convergence techniques to analyze the long-time behavior of transition probabilities toward periodic measures.
• Applies Hölder’s inequality and exponential moment estimates to bound the difference between transition probabilities and invariant measures.
• Employs a density argument via approximating indicator functions by bounded Lipschitz functions to extend convergence results to general Borel sets.

Experimental results

Research questions

• RQ1 Under what conditions does a stochastic differential equation on R^d admit a stable random periodic solution?
• RQ2 How can ergodicity be established in the random periodic regime, analogous to classical ergodicity for stationary measures?
• RQ3 What conditions ensure the convergence of the Krylov–Bogolyubov scheme for periodic measures in the presence of time-periodic noise?
• RQ4 How can the two-point generator and strong Feller property be used to analyze the long-term statistical behavior of SDEs with random periodicity?
• RQ5 What role does the integrability of the drift coefficient play in ensuring the existence and uniqueness of periodic measures?

Key findings

• The paper proves that under a Lyapunov-type condition and integrability of the drift coefficient, the Krylov–Bogolyubov scheme for periodic measures converges exponentially fast.
• A uniform exponential decay rate β^nτ with 0 < β < 1 is established for the convergence of the Krylov–Bogolyubov scheme, ensuring statistical stability in the long run.
• The existence of a random periodic solution is guaranteed when the two-point generator satisfies certain boundedness and decay conditions, ensuring pathwise invariance under the random time shift θτ.
• The strong Feller property is used to ensure that transition probabilities are smooth, enabling the application of weak convergence techniques to periodic measures.
• The convergence of the Krylov–Bogolyubov scheme is shown for all Borel sets A ⊂ L_s, extending the result beyond bounded Lipschitz functions via a density argument.
• The paper establishes that periodic measures are invariant under the Markov evolution (T_{s+kτ,s})_k∈ℕ, confirming their role as long-term statistical attractors in the random periodic regime.