[Paper Review] Ergodicity for Neutral Type SDEs with Infinite Length of Memory

Jianhai Bao, Feng‐Yu Wang+1 more|arXiv (Cornell University)|May 9, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance16 references4 citations

TL;DR

This paper establishes exponential ergodicity for neutral-type stochastic differential equations (SDEs) with infinite memory using a Wasserstein coupling approach based on the weak Harris theorem. By constructing a coupling via change of measure and leveraging a reference norm with exponential decay, the authors prove geometric ergodicity in the Wasserstein distance for functional solutions on path space, even under high degeneracy and infinite-dimensional dynamics.

ABSTRACT

In this paper, the weak Harris theorem developed in \cite{HMS11} is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

Motivation & Objective

• To establish exponential ergodicity for neutral-type SDEs with infinite memory, where solutions are non-Markovian and the path space is infinite-dimensional.
• To overcome the limitations of classical tools—such as functional inequalities, Lyapunov conditions, and standard Harris’ theorem—due to the lack of Dirichlet forms and infinitesimal generators on path space.
• To develop a framework applicable to highly degenerate, path-dependent SDEs where total variation and strong Feller properties are unavailable due to mutual singularity of laws.
• To extend the weak Harris theorem to infinite-memory systems using a Wasserstein coupling constructed via change of measure.
• To provide a concrete example demonstrating the applicability of the theoretical framework to a specific class of neutral SDEs with exponentially decaying memory influence.

Proposed method

• Adapts the weak Harris theorem from [19] to infinite-memory neutral SDEs using a Wasserstein coupling constructed via change of measure.
• Introduces a weighted norm $\|\cdot\|_r$ on the path space $\mathscr{C}_r$ to model exponential decay of historical influence, with $r>0$ controlling the decay rate.
• Defines the segment process $X_t(\theta) = X(t+\theta)$ for $\theta \in (-\infty,0]$, transforming the infinite-dimensional SDE into a Markov process on $\mathscr{C}_r$.
• Applies Itô's formula to the squared norm of the coupling difference $\Lambda^X(t)$, incorporating drift and diffusion terms with memory dependence.
• Uses BDG inequality to control the martingale term in the coupling dynamics, ensuring integrability and moment bounds.
• Imposes conditions (H1), (A1), (A2), and (1.10) to ensure contraction and uniform integrability, leading to exponential decay of the Wasserstein distance between laws.

Experimental results

Research questions

• RQ1 Can exponential ergodicity be established for neutral-type SDEs with infinite memory, where classical tools fail due to infinite-dimensional and degenerate structure?
• RQ2 How can the weak Harris theorem be adapted to path-dependent SDEs with infinite memory, especially when total variation and strong Feller properties are not available?
• RQ3 What conditions on the drift, diffusion, and delay kernels ensure that the functional solution converges geometrically in Wasserstein distance?
• RQ4 Can a Wasserstein coupling via change of measure be constructed to yield exponential contraction for such infinite-memory systems?
• RQ5 What role does the exponential decay norm $\|\cdot\|_r$ play in ensuring moment stability and convergence?

Key findings

• The functional solution to the neutral-type SDE (1.1) is exponentially ergodic in the Wasserstein distance under the proposed conditions.
• The convergence rate is geometric, with $\mathbb{E}[\text{W}_2^2(P_t(x,\cdot),\mu)] \leq C e^{-\kappa t}$ for some $C>0$, $\kappa>0$, where $\mu$ is the invariant measure.
• The key condition $\lambda_1 - 2r\beta_1 - \lambda_2\delta_r(\mu_0) > 0$ ensures sufficient contraction in the coupling dynamics.
• The example in Section 3.1 verifies all required conditions: (H1) with $\alpha = \gamma_1 \sqrt{\delta_r(\mu_0)} < 1$, (A1)–(A2), and (1.10) via Hölder and Young’s inequalities.
• The moment bound $\mathbb{E}[\sup_{s\leq t} e^{2rs}|\Lambda^X(s)|^2] \leq c_5((1+t)\|\xi\|_r^2 + e^{2rt} + \int_0^t e^{2rs}\mathbb{E}|X(s)|^2 ds)$ confirms stability and supports exponential decay.
• The proof shows that the Wasserstein coupling via change of measure yields a contraction that leads to exponential ergodicity despite the absence of strong Feller or irreducibility in total variation.