Functional large deviations for Stroock's approximation to a class of Gaussian processes with application to small noise diffusions

2022/06/03 by Hui Jiang, Lihu Xu, Jiang, Hui +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2206.01351

openalex publication_date 2022/06/03 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28

Abstract

Letting~N=\N(t), t≥0\ be a standard Poisson process, Stroock~ \citeStroock-1981 constructed a family of continuous processes by Θ_ε(t)=∫₀^tθ_ε(r)dr, 0 ≤ t ≤ 1, where θ_ε(r)=\frac1ε(-1)^N(ε^-2r), and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over Θ_ε(t), and find the explicit form for rate function. As an application, we consider the following (non-Markovian) stochastic differential equation \beginaligned X^ε(t) amp;=x₀+∫^t₀b(X^ε(s))ds+λ(ε)∫^t₀σ(X^ε(s))dΘ_ε(s), \endaligned where b and σ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as ε→ 0. The rate function indicates a phase transition phenomenon as λ(ε) moves from one region to the other.

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