Smoothness of Flow and Path-by-Path Uniqueness in Stochastic\n Differential Equations

2017/09/07 by Siva Athreya, Athreya, Siva, Suprio Bhar +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1709.02115

openalex publication_date 2017/09/07 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider the stochastic differential equation X_t = x₀ +
int₀^t\nf(X_s)ds +
int₀^t
sigma(X_s)dB^H_s, with x₀ \∈ \ℝ^d, d \≥\n1, f: \ℝ^d \→ \ℝ^d is bounded continuous, \σ:\n\ℝ^d \→ \ℝ^{d\× d} is a uniformly elliptic,\nbounded, twice continuously differentiable conservative vector field and B^H\nis fractional Brownian motion with H \∈ (\(1)/(3), \(1)/(2)]. When\nd=1, H= \(1)/(2), and f is H "older continuous, in the spirit of Davie\n[D07], we establish the existence of a null set \N depending only on\nf, \σ such that for all x₀\∈ \ℝ and \ω \∈\n\Ω\∖ \N, the above equation admits a path-by-path unique\nsolution. Our proof is based on establishing the uniform continuous\ndifferentiability of the flow associated with the equation. We also establish\nthe path-by-path uniqueness for d \≥ 1 and H \∈ (\(1)/(3),\n\(1)/(2)], but the null set may depend on x₀, thus extending a result of\nCatellier-Gubinelli [CG12].\n

Citations

• Averaging along irregular curves and regularisation of ODEs

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