A singular stochastic differential equation driven by fractional Brownian motion

Abstract

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0.

Existence and Uniqueness of Solution for Generalization of Fractional Bessel Type Process

“…By following similar arguments in the proof of Theorem 2.3 in[11], for all s, t ∈[0, T ], s ≤ t such that t − s ≤ ∆, we have z s,t,∞ ≤ 2|z(s)| + 4γ(k + C)T + 4T β . ∨(8γ(k+C)+8γC)∨(8σγR B 0,T ,β ) 1/β +1 × × |z(0)| + 4γ(k + C)T + 4T β .…”

Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion

“…For any p ≥ 1 and ε > 0, similar to [16,Proposition 3.4], the chain rule applied to 1 (ε+Z(t)) p yields…”

“…We refer to [1,2,6,24] for the study of the standard Brownian case. For the fractional Brownian case with H ∈ ( 1 2 , 1), we give the Malliavin derivative of the exact solution and the boundedness of the inverse moments of the exact solution, utilizing the techniques in [16]. Based on these a priori estimates, we prove that the backward Euler scheme applied to (2), as well as the corresponding numerical approximation to (1), converges with order one in the uniformly strong sense.…”

Integration by Parts Formula and Applications for SDEs Driven by Fractional Brownian Motions

“…For instance, based on the approach of [27], Nualart and Rȃşcanu [22] proved the existence and uniqueness result with Hurst parameter H > 1/2; Coutin and Qian [6] also derived the existence and uniqueness result for H ∈ (1/4, 1/2) via the theory of rough path analysis introduced in [18]; in [14,15] and [24], the authors studied the ergodicity and Talagrand's transportation inequalities for the solutions, respectively; Fan [10,11] established Harnack inequalities for the solution with H < 1/2 and H > 1/2, respectively. As for the regularities, the readers may refer to [3,17,20,23] and references therein. However, as far as we know, the study of the existence of density of the solution mainly depends on the Malliavin calculus.…”