Flow properties of differential equations driven by fractional Brownian motion. (English) Zbl 1135.60034

Decreusefond, Laurent; Nualart, David

Baxendale, Peter (ed.) et al., Stochastic differential equations: theory and applications. A volume in honor of Professor Boris L. Rozovskii. Hackensack, NJ: World Scientific (ISBN 978-981-270-662-1/hbk). Interdisciplinary Mathematical Sciences 2, 249-262 (2007).

D. Nualart and A. Răşcanu [Collect. Math. 53, No.1, 55-81 (2002; Zbl 1018.60057)] studied stochastic differential equations (SDEs) \(dX_s^{t,x}=\sigma(s,X_s^{t,x})dB_s^H+b(s,X_s^{t,x})ds,\, s\in[t,T],\, X_t^{t,x}=x\in \mathbb{R}^d,\) driven by an \(m\)-dimensional fractional Brownian motion \(B^H\). Their assumption on the Hurst parameter \(H>1/2\) guarantees that the stochastic integral occurring in the SDE can be defined pathwise in the Riemann-Stieltjes sense. They proved a global existence and uniqueness result for the solutions of such SDEs, and they also got pathwise estimates of the solution process. The authors of the present paper use these results of Nualart and Rascanu, and they study the flow and homeomorphic properties of \(X_s^{t,x}\) as a function of \(x\). Since the equation is a pathwise one, deterministic methods are used here, in particular a polygonal approximation of the fractional Brownian motion and pathwise estimates of the solution process depending on its initial condition.
For the entire collection see [Zbl 1117.60001].

Reviewer: Rainer Buckdahn (Brest)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals

Keywords:

fractional Brownian motion; stochastic differential equation; stochastic flow

Citations:

Zbl 1018.60057