Asymptotic expansions of solutions of stochastic differential equations driven by multivariate fractional Brownian motions having Hurst indices greater than 1/3. Issue 6 (3rd December 2018)

• Record Type: Journal Article
• Authors: Vaskouski, Maksim; Kachan, Ilya
• Is Part Of: Stochastic analysis and applications. Volume 36:Issue 6(2018)
• Page Start: 909
• Page End: 931
• Publication Date: 2018-12-03
• Subjects: Asymptotic expansions -- multivariate fractional Brownian motion -- rough path integral -- stochastic differential equation; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2018.1483247
• Languages: English
• ISSNs: 0736-2994

Abstract

Abstract: In this paper, we consider n -dimensional stochastic differential equations driven by multivariate fractional Brownian motion with Hurst indices greater than 1 3 . Using a Taylor type development we obtain an expansion of expectations P t g = E g ( X t x ) for small t, where X t x denotes the solution of the mentioned stochastic differential equation with initial value x ∈ R n, and g : R n → R is a sufficiently smooth function. Also we compute coefficients appearing in the expansion of P t g in the case when the fractional Brownian motions have the Hurst indices greater than 1 2 . Finally, we consider the case of commutative vector fields getting Kolmogorov's backward partial differential equation for the function ( P t g ) ( x ) .