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Trees and asymptotic expansions for fractional stochastic differential equations. (English) Zbl 1172.60017
Summary: We consider an \(n\)-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter \(H>1/3\). We derive an expansion for \(E[f(X_t)]\) in terms of \(t\), where \(X\) denotes the solution to the SDE and \(f:\mathbb R^n\to\mathbb R\) is a regular function. Comparing to F. Baudoin and L. Coutin [Stochastic Processes Appl. 117, No. 5, 550–574 (2007; Zbl 1119.60043)], where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case \(H>1/2\).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G17 Sample path properties
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
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