Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion. (English) Zbl 1226.60091

Unterberger, Jeremie M.

Electron. Commun. Probab. 15, 411-417 (2010).

Summary: As a general rule, differential equations driven by a multi-dimensional irregular path \(\Gamma \) are solved by constructing a rough path over \(\Gamma \). The domain of definition – and also the estimates – of the solutions depend on upper bounds for the rough path; these general, deterministic estimates are too crude to apply them, e.g., to the solutions of stochastic differential equations with linear coefficients driven by a Gaussian process with Hölder regularity \(\alpha <1/2\). We prove here (by showing convergence of Chen’s series) that linear stochastic differential equations driven by an analytic fractional Brownian motion [S. Tindel and the author, Collect. Math. 62, No. 2, 197–223 (2011; Zbl 1220.60022), the author, Ann. Probab. 37, No. 2, 565–614 (2009; Zbl 1172.60007)] with arbitrary Hurst index \(\alpha \in \)(0,1) may be solved on the closed upper half-plane, and that the solutions have finite variance

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals
60G22	Fractional processes, including fractional Brownian motion
60G12	General second-order stochastic processes

Keywords:

stochastic differential equations; fractional Brownian motion; analytic fractional Brownian motion; rough paths; Hölder continuity; Chen series

Citations:

Zbl 1220.60022; Zbl 1172.60007