Taylor Expansions and Castell Estimates for Solutions of Stochastic Differential Equations Driven by Rough Paths

Abstract

1.1 Motivation and development when the driving process is fractional Brownian motion with Hurst parameter H . • Resolution :H > 1/2M.Zahle-1998;H ∈ (1/4, 1/2): see P.Friz, N.victoir-2010 •H > 1/2 existence and smoothness of the density: elliptic condition:D.Nualart, B.Saussereau-2009; Y.Hu,D.Nualart-2007. Hormander condition: F.Baudoin, M.Hairer-2007. •H ∈ (1/4, 1/2), smoothness of the density: elliptic condition: T.Cass, C. Litterer, T.Lyons,2013. Hypoelliptic condition: T.Cass, P.Friz-2010, M.Hairer, N.S.Pillai-2013, Y.Hu, S.Tindel-2013 Hormander’s type critetion: T.Cass, M.Hairer, C.Litterer, S.Tindel-2012 •H > 1/2 gradient bound-F.Baudoin, C.Ouyang,2013 upper bound in a skew-symmetric situation:F.Baudoin, C.Ouyang, S.Tindel-To appear. •H > 1/4 strict positivity and upper bounds for the density.-F.baudoin, E.Nualart, C.Ouyang, S. Tindel-2014 Stochastic Taylor series • Brownian motion case: first introduced by R. Azencott (1982) and G. Ben Arous (1989). • convergence and deterministic estimate for fBM with H > 1/2-F.Baudoin, X.J. Zhang-2012 •Asymptotic expansion (Castell estimate) for BM, Castell -1993.