Smoothness of the density for solutions to Gaussian rough differential equations

TY - JOUR

T1 - Smoothness of the density for solutions to Gaussian rough differential equations

AU - Cass, Thomas

AU - Hairer, Martin

AU - Litterer, Christian

AU - Tindel, Samy

PY - 2015/2

Y1 - 2015/2

N2 - We consider stochastic differential equations of the form dYt = V (Yt)dXt + V0(Yt)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V = (V1, . . . , Vd ) satisfy Hörmander's bracket condition, we demonstrate that Yt admits a smooth density for any t ε (0,T ], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H >1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.

AB - We consider stochastic differential equations of the form dYt = V (Yt)dXt + V0(Yt)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V = (V1, . . . , Vd ) satisfy Hörmander's bracket condition, we demonstrate that Yt admits a smooth density for any t ε (0,T ], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H >1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.

KW - Gaussian processes

KW - Malliavin calculus

KW - Rough path analysis

UR - https://www.scopus.com/pages/publications/84920262357

U2 - 10.1214/13-AOP896

DO - 10.1214/13-AOP896

M3 - Article

AN - SCOPUS:84920262357

SN - 0091-1798

VL - 43

SP - 188

EP - 239

JO - Annals of Probability

JF - Annals of Probability

IS - 1

ER -