Smoothness of the density for solutions to Gaussian rough differential equations

Abstract

We consider stochastic differential equations of the form $d{Y}_t=V(Y_t)d{X}_t+V_0(Y_t)dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\dots ,V_d)$ satisfy Hörmander's bracket condition, we demonstrate that $Y_t$ admits a smooth density for any $t\in (0,T]$, 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 $T$.

Published in at http://dx.doi.org/10.1214/13-AOP896 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Integrability and tail estimates for Gaussian rough differential equations
• Doob--Meyer for rough paths
• On Malliavin's proof of Hörmander's theorem
• Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions

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• Malliavin regularity of solutions to mixed stochastic differential equations
• Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions
• Constrained Rough Paths
• A flow-based approach to rough differential equations
• Path-by-path regularisation through multiplicative noise in rough, Young, and ordinary differential equations
• The Hörmander condition for delayed stochastic differential equations
• Penalisation techniques for one-dimensional reflected rough differential equations
• A support and density theorem for Markovian rough paths
• Precise Local Estimates for Hypoelliptic Differential Equations driven by Fractional Brownian Motions
• Regularity of the Ito-Lyons map
• Density of the signature process of fBm
• On the Variational Regularity of Cameron-Martin paths
• Positivity of the density for rough differential equations
• The Uniqueness of Signature Problem in the Non-Markov Setting
• Rough path theory
• A Version of Hörmander's Theorem for Markovian Rough Paths
• A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths
• Smoothness of densities for path-dependent SDEs under Hörmander's condition
• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions