The inverse problem for rough controlled differential equations

We provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable, to some order, from observing the resulting controlled evolution. Physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result.Comment: added section on rough path theor

Controlled viscosity solutions of fully nonlinear rough PDEs

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers

Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths

We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy H\"{o}rmander's celebrated "Lie bracket condition," we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure. Our second main result states that under H\"{o}rmander's condition, the solutions to rough differential equations driven by fractional Brownian motion with $H\in(\frac{1}{3},\frac{1}{2}]$ enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not "look into the future."Comment: Published in at http://dx.doi.org/10.1214/12-AOP777 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form $dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\ldots,V_d)$ satisfy H\"{o}rmander'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 $T$.Comment: 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

Rough linear transport equation with an irregular drift

We study the linear transport equation $$\frac{\partial}{\partial t} u ( t,x ) +b ( t,x ) \cdot \nabla u ( t,x ) + \nabla u ( t,x ) \cdot \frac{\partial}{\partial t} X ( t ) =0, \hspace{2em} u ( 0,x ) =u_{0} ( x )$$ where $b$ is a vectorfield of limited regularity and $X$ a vector-valued H\"older continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the PDE and solve that equation for smooth vectorfields $b$. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed