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

Integrability and tail estimates for Gaussian rough differential equations

We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems.Comment: Published in at http://dx.doi.org/10.1214/12-AOP821 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A dual algorithm for stochastic control problems : Applications to Uncertain Volatility Models and CVA

We derive an algorithm in the spirit of Rogers [SIAM J. Control Optim., 46 (2007), pp. 1116--1132] and Davis and Burstein [Stochastics Stochastics Rep., 40 (1992), pp. 203--256] that leads to upper bounds for stochastic control problems. Our bounds complement lower biased estimates recently obtained in Guyon and Henry-Labordère [J. Comput. Finance, 14 (2011), pp. 37--71]. We evaluate our estimates in numerical examples motivated by mathematical finance. Read More: http://epubs.siam.org/doi/10.1137/15M101994

Rough Paths on Manifolds

We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds

On the integration of weakly geometric rough paths

We close a gap in the theory of integration for weakly geometric rough paths in the in…nite-dimensional setting. We show that the integral of a weakly geometric rough path against a su¢ ciently regular one form is, once again, a weakly geometric rough path

A combinatorial approach to geometric rough paths and their controlled paths

We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [CDL15] to the case of arbitrary $p$