Non-degeneracy of Wiener functionals arising from rough differential equations

By T. Cass, Peter K. Friz, Nicolas Victoir. 2007

64 citations recorded by the source corpus.

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of the solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/268036e4ae7607ec4b35af534d3ec75db3f5aa28

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Citations in the corpus

Integrability of (Non-)Linear Rough Differential Equations and Integrals (2011, background): “H →֒ C where 1/p+ 1/q > 1 and sample paths have finite p-variation, a natural condition, in particular in the context of Malliavin calculus, whose importance was confirmed in a number of papers, [11], [10], [3], [2], see also [12].”
The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory (2013, background): “CYR has many consequences: for instance, it allows for a Malliavin calculus [3, 4], [20], Chapters 15, 20, w.”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, unknown): “3 is taken from [22].”
Densities for rough differential equations under Hormander's condition (2007, background): “The proof of Theorem 1 is based on the fact [3] that RDE solutions driven by Gaussian signals are “H-differentiable,” i.”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “A theory of such Gaussian rough paths has been developed by a succession of authors [3, 5, 7, 10], and we will mostly work within their framework.”
Differential equations driven by Gaussian signals (2007, methodology): “Relatedly, solutions to (rough) differential equations driven by Gaussian signals are H-differentiable which allows to establish density results using Malliavin calculus, to be discussed in [5].”
A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, background): “concerning the law of the solution [1, 4, 29] or its ergodic properties [20], the picture here is far from being complete.”
Partial differential equations driven by rough paths (2008, background): “� t and then imposing the necessary non-degeneracy conditions on φ. The existence of a density for the solution of an RDE driven by a Gaussian signal was proved in [ 1 ] under the following assumptions on the vector fields and the driving signal.”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “7) for the Cameron–Martin space, for any 1/p+1/q > 1, [7], Proposition 2.”
Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion (2009, unknown): “See also [2,3] for other recent results.”