Densities for rough differential equations under Hormander's condition

By T. Cass, Peter K. Friz. 2007

152 citations recorded by the source corpus.

We consider stochastic differential equations dY = V (Y) dX driven by a multidimensional Gaussian process X in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215-310]. Using Malliavin Calculus we show that Y t admits a density for t ∈ (0, T] provided (i) the vector fields V = (V 1 , ... , V d ) satisfy Hormander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1/4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/ad44e35143be7b1ecc1ab72061960e37f8a93efc

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Citations in the corpus

Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “Compared to the setting of [6], we have to impose some additional assumptions on R(s, t), but our more quantitative approach allows us in return to relax the zero–one law condition required in this paper.”
Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion (2009, unknown): “See also [2,3] for other recent results.”
Convergence rates for the full Gaussian rough paths (2011, background): “This framework has since proven useful in a variety of different applications ranging from non-Markovian Hörmander theory [2] to non-linear PDEs perturbed by space-time white-noise [12].”
The Signature of a Rough Path: Uniqueness (2014, background): “Friz [7], to support theorem and large deviations by M.”
Short-time near-the-money skew in rough fractional volatility models (2017, methodology): “A similar situation was in fact encountered in Cass and Friz (2010), where fractional scaling at times near zero was important. To quantify the perturbation, the authors of Cass and Friz (2010) introduced an easy to verify coupling condition (see Corollary 2 in Cass and Friz 2010).”
Differential equations driven by Gaussian signals (2007, background): “Nonetheless, the support description of Gaussian rough path (and resultingly: of iterated integrals up to any given order) has been a key ingredient in establishing non-degeneracy of the Malliavin convariance matrix under H¨ormander’s condit i on [2].”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “Our discussion of RDEs under Hörmander’s condition follows closely [21].”
The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory (2013, methodology): “with F being concave and F(t)=O(t2H ), already the CYR is unclear in the aforementioned references [3, 5].”
Integrability Estimates for Gaussian Rough Differential Equations (2011, methodology): “Worse still, the work Oberhauser and Friz [7] shows that the inequality (1.2) can actually be saturated for a (deterministic) choice of diﬀerential equation and driving rough path.”