A version of Hörmander’s theorem for the fractional Brownian motion

By Fabrice Baudoin, Martin Hairer. 2007

97 citations recorded by the source corpus.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/d2c0a9d6c2ee6f82ea26be11f4d11aaf25c24b5d

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Citations in the corpus

Densities for rough differential equations under Hormander's condition (2007, background): “Using deterministic estimates for the Jacobian of the flow this density was then shown to be smooth [20]; building on the same estimates the Hörmander case was obtained in [1].”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “Suppose R has finite twodimensional ρ-variation for some ρ in [1,2).”
Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion (2009, methodology): “The regularity results obtained here have been used in a recent paper by Baudoin and Hairer [1] to show the smoothness of the density under a hypoellipticity Hörmander’s condition.”
On probability laws of solutions to differential systems driven by a fractional Brownian motion (2014, methodology): “c1 min[0,1] |f |4 ‖f ‖2∞ + ‖f ‖2γ ≤ ‖f ‖(2)H is proved in [5], Lemma 4.”
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths (2011, background): “tely from Lemma 1 the bound khϕ,ZVi(·)ikCγ ≤MR a (5.7) z for some exponent a. Combining this with (5.5), the claim follows at once. The next lemma involves an iterative argument (similar in spirit to [3, 29, 38]) to show that a similar bound holds with Vj replaced by any vector ﬁeld obtained by taki”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “Existence of a smooth density under Hörmander’s condition was then obtained in [9], relying on some specific properties of fractional Brownian motion.”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “Then if R is of finite ρ-variation for some ρ ∈ [1,2) we can embed H in the space Cρ-var(I,Rd); in fact, |h|H ≥ |h|ρ- var;I √ Vρ(R; I × I) .”
Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion (2012, background): “Regularity (in the sense of Malliavin calculus) and absolute continuity of the law of the random variables Xt have since been investigated in [2, 19, 23, 26].”
Controlled differential equations as Young integrals: A simple approach (2010, background): “In addition, this result may serve as a base for dealing with Malliavin calculus for SDE driven by fractional Brownian motion [ 2 , 32].”