arXiv subjects
Publications and source records attributed to Eija Laukkarinen.
We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is a pure jump Lévy process and $f$ is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f(X_1)$ depend both on the regularity of $f$ and the Blumenthal-Getoor index of the Lévy measure.
We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying Lévy process is a compound Poisson process on a finite time interval.
Assume a Lévy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.
The Malliavin derivative for a Lévy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$