555 research outputs found
Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach
We investigate the problem of finding necessary and sufficient conditions for
convergence in distribution towards a general finite linear combination of
independent chi-squared random variables, within the framework of random
objects living on a fixed Gaussian space. Using a recent representation of
cumulants in terms of the Malliavin calculus operators (introduced
by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to
random variables living in a finite sum of Wiener chaoses. As an important
by-product of our analysis, we shall derive a new proof and a new
interpretation of a recent finding by Nourdin and Poly \cite{n-po-1},
concerning the limiting behaviour of random variables living in a Wiener chaos
of order two. Our analysis contributes to a fertile line of research, that
originates from questions raised by Marc Yor, in the framework of limit
theorems for non-linear functionals of Brownian local times
Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion
We prove an existence and uniqueness theorem for solutions of
multidimensional, time dependent, stochastic differential equations driven
simultaneously by a multidimensional fractional Brownian motion with Hurst
parameter H>1/2 and a multidimensional standard Brownian motion. The proof
relies on some a priori estimates, which are obtained using the methods of
fractional integration, and the classical Ito stochastic calculus. The
existence result is based on the Yamada-Watanabe theorem.Comment: 21 page
Mixed fractional stochastic differential equations with jumps
In this paper, we consider a stochastic differential equation driven by a
fractional Brownian motion (fBm) and a Wiener process and having jumps. We
prove that this equation has a unique solution and show that all its moments
are finite
A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution
In this paper, we establish a version of the Feynman-Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman-Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.postprin
Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index
We consider a mixed stochastic differential equation driven by possibly
dependent fractional Brownian motion and Brownian motion. Under mild regularity
assumptions on the coefficients, it is proved that the equation has a unique
solution
Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise
We consider the family of stochastic partial differential equations indexed
by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) =
\eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*}
(t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation,
is a second-order partial differential operator with constant coefficients,
and are smooth functions and is a Gaussian noise, white
in time and with a stationary correlation in space. Let p^\eps_{t,x} denote
the density of the law of u^\eps(t,x) at a fixed point
(t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0}
\eps^2\log p^\eps_{t,x}(y) for a fixed . The results apply to a class
of stochastic wave equations with and to a class of stochastic
heat equations with .Comment: 39 pages. Will be published in the book " Stochastic Analysis and
Applications 2014. A volume in honour of Terry Lyons". Springer Verla
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