230 research outputs found
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
Intersection local times of independent fractional Brownian motions as generalized white noise functionals
In this work we present expansions of intersection local times of fractional
Brownian motions in , for any dimension , with arbitrary Hurst
coefficients in . The expansions are in terms of Wick powers of white
noises (corresponding to multiple Wiener integrals), being well-defined in the
sense of generalized white noise functionals. As an application of our
approach, a sufficient condition on for the existence of intersection local
times in is derived, extending the results of D. Nualart and S.
Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional
Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and
more general Hurst coefficients.Comment: 28 page
On small time asymptotics for rough differential equations driven by fractional Brownian motions
We survey existing results concerning the study in small times of the density
of the solution of a rough differential equation driven by fractional Brownian
motions. We also slightly improve existing results and discuss some possible
applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of
Peter Laurenc
Invariant densities for dynamical systems with random switching
We consider a non-autonomous ordinary differential equation on a smooth
manifold, with right-hand side that randomly switches between the elements of a
finite family of smooth vector fields. For the resulting random dynamical
system, we show that H\"ormander type hypoellipticity conditions are sufficient
for uniqueness and absolute continuity of an invariant measure.Comment: 16 pages; we replaced our original article to point out and close a
gap in the discussion of the Lorenz system in Section 7 (see Remark 2); this
gap is only present in the journal version of this article --- it wasn't
present in the previous arxiv versio
On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
We show a concise extension of the monotone stability approach to backward
stochastic differential equations (BSDEs) that are jointly driven by a Brownian
motion and a random measure for jumps, which could be of infinite activity with
a non-deterministic and time inhomogeneous compensator. The BSDE generator
function can be non convex and needs not to satisfy global Lipschitz conditions
in the jump integrand. We contribute concrete criteria, that are easy to
verify, for results on existence and uniqueness of bounded solutions to BSDEs
with jumps, and on comparison and a-priori -bounds. Several
examples and counter examples are discussed to shed light on the scope and
applicability of different assumptions, and we provide an overview of major
applications in finance and optimal control.Comment: 28 pages. Added DOI
https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final
publication, corrected typo (missing gamma) in example 4.1
Fractional backward stochastic differential euqations and fractional backward variational inequalities
In the framework of fractional stochastic calculus, we study the existence
and the uniqueness of the solution for a backward stochastic differential
equation, formally written as: [{[c]{l}% -dY(t)=
f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t), \quad t\in[0,T], Y(T)=\xi,.]
where is a stochastic process given by , , and is a
fractional Brownian motion with Hurst parameter greater than 1/2. The
stochastic integral used in above equation is the divergence-type integral.
Based on Hu and Peng's paper, \textit{BDSEs driven by fBm}, SIAM J Control
Optim. (2009), we develop a rigorous approach for this equation. Moreover, we
study the existence of the solution for the multivalued backward stochastic
differential equation [{[c]{l} -dY(t)+\partial\varphi(Y(t))dt\ni
f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t),\quad t\in[0,T], Y(T)=\xi,.] where
is a multivalued operator of subdifferential type associated
with the convex function .Comment: 41 page
Malliavin calculus for fractional heat equation
In this article, we give some existence and smoothness results for the law of
the solution to a stochastic heat equation driven by a finite dimensional
fractional Brownian motion with Hurst parameter . Our results rely on
recent tools of Young integration for convolutional integrals combined with
stochastic analysis methods for the study of laws of random variables defined
on a Wiener space.Comment: Dedicated to David Nualart on occasion of his 60th birthda
Stein's method on Wiener chaos
We combine Malliavin calculus with Stein's method, in order to derive
explicit bounds in the Gaussian and Gamma approximations of random variables in
a fixed Wiener chaos of a general Gaussian process. We also prove results
concerning random variables admitting a possibly infinite Wiener chaotic
decomposition. Our approach generalizes, refines and unifies the central and
non-central limit theorems for multiple Wiener-It\^o integrals recently proved
(in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre,
Peccati and Tudor. We apply our techniques to prove Berry-Ess\'een bounds in
the Breuer-Major CLT for subordinated functionals of fractional Brownian
motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck
semigroups, we also recover a technical result recently proved by Chatterjee,
concerning the Gaussian approximation of functionals of finite-dimensional
Gaussian vectors.Comment: 39 pages; Two sections added; To appear in PTR
Fractional smoothness and applications in finance
This overview article concerns the notion of fractional smoothness of random
variables of the form , where is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages
Shell Model for Time-correlated Random Advection of Passive Scalars
We study a minimal shell model for the advection of a passive scalar by a
Gaussian time correlated velocity field. The anomalous scaling properties of
the white noise limit are studied analytically. The effect of the time
correlations are investigated using perturbation theory around the white noise
limit and non-perturbatively by numerical integration. The time correlation of
the velocity field is seen to enhance the intermittency of the passive scalar.Comment: Replaced with final version + updated figure
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