65 research outputs found
The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion
The paper focuses on discrete-type approximations of solutions to
non-homogeneous stochastic differential equations (SDEs) involving fractional
Brownian motion (fBm). We prove that the rate of convergence for Euler
approximations of solutions of pathwise SDEs driven by fBm with Hurst index
can be estimated by ( is the diameter of
partition). For discrete-time approximations of Skorohod-type quasilinear
equation driven by fBm we prove that the rate of convergence is .Comment: 21 pages, (incorrect) weak convergence result removed, to appear in
Stochastic
Stochastic evolution equations driven by Liouville fractional Brownian motion
Let H be a Hilbert space and E a Banach space. We set up a theory of
stochastic integration of L(H,E)-valued functions with respect to H-cylindrical
Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in
the interval (0,1). For Hurst parameters in (0,1/2) we show that a function
F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical
Liouville fBm if and only if it is stochastically integrable with respect to an
H-cylindrical fBm with the same Hurst parameter. As an application we show that
second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by
space-time noise which is white in space and Liouville fractional in time with
Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous
both and space.Comment: To appear in Czech. Math.
Regularity of the Solutions to SPDEs in Metric Measure Spaces
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernel estimates. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4
A parabolic stochastic differential equation with fractional Brownian motion input
An existence and uniqueness theorem is proved for a quasilinear stochastic evolution equation with an additive noise in the form of a stochastic integral with respect to a Hilbert space-valued fractional Borwnian motion. Ideas of the finite-dimensional approximation by the Galerkin method are used.Fractional Brownian motion Rigged Hilbert spaces Stochastic evolution equation
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