118 research outputs found
On the law of the solution to a stochastic heat equation with fractional noise in time
We study the law of the solution to the stochastic heat equation with
additive Gaussian noise which behaves as the fractional Brownian motion in time
and is white in space. We prove a decomposition of the solution in terms of the
bifractional Brownian motion
Sample Paths of the Solution to the Fractional-colored Stochastic Heat Equation
Let u = {u(t, x), t [0, T ], x R d } be the solution to the
linear stochastic heat equation driven by a fractional noise in time with
correlated spatial structure. We study various path properties of the process u
with respect to the time and space variable, respectively. In particular, we
derive their exact uniform and local moduli of continuity and Chung-type laws
of the iterated logarithm
Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process
The purpose of this paper is to estimate the self-similarity index of the
Rosenblatt process by using the Whittle estimator. Via chaos expansion into
multiple stochastic integrals, we establish a non-central limit theorem
satisfied by this estimator. We illustrate our results by numerical
simulations
The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution
In this article we consider the stochastic heat equation in (0,T) \times \bR^d, with vanishing initial conditions, driven by a
Gaussian noise which is fractional in time, with Hurst index , and colored in space, with spatial covariance given by a function
. Our main result gives the necessary and sufficient condition on for
the existence of the process solution. When is the Riesz kernel of order
this condition is , which is a relaxation of
the condition encountered when the noise is white in space.
When is the Bessel kernel or the heat kernel, the condition remains
Statistical aspects of the fractional stochastic calculus
We apply the techniques of stochastic integration with respect to fractional
Brownian motion and the theory of regularity and supremum estimation for
stochastic processes to study the maximum likelihood estimator (MLE) for the
drift parameter of stochastic processes satisfying stochastic equations driven
by a fractional Brownian motion with any level of H\"{o}lder-regularity (any
Hurst parameter). We prove existence and strong consistency of the MLE for
linear and nonlinear equations. We also prove that a version of the MLE using
only discrete observations is still a strongly consistent estimator.Comment: Published at http://dx.doi.org/10.1214/009053606000001541 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Martingale structure of Skorohod integral processes
Let the process Y(t) be a Skorohod integral process with respect to Brownian
motion. We use a recent result by Tudor (2004), to prove that Y(t) can be
represented as the limit of linear combinations of processes that are products
of forward and backward Brownian martingales. Such a result is a further step
towards the connection between the theory of continuous-time (semi)martingales,
and that of anticipating stochastic integration. We establish an explicit link
between our results and the classic characterization, due to Duc and Nualart
(1990), of the chaotic decomposition of Skorohod integral processes. We also
explore the case of Skorohod integral processes that are time-reversed Brownian
martingales, and provide an "anticipating" counterpart to the classic Optional
Sampling Theorem for It\^{o} stochastic integrals.Comment: To appear in The Annals of Probabilit
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