Volume 15, Issue 1

In this paper we use the chaos expansion method to define a derivative operator and the corresponding Skorokhod integral for Levy processes with no drift and Brownian part. The main tool in establishing the derivation property is the product formula for two multiple integrals.

A paper is devoted to new expansions of random processes in the form of series. In particular case the expansions in series of stationary stochastic processes with absolutely continuous spectral function and the expansions with respect to some functions which generate wavelet basis are obtained. These results are used for model construction of stochastic processes in such way that the model approximates the process with given reliability and accuracy in some Banach spaces. The conditions of uniform convergence of Gaussian random series with independent summands are also given.

We obtain a formula for the distribution of the first hitting time by the Brownian motion of an one-sided curved boundary f ( t ) which gives a new characterization of the first-passage density p ( t ). It is shown that p ( t ) is a product of the known density for reaching the level y = f (0) at time t and a function that is a solution to the Cauchy problem for some parabolic operator. The boundary-independent part of this operator is just a part of the Kolmogorov's equation for a Brownian excursion transition density.

This article studies the existence of weak solutions for a stochastic version of the FitzHugh-Nagumo equation in a time dependent domain Q , where Q is the image of a cylinder C of n +1 . The random elements are introduced through initial values and forcing terms of associated Cauchy problem, which may be a white noise in the time.

This paper shows that discretization after the application of Itô formula in the Girsanov likelihood produces estimators of the drift which have faster rates of convergence than the Euler estimator for stationary ergodic diffusions and is free of approximating the stochastic integral. The discretization schemes are related to the Hausdorff moment problem. Interalia I introduce a new stochastic integral which will be of independent interest. I show strong consistency, asymptotic normality and a Berry-Esseen bound for the corresponding approximate maximum likelihood estimators of the drift parameter from high frequency data observed over a long time period.

The distribution of a random variable ξ = 2 −k τ k , where τ k are independent random variables taking values 0, 1, 2 with probabilities p ik , i = 0, 1, 2, is investigated. The behavior of the characteristic function of the r.v. ξ at infinity is studied in details, and necessary and sufficient conditions for singularity resp. absolute continuity of the distribution of the r.v. ξ are found.