Alexey Kulik

Rates of approximation of nonsmooth integral-type functionals of Markov processes

We provide strong

NONLINEAR TRANSFORMATIONS OF SMOOTH MEASURES ON INFINITE-DIMENSIONAL SPACES

L_p

Asymptotic Behaviour of the Distribution Density of the Fractional Lévy Motion

-rates of approximation of nonsmooth integral-type functionals of Markov processes by integral sums. Our approach is, in a sense, process insensitive and is based on a modification of some well-developed estimates from the theory of continuous additive functionals of Markov processes.

Optimal stopping of random sequences and processes

We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.

Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures

We investigate the distribution properties of the fractional Levy motion defined by the Mandelbrot-Van Ness representation:

Basic functionals of the risk theory

\displaystyle{Z_{t}^{H}:=\int _{ \mathbb{R}}f(t,s)dZ_{s},}

Markov and diffusion processes

where Z s , \(s \in \mathbb{R}\), is a (two-sided) real-valued Levy process, and