Kulik A. M.
Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with levy noise
Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 435-447
A sufficient condition is obtained for smoothness of the density of distribution for a multidimensional Levy-driven Ornstein-Uhlenbeck process, i.e., a solution to a linear stochastic differential equation with Levy noise.
Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1208-1224
For additive functionals defined on a sequence of Markov chains that approximate a Markov process, we establish the convergence of functionals under the condition of local convergence of their characteristics (mathematical expectations).
Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 340–381
For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given.
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise
Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1261–1283
The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise.
Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition
Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 170–186
For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.
On the solution of a one-dimensional stochastic differential equation with singular drift coefficient
Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 642–655
We determine generalized diffusion coefficients and describe the structure of local times for a process defined as a solution of a one-dimensional stochastic differential equation with singular drift coefficient.
Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 323-331
We study the classes C(α, β) and C H(α, β) of logarithmically convex measures that are a natural generalization of the notion of Boltzmann measure to an infinite-dimensional case. We prove a theorem on the characterization of these classes in terms of finite-dimensional projections of measures and describe some applications to the theory of random series.
Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes
Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 216-226
We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1226-1250
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.
Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 288
Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 936–945
We give a sequence of stochastic integro-differential equations that approximates a stochastic differential equation with an anticipating initial condition and localized Skorokhod stochastic integral. A sequence of solutions of these equations is obtained. The convergence of this sequence to a certain process implies that this process is a solution (generally speaking, local) of the original equation.
Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 769–775
We give the complete description of the set of partial limits for a large class of sequences of weighted sums of independent random variables with triangular matrices of coefficients.
Approximations of continuous periodic functions that are differentiate along the trajectories of dynamical systems
Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 111–114