2019
Том 71
№ 11

All Issues

Shevchenko H. M.

Articles: 3
Article (Ukrainian)

Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations

Ral’chenko K. V., Shevchenko H. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1256–1268

We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.

Article (Ukrainian)

Euler Approximations of Solutions of Abstract Equations and Their Applications in the Theory of Semigroups

Mishura Yu. S., Shevchenko H. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 399-410

Using the Euler approximations of solutions of abstract differential equations, we obtain new approximation formulas for C 0-semigroups and evolution operators.

Article (Ukrainian)

On Multidimensional Generalized Diffusion Processes

Shevchenko H. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1291-1296

We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Hölder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.