2019
Том 71
№ 6

Malovichko T. V.

Articles: 3
Article (Russian)

Girsanov theorem for stochastic flows with interaction

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 384-390

We prove an analog of the Girsanov theorem for the stochastic differential equations with interaction $$dz(u,t) = a(z(u,t),μt)dt + ∫R f(z(u,t)−p)W(dp,dt),$$ where $W$ is a Wiener sheet on $ℝ × [0; +∞)$ and $a(∙)$ is a function of special type.

Article (Russian)

Convergence of solutions of stochastic differential equations to the Arratia flow

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1529–1538

We consider the solution $x_{\varepsilon}$ of the equation $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt),$$ $$x_{\varepsilon}(u,0) = u,$$ where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$ For the case where $\varphi_{\varepsilon}^2$ converges to $p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ i.e., where a boundary function describing the influence of a random medium is singular more than at one point, we prove that the weak convergence of $\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ to $\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$ takes place as $\varepsilon\rightarrow0_+$ (here, $X$ is the Arratia flow).

Article (Ukrainian)

Properties of a wiener process with coalescence

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 489–504

A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue measure. The behavior of a semigroup of a Wiener process with coalescence in the two-dimensional case and properties of a Wiener flow with coalescence are studied.