# Dorogovtsev A. A.

### Clark representation for local times of self-intersection of Gaussian integrators

Dorogovtsev A. A., Izyumtseva O. L., Salhi N.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 12. - pp. 1587-1614

We prove the existence of a multiple local time of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernel, describe its Ito – Wiener expansion and establish the Clark representation.

### Mykola Ivanovych Portenko (on his 75th birthday)

Dorogovtsev A. A., Kopytko B.I., Osipchuk M. M.

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1714-1716

### Local times of self-intersection

Dorogovtsev A. A., Izyumtseva O. L.

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 290-340

This survey article is devoted to the local times of self-intersection as the most important geometric characteristics of random processes. The trajectories of random processes are, as a rule, very nonsmooth curves. This is why to characterize the geometric shape of the trajectory it is impossible to use the methods of differential geometry. Instead of this, one can consider the local times of self-intersection showing how much time the process stays in “small” vicinities of its self-crossing points. In our paper, we try to describe the contemporary state of the theory of local times of self-intersection for Gaussian and related processes. Different approaches to the definition, investigation, and application of the local times of self-intersection are considered.

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### Some remarks on a Wiener flow with coalescence

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1327–1333

We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time.

### Smoothing Problem in Anticipating Scenario

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1218–1234

We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration.

### On one condition of weak compactness of a family of measure-valued processes

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 885–891

We present a criterion for the weak compactness of continuous measure-valued processes in terms of the weak compactness of families of certain space integrals of these processes.

### On random measures on spaces of trajectories and strong and weak solutions of stochastic equations

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 625–633

We investigate stationary random measures on spaces of sequences or functions. A new definition of a strong solution of a stochastic equation is proposed. We prove that the existence of a weak solution in the ordinary sense is equivalent to the existence of a strong measure-valued solution.

### Mykola Ivanovych Portenko (On His 60th Birthday)

Dorogovtsev A. A., Kopytko B.I., Korolyuk V. S., Mitropolskiy Yu. A., Samoilenko A. M., Skorokhod A. V., Sytaya G. N.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 147-148

### Measure-Valued Markov Processes and Stochastic Flows

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 178-189

We consider a new class of Markov processes in the space of measures with constant mass. We present the construction of such processes in terms of probabilities that control the motion of individual particles. We study additive functionals of such processes and give examples related to stochastic flows with interaction.

### Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1194-1204

We consider the structure of orthogonal polynomials in the space *L* _{2}(*B*, μ) for a probability measure μ on a Banach space *B*. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (*B*, μ).

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Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1062-1074

We consider integrals of random mappings with respect to consistent random measures in *C*([0; 1]).

### Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 123–127

By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.

### Stochastic integration and one class of Gaussian random processes

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 485–495

We consider one class of Gaussian random processes that are not semimartingales but their increments can play the role of a random measure. For an extended stochastic integral with respect to the processes considered, we obtain the Itô formula.

### Some characteristics of sequences of iterations with random perturbations

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1047-1063

For a sequence of random iterations, we study the set of partial limits and the frequency of visiting their neighborhoods.

### Stochastic analysis and one minimization problem

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1568–1571

We consider the problem of finding the form of a functional of an infinite-dimensional argument for which a certain given expression takes the minimum value for a fixed value of the parameter. The equation obtained for an unknown functional resembles equations with extended stochastic integral.

### Periodic solutions of differential equations perturbed by stochastic processes

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1642–1648

### Generalized stochastic integrals for smooth functionals of the white noise

Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1460–1466

### Analogs of convolution type equations

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1413–1416

### Characteristic functional of a quadratic form in gaussian elements in a Hilbert algebra

Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 230–233