Vadym Radchenko

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• Doctor of Sciences
• Professor (Full) at Taras Shevchenko National University of Kyiv

Random functions $\mu (x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu (x)$, $x\in {[0,1]^{d}}$, is proved. Fourier series expansion of $\mu (x)$, $x\in [0,2\pi ]$, is obtained. These results are proved under weaker conditions than similar results in previous papers.

Random functions $\mu(x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu(x)$, $x\in[0,1]^d$ is proved. Fourier series expansion of $\mu(x)$, $x\in[0,2\pi]$ is obtained. These results are proved under weaker conditions than similar results in previous papers.

We study the class of one-dimensional equations driven by a stochastic measure $\mu$. For $\mu$ we assume only $\sigma$-additivity in probability. This class of equations include the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.

We consider the stochastic transport equation where the randomness is given by the symmetric integral with respect to stochastic measure. For stochastic measure, we assume only $\sigma$-additivity in probability and continuity of paths. The existence and uniqueness of the weak solution to the equation are proved.

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

We study the one-dimensional equation driven by a stochastic measure μ \mu . For μ \mu we assume only σ \sigma -additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equatio...

The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.

The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.

This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes....

This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes....

This chapter presents the solutions to the equations listed as exercises at the end of the each chapter of this book devoted to the study of Stochastic Measures (SMs). The book defines an SM as an σ‐additive in probability random function defined on an σ‐algebra of sets. It studies general stochastic measures. For many important classes of random p...

This chapter considers the definition of the integral of a random function and studies the equation with such an integral. It aims to construct an integral of a random function with respect to stochastic measures (SM). The chapter defines the “symmetric” integral as the limit in probability of midpoint integral sums. It then provides the definition...

General Stochastic Measures provides a comprehensive theoretical overview of stochastic measures (SMs), including the basic properties of the integrals of real functions with respect to SMs. An SM is a σ‐additive in probability random function, defined on a σ‐algebra of sets. This chapter considers equations driven by SMs and the solution to the st...

General Stochastic Measures provides a comprehensive theoretical overview of stochastic measures (SMs), including the basic properties of the integrals of real functions with respect to SMs. An SM is a σ‐additive in probability random function, defined on a σ‐algebra of sets. This chapter defines the integral of a Hilbert‐space‐valued function with...

General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. This chapter provides the averaging principle for the one‐dimensional heat equation driven by an SM and a stochastic equation with the symmetric integral. In both cases, the rate of...

General Stochastic Measures provides a comprehensive theoretical overview of stochastic measures (SMs), including the basic properties of the integrals of real functions with respect to SMs. An SM is a σ‐additive in probability random function, defined on a σ‐algebra of sets. The chapter focuses on the study of a Riemann‐type integral. With this de...

General Stochastic Measures provides a comprehensive theoretical overview of stochastic measures (SMs), including the basic properties of the integrals of real functions with respect to SMs. An SM is a σ‐additive in probability random function, defined on a σ‐algebra of sets. This chapter proves that the paths of SMs belong to the isotropic Besov s...

This chapter considers some partial differential equations driven by Stochastic Measures (SMs). The existence, uniqueness and Holder regularity of the solutions are obtained. It explores separately the cases of SM dependent on the spatial variable and SM dependent on the time variable. It explains the mild solutions of the equations and the weak so...

We consider the stochastic heat equation on [0; T] x R in the mild form driven by a general stochastic measure m, for m we assume only sigma-additivity in probability. The time-averaging of the equation is studied, we estimate the rate of uniform a. s. convergence to the solution of the averaged equation.

Stochastic parabolic equation driven by a σ-finite stochastic measure in the interval [0,T] × R is studied. The only condition imposed on the stochastic integrator is its σ-additivity in probability on bounded Borel sets. The existence, uniqueness, and Hölder continuity of a mild solution are proved. These results generalize those known earlier for...

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fej\`{e}r sums we obtain the approximations of mild solution of the equation.

Equation with the symmetric integral with respect to stochastic measure is considered. For the integrator, we assume only $\sigma$-additivity in probability and continuity of the paths. It is proved that the averaging principle holds for this case, the rate of convergence to the solution of the averaged equation is estimated.

We study the one-dimensional stochastic heat equation in the mild form driven by a general stochastic measure $\mu$, for $\mu$ we assume only $\sigma$-additivity in probability. The time averaging of the equation is considered, uniform a. s. convergence to the solution of the averaged equation is obtained.

We study the one-dimensional stochastic heat equation in the mild form driven by a general stochastic measure $\mu$, for $\mu$ we assume only $\sigma$-additivity in probability. The time averaging of the equation is considered, uniform a. s. convergence to the solution of the averaged equation is obtained.

The Fourier series and Fourier–Haar series are introduced for general stochastic measures. The convergence of partial sums of these series and the absolute continuity of a stochastic measure are studied. An application is given for the convergence of solutions of the stochastic heat equation.

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.

We consider processes of the form μ(t) = μ((0,t]), where μ is a σ-additive in probability stochastic set function. Convergence of a random Fourier series to μ(t) is proved, and the approximation of integrals with respect to μ using Fejèr sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by μ...

The Fourier transform is defined for general stochastic measures in ℝd. The inversion theorem for this transform is proved and a connection to the convergence of stochastic integrals is established. An example of applications of this result is considered for the convergence of solutions of the stochastic heat equation.

Let $\mu$ be a general stochastic measure, where we assume for $\mu$ only $\sigma$-additivity in probability and continuity of paths. We prove that the symmetric integral $\int_{[0,T]}f(\mu_t, t)\circ\,{\rm d}\mu_t$ is well defined. For stochastic equations with this integral, we obtain the existence and uniqueness of a solution.

Let $\mu$ be a general stochastic measure, where we assume for $\mu$ only $\sigma$-additivity in probability and continuity of paths. We prove that the symmetric integral $\int_{[0,T]}f(\mu_t, t)\circ\,{\rm d}\mu_t$ is well defined. For stochastic equations with this integral, we obtain the existence and uniqueness of a solution.

For random functions with trajectories from Besov space, the integral with respect to general stochastic measure is defined. For some stochastic equations with such integrals, the statements on existence and uniqueness of solutions are obtained.

We consider stochastic evolution equations in Hilbert space driven by general stochastic measures. For stochastic measures in the equations we assume σ-additivity in probability only. The integrals of deterministic functions with respect to stochastic measures in Hilbert space are defined. Existence and continuity of the mild solutions of the equat...

A stochastic heat equation on $[0,T]\times{\mathbb{R}}$ driven by a general stochastic measure $d\mu(t)$ is investigated in this paper. For the integrator $\mu$, we assume the $\sigma$-additivity in probability only. The existence, uniqueness, and H\"{o}lder regularity of the solution are proved.

Stochastic integrals of real-valued functions with respect to general stochastic measures are considered in the chapter. For the integrator we assume the σ-additivity in probability only. The chapter contains a review of recent results concerning Besov regularity of stochastic measures, continuity of paths of stochastic integrals, and solutions of...

We prove that continuous paths of \sigma-additive in probability set function belong to Besov space.

A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3.

For stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of such integrals are studied in the paper.

We consider the stochastic cable equation that involves an integral with respect to a general random measure. We prove that the paths of the mild solution of the equation are Hölder continuous.

We prove a theorem on the continuity with respect to a parameter and an analogue of Fubini's theorem for integrals with respect to a general stochastic measure defined on Borel subsets of $ \mathbb{R}$. These results are applied to study the stochastic heat equation considered in a mild as well as in a weak form.

This paper considers stochastic measures, i.e., sets of functions given on the Borel sigma-algebra in [0, 1]d sigma-additive with respect to probability. It is shown that realizations of continuous random functions generated by stochastic measures belong to the Besov spaces under some general sufficiently assumptions.

The stochastic heat equation on [0, T] x R driven by a general stochastic measure is investigated. Existence and uniqueness of the solution is established. Hölder regularity of the solution in time and space variables is proved.

We consider the heat equation and wave equation with constant coefficients that contain a term given by an integral with respect to a random measure. Only the condition of sigma-additivity in probability is imposed on the random measure. Solutions of these equations are presented. For each equation, we prove that its solutions coincide under certai...

We study integrals of real functions considered with respect to general random measures. The integrals are assumed to depend on a parameter. We obtain sufficient conditions for the existence of a continuous version of random functions and sufficient conditions such that a random measure generated by increments of these random functions exists.

We consider a model where the price of an option is driven by a Wiener process and changes randomly at the moments determined by a homogeneous Poisson process. The formula for the minimum variance hedging strategy is obtained for a European type call option. The derivation of the formula is based on the Föllmer-Schweizer decomposition of a continge...

Let μ be a random σ-additive in probability set function defined on Borel subsets of [a,b]. We prove that if the process , has continuous paths, then they belong a.s. to the Besov space for all .

We consider a model in which the asset price is driven by the Wiener process and, in addition, has random changes at earlier known nonrandom time moments. The explicit form of the variance-minimizing hedging strategy for the European call option is derived. The results are based on the Föllmer-Schweizer decomposition of contingent claims.

The product of a random measure $X$ and a real measure $Y$ is defined as a random measure on $X\times Y$. We obtain conditions under which the integral of a real function with respect to the product measure equals the iterated integrals of this function.

Some information on the XLI all-Ukrainian olympiad of pupils (2001, Ternopil') is given. Pupils of 8-11 forms took part at the olympiad. Olympiad's problems with solutions are presented.

We obtain conditions for the convergence of expressions$$(\mu (A))^{ - 1} \smallint _A fd\mu $$ inL 0 as the setA decreases.

For random functions that are sums of random functional series, we determine an integral over a general random measure and prove limit theorems for this integral. We consider the solution of an integral equation with respect to an unknown random measure.

$\sm$-additive random measures and integrals with respect to them of real valued functions are considered in the most general setting. The statement of convergence of $\int f\,d\mu_n\tlp\int f\,d\mu$, $\ny$, is proved under conditions similar to uniform integrability. An analogue of the Valle--Poussin theorem is established. A criterion is given fo...

σ-additive random measures and integrals with respect to them of real valued functions are considered in the most general setting. The statement of convergence of (Latin small letter esh) f dμn →P (Latin small letter esh) f dμ, n → ∞, is proved under conditions similar to uniform integrability. An analogue of the Valle-Poussin theorem is establishe...

The integral of a measurable random function ξ(x) with respect to a nonnegative measure m is considered. It is shown that such an integral may be introduced as a limit in the probability of integrals of simple random functions. An analogue of the Lebesgue theorem for convergence in probability ∫ξ n (x)dm, n→∞, is proved. A criterion that this rando...

We study integrals ∫fdμ of real functions over L 0-valued measures. We give a definition of convergence of real functions in quasimeasure and, as a special case, in L 0-measure. For these types of convergence, we establish conditions of convergence in probability for integrals over L 0-valued measures, which are analogous to the conditions of unifo...

Integrals of real functions with respect to Lo-valued measures are considered. It is proved that if the functions fn converge in measure to f, then ∫fndμ ∝fdμ if and only if some condition holds for fn, similar to the condition of uniform integrability for integrals with respect to scalar measures.