2019
Том 71
№ 6

All Issues

Gusak D. V.

Articles: 31
Article (Ukrainian)

On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains

Gusak D. V., Herych M. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1034-1049

For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps.

Anniversaries (Ukrainian)

Volodymyr Semenovych Korolyuk (on his 90th birthday)

Bratiichuk N. S., Gusak D. V., Kovalenko I. N., Lukovsky I. O., Makarov V. L., Samoilenko A. M., Samoilenko I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1151-1152

Brief Communications (Ukrainian)

Conditions for balance between survival and ruin

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 988-993

Let $\xi_t$ be a classic risk process or a risk process with stochastic premiums. We establish conditions for balance between ruin and survival in the case of zero initial capital $u = 0$ (ruin probability $q_{+} = \psi(0) = 1/2$, survival probability $p_{+} = 1 — q_{+} = 1/2$) and determine premium estimates under these conditions.

Article (Ukrainian)

Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1021-1029

Let $\xi(t)$ be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps $\xi_k : \textbf{E}[z^{\xi_k} / \xi_k < 0] = \frac{1 − b}{z − b},\quad 0 ≤ b < 1.$ For the moment generating function of the sojourn time of $\xi(t)$ in a fixed state, we obtain relations in terms of the roots $z_s < 1 < \widehat{z}_s$ of the Lundberg equation. By passing to the limit $(s → 0)$ in the obtained relations, we determine the distributions of $l_r(\infty)$.

Article (Ukrainian)

Behavior of risk processes with random premiums after ruin and a multivariate ruin function

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1473–1484

We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function.

Article (Ukrainian)

Behavior of classical risk processes after ruin and a multivariate ruin function

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1339–1352

We establish relations for the distribution of functionals associated with the behavior of a classical risk process after ruin and a multivariate ruin function.

Anniversaries (Ukrainian)

Volodymyr Semenovych Korolyuk (the 80th anniversary of his birth)

Bratiichuk N. S., Gusak D. V., Kovalenko I. N., Portenko N. I., Samoilenko A. M., Skorokhod A. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1155-1157

Article (Ukrainian)

On the Exit of One Class of Random Walks from an Interval

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1209–1217

We consider the random walk $S_n = \sum_{k\leqn}\xi_k \quad (S_n = 0)$ whose characteristic function of jumps $\xi_k$ satisfies the condition of almost semicontinuity. We investigate the problem of the exit of such $S_n$ from a finite interval.

Obituaries (Ukrainian)

Anatolii Yakovych Dorogovtsev

Buldygin V. V., Gorodnii M. F., Gusak D. V., Korolyuk V. S., Samoilenko A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1151-1152

Article (Ukrainian)

Compound Poisson Processes with Two-Sided Reflection

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1616-1625

We consider a compound oscillating Poisson process with two-sided reflection. This process is defined by an upper-semicontinuous compound Poisson process ξ(t) and its functionals, namely the first-exit time of ξ(t) from an interval and the first-exit time of ξ(t) across the upper and lower levels. We study the main characteristics of this oscillating process in terms of the potential and resolvent of the process ξ(t) introduced by Korolyuk. For this purpose, we refine the Pecherskii identities and some other results for upper-semicontinuous Poisson processes.

Article (Ukrainian)

Distribution of Overjump Functionals of a Semicontinuous Homogeneous Process with Independent Increments

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 303-321

We establish relations for the distributions of functionals associated with an overjump of a process ξ(t) with continuously distributed jumps of arbitrary sign across a fixed level x > 0 (including the zero level x = 0 and infinitely remote level x → ∞). We improve these relations in the case where the distributions of maxima and minima of ξ(t) may have an atom at zero. The distributions of absolute extrema of semicontinuous processes are defined in terms of these atomic probabilities and the cumulants of the corresponding monotone processes.

Anniversaries (Ukrainian)

On the 75th Birthday of Vladimir Semenovich Korolyuk

Gusak D. V., Kovalenko I. N., Samoilenko A. M., Skorokhod A. V., Yadrenko M. I.

Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1011-1013

Article (Ukrainian)

On the Creative Contribution of V. S. Korolyuk to the Development of Probability Theory

Bratiichuk N. S., Gusak D. V., Svishchuk A. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1014-1030

We present a brief survey of the main results obtained by V. S. Korolyuk in probability theory and mathematical statistics.

Article (Ukrainian)

Ruin problem for an inhomogeneous semicontinuous integer-valued process

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 208-219

For a process ξ(t = ξ1(t)+χ(t), t≥0, ξ(0) = 0, inhomogeneous with respect to time, we investigate the ruin problem associated with the corresponding random walk in a finite interval, (here, ξ1 (t) is a homogeneous Poisson process with positive integer-valued jumps and χ(t) is an inhomogeneous lower-semicontinuous process with integer-valued jumps ξ n ≥-1).

Chronicles (Ukrainian)

The third ukrainian-scandinavian conference on probability theory and mathematical statistics

Gusak D. V., Kulik A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 288

Brief Communications (Ukrainian)

Limit behavior of the distribution of the ruin moment of a modified risk process

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 847–853

For modified risk process with instantaneous reflection at the point $B > 0$ under which the considered process $$\zeta(t) = \zeta_{B, \mu}(t),\; \zeta(0) = u,\; 0 \leq u \leq B,$$ returns in the initial state $u$, we investigate the limit behavior of generating function of the first ruin moment as $u \rightarrow B$ and $B \rightarrow \infty$.

Brief Communications (Ukrainian)

On the first ruin moment for a modified risk process with immediate reflection

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1419–1425

For a modified risk process with immediate reflection downward, we establish relations for an integral transformation of its characteristic function and the corresponding transformation of the limit distribution of the considered process under ergodicity conditions. The distribution is obtained for the first ruin moment of the introduced risk process.

Article (Ukrainian)

Basic identities for additive continuously distributed sequences

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1651-1660

For an additive sequence ξ(n), we establish basic factorization identities and express the distributions of limiting Junctionals (extremum values of ξ(n), the time and value of the first jump over a fixed level, etc.) in terms of the components of factorization.

Article (Ukrainian)

Oscillating processes with independent increments and nondegenerate Wiener component

Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1415–1421

Article (Ukrainian)

Lattice semicontinuous poisson processes on Markov chains

Gusak D. V., Tureniyazova A. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 707-711

Article (Ukrainian)

Ergodic distribution of an oscillating process with independent increments

Bratiichuk N. S., Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 547–554

Article (Ukrainian)

Vladimir Semenovich Korolyuk (on his sixtieth birthday)

Gusak D. V., Mitropolskiy Yu. A., Skorokhod A. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1985. - 37, № 4. - pp. 488–489

Article (Ukrainian)

How often is the sum of independent random variables larger than a given number?

Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1982. - 34, № 3. - pp. 289—295

Article (Ukrainian)

Intersection of a level by a homogeneous process with independent increments and a nondegenerate wiener component

Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 373 – 378

Article (Ukrainian)

The time spent above a fixed level by a class of controlled random processes

Gusak D. V., Peresypkina S. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1978. - 30, № 3. - pp. 352–357

Article (Ukrainian)

Asymptotic method for probability problems

Gusak D. V., Mitropolskiy Yu. A., Skorokhod A. V., Turbin A. F.

Full text (.pdf)

Ukr. Mat. Zh. - 1975. - 27, № 4. - pp. 471–476

Article (Ukrainian)

Distribution of the exit time and value for homogeneous processes with independent increments given on a finite Markov chain

Gusak D. V., Peresypkina S. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1974. - 26, № 3. - pp. 291–299

Article (Ukrainian)

A class of processes with independent increments on a finite Markov chain

Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1973. - 25, № 2. - pp. 170—178

Chronicles (Russian)

The work of the Sixth Mathematical School

Demenin A. N., Gusak D. V., Korolyuk V. S.

Full text (.pdf)

Ukr. Mat. Zh. - 1969. - 21, № 2. - pp. 281

Article (Russian)

On the asymptotic of time distribution or the first yield of a homogeneous process with independent increments

Gusak D. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1964. - 16, № 4. - pp. 463-474

Article (Russian)

On the asymptoticity of distributions of maximum deviation in a Poisson process

Gusak D. V., Korolyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1962. - 14, № 2. - pp. 138-144

The author discusses the algorithm of asymptotic expansions for the distribution of maximum deviations in a Poisson process leading to equations for the terms of the asymptotic.