# Gusak D. V.

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

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.

### 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.

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

### Conditions for balance between survival and ruin

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.

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

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 $l_r(\infty)$.

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

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.

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

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.

### 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.

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

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

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.

### Anatolii Yakovych Dorogovtsev

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

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

### Compound Poisson Processes with Two-Sided Reflection

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.

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

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.

### On the 75th Birthday of Vladimir Semenovich Korolyuk

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

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

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

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

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.

### Ruin problem for an inhomogeneous semicontinuous integer-valued process

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

For a process ξ(*t* = ξ_{1}(*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, ξ_{1} (*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).

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

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

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

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 $B \rightarrow \infty$.

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

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.

### Basic identities for additive continuously distributed sequences

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.

### On crossing of a level by processes defined by sums of a random number of terms

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 897–914

We study the joint distribution of boundary functionals related to the crossing of a positive (negative) level by a process consisting of a homogeneous Poisson process and a process defined by sums of a random number of continuously distributed terms.

### Oscillating processes with independent increments and nondegenerate Wiener component

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

### Lattice semicontinuous poisson processes on Markov chains

Gusak D. V., Tureniyazova A. I.

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

### Ergodic distribution of an oscillating process with independent increments

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

### Vladimir Semenovich Korolyuk (on his sixtieth birthday)

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

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

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

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

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

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

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

Gusak D. V., Peresypkina S. I.

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

### Asymptotic method for probability problems

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

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

### 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.

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

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

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

### The work of the Sixth Mathematical School

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

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

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

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

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

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.