2019
Том 71
№ 11

### All Issues

Articles: 4
Article (Russian)

### Two-boundary problems for a random walk

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1485–1509

We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps.

Article (Russian)

### Two-boundary problems for a Poisson process with exponentially distributed component

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 922–953

For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the interval.

Article (Russian)

### On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1359–1384

For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution of the supremum, infimum, and value of the process.

Article (Russian)

### Boundary Functionals of a Semicontinuous Process with Independent Increments on an Interval

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 381-398

We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered.