For a semicontinuous homogeneous process ξ(t) with independent increments the distribution of the its total duration of stay in an interval is obtained. In the case E ξ(1) = 0, E ξ(1) 2 < ∞, the limit theorem on a weak convergence of the time of duration of stay in an interval of the process to distribution of the time of duration of stay of Wiener process in the interval(0, 1) is obtained. ForWiener process the distribution of the total duration of stay in an interval is found.
Abstract. Two-boundary problems for a random walk with negative geometric jumps are considered, and the corresponding results for a usual semicontinuous random walk are generalized for them. The following results are obtained: the probability distribution of ruin is found and expressed in terms of the lower and upper boundaries; formulas are given for the joint distribution of the infimum, supremum, and the walk itself at an arbitrary time instance; the transient probabilities and ergodic distribution are evaluated for the process describing the evolution of the random walk with two boundaries.Two-boundary problems for random walks and stochastic processes have several applications in the queue theory, storage and inventory theories, reliability theory, and in many other fields.Two-boundary problems have been studied for semicontinuous random walks and for semicontinuous stochastic processes. In this paper we solve two-boundary problems for random walks with negative geometric jumps. This model is a generalization of a usual model of semicontinuous random walks.
Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.
IntroductionWe assume that all random variables and stochastic processes are defined on (Ω, F, {F t }, P ), a filtered probability space, where the filtration {F t } satisfies the usual conditions of rightcontinuity and completion. A Lévy process is a F -adapted stochastic process {ξ(t); t ≥ 0} which has independent and stationary increments and whose paths are right-continuous with left limits [1]. Under the assumption that ξ(0) = 0 the Laplace transform of the process {ξ(t); t ≥ 0} has the form E[e −pξ(t) ] = e t k(p) , Re p = 0, where the function k(p) is called the Laplace exponent and is given by the formula ([6])Here α, σ ∈ R and Π(·) is a measure on the real line. The introduced process is a space homogeneous, strong Markov process. Note, that the distribution of the first exit time from an interval plays a crucial role in applications and its knowledge also allows to solve a number of other two-boundary problems. Let us fix B > 0 and define the variable
For a process ξ(t) with independent increments an integral transformations for the joint distribution of the moment of the first exittime of process from an interval and the value of overjump of process through border at the moment of this exittime are obtained.
1.Let ξ(t) ∈ R, t ≥ 0, ξ(0) = 0 be homogeneous process with independent increments [1], with the cumulantand continuous from the right sample trajectories. Note that this process is strictly Markov process.([2]) Let B > 0 be fixed, x ∈ (0, B), y = B − x, and introduce random variableis the moment of the first exittime from the interval (−y, x) of the process ξ(t), t ≥ 0. The moment of the first exittime of this process from an interval is a Markov moment.which are a value of exittime of process through upper bound at the moment of the first exittime of the process from an interval through upper bound and a value of overjump of process through lower bound at the moment of the first exittime of the process from an interval through lower bound. The goal of this paper is the finding of the integral transformations1 Translated by A. I. Vladimirova
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