The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.

A recent result of Takács (1995) gives explicitly the density of the time spent before t above a level x ≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

In the framework of discounted Markov decision processes, we consider the case that the transition probability varies in some given domain at each time and its variation is unknown or unobservable.

To this end we introduce a new model, named controlled Markov set-chains, based on Markov set-chains, and discuss its optimization under some partial order.

Also, a numerical example is given to explain the theoretical results and the computation.

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

Let {(Xn,Jn)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {Fij}, Fij(B) = ℙ[X1 ∈ B, J1 = j | J0 = i], B ∈ B(ℝ), i, j ∈ E. If Fij([x,∞))/(1-H(x)) → Wij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G+(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼Xn < 0, at least one Wij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[supn≥0Sn > x] → (−𝔼Xn)−1 ∫x∞ ℙ[Xn > u]du as x → ∞, where Sn = ∑1nXk, S0 = 0. Two general queueing applications of this result are given.

First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

A stochastic gradient descent method is combined with a consistent auxiliary estimate to achieve global convergence of the recursion. Using step lengths converging to zero slower than 1/n and averaging the trajectories, yields the optimal convergence rate of 1/√n and the optimal variance of the asymptotic distribution. Possible applications can be found in maximum likelihood estimation, regression analysis, training of artificial neural networks, and stochastic optimization.

The Robbins–Monro algorithm with randomly varying truncations for measurements with non-additive noise is considered. Assuming that the function under observation is locally Lipschitz-continuous in its first argument and that the noise is a φ-mixing process, strong consistency of the estimate is shown. Neither growth rate restriction on the function, nor the decreasing rate of the mixing coefficients are required.

We introduce the concept of the absorbing process for analysing a state process. Our aim is to show the existence of the absorbing process with probability one. This process is shown to be stationary, asymptotically stationary, periodic or a.m.s., if the input distribution has such properties. The real process is absorbed into this process so that its stability and some other properties are easily derived.

The autopsy data of a coherent system is the pair (T, I), where T is the time to the failure of the system, and I is the set of components failed by that time. A basic question is under which conditions the distribution of the autopsy data determines the joint life distribution of the components. This problem has been studied under various distributional and structural assumptions (always under the assumption of independent component life distributions) by Meilijson (1981), Nowik (1990) and Antoine, Doss and Hollander (1993). In this paper we study the same problem in the situation when the autopsy data is supplemented with data from lifemonitoring and conditional lifemonitoring of some components, as introduced by Meilijson (1994).

A variety of convergence results for genealogical and line-of-descendent processes are known for exchangeable neutral population genetics models. A general convergence-to-the-coalescent theorem is presented, which works not only for a larger class of exchangeable models but also for a large class of non-exchangeable population models. The coalescence probability, i.e. the probability that two genes, chosen randomly without replacement, have a common ancestor one generation backwards in time, is the central quantity to analyse the ancestral structure.

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.

Let X = (X1, …, Xn) be a random binary vector, with a known joint distribution P. It is necessary to inspect the coordinates sequentially in order to determine if Xi = 0 for every i, i = 1, …, n. We find bounds for the ratio of the expected number of coordinates inspected using optimal and greedy searching policies.

The traffic equations are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the traffic equations are therefore of great importance. This note shows that the new condition stating that each transition is covered by a minimal closed support T-invariant, is necessary and sufficient for the existence of a solution for the traffic equations for batch routing queueing networks and stochastic Petri nets.

An infinite dam with input formed by a compound Poisson process is considered. As an output policy, we adopt the PλM-policy. The stationary distribution and expectation of the level of water in the reservoir are obtained.

The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.