In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.

We consider a machine which generates income at rate I during its operating time. The machine is composed of N independent stochastically failing units. A failure of each one of the units causes a breakdown of the machine.

The machine's status (good or failed) is observed continuously, by a controller, at zero cost. In the event of a breakdown, exactly one unit is failed, and a series of inspections is performed, in order to identify the failed unit and the reasons for its failure. At any time along the inspection process only one unit can be tested.

An inspection of a given unit is characterized by its cost rate and inspection time. At the end of the inspection process, the failed unit is repaired at a known cost, and a new operating cycle is started.

Our objective is to formulate an optimal inspection strategy under two optimality criteria: long-run average net income, and total expected discounted net income.

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

We consider, for t in the boundary of a Galton–Watson tree $(\partial \textsf{T})$, the covering number $(\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (sn), we almost surely establish the Hausdorff dimension of the set $\{ t \in \partial {\textsf{T}}:{{\textsf{N}}_n}(t) - nb \ {\sim} \ {s_n}\} $ for b ∈ I.

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics.

In this paper we consider the long term behaviour till absorption of multitype Galton-Watson processes using the theory of quasi-stationary distributions.

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R2, is given.

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i(t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γm are independent of Mk = (Mk,1, · ··, Mk,m), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M1 and M2 are transformed into some stochastic orders between Z1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γm are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects Wv ⊆ W such that w ∈ Wv with probability p, independently for all v ∈ V and all w ∈ W. Given two vertices v1, v2 ∈ V, we set v1 ∼ v2 if and only if Wv1 ∩ Wv2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

In this paper we present simple extensions of earlier works on the optimal time to exchange one basket of log Brownian assets for another. A superset and subset of the optimal stopping region in the case where both baskets consist of multiple assets are obtained. It is also shown that a conjecture of Hu and Øksendal (1998) is false except in the trivial case where all the assets in a basket are the same processes.

We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution, where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations are typically more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Zn; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An ‘environment’ in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here ‘term-time forcing’ refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {Λ(t)}t≥0. Threshold conditions in terms of R⋆ are obtained, such that R⋆>1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.

In this paper we build the increasing convex (concave) order for the scalar product of random vectors with an upper (lower) tail permutation decreasing joint density. As applications, we revisit allocations of portfolio risks in financial engineering and of coverage limits and deductibles in insurance. Some related results in the literature are substantially updated.

In this article, we consider the limit behavior of the hazard rate function of mixture distributions, assuming knowledge of the behavior of each individual distribution. We show that the asymptotic baseline function of the hazard rate function is preserved under mixture.

Let Xi ≥ 0 be independent, i = 1,…, n, with known distributions and let Xn*= max(X1,…,Xn). The classical ‘ratio prophet inequality’ compares the return to a prophet, which is EXn*, to that of a mortal, who observes the Xis sequentially, and must resort to a stopping rule t. The mortal's return is V(X1,…,Xn) = max EXt, where the maximum is over all stopping rules. The classical inequality states that EXn* < 2V(X1,…,Xn). In the present paper the mortal is given k ≥ 1 chances to choose. If he uses stopping rules t1,…,tk his return is E(max(Xt1,…,Xtk)). Let t(b) be the ‘simple threshold stopping rule’ defined to be the smallest i for which Xi ≥ b, or n if there is no such i. We show that there always exists a proper choice of k thresholds, such that EXn* ≤ ((k+1)/k)Emax(Xt1,…,Xtk)), where ti is of the form t(bi) with some added randomization. Actually the thresholds can be taken to be thej/(k+1) percentile points of the distribution of Xn*, j = 1,…,k, and hence only knowledge of the distribution of Xn* is needed.