We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also determined and the paper concludes with exact distribution results concerning exits from bounded intervals. The proofs use simple martingale techniques together with some classical expansions of polynomials and Rouché's theorem from complex function theory.

This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.

For a bivariate Lévy process (ξt ,ηt )t≥ 0 and initial value V 0 define the generalised Ornstein–Uhlenbeck (GOU) process V t :=eξ t (V 0+∫t 0 e-ξ s- dηs ), t≥0, and the associated stochastic integral process Z t :=∫0 t e-ξ s- dηs , t≥0. Let T z :=inf{t>0: V t <0|V 0=z} and ψ(z):=P(T z <∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of T z as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.

A stochastic perpetuity takes the form D∞=∑n=0 ∞ exp(Y 1+⋯+Y n )Bn , where Y n :n≥0) and (B n :n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by D n+1=A n D n +B n , n≥0, where A n =eY n ; D ∞ then satisfies the stochastic fixed-point equation D ∞D̳AD ∞+B, where A and B are independent copies of the A n and B n (and independent of D ∞ on the right-hand side). In our framework, the quantity B n , which represents a random reward at time n, is assumed to be positive, unbounded with EB n p <∞ for some p>0, and have a suitably regular continuous positive density. The quantity Y n is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D ∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D ∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.

Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large populations. The transition occurs when the population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or grow quickly towards the carrying capacity, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.

We apply the known formulae of the RESTART problem to Markov models of software (and many other) systems, and derive new equations. We show how checkpoints might be included, with their resultant performance under RESTART. The result is a complete procedure for finding the mean, variance, and tail behavior of the job completion time as a function of the failure rate. We also provide a detailed example.

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

Let {M n }n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form Q A (x) = limn→∞P(M n ≤ x | M 0 ≤ A,M 1 ≤ A, …, M n ≤ A). Suppose that M 0 has distribution Q A , and define T A Q A = min{n | M n > A, n ≥ 1}, the first time when M n exceeds A. We provide sufficient conditions for Q A (x) to be nonincreasing in A (for fixed x) and for T A Q A to be stochastically nondecreasing in A.

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒ t,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e , t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

We investigate the maximal number M k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k -1 M k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x 0 > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe6βt ] for positive constants k and p for times t exceeding some positive value u. We obtain explicit expressions for k, p, and u in terms of β, σ, x 0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In the associated Fleming-Viot process N particles evolve independently in Λ with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 / N.

We consider the level hitting times τy = inf{t ≥ 0 | X t = y} and the running maximum process M t = sup{X s | 0 ≤ s ≤ t} of a growth-collapse process (X t )t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of X t and give a power series expansion of the reciprocal of Ee−sτ y . We prove asymptotic results for τy and M t : for example, if m(y) = Eτy is of rapid variation then M t / m -1(t) →w 1 as t → ∞, where m -1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and X t is ergodic, then M t / m -1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

In this paper we introduce conditionally independent increment point processes, that is, processes that are conditionally independent inside and outside a bounded set A given N(A), the number of points in A. We show that these point processes can be characterized by means of the avoidance function of a multinomial ‘support process’, the solution of a suitably defined linear system of equations, and, finally, the infinitesimal matrix of a continuous-time Markov chain.

We present a number of new solutions to an integral equation arising in the limiting theory of Bellman-Harris processes. The argument proceeds via straightforward analysis of Mellin transforms. We also derive a criterion for the analyticity of the Laplace transform of the limiting distribution on Re(u) ≥ -c for some c > 0.

We supply some relations that establish intertwining from duality and give a probabilistic interpretation. This is carried out in the context of discrete Markov chains, fixing up the background of previous relations established for monotone chains and their Siegmund duals. We revisit the duality for birth-and-death chains and the nonneutral Moran model, and we also explore the duality relations in an ultrametric-type dual that extends the Siegmund kernel. Finally, we discuss the sharp dual, following closely the Diaconis-Fill study.

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.