Hanski's incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.

We construct a flow of continuous-time and discrete-state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses, over the positive half line, studied in Li (2014).

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function f and our current net worth X(t) for any t, we invest an amount f(X(t)) in the market. We need a fortune of M 'superdollars' to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Itô process dX(t) = (1 + f(X(t)))dt + f(X(t))dW(t). We show how to choose the optimal f = f₀ and show that the choice of f₀ is optimal among all nonanticipative investment strategies, not just among Markovian ones.

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

We consider an r component system having an arbitrary binary monotone structure function. We suppose that shocks occur according to a point process and that, independent of what has already occurred, each new shock is one of r different types, with respective probabilities p₁, . . . , p_r. We further suppose that there are given integers n₁, . . . , n_r such that component i fails (and remains failed) when there have been a total of n_i type-i shocks. Letting L be the time at which the system fails, we are interested in using simulation to estimate E[L], E[L²], and P(L > t). We show how to efficiently accomplish this when the point process is (i) a Poisson, (ii) a renewal, and (iii) a Hawkes process.

Environmental stress screening (ESS) of manufactured items is used to reduce the occurrence of future failures that are caused by latent defects by eliminating the items with these defects. Some practical descriptions of the relevant ESS procedures can be found in the literature; however, the appropriate stochastic modeling and the corresponding thorough analysis have not been reported. In this paper we develop a stochastic model for the ESS, analyze the effect of this operation on the population characteristics of the screened items, and also consider the relevant optimization issues.

Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D(a₀, . . . , a_d), Pr(B = (0, . . . , 0, 1, 0, . . . , 0)) = a_i / a with a = ∑_i=0^da_i, and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution B_k(a₀, . . . , a_d) with k an integer such that the above result holds when B follows B_k(a₀, . . . , a_d) and when Y ~ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a₀ = · · · = a_d = 1.

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

In this paper we study a reinsurance game between two insurers whose surplus processes are modeled by arithmetic Brownian motions. We assume a minimax criterion in the game. One insurer tries to maximize the probability of absolute dominance while the other tries to minimize it through reinsurance control. Here absolute dominance is defined as the event that liminf of the difference of the surplus levels tends to -∞. Under suitable parameter conditions, the game is solved with the value function and the Nash equilibrium strategy given in explicit form.

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n^-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, . . . , m}, and let S_T,i be the stopped sum denoting the number of appearances of outcome 'i' in X₁, . . . , X_T, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S_T,0, S_T,1, . . . , S_T,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set-valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.

In Chaudhuri and Dasgupta's 2006 paper a certain stochastic model for 'replicating character strings' (such as in DNA sequences) was studied. In their model, a random 'input' sequence was subjected to random mutations, insertions, and deletions, resulting in a random 'output' sequence. In this paper their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that, under certain conditions, strict stationarity of the 'input' sequence would be preserved by the 'output' sequence, and they proved a similar 'preservation' result for the property of strong mixing with exponential mixing rate. In our setup, we will in spirit slightly extend their 'preservation of stationarity' result, and also prove a 'preservation' result for the property of absolute regularity with summable mixing rate.

As the name suggests, the family of general error distributions has been used to model nonnormal errors in a variety of situations. In this article we show that the asymptotic distribution of linearly normalized partial maxima of random observations from the general error distributions is Gumbel when the parameter of these distributions lies in the interval (0, 1). Our result fills a gap in the literature. We also establish the corresponding density convergence, obtain an asymptotic distribution of the partial maxima under power normalization, and state and prove a strong law. We also study the asymptotic behaviour of observations near the partial maxima and the sum of such observations.

We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n - 1) · · · X(1) of independent and identically distributed copies X(k) of X converge almost surely as n → ∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

We consider a system of urns of Pólya type, containing balls of two colors; the reinforcement of each urn depends on both the content of the urn and the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely as time tends to ∞ to the same limit for all urns. A normal approximation for a large number of urns is also obtained.

We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

We propose a two-urn model of Pólya type as follows. There are two urns, urn A and urn B. At the beginning, urn A contains r_A red and w_A white balls and urn B contains r_B red and w_B white balls. We first draw m balls from urn A and note their colors, say i red and m - i white balls. The balls are returned to urn A and bi red and b(m - i) white balls are added to urn B. Next, we draw ℓ balls from urn B and note their colors, say j red and ℓ - j white balls. The balls are returned to urn B and aj red and a(ℓ - j) white balls are added to urn A. Repeat the above action n times and let X_n be the fraction of red balls in urn A and Y_n the fraction of red balls in urn B. We first show that the expectations of X_n and Y_n have the same limit, and then use martingale theory to show that X_n and Y_n converge almost surely to the same limit.