Consider a geometric Brownian motion Xt(ω) with drift. Suppose that there is an independent source that sends signals at random times τ1 < τ2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward Sτ, where St = max(max0≤u≤tXu, s) for some constant s ≥ X0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−rτiSτi | X0 = x, S0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ* is governed by a threshold a* such that τ* = inf{τn: Xτn≤a*Sτn}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ* = τ1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

The aim of this paper is to evaluate the performance of the optimal policy (the Gittins index policy) for open tax problems of the type considered by Klimov in the undiscounted limit. In this limit, the state-dependent part of the cost is linear in the state occupation numbers for the multi-armed bandit, but is quadratic for the tax problem. The discussion of the passage to the limit for the tax problem is believed to be largely new; the principal novelty is our evaluation of the matrix of the quadratic form. These results are confirmed by a dynamic programming analysis, which also suggests how the optimal policy should be modified when resources can be freely deployed only within workstations, rather than system-wide.

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

For two-person zero-sum games, where the probability of each player winning is a continuous function of time and is known to both players, the mutually optimal strategy for proposing and accepting a doubling of the game value is known. We present an algorithm for deriving the optimal doubling strategy of a player who is aware of the suboptimal strategy followed by the opponent. We also present numerical results about the magnitude of the benefits; the results support the claim that repeated application of the algorithm by both players leads to the mutually optimal strategy.

We consider a classic competing-species model with the rates changed to include Gaussian white noise. We show that if the noise is not too large, then the stochastic version is ergodic. An explicit relation between the noise and the original competing-species parameters gives a sufficient condition for ergodicity.

We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝn. Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

Let {Xk, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1nwkXk and the maximum of weighted sums max1≤m≤n∑k=1mwkXk, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk.

Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

In this paper, we consider a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The solutions to the equation are provided in the cases of both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second- and higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operators.

We study the asymptotic tail behavior of the conditional probability distributions of rt+k and rt+1+⋯+rt+k when (rt)t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.

Higher-order asymptotics are also obtained when considering asemiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.

In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.

Suppose that a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume that whenever the gambler stakes an amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 − w, where w < ½. Dubins and Savage showed that the optimal strategy, which they called ‘bold play’, is always to bet min{f, 1 − f}, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than ℓ at one time. We show that the bold strategy of always betting min{ℓ, f, 1 − f} is not optimal if ℓ is irrational, extending a result of Heath, Pruitt, and Sudderth.

We consider solutions of Burgers' equation with linear or quadratic external potential and stationary random initial conditions of Ornstein-Uhlenbeck type. We study a class of limit laws that correspond to a scale renormalization of the solutions.

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing the stake with probability 1 - w (r > 0, 0 < w < 1). Mr. G is permitted to gamble at the casino, but each time he places a bet, the amount of his debt is increased by a factor of 1 + α (α ≥ 0). How should Mr. G gamble to maximize his chance of reaching his (moving) target and thereby surviving? Dubins and Savage showed that an optimal strategy is to stake boldly if the primitive casino is subfair or fair (i.e. w(1 + r) ≤ 1) and the inflation rate α is 0. Intuitively, a positive inflation rate would motivate Mr. G to try to reach his goal as quickly as possible, so it seems plausible that the bold strategy is optimal. However, Chen, Shepp, and Zame found that, surprisingly, the bold strategy is no longer optimal for subfair primitive casinos with inflation if both r > 1 and α satisfies 1/r ≤ α < r. They also conjectured that the bold strategy is optimal for subfair primitive casinos with inflation if r < 1. It is shown in the present paper that this conjecture is true provided that w ≤ ½. Furthermore, by introducing an interesting notion of sharp strategy, additional results are obtained on optimality of the bold strategy.