This paper introduces a benchmark approach for the modelling of continuous, complete financial markets, when an equivalent risk-neutral measure does not exist. This approach is based on the unique characterization of a benchmark portfolio, the growth optimal portfolio, which is obtained via a generalization of the mutual fund theorem. The discounted growth optimal portfolio with minimum variance drift is shown to follow a Bessel process of dimension four. Some form of arbitrage can be explicitly modelled by arbitrage amounts. Fair contingent claim prices are derived as conditional expectations under the real world probability measure. The Heath-Jarrow-Morton forward rate equation remains valid despite the absence of an equivalent risk neutral measure.

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

The goal of this paper is to investigate properties of statistical procedures based on numbers of different patterns by using generating functions for the probabilities of a prescribed number of occurrences of given patterns in a random text. The asymptotic formulae are derived for the expected value of the number of words occurring a given number of times and for the covariance matrix. The form of the optimal linear test based on these statistics is established. These problems appear in testing for the randomness of a string of binary bits, DNA sequencing, source coding, synchronization, quality control protocols, etc. Indeed, the probabilities of repeated (overlapping) patterns are important in information theory (the second-order properties of relative frequencies) and molecular biology problems (finding patterns with unexpectedly low or high frequencies).

In this paper, we study a fluid model with partial message discarding and early message discarding, in which a finite buffer receives data (or information) from N independent on/off sources. All data generated by a source during one of its on periods is considered as a complete message. Our discarding scheme consists of two parts: (i) whenever some data belonging to a message has been lost due to overflow of the buffer, the remaining portion of this message will be discarded, and (ii) as long as the buffer content surpasses a certain threshold value at the instant an on period starts, all information generated during this on period will be discarded. By applying level-crossing techniques, we derive the equations for determining the system's stationary distribution. Further, two important performance measures, the probability of messages being transmitted successfully and the goodput of the system, are obtained. Numerical results are provided to demonstrate the effect of control parameters on the performance of the system.

Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive or negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positively or negatively dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w1), …, N(S,wd)), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O((log n) / √n) in the i.i.d. case and O((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

Distributional fixed points of a Poisson shot noise transform (for nonnegative and nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than −1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

We consider a stochastic version of the Ricker model describing the density of an unstructured isolated population. In particular, we investigate the effects of independently varying the per capita growth rate and the parameter governing density dependent feedback. We derive conditions on the distributions sufficient to guarantee different forms of stochastic stability such as null recurrence or positive recurrence. We find, for example, that null recurrence appears in two widely different scenarios: when there is a mean-zero growth rate or via a growth-catastrophe behaviour.

Let (Sk)k≥0 be a Markov chain with state space E and (ξx)x∊E be a family of ℝp-valued random vectors assumed independent of the Markov chain. The ξx could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Consider a real-valued Itô process X(t) = x + ∫0tμ(s)ds + ∫0tσ(s)dW(s) + A(t) driven by a Brownian motion {W(t) : t > 0}. The controller chooses the real-valued progressively measurable processes μ, σ and A subject to constraints |μ(t)| ≤ μ0(X(t-)) and |σ(t)| ≥ σ0(X(t-)), where the functions μ0 and σ0 are given. The process A is a bounded variation process and |A|(t) represents its total variation on [0,t]. The objective is to minimize the long-term average cost lim supT→∞(1/T)E[|A|(T) + ∫0Th(X(s))ds], where h is a given nonnegative continuous function. An optimal process X* is determined. It turned out that X* is a reflecting diffusion process whose state space is a finite interval [a*, b*]. The optimal drift and diffusion controls are explicitly derived and the optimal bounded variation process A* is determined in terms of local-time processes of X* at the points a* and b*.

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞2 otherwise, both contradicting the usual 𝒞1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesney et al. (1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

We extend classical renewal theorems to the weighted case. A hierarchical chain of successive sharpenings of asymptotic statements on the weighted renewal functions is obtained by imposing stronger conditions on the weighting coefficients.

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

We study the long-run behaviour of interactive Markov chains on infinite product spaces. The behaviour at a single site is influenced by the local situation in some neighbourhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the Dobrushin-Vasserstein contraction technique, we show that macroscopic convergence implies that the underlying microscopic process has local asymptotic loss of memory.

An important class of queueing networks is characterized by the following feature: in contrast with ordinary units, a disaster may remove all work from the network. Applications of such networks include computer networks with virus infection, migration processes with mass exodus and serial production lines with catastrophes. In this paper, we deal with a two-stage tandem queue with blocking operating under the presence of a secondary flow of disasters. The arrival flows of units and disasters are general Markovian arrival processes. Using spectral analysis, we determine the stationary distribution at departure epochs. That distribution enables us to derive the distribution of the number of units which leave the network at a disaster epoch. We calculate the stationary distribution at an arbitrary time and, finally, we give numerical results and graphs for certain probabilistic descriptors of the network.

We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.