We consider the stability of a network serving a patchwork of overlapping regions where customers from a local region are assigned to a collection of local servers.These customers join the queue of the local server with the shortest queue of waiting customers.We then describe how the backlog in the network overloads.We do this in the simple case of two servers each of which receives a dedicated stream of customers in addition to customers from a stream of smart customers who join the shorter queue. There are three distinct ways the backlog can overload. If one server is very fast, then that server takes all the smart customers along with its dedicated customers and keeps its queue small while the dedicated customers at the other server cause the overload.We call this the unpooled case. If the proportion of smart customers is large, then the two servers overload in tandem.We call this the strongly pooled case. Finally, there is the weakly pooled case where both queues overload but in different proportions. The fact that strong pooling can be attained based on a local protocol for overlapping regions may have engineering significance. In addition, this paper extends the methodology developed in McDonald (to appear The Annals of Applied Probability) to cover periodicities. The emphasis here is on sharp asymptotics, not rough asymptotics as in large deviation theory. Moreover, the limiting distributions are for the unscaled process, not for the fluid limit as in large deviation theory. In the strongly pooled case, for instance, we give the limiting distribution of the difference between the two queues as the backlog grows.We also give the exact asymptotics of the mean time until overload.

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traffic limit and the limiting cost is the same as the optimal cost in the Brownian control problem. The techniques developed here are expected to be useful for analyzing the performance of threshold-type policies in more complex multiserver systems.

We consider a Jackson-type network, each of whose nodes contains N identical channels with a single server. Upon arriving at a node, a task selects m of the channels at random and joins the shortest of the m queues observed.We fix a collection of channels in the network, and analyze how the queue-length processes at these channels vary as $N \to \infty$. If the initial conditions converge suitably, the distribution of these processes converges in local variation distance to a limit under which each channel evolves independently.We discuss the limiting processes which arise, and in particular we investigate the point processes of arrivals and departures at a channel when the networks are in equilibrium, for various values of the system parameters.

This paper examines the continuous-time portfolio-consumption problem of an agent with a recursive utility in the presence of nonlinear constraints on the wealth.Using backward stochastic differential equations, we state a dynamic maximum principle which generalizes the characterization of optimal policies obtained by Duffie and Skiadas [J.Math Econ. 23, 107 –131 (1994)] in the case of a linear wealth. From this property, we derive a characterization of optimal wealth and utility processes as the unique solution of a forward-backward system. Existence of an optimal policy is also established via a penalization method.

This paper accompanies a previous one by D.Kramkov and the present author. While in [17 ] we considered utility functions $U : \mathbb{R}_+ \to \mathbb{R}$ satisfying the Inada conditions $U'(0) = \infty$ and $U'(\infty) = 0$, in the present paper we consider utility functions $U : \mathbb{R} \to \mathbb{R}$, which are finitely valued, for all $x\epsilon\mathbb{R}$ and satisfy $U'(-\infty) = \infty$ and $U'(\infty) = 0. A typical example of this situation is the exponential utility $U(x) = -e^{-x}$.

In the setting of [17 ] the following crucial condition on the asymptotic elasticity of U, as x tends to $+\infty$, was isolated: $lim sup_{x\to+\infty}\frac{xU'(x)}{U(x)}<1$. This condition was found to be necessary and sufficient for the existence of the optimal investment as well as other key assertions of the related duality theory to hold true, if we allow for general semi-martingales to model a (not necessarily complete) financial market.

In the setting of the present paper this condition has to be accompanied by a similar condition on the asymptotic elasticity of U, as x tends to $-\infty$, namely, $\lim \inf_{x\to-\infty\}frac{xU'(x)}{U(x)}>1$. If both conditions are satisfied —we then say that the utility function U has reasonable asymptotic elasticity —we prove an existence theorem for the optimal investment in a general locally bounded semi-martingale model of a financial market and for a utility function $U : \mathbb{R} \to \mathbb{R}$, which is finitely valued on all of $\mathbb{R}$; this theorem is parallel to the main result of [17 ].We also give examples showing that the reasonable asymptotic elasticity of U also is a necessary condition for several key assertions of the theory to hold true.

Given a particular market variable, which could be finite dimensional (e.g., a price vector of a collection of stocks) or infinite dimensional (e.g., a price trajectory of some security over some period of time), we find the unique optimal European claim contingent on that variable in the sense that, for a given price and risk tolerance level, this claim has the highest expected return possible. The optimal contingent claims seem to be attractive investment instruments and are proposed for trade in derivatives markets.

We analyze the intertemporal utility maximization problem under uncertainty for the preferences proposed by Hindy, Huang and Kreps. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including both complete and incomplete markets. For the complete market setting, we prove an infinite-dimensional version of the Kuhn –Tucker theorem which implies necessary and sufficient conditions for optimality. Using this characterization, we show that optimal plans prescribe consuming just enough to keep the induced level of satisfaction always above some stochastic lower bound. When uncertainty is generated by a Lévy process and agents exhibit constant relative risk aversion, we derive solutions in closed form. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or in a singular way.

We obtain a strong approximation for the empirical process of n observe delements of a GARCH sequence. The weak convergence of the empirical process and the law of the iterated logarithm are immediate consequences.

We introduce backoff processes, an idealized stochastic model of browsing on the World Wide Web, which incorporates both hyperlink traversals and use of the “back button.” With some probability the next state is generated by a distribution over out-edges from the current state, as in a traditional Markov chain.With the remaining probability, however, the next state is generated by clicking on the back button and returning to the state from which the current state was entered by a “forward step.” Repeated clicks on the back button require access to increasingly distant history. We show that this process has fascinating similarities to and differences from Markov chains. In particular, we prove that, like Markov chains, back-off processes always have a limit distribution, and we give algorithms to compute this distribution. Unlike Markov chains, the limit distribution may depend on the start state.

We consider a sequential interval packing process similar to Rényi ’s “car parking problem” but with a generator of random intervals which allows for arbitrarily small lengths. Embedding the process in continuous time, we view it as a self-similar interval splitting process.We determine the asymptotical behavior of the quantities such as the number of intervals packed to some instant and obtain convergence results in the context of the more general splitting model.

We study long strange intervals in a linear stationary stochastic process with regularly varying tails. It turns out that the length of the longest strange interval grows, as a function of the sample size, at different rates in different parts of the parameter space.We argue that this phenomenon may be viewed in a fruitful way as a phase transition between short-and long-range dependence.We prove a limit theorem that may form a basis for statistical detection of long-range dependence.

In this paper, we investigate the existence and uniqueness of the solution for a class of stochastic wave equations in two space-dimensions containing a non-linearity of polynomial type. The method used in the proofs combines functional analysis arguments with probabilistic tools, and furher estimates for the Green function associated with the classical wave equation.

A stochastic analogue of the Lotka –Volterra model for predator–prey relationshipis obtained when the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order $\varepsilon^2$, where $\varepsilon>0$ is a small parameter.The evolution of this system is studied on large time intervals of $O(1/\varepsilon^2)$. It is shown that for small initial population sizes the stochastic model is adequate, whereas for large initial population sizes it is not as suitable, because it leads to ever-increasing fluctuations in population sizes, although it still precludes extinction. New results for the classical deterministic Lotka–Volterra model are obtained by a probabilistic method; we show in particular that large population sizes of predator and prey coexist only for a very short time, and most of the time one of the populations is exponentially small.

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecu- lar sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration.We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods.