We consider a multivariate .nancial market withtransaction costs as in Kabanov. We study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. We prove that the value of this stochastic control problem is given by the cost of the cheapest buy-and-hold strategy. This is an extension of the already known result in the one-dimensional case. An important feature of our analysis is that we do not make use of the dual formulation of the problem, as in the previous literature.

This study is motivated by problems of molecular sequence comparisons for biological traits conserved or lost over evolution time.A marker of interest is distributed in the genome of the ancestor and inherited among $l$ offspring species which descend from this common ancestor. Each marker will be retained or lost during the evolution of the descendent species. The objective of the analysis here is to ascertain probabilities of clustering or overdispersion of the marker array among the sequences of the descendent species. Limiting distributions for the extremal $r$-scan statistics (defined in text) of the trait distributed among the $l$ dependent offspring processes are derived by adapting the Chen–Stein Poisson approximation method. Results that accommodate new occurrences of the trait (gene) arising from duplications and transposition occurrences are also described.The $r$-scan statistical analysis is further applied to a multi sequence combined Poisson model where ${B_1,\dots, B_l}$ are generated from $m$ independent Poisson processes ${A_1,\dots, A_m}$ such that $B_k = \bigcup_{i\epsilonZ_k}A_i$, where ${Z_k}_1\leqk\leql$ are subsets of ${1, 2,\dots,m}$.

We analyze the convergence to stationarity of a simple nonreversible Markov chain that serves as a model for several nonreversible Markov chain sampling methods that are used in practice. Our theoretical and numerical results show that nonreversibility can indeed lead to improvements over the diffusive behavior of simple Markov chain sampling schemes. The analysis uses both probabilistic techniques and an explicit diagonalization.

Several studies of file sizes either being downloaded or stored in the World Wide Web have commented that tails can be so heavy that not only are variances infinite, but so are means. Motivated by this fact, we study the infinite node Poisson model under the assumption that transmission times are heavy tailed with infinite mean. The model is unstable but we are able to provide growth rates. Self-similar but nonstationary Gaussian process approximations are provided for the number of active sources, cumulative input, buffer content and time to buffer overflow.

We give an explicit analytic series expansion of the (max, plus)-Lyapunov exponent $\gamma(p)$ of a sequence of independent and identically distributed randommatrices, generated via a Bernoulli scheme depending on a small parameter $p$. A key assumption is that one of the matrices has a unique normalized eigenvector. This allows us to obtain a representation of this exponent as the mean value of a certain random variable.We then use a discrete analogue of the so-called light-traffic perturbation formulas to derive the expansion.We show that it is analytic under a simple condition on $p$. This also provides a closed formexpression for all derivatives of $\gamma(p)$ at $p = 0$ and approximations of $\gamma(p)$ of any order, together with an error estimate for finite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters $(p_1,\dots, p_m)$ and expansions for exponents associated with iterates of a class of random operators which includes the class of nonexpansive and homogeneous operators. Several examples pertaining to computer and communication sciences are investigated: timed event graphs, resource sharing models and heap models.

This paper derives the strong approximation for a multiclass queueing network,where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival; that is, a first-in first-out (FIFO) discipline is in force, and among groups, jobs are served under a preassigned preemptive priority discipline. We obtain the strong approximation for the network through an inductive application of an input–output analysis for a single-station queue. Specifically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are proposed to approximate the stationary distribution of various performance measures of the queueing network. Our work extends and complements the existing work of Peterson and Harrison and Williams on the feedforward queueing network. The numeric examples show that strong approximation provides a better approximation than that suggested by a straightforward interpretation of the heavy traffic limit theorem.

We utilize and develop elements of the recent achievable region account of Gittins indexation by Bertsimas and Niño-Mora to design index-based policies for discounted multi-armed bandits on parallel machines. The policies analyzed have expected rewards which come within an $O(\alpha)$ quantity of optimality, where $\alpha > 0$ is a discount rate. In the main, the policies make an initial once for all allocation of bandits to machines, with each machine then handling its own workload optimally. This allocation must take careful account of the index structure of the bandits. The corresponding limit policies are average-overtaking optimal.

This paper describes a general approach for dynamic control of stochastic networks based on fluid model analysis, where in broad terms, the stochastic network is approximated by its fluid analog, an associated fluid control problem is solved and, finally, a scheduling rule for the original system is defined by interpreting the fluid control policy. The main contribution of this paper is to propose a general mechanism for translating the solution of the fluid optimal control problem into an implementable discrete-review policy that achieves asymptotically optimal performance under fluid scaling, and guarantees stability if the traffic intensity is less than one at each station. The proposed policy reviews system status at discrete points in time, and at each such point the controller formulates a processing plan for the next review period,based on the queue length vector observed, using the optimal control policy of the associated fluid optimization problem. Implementation of such a policy involves enforcement of certain safety stock requirements in order to facilitate the execution of the processing plans and to avoid unplanned server idleness. Finally, putting aside all considerations of system optimality, the following generalization is considered: every initial condition is associated with a feasible fluid trajectory that describes the desired system evolution starting at that point. A discrete-review policy is described that asymptotically tracks this target specification; that is, it achieves the appropriate target trajectory as its fluid limit.

One of the successes of the Brownian approximation approach to dynamic control of queueing networks is the design of a control policy for closed networks with two servers by Harrison and Wein. Adopting a Brownian approximation with only heuristic justification, theyinterpret the optimal control policy for the Brownian model as a static priority rule and conjecture that this priority rule is asymptotically optimal as the closed networks’s population becomes large. This paper studies closed queueing networks with two servers that are balanced, that is, networks that have the same relative load factor at each server. The validity of the Brownian approximation used by Harrison and Wein is established by showing that, under the policy they propose, the diffusion-scaled workload imbalance process converges weakly in the infinite population limit to the diffusion predicted by the Brownian approximation. This is accomplished by proving that the fluid limits of the queue length processes undergo state space collapse in finite time under the proposed policy, thereby enabling the application of a powerful new technique developed by Williams and Bramson that allows one to establish convergence of processes under diffusion scaling by studying the behavior of limits under fluid scaling. A natural notion of asymptotic optimality for closed queueing networks is defined in this paper.The proposed policy is shown to satisfy this definition of asymptotic optimality by showing that the performance under the proposed policy approximates bounds on the performance under every other policy arbitrarily well as the population increases without bound.

The problem of large deviations for a Jackson network is analyzed in detail. A new representation of the rate function is given and a simple procedure is proposed to get its closed form expression. The methods used rely on twisted distributions, localized processes, fluid limits and a careful analysis of some functions.

or a particular conditionally heteroscedastic nonlinear (ARCH) process for which the conditional variance of the observable sequence $r_t$ is the square of an inhomogeneous linear combination of $r_s, s < t$, we give conditions under which, for integers $l \geq 2, r_t^l$ has long memory autocorrelation and normalized partial sums of $r_t^l$ converge to fractional Brownian motion.

Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk.When the stationary process is heavy-tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy-tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke.We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example,by a linear process.