scientific non-commercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a first-in-first-out basis, where the processing of a given job may be performed by any server from a given (class-dependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected 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. In a prior work [40], the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an “interpretation ” of the analytic solution to an associated Brownian control problem (formal heavy

Communications systems often have many types of users. Since they share the same resource, there is a conflict in their needs. This conflict leads to the imposition of controls on admission or elsewhere. In this paper, there are two types of customers, GP (Guaranteed Performance) and BE (Best Effort). We consider an admission control of GP customer which has two roles. First, to guarantee the performance of the existing GP customers, and second, to regulate the congestion for the BE users. The optimal control problem for the actual physical system is difficult. A heavy traffic approximation is used, with optimal or nearly optimal controls. It is shown that the optimal values for the physical system converge to that for the limit system and that good controls for the limit system are also good for the physical system. This is done for both the discounted and average cost per unit time cost criteria. Additionally, asymptotically, the pathwise average (not mean) costs for the physical system are nearly

The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The information that the assignment can be based on is quite flexible, and several information structures (data on which the control is based) are considered. The assignment can be made on arrival or when the job is to be processed. There can be bursty arrivals (the bursts depending on randomly varying environmental factors), rather general nonlinear cost functions and other complications. It is shown, under reasonably general conditions, that the optimal costs for the physical systems converge to the optimal cost for the heavy traffic limit problem, as the heavy traffic parameter goes to its limit. Numerical data is presented to illustrate some of the potential uses of the limit process for obtain...

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The ...

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed.

We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems [14], associated with a broad family of stochastic networks, follows as a consequence.

We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis [19] are studied. The main result of the paper shows that the value function of the Markov Decision Problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet-Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with

The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is little idle time and the queue length processes can be approximated by reected diffusion processes under suitable scaling. Three principal problems are considered: the multiplexer system, controlled admission in multiserver systems such as ISDN, and the polling or scheduling problem.

An efficient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska,

In this paper we examine the problem of finding investors ’ reservation option prices and corresponding early exercise policies of Americanstyle options in the market with proportional transaction costs using the utility based approach proposed by Davis and Zariphopoulou (1995). We present a model, where investors have a CARA utility, and derive some properties of reservation option prices. We discuss the numerical algorithm and propose a new formulation of the problem in terms of quasi-variational HJB inequalities. Based on our formulation, we suggest original discretization schemes for computing reservation prices of American-style option. The discretization schemes are then implemented for computing prices of American put and call options. We examine the effects on the reservation option prices and the corresponding early exercise policies of varying the investor’s ARA and the level of transaction costs. We find that in the market with transaction costs the holder of an American-style option exercises this option earlier as compared to the case with no transaction costs. This phenomenon concerns both put and call options written on a non-dividend paying stock. The higher level the transaction costs is, or the higher risk avers the option holder is, the earlier an American option is exercised. Key words: option pricing, transaction costs, stochastic control, optimal stopping, Markov chain approximation.