Equilibrium States of Random Economies with Locally Interacting Agents and Solutions to Stochastic Variational Inequalities in 〈L1,L∞〉

We study a stochastic model of an economy with locally interacting agents. The basis of the study is a deterministic model of dynamic economic equilibrium proposed by Polterovich. We generalize Polterovich's theory, in particular, in two respects. We introduce stochastics and consider a version of the model with local interactions between the agents. The structure of the interactions is described in terms of random fields on a directed graph. Equilibrium states of the system are solutions to certain variational inequalities in spaces of random vectors. By analyzing these inequalities, we establish an existence theorem for equilibrium, which generalizes and refines a number of previous results.     

Deterministic equivalent for a continuous-time linear-convex stochastic control problem

We consider a finite-horizon control model with additive input. There are two convex functions which describe the running cost and the terminal cost within the system. The cost of input is proportional to the input and can take both positive and negative values. It is shown that there exists a deterministic control problem whose optimal cost is the same as the one in the stochastic control problem. The optimal policy for the stochastic problem consists of keeping the process as close to the optimal deterministic trajectory as possible.This research is supported by NSERC Grant A4619, MRCO, NSF Grant DMS-86-01510, and AFOSR Grant 87-0278.     

Infinite-horizon investment consumption model with a nonterminal bankruptcy

This paper solves a general continuous-time consumption and portfolio decision problem for a single agent for whom there exists, upon bankruptcy, a possibility of recovery from his bankruptcy. The main contribution of the paper is in the modeling of the recovery process. Moreover, it is shown that the model with recovery has a one-to-one correspondence with the model with terminal bankruptcy treated in the literature.This research was supported by Grants SSHRC-410-83-9888 and NSERC-A4619 to the first author and by Grants NSF-DMS-86-01510 and AFOSR-88-0183 to the second author. Comments from E. Presman are gratefully acknowledged.     

Diffusion approximation forGI/G/1 controlled queues

A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost.Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called reflected follower problem. The optimal policy in this problem is characterized by a single numberz ^* so that the optimal process is a reflected diffusion in [0,z ^*]. To obtainz ^* one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz ^* is approximately optimal.     

Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth

This paper studies stochastic inventory problems with unbounded Markovian demands, ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are lower semicontinuous with polynomial growth. Finite-horizon problems, stationary and nonstationary discounted-cost infinite-horizon problems, and stationary long-run average-cost problems are addressed. Existence of optimal Markov or feedback policies is established. Furthermore, optimality of (s, S)-type policies is proved when, in addition, the ordering cost consists of fixed and proportional cost components and the surplus cost is convex.     

Optimal non-proportional reinsurance control and stochastic differential games

We study stochastic differential games between two insurance companies who employ reinsurance to reduce risk exposure. We consider competition between two companies and construct a single payoff function of two companies’ surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize the payoff function while its opponent is simultaneously choosing a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of the game and prove a verification theorem for a general payoff function. For the payoff function being the probability that the difference between two surplus reaches an upper bound before it reaches a lower bound, the game is solved explicitly.     

On absolute ruin minimization under a diffusion approximation model

In this paper, we assume that the surplus process of an insurance entity is represented by a pure diffusion. The company can invest its surplus into a Black-Scholes risky asset and a risk free asset. We impose investment restrictions that only a limited amount is allowed in the risky asset and that no short-selling is allowed. We further assume that when the surplus level becomes negative, the company can borrow to continue financing. The ultimate objective is to seek an optimal investment strategy that minimizes the probability of absolute ruin, i.e. the probability that the liminf of the surplus process is negative infinity. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is analyzed and a verification theorem is proved; applying the HJB method we obtain explicit expressions for the S-shaped minimal absolute ruin function and its associated optimal investment strategy. In the second part of the paper, we study the optimization problem with both investment and proportional reinsurance control. There the minimal absolute ruin function and the feedback optimal investment-reinsurance control are found explicitly as well.     

Optimal Production and Setup Scheduling: A One-Machine, Two-Product System

This paper studies the scheduling problem for two products on a single production facility. The objective is to specify a production and setup policy that minimizes the average inventory, backlog, and setup costs. Assuming that the production rate can be adjusted during the production runs, we provide a close form for an optimal production and setup schedule. Dynamic programming and Hamilton–Jacobi–Bellman equation is used to verify the optimality of the obtained policy.