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One of the fundamental issues in sensor networks is the coverage problem, which reflects how well a sensor network is monitored or tracked by sensors. In this paper, we formulate this problem as a decision problem, whose goal is to determine whether every point in the service area of the sensor network is covered by at least k sensors, where k is a given parameter. The sensing ranges of sensors can be unit disks or non-unit disks. We present polynomial-time algorithms, in terms of the number of sensors, that can be easily translated to distributed protocols. The result is a generalization of some earlier results where only k = 1 is assumed. Applications of the result include determining insufficiently covered areas in a sensor network, enhancing fault-tolerant capability in hostile regions, and conserving energies of redundant sensors in a randomly deployed network. Our solutions can be easily translated to distributed protocols to solve the coverage problem.

We provide sufficient conditions for a dynamic consumption-portfolio problem in continuous time to have a solution for a class of utility functions, when the price system follows a diffusion process and when the space of admissible policies is a linear space. Besides a regularity condition, it suffices to check whether a uniform growth and a local Lipschitz condition are satisfied by the parameters of a system of stochastic differential equations, which is completely derived from the price system. The class of utility functions includes concave functions that are, roughly, either bounded from below or strictly concave, and whose coefficients of relative risk aversion have nonzero limit infima as consumption/wealth goes to infinity. 'The authors would like to acknowledge helpful conversations with Sergiu Hart, Andreu Mas Colell, and Ho-Mou Wu. The expositions of this paper were improved by comments from two anonymous referees. Various comments were made by seminar participants at

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This paper develops a model of competitive bidding with a resale market. The primary market is modelled as a common-value auction, in which bidders participate for the purpose of resale. After the auction the winning bidders sell the objects in a secondary market and the buyers on the secondary market receive information about the bids submitted in the auction.

We study a model of optimal consumption and portfolio choice which captures, in two different interpretations, the notions of local substitution and irreversible purchases of durable goods. The class of preferences we consider excludes all nonlinear time-additive and nearly all the non-time-additive utility functions used in the literature. We discuss heuristically necessary conditions and provide sufficient conditions for a consumption and portfolio policy to be optimal. Furthermore, we demonstrate our general theory by solving in a closed form the optimal consumption and portfolio policy for a particular felicity function when the prices of the assets follow a geometric Brownian motion process. The optimal consumption policy in our solution consists of a possible initial "gulp" of consumption, or a period of no consumption, followed by a process of accumulated consumption with singular sample paths. In almost all states of nature, the agent consumes periodically and invests more in the risky assets than an agent with time-additive utility whose felicity function has the same curvature and the same time-discount parameter. We compute the equilibrium risk premium in a representative investor economy with a single physical production technology whose rate of return follows a Brownian motion.In addition, we provide some simulation results that demonstrate the properties of the purchase series for durable goods with different half-lives.

We provide sufficient conditions for the existence of a solution to a consumption and portfolio problem in continuous time under uncertainty with an infinite horizon. \N'hen the price processes for securities are diffusion processes, optimal policies can be computed by solving a linear partial differential equation. We also provide conditions under which the solution to an infinite horizon problem is the limit of the solutions to finite horizon problems when the horizon increases to infinity.'The authors would like to thank conversations with Robert Merton.

A restriction to nonnegative wealth is sufficient to preclude all arbitrage opportunities in financial models that have risk neutral probabilities that are valid for all simple strategies.Imposing nonnegative wealth does not constrain agents from making the choice they would make under the standard integrability condition. This conclusion does not depend on whether the markets are complete.'We are grateful for useful convereatione with