Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs
In this paper we propose a continuous time stochastic inventory model for a traded commodity whose supply purchase in the spot market is affected by price and demand uncertainty. A firm aims at meeting a random demand of the commodity at a random time by maximizing total expected profits. We model the firm's optimal procurement problem as a singular stochastic control problem in which a nondecreasing control policy represents the cumulative investment made by the firm in the spot market (that is, a so-called stochastic "monotone follower problem"). We assume a general exponential L\'evy process for the commodity's spot price, contrary to the common use of a Brownian setting, and we model the holding cost by a general convex function. We obtain sufficient and necessary first order conditions for optimality and we provide the optimal procurement policy in terms of a "base inventory" process; that is, a minimal time-dependent desirable inventory level that the firm's manager must reach at any time. In the case of linear holding costs and exponentially distributed random demand, we are able to provide an explicit analytic solution. The paper is completed by some computer drawings showing the behaviour of the optimal inventory for spot prices given by a geometric Brownian motion, an exponential jump-diffusion, or an exponential Ornstein-Uhlenbeck process.
Year of publication: |
2014-09
|
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Authors: | Chiarolla, Maria B. ; Ferrari, Giorgio ; Stabile, Gabriele |
Institutions: | arXiv.org |
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