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In this paper we study a continuous time stochastic inventory model for acommodity traded in the spot market and whose supply purchase is affected byprice and demand uncertainty. A firm aims at meeting a random demand of thecommodity at a random time by maximizing total expected profits. We model thefirm's optimal procurement problem as a singular stochastic control problem inwhich controls are nondecreasing processes and represent the cumulativeinvestment made by the firm in the spot market (a so-called stochastic"monotone follower problem"). We assume a general exponential L\'evy processfor the commodity's spot price, rather than the commonly used geometricBrownian motion, and general convex holding costs.  We obtain necessary and sufficient first order conditions for optimality andwe provide the optimal procurement policy in terms of a "base inventory"process; that is, a minimal time-dependent desirable inventory level that thefirm's manager must reach at any time. In particular, in the case of linearholding costs and exponentially distributed demand, we are also able to obtainthe explicit analytic form of the optimal policy and a probabilisticrepresentation of the optimal revenue. The paper is completed by some computerdrawings of the optimal inventory when spot prices are given by a geometricBrownian motion and by an exponential jump-diffusion process. In the first casewe also make a numerical comparison between the value function and the revenueassociated to the classical static "newsvendor" strategy.

Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs