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 .
In this paper we study a continuous time stochastic inventory model for a commodity traded in the spot market and whose supply purchase 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 controls are nondecreasing processes and represent the cumulative investment made by the firm in the spot market ( a so-called stochastic "monotone follower problem"). We assume a general exponential L\'evy process for the commodity's spot price, rather than the commonly used geometric Brownian motion, and general convex holding costs . We obtain necessary and sufficient 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 particular, in the case of linear holding costs and exponentially distributed demand, we are also able to obtain the explicit analytic form of the optimal policy and a probabilistic representation of the optimal revenue . The paper is completed by some computer drawings of the optimal inventory when spot prices are given by a geometric Brownian motion and by an exponential jump-diffusion process. In the first case we also make a numerical comparison between the value function and the revenue associated to the classical static "newsvendor" strategy .
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propose
study
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traded commodity whose supply purchase
commodity traded
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and whose supply purchase
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a nondecreasing control policy represents
controls are nondecreasing processes and represent
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that is,
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contrary to the common use of a Brownian setting, and we model the holding cost by a general convex function
rather than the commonly used geometric Brownian motion, and general convex holding costs
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sufficient and necessary
necessary and sufficient
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particular, in
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random
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able to provide an explicit analytic solution
also able to obtain the explicit analytic form of the optimal policy and a probabilistic representation of the optimal revenue
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showing the behaviour
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for spot prices
when spot prices are
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,
and by
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, or an exponential Ornstein-Uhlenbeck process
process. In the first case we also make a numerical comparison between the value function and the revenue associated to the classical static "newsvendor" strategy