38 research outputs found
Erratum
No full textThe discussion, from Maria B. Chiarolla and Ulrich G. Haussmann, represents the correction of an error made in Theorem 5.1 of Chiarolla and Haussmann. 2005. Explicit solution of a stochastic, irreversible investment problem and its moving threshold. Mathematics of Operations Research 30(1) 91–108
Singular Stochastic Control of a Singular Diffusion Process
No full textThis paper studies the monotone follower problem for a one-dimensional singular diffusion process. The dynamic programming principle is established. It is shown that the value function is continuous and satisfies the Hamilton-Jacobi-Bellman equation in the viscosity sens
Optimal dynamic procurement policies for a storable commodity with Lévy prices and convex holding costs
No full textIn 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 Lévy 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
A unifying view on the irreversible investment exercise boundary in a stochastic, time-inhomogeneous capacity expansion problem
No full textThis paper studies the investment exercise boundary erasing in a stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval and a state dependent scrap value associated with the production facility at the finite horizon . The capacity process is a time-inhomogeneous diffusion in which a monotone nondecreasing, possibly singular, process representing the cumulative investment enters additively. The levels of capacity, employment and operating capital contribute to the firm's production and are optimally chosen in order to maximize the expected total discounted profits. Two different approaches are employed to study and characterize the boundary. From one side, some first order condition are solved by using the Bank and El Karoui Representation Theorem, and that sheds further light on the connection between the threshold which the optimal policy of the singular stochastic control problem activates at and the optional solution of Representation Theorem. Its application in the presence of the scrap value is new. It is accomplished by a suitable fictitious extension to of the firm's horizon and a devise to overcome the difficulties due to the presence of a non integral term in the maximizing functional. The optimal investment process is shown to become active at the so-called ``base capacity'' level, which is given as the unique solution of an integral equation. On the other hand, when the coefficients of the uncontrolled capacity process are deterministic, the optimal stopping problem classically associated to the original capacity problem is resumed and, without invoking variational teckniques but only by means of probabilistic methods, some essential properties of the investment exercise boundary, the ``free boundary'' of its continuation region, are obtained. Despite the lack of knowledge of boundary's continuity, the optimal investment process is proved to be continuous, except for a possible initial jump. Finally, unifying approaches and views, the exercise boundary is shown to coincide with the base capacity, and hence it is characterized by an integral equation not requiring any a priori regularity
On the Lack of Optimal Classical Stochastic Controls in a Capacity Expansion Problem
No full textThe stochastic control problem of a firm aiming to optimally expand the production capacity, through irreversible investment, in order to maximize the expected total profits on a finite time interval has been widely studied in the literature when the firm’s capacity is modeled as a controlled Itˆo process in which the control enters additively and it is a general nondecreasing stochastic process, possibly singular as a function of time, representing the cumulative investment up to time t. This note proves that there is no solution when the problem falls in the so-called classical control setting; that is, when the control enters the capacity process as the rate of real investment, and hence the cumulative investment up to time t is an absolutely continuous process (as a function of time). So, in a sense, this note explains the need for the larger class of nondecreasing control processes appearing in the literature
Analytical Pricing of American Put Options on a Zero Coupon Bond in the Heath-Jarrow-Morton Model
No full textWe study the optimal stopping problem of pricing an American Put option on a Zero Coupon Bond (ZCB) in Musiela’s parametrization of the Heath–Jarrow–Morton (HJM) model for forward interest
rates. First we show regularity properties of the price function by probabilistic methods. Then we find an infinite dimensional variational formulation of the pricing problem by approximating the original optimal stopping problem by finite dimensional ones, after a suitable smoothing of the payoff. As expected, the first time the price of the American bond option equals the payoff is shown to be optimal. c 2014 Elsevier B.V. All rights reserved
Controlling Inflation: the infinite horizon case
No full textThis paper studies the two-dimensional singular stochastic control problem over an infinite time-interval arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate. It is shown that this problem admits a variational formulation which can be differentiated (in some sense) to lead to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the Bank. Substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established when the regularity of the free boundary is strengthened slightly, and it is shown that the optimal process is a diffusion reflected at the boundary
