A unifying view on the irreversible investment exercise boundary in a stochastic, time-inhomogeneous capacity expansion problem

This paper studies the investment exercise boundary erasing in a stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$ and a state dependent scrap value associated with the production facility at the finite horizon $T$. 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 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, 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 some essential properties of the investment exercise boundary are obtained. The optimal investment process is proved to be continuous. Unifying approaches and views, the exercise boundary is shown to coincide with the base capacity, hence it is characterized by an integral equation not requiring any a priori regularity

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

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.Comment: 28 pages, 3 figures; improved presentation, added new results and section

Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources

In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with N firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding Social Planner optimal policy. Our conditions are a stochastic infinite-dimensional generalization of the Kuhn-Tucker Theorem. The Lagrange multiplier takes the form of a nonnegative optional random measure on [0,T] which is flat off the set of times for which the constraint is binding, i.e. when all the fuel is spent. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank (2005). In the infinite-horizon case, with operating profit functions of Cobb-Douglas type, our method allows the explicit calculation of the optimal policy in terms of the `base capacity' process, i.e. the unique solution of the Bank and El Karoui representation problem (2004).Comment: 25 page

The Management of Decumulation Risks in a Defined Contribution Pension Plan

The aim of the paper is to lay the theoretical foundations for the construction of a flexible tool that can be used by pensioners to find optimal investment and consumption choices in the distribution phase of a defined contribution pension plan. The investment/consumption plan is adopted until the time of compulsory annuitization, taking into account the possibility of earlier death. The effect of the bequest motive and the desire to buy a higher annuity than the one purchasable at retirement are included in the objective function. The mathematical tools provided by dynamic programming techniques are applied to find closed-form solutions: numerical examples are also presented. In the model, the tradeoff between the different desires of the individual regarding consumption and final annuity can be dealt with by choosing appropriate weights for these factors in the setting of the problem. Conclusions are twofold. First, we find that there is a natural time-varying target for the size of the fund, which acts as a sort of safety level for the needs of the pensioner. Second, the personal preferences of the pensioner can be translated into optimal choices, which in turn affect the distribution of the consumption path and of the final annuity

A unifying view on the irreversible investment exercise boundary in a stochastic, time-inhomogeneous capacity expansion problem

This paper studies the investment exercise boundary erasing in a stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$ and a state dependent scrap value associated with the production facility at the finite horizon $T$. 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 $+ \infty$ 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

The 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

An Algorithm for Equilibrium in a Dynamic Stochastic Monetary Economy

This paper establishes an algorithm for the equilibrium in a stochastic continuous time economy model, on a finite time interval, including a representative agent maximizing her expected total utility of consumption, leisure, and money, and a single firm that optimally produces the consumption good and maximizes its expected total profits based on employment rate and money held. First, under the assumption of equilibrium, a link between the firm’s control problem and the representative agent’s optimal expected total utility is obtained. Then such link is exploited to establish an algorithm for equilibrium

IDENTIFYING THE FREE BOUNDARY OF A STOCHASTIC, IRREVERSIBLE INVESTMENT PROBLEM VIA THE BANK-EL KAROUI REPRESENTATION THEOREM

We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Itô diffusion controlled by a nondecreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality, and we characterize the optimal solution in terms of the base capacity process l(t), i.e., the unique solution of a representation problem in the spirit of Bank and El Karoui [P. Bank and N. El Karoui, Ann. Probab., 32(2004), pp. 1030-1067]. We show that the base capacity is deterministic and it is identified with the free boundary ŷ(t) of the associated optimal stopping problem when the coefficients of the controlled diffusion are deterministic functions of time. This is a novelty in the literature on finite horizon singular stochastic control problems. As a subproduct this result allows us to obtain an integral equation for the free boundary, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process. © 2014 Society for Industrial and Applied Mathematics