Irreversible Investment Problem Research Articles

Irreversible investment with random delay and partial prepayment

Classical irreversible investment problem admits an optimal strategy of threshold type. But there is no consensus on how the investor should adjust the threshold, if there is an implementation delay. By formulating a general problem with random delay and partial prepayment, we find that the effect of delay can be opposite for different prepayment rates. Besides, although the constant delay model is commonly used, we argue that it is a reasonable approximation only if the discount rate is small.

An irreversible investment problem with demand on a finite horizon: The optimal investment boundary analysis

This paper studies a continuous-time, finite horizon, irreversible investment problem where a social planner aims to minimize total expected costs of production capacity and demand. Our model allows for general diffusion dynamics on the demand as well as production capacity process controlled by a nondecreasing process representing the cumulative investment. Mathematically, it is a singular stochastic control problem whose value function satisfies a two-dimensional parabolic variational inequality with gradient constraint. The problem gives rise to a free boundary which stands for the optimal investment boundary. We use partial differential equation (PDE) approach to characterize some features of the free boundary and prove the C2,1 regularity of the value function. To the best of our knowledge, the method to study the monotonicity of free boundary about the time direction is an innovation.

On an irreversible investment problem with two-factor uncertainty

We consider a real options model for the optimal irreversible investment problem of a profit-maximizing company. The company has the opportunity to invest in a production plant capable of producing two products, of which the prices follow two independent geometric Brownian motions. After paying a constant sunk investment cost, the company sells the products on the market and thus receives a continuous stochastic revenue flow. This investment problem is set as a two-dimensional optimal stopping problem. We find that the optimal investment decision is triggered by a convex curve, which we characterize as the unique continuous solution to a nonlinear integral equation. Furthermore, we provide analytical and numerical comparative statics results of the dependency of the project's value and investment decision with respect to the model's parameters.

A Knightian irreversible investment problem

In this paper, we study an irreversible investment problem under Knightian uncertainty. In a general framework, in which Knightian uncertainty is modeled through a set of multiple priors, we prove existence and uniqueness of the optimal investment plan, and derive necessary and sufficient conditions for optimality. This allows us to construct the optimal policy in terms of the solution to a stochastic backward equation under the worst-case scenario. In a time-homogeneous setting – where risk is driven by a geometric Brownian motion and Knightian uncertainty is realized through a so-called “κ-ignorance” – we are able to provide the explicit form of the optimal irreversible investment plan.

Impact of Plant Utilization on Irreversible Investment Under Uncertainty with Application to Refinery Investment

Why are some regions preferred when investors consider irreversible investment? This study offers an explanation to this question and suggests improvements that will assist disadvantaged regions improve on their bid for funds. The paper considers irreversible investment under uncertainty when installed capacity utilization is incorporated. We develop a normative model for irreversible investment problem under uncertainty using real options approach. Capacity utilization was not a major consideration by previous authors who assumed that installed capacity would be fully utilized. Variations in capacity utilization may be attributed to disruption in input supply or infrastructural bottlenecks that limit firms to get their products to customers. This study modifies the geometric Brownian motion for the value of a project to account for capacity utilization in the derivation of irreversible investment decision rule. The proposed model provides a theoretical explanation of how utilization affects irreversible investment decisions. Data on petroleum refinery margins is used to illustrate application of the proposed model to refinery investment. The study reveals that capacity utilization has an inverse effect on the investment trigger, and so, links irreversible investment decisions to plant utilization. We recommend optimal utilization of installed plant capacity for regions seeking funds for irreversible investment.

An Integral Equation Approach to the Irreversible Investment Problem with a Finite Horizon

This paper studies an irreversible investment problem under a finite horizon. The firm expands its production capacity in irreversible investments by purchasing capital to increase productivity. This problem is a singular stochastic control problem and its associated Hamilton–Jacobi–Bellman equation is derived. By using a Mellin transform, we obtain the integral equation satisfied by the free boundary of this investment problem. Furthermore, we solve the integral equation numerically using the recursive integration method and present the graph for the free boundary.

Ratcheting with a bliss level of consumption

In this paper we study the portfolio selection problem of an agent who has a bliss point of consumption and does not tolerate a decline in consumption. We show that the optimal consumption process exhibits ratcheting and, as time elapses, the agent’s consumption approaches but never reaches the bliss level and her/his optimal investment in the risky asset approaches zero if the initial wealth level is not sufficient to maintain it. We use the martingale method and study the dual problem, which is similar to an incremental irreversible investment problem. We transform the dual problem into an optimal stopping problem, which has the same characteristic as finding the optimal exercise time of a perpetual American put option. We recover the value function by establishing a duality relationship and obtain a closed-form solution for the optimal consumption and portfolio strategy.

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity, as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a reflected diffusion at a suitable boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems, and it is characterized in terms of the family of unique continuous solutions to parameter-dependent, nonlinear integral equations of Fredholm type.

Convex duality for stochastic singular control problems

We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer [Ann. Appl. Probab. 9 (1999) 904–950] for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the formulation of a suitable duality framework, the identification of the problem’s dual functional as well as the full duality for the primal and dual value functions and their optimizers. The scope of our results is illustrated by an irreversible investment problem and the Hindy–Huang–Kreps utility maximization problem for incomplete financial markets.

An irreversible investment problem with maintenance expenditure on a finite horizon: Free boundary analysis

This paper concerns a continuous-time, finite horizon, optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty. We assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity. The objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint. The problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies, respectively. We investigate behaviors of free boundaries, study regularities of the value function, and give optimal investment and maintenance policies. As we know, this is a first integral result for an investment–maintenance problem with a finite time horizon due to use of partial differential equation (PDE) technique.

Irreversible investment under Lévy uncertainty: an equation for the optimal boundary

AbstractWe derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point inRwith positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.

Irreversible Investment under LLvy Uncertainty: An Equation for the Optimal Boundary

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Levy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener-Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Levy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of (i) Cobb-Douglas type and (ii) CES type. In the first case the function is separable and in the second case non-separable.

On the Optimal Boundary of a Three-Dimensional Singular Stochastic Control Problem Arising in Irreversible Investment

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem.We provide the optimal control as the solution of a Skorohod reflection problem at a suitable free-boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems and it is characterized in terms of the family of unique continuous solutions to parameter-dependent nonlinear integral equations of Fredholm type.

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 $\hat{y}(t)$ 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.

An Irreversible Investment Problem with Maintenance Expenditure

This paper studies the optimal investment problem with maintenance expenditure of a firm under uncertainty. We assume that the investment is irreversible and the maintenance cost of the firm has positive effects on its stochastically growing capital productivity. The objective of the firm is to construct optimal investing and maintenance policies to maximize its expected total cash flow over the infinite time horizon. We adopt the penalty method to obtain a solution of the variational inequality associated with the optimization problem. The optimal solution of barrier type for the investment-maintenance problem is shown to exist.

Incomplete markets, ambiguity, and irreversible investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of ambiguity over the appropriate no-arbitrage discount factor. The maxmin utility over multiple priors framework is used to model and solve the irreversible investment problem. Multiple priors are modeled using the notion of κ ‐ignorance . This set-up is used to analyze finitely lived options. For infinitely lived options the notion of constant κ ‐ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motion. It is argued that an increase in the set of priors delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect on investment.

Irreversible investment with Cox–Ingersoll–Ross type mean reversion

We solve a Dixit and Pindyck type irreversible investment problem in continuous time under the assumption that the project value follows a Cox–Ingersoll–Ross process. This setup works well for modeling foreign direct investment in the framework of real options, when the exchange rate is uncertain and the project value fixed in a foreign currency. We indicate how the solution qualitatively differs from the two classical cases: geometric Brownian motion and geometric mean reversion. Furthermore, we discuss analytical properties of the Cox–Ingersoll–Ross process and demonstrate potential advantages of this process as a model for the project value with regard to the classical ones.

Incomplete Markets, Knightian Uncertainty, and Irreversible Investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of Knightian uncertainty over the appropriate discount factor. Maxmin utility over multiple priors is used to solve the irreversible investment problem. The notion of priors with kappa-ignorance are used to analyse finitely lived options. For infinitely lived options the notion of constant kappa-ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motions. It is argued that an increase in the level of ambiguity delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect.

Irreversible Investment with Cox-Ingersoll-Ross Type Mean Reversion

We solve a Dixit and Pindyck type irreversible investment problem in continuous time under the assumption that the project value follows a Cox-Ingersoll-Ross process. We indicate how the solution qualitatively differs from the two classical cases geometric Brownian motion and geometric mean reversion. Furthermore we discuss analytical properties of the Cox-Ingersoll-Ross process and demonstrate potential advantageous of this process as a model of the project value with regards to the classical ones.

Erratum

The discussion below, from Maria B. Chiarolla and Ulrich G. Haussmann, represents the correction of an error made in Theorem 5.1 of Chiarolla, Maria B., Ulrich G. Haussmann. 2005. Explicit solution of a stochastic, irreversible investment problem and its moving threshold. Mathematics of Operations Research 30(1) 91–108.