A model for the long-term optimal capacity level of an investment project
21 Nov 2011-International Journal of Theoretical and Applied Finance (World Scientific Publishing Co. Pte. Ltd.)-Vol. 14, Iss: 02, pp 187-196
TL;DR: In this article, the authors consider an investment project that produces a single commodity and determine a capacity expansion strategy that maximizes the ergodic or long-term average payoff resulting from the project's management.
Abstract: We consider an investment project that produces a single commodity. The project’s operation yields payoff at a rate that depends on the project’s installed capacity level and on an underlying economic indicator such as the output commodity’s price or demand, which we model by an ergodic, one-dimensional Itˆo diffusion. The project’s capacity level can be increased dynamically over time. The objective is to determine a capacity expansion strategy that maximizes the ergodic or long-term average payoff resulting from the project’s management. We prove that it is optimal to increase the project’s capacity level to a certain value and then take no further actions. The optimal capacity level depends on both the long-term average and the volatility of the underlying diffusion.
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TL;DR: In this paper, a reversible investment problem is studied where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. But the resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem for which explicit solution is not available in general and the standard verification approach cannot be applied a priori.
Abstract: This paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. Our model allows for general diffusion dynamics on the demand as well as general cost functional. The resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem, for which explicit solution is not available in general and the standard verification approach cannot be applied a priori. We use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution. In the quadratic cost case, we are able to prove a smooth fit $C^2$ property, which gives rise to a full characterization of the optimal boundaries and value function.
33 citations
TL;DR: In this article, the authors established a convergence theorem for multi-dimensional stochastic approximation when the innovations satisfy some light averaging properties in the presence of a pathwise Lyapunov function.
Abstract: The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the ''innovations'' satisfy some ''light'' averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by Finance.
33 citations
TL;DR: In this article, the authors consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator, and the associated optimisation problem takes the form of a singular control problem that admits an explicit solution.
Abstract: We consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator. The associated optimisation problem takes the form of a singular stochastic control problem that admits an explicit solution. A special characteristic of this stochastic control problem is that changes of the state process due to control action depend on the state process itself in a proportional way.
19 citations
Cites methods from "A model for the long-term optimal c..."
...Related models that have been studied in the mathematics literature include Davis, Dempster, Sethi and Vermes [13], Arntzen [4], Øksendal [42], Wang [48], Chiarolla and Haussmann [11], Bank [6], Alvarez [2,3], Løkka and Zervos [35], Steg [45], Chiarolla and Ferrari [9], De Angelis, Federico and Ferrari [15], and references therein....
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...The random variable X0t can represent an economic indicator such as the price B Mihail Zervos mihalis.zervos@gmail.com Hessah Al Motairi almotairi.h@gmail.com 1 Department of Mathematics, College of Science, Kuwait University, Kuwait City, Kuwait 2 Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK of or the demand for one unit of a given investment project’s output at time t ....
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...At this point, it is worth noting that Guo and Zervos [25] have considered the same state dynamics in the optimal execution problem that they study....
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...Furthermore, capacity expansion models with costly reversibility were introduced by Abel and Eberly [1], and were further studied by Guo and Pham [22], Merhi and Zervos [40], Guo and Tomecek [23,24], Guo, Kaminsky, Tomecek and Yuen [21], Løkka and Zervos [36], De Angelis and Ferrari [16], and Federico and Pham [19]....
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...Apart from references that we have discussed in the context of capacity expansion models, Bahlali et al. [5] Chiarolla and Haussmann [10], Chow, Menaldi andRobin [12], Davis andZervos [14], Fleming and Soner [20, ChapterVIII], Haussmann and Suo [27,28], Harrison and Taksar [26], Jack, Johnson and Zervos [29], Jacka [30,31], Karatzas [32],Ma [37],Menaldi and Robin [39], Øksendal [42], Shreve et al. [43], Soner and Shreve [44], Sun [46] and Zhu [49], provide an alphabetically ordered list of further contributions....
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TL;DR: This work solves the genuinely two-dimensional stochastic control problem by constructing an explicit solution to an appropriate Hamilton--Jacobi--Bellman equation and by fully characterizing an optimal investment strategy.
Abstract: We consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment. In particular, we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project's output commodity. We model this economic indicator by means of a general one-dimensional ergodic diffusion. At any time, the project's capacity level can be increased or decreased at given proportional costs. The aim is to maximize an ergodic performance criterion that reflects the long-term average payoff resulting from the project's management. We solve this genuinely two-dimensional stochastic control problem by constructing an explicit solution to an appropriate Hamilton--Jacobi--Bellman equation and by fully characterizing an optimal investment strategy.
17 citations
TL;DR: This work provides an optimal trading rule that allows buying and selling an asset sequentially over time using a switchable mean-reversion model with a Markovian jump to determine a sequence of trading times to maximize an overall return.
Abstract: This work provides an optimal trading rule that allows buying and selling an asset sequentially over time. The asset price follows a switchable mean-reversion model with a Markovian jump. Such a model can be applied to assets with a "staircase" price behavior and yet is simple enough to allow an analytic solution. The objective is to determine a sequence of trading times to maximize an overall return. The corresponding value functions are characterized by a set of quasi-variational inequalities. A closed-form solution is obtained under suitable conditions. The sequence of trading times can be given in terms of a set of threshold levels. Finally, numerical examples are given to demonstrate the results.
9 citations
References
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Book•
01 Jan 1987
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
Abstract: 1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- A. Fundamental inequalities.- B. Convergence results.- C. The optional sampling theorem.- 1.4. The Doob-Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- A. The consistency theorem.- B. The Kolmogorov-?entsov theorem.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- A. Weak convergence.- B. Tightness.- C. Convergence of finite-dimensional distributions.- D. The invariance principle and the Wiener measure.- 2.5. The Markov Property.- A. Brownian motion in several dimensions.- B. Markov processes and Markov families.- C. Equivalent formulations of the Markov property.- 2.6. The Strong Markov Property and the Reflection Principle.- A. The reflection principle.- B. Strong Markov processes and families.- C. The strong Markov property for Brownian motion.- 2.7. Brownian Filtrations.- A. Right-continuity of the augmented filtration for a strong Markov process.- B. A "universal" filtration.- C. The Blumenthal zero-one law.- 2.8. Computations Based on Passage Times.- A. Brownian motion and its running maximum.- B. Brownian motion on a half-line.- C. Brownian motion on a finite interval.- D. Distributions involving last exit times.- 2.9. The Brownian Sample Paths.- A. Elementary properties.- B. The zero set and the quadratic variation.- C. Local maxima and points of increase.- D. Nowhere differentiability.- E. Law of the iterated logarithm.- F. Modulus of continuity.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- A. Simple processes and approximations.- B. Construction and elementary properties of the integral.- C. A characterization of the integral.- D. Integration with respect to continuous, local martingales.- 3.3. The Change-of-Variable Formula.- A. The Ito rule.- B. Martingale characterization of Brownian motion.- C. Bessel processes, questions of recurrence.- D. Martingale moment inequalities.- E. Supplementary exercises.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- A. Continuous local martingales as stochastic integrals with respect to Brownian motion.- B. Continuous local martingales as time-changed Brownian motions.- C. A theorem of F. B. Knight.- D. Brownian martingales as stochastic integrals.- E. Brownian functionals as stochastic integrals.- 3.5. The Girsanov Theorem.- A. The basic result.- B. Proof and ramifications.- C. Brownian motion with drift.- D. The Novikov condition.- 3.6. Local Time and a Generalized Ito Rule for Brownian Motion.- A. Definition of local time and the Tanaka formula.- B. The Trotter existence theorem.- C. Reflected Brownian motion and the Skorohod equation.- D. A generalized Ito rule for convex functions.- E. The Engelbert-Schmidt zero-one law.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- A. The mean-value property.- B. The Dirichlet problem.- C. Conditions for regularity.- D. Integral formulas of Poisson.- E. Supplementary exercises.- 4.3. The One-Dimensional Heat Equation.- A. The Tychonoff uniqueness theorem.- B. Nonnegative solutions of the heat equation.- C. Boundary crossing probabilities for Brownian motion.- D. Mixed initial/boundary value problems.- 4.4. The Formulas of Feynman and Kac.- A. The multidimensional formula.- B. The one-dimensional formula.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- A. Definitions.- B. The Ito theory.- C. Comparison results and other refinements.- D. Approximations of stochastic differential equations.- E. Supplementary exercises.- 5.3. Weak Solutions.- A. Two notions of uniqueness.- B. Weak solutions by means of the Girsanov theorem.- C. A digression on regular conditional probabilities.- D. Results of Yamada and Watanabe on weak and strong solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- A. Some fundamental martingales.- B. Weak solutions and martingale problems.- C. Well-posedness and the strong Markov property.- D. Questions of existence.- E. Questions of uniqueness.- F. Supplementary exercises.- 5.5. A Study of the One-Dimensional Case.- A. The method of time change.- B. The method of removal of drift.- C. Feller's test for explosions.- D. Supplementary exercises.- 5.6. Linear Equations.- A. Gauss-Markov processes.- B. Brownian bridge.- C. The general, one-dimensional, linear equation.- D. Supplementary exercises.- 5.7. Connections with Partial Differential Equations.- A. The Dirichlet problem.- B. The Cauchy problem and a Feynman-Kac representation.- C. Supplementary exercises.- 5.8. Applications to Economics.- A. Portfolio and consumption processes.- B. Option pricing.- C. Optimal consumption and investment (general theory).- D. Optimal consumption and investment (constant coefficients).- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Levy's Theory of Brownian Local Time.- 6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- A. The process of passage times.- B. Poisson random measures.- C. Subordinators.- D. The process of passage times revisited.- E. The excursion and downcrossing representations of local time.- 6.3. Two Independent Reflected Brownian Motions.- A. The positive and negative parts of a Brownian motion.- B. The first formula of D. Williams.- C. The joint density of (W(t), L(t), ? +(t)).- 6.4. Elastic Brownian Motion.- A. The Feynman-Kac formulas for elastic Brownian motion.- B. The Ray-Knight description of local time.- C. The second formula of D. Williams.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.
8,639 citations
Book•
01 Jan 1993
TL;DR: In this article, the authors present a new approach to problems of evaluating and optimizing the performance of continuous-time stochastic systems, based on the use of a family of Markov processes called Piecewise-Deterministic Processes (PDPs) as a general class of stocha- system models.
Abstract: This book presents a radically new approach to problems of evaluating and optimizing the performance of continuous-time stochastic systems. This approach is based on the use of a family of Markov processes called Piecewise-Deterministic Processes (PDPs) as a general class of stochastic system models. A PDP is a Markov process that follows deterministic trajectories between random jumps, the latter occurring either spontaneously, in a Poisson-like fashion, or when the process hits the boundary of its state space. This formulation includes an enormous variety of applied problems in engineering, operations research, management science and economics as special cases; examples include queueing systems, stochastic scheduling, inventory control, resource allocation problems, optimal planning of production or exploitation of renewable or non-renewable resources, insurance analysis, fault detection in process systems, and tracking of maneuvering targets, among many others.The first part of the book shows how these applications lead to the PDP as a system model, and the main properties of PDPs are derived. There is particular emphasis on the so-called extended generator of the process, which gives a general method for calculating expectations and distributions of system performance functions. The second half of the book is devoted to control theory for PDPs, with a view to controlling PDP models for optimal performance: characterizations are obtained of optimal strategies both for continuously-acting controllers and for control by intervention (impulse control). Throughout the book, modern methods of stochastic analysis are used, but all the necessary theory is developed from scratch and presented in a self-contained way. The book will be useful to engineers and scientists in the application areas as well as to mathematicians interested in applications of stochastic analysis.
1,255 citations
"A model for the long-term optimal c..." refers methods in this paper
...[5] M. H. A. Davis, M. A. H. Dempster, S. P. Sethi and D. Vermes (1987), Optimal capacity expan- sion under uncertainty, Advances in Applied Probability , vol. 19, pp.156–176....
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...[4] M. H. A. Davis (1993), Markov models and optimization, Chapman & Hall....
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...A number of other related models have been studied by Abel and Eberly [1], Davis, Dempster, Sethi and Vermes [5] (see also Davis [4]), Kobila [7], Øksendal [11], Wang [14], Bank [2], Chiarolla and Haussmann [3], Merhi and Zervos [10], Guo and Tomecek [6] and in references therein....
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TL;DR: In this paper, the Jorgensonian concept of user cost of capital was extended to the case of irreversible investment and the authors defined and calculated the user costs of capital associated with the purchase and sale of capital, respectively.
Abstract: price and sell capital at a lower price We solve for the optimal investment of a firm that faces costly reversibility under uncertainty and we extend the Jorgensonian concept of the user cost of capital to this case We define and calculate cv and cL as the user costs of capital associated with the purchase and sale of capital, respectively Optimality requires the firm to purchase and sell capital as needed to keep the marginal revenue product of capital in the closed interval [CL, cu] This prescription encompasses the case of irreversible investment as well as the standard neoclassical case of costlessly reversible investment
591 citations
"A model for the long-term optimal c..." refers methods in this paper
...References [1] B. A. Abel and J. C. Eberly (1996), Optimal investment with costly reversibility, Review of Economic Studies, vol. 63, pp.581–593....
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...A number of other related models have been studied by Abel and Eberly [1], Davis, Dempster, Sethi and Vermes [5] (see also Davis [4]), Kobila [7], Øksendal [11], Wang [14], Bank [2], Chiarolla and Haussmann [3], Merhi and Zervos [10], Guo and Tomecek [6] and in references therein....
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Posted Content•
TL;DR: This paper reviews models of capacity investment under uncertainty in three settings and reviews how to incorporate risk aversion in capacity investment and contrasts hedging strategies involving financial versus operational means.
Abstract: This article reviews the literature on strategic capacity management concerned with determining the sizes, types, and timing of capacity investments and adjustments under uncertainty. Specific attention is given to recent developments to incorporate multiple decision makers, multiple capacity types, hedging and risk aversion. Capacity is a measure of processing abilities and limitations and is represented as a vector of stocks of various processing resources, while investment is the change of capacity and includes expansion and contraction. After discussing general issues in capacity investment problems, the article reviews models of capacity investment under uncertainty in three settings: The first reviews optimal capacity investment by single and multiple risk-neutral decision makers in a stationary environment where capacity remains constant. Allowing for multiple capacity types, the associated optimal capacity portfolio specifies the amounts and locations of safety capacity in a processing network. Its key feature is that it is unbalanced, i.e., regardless of how uncertainties are realized, one typically will never fully utilize all capacities. The second setting reviews the adjustment of capacity over time and the structure of optimal investment dynamics. The article ends by reviewing how to incorporate risk-aversion in capacity investment and contrasts hedging strategies involving financial versus operational means.
519 citations
"A model for the long-term optimal c..." refers background in this paper
...Capacity expansion models have attracted considerable interest in the literature and can be traced beck to Manne [9]; see Van Mieghem [13] for a recent survey....
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...[13] J. A. Van Mieghem (2003), Commissioned paper: capacity management, investment, and hedging: review and recent developments, Manufacturing & Service Operations Management , vol. 5, pp. 269– 302....
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TL;DR: In this paper, the authors review the literature on strategic capacity management concerned with determining the sizes, types, and timing of capacity investments and adjustments under uncertainty, and incorporate risk aversion in capacity investment and contrast hedging strategies involving financial versus operational means.
Abstract: This paper reviews the literature on strategic capacity management concerned with determining the sizes, types, and timing of capacity investments and adjustments under uncertainty. Specific attention is given to recent developments to incorporate multiple decision makers, multiple capacity types, hedging, and risk aversion. Capacity is a measure of processing abilities and limitations and is represented as a vector of stocks of various processing resources, while investment is the change of capacity and includes expansion and contraction. After discussing general issues in capacity investment problems, the paper reviews models of capacity investment under uncertainty in three settings:The first reviews optimal capacity investment by single and multiple risk-neutral decision makers in a stationary environment where capacity remains constant. Allowing for multiple capacity types, the associated optimal capacity portfolio specifies the amounts and locations of safety capacity in a processing network. Its key feature is that it is unbalanced; i.e., regardless of how uncertainties are realized, one typically will never fully utilize all capacities. The second setting reviews the adjustment of capacity over time and the structure of optimal investment dynamics. The paper ends by reviewing how to incorporate risk aversion in capacity investment and contrasts hedging strategies involving financial versus operational means.
498 citations