scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Irreversible Capital Accumulation with Economic Impact

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.

Content maybe subject to copyright    Report

Citations
More filters
Proceedings ArticleDOI
01 Dec 1984
TL;DR: In this paper, the authors consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients, and their objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process.
Abstract: We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

46 citations

Journal ArticleDOI
TL;DR: The dynamic programming principle obtains an exactly-solvable variational inequality that governs a sustainable, threshold-type optimal management policy that maximizes a performance index that covers the cost of countermeasures, the loss of riverine fish, and the ecosystem services provided by the bird.

25 citations

Journal ArticleDOI
TL;DR: A stochastic control model for finding an ecologically sound, fit-for-purpose dam operation policy to suppress bloom of attached algae in its downstream is presented and solution behaviour of the Hamilton–Jacobi–Bellman equation near the origin, namely at the early stage of algae growth.
Abstract: A stochastic control model for finding an ecologically sound, fit-for-purpose dam operation policy to suppress bloom of attached algae in its downstream is presented. A singular exactly solvable and a more realistic regular-singular cases are analysed in terms of a Hamilton-Jacobi-Bellman equation. Regularity and consistency of the value function are analysed and its classical verification theorem is established. Practical implications of the mathematical analysis results are discussed focusing on parameter dependence of the optimal controls. An asymptotic analysis with a numerical computation reveals solution behaviour of the Hamilton-Jacobi-Bellman equation near the origin, namely at the early stage of algae growth.

22 citations


Cites background or methods from "Irreversible Capital Accumulation w..."

  • ...Related SDEs with a not usual but simpler singular control terms multiplied by state variables have been mathematically and numerically analysed [2,22,38,41,48]....

    [...]

  • ...The parameter l represents the effectiveness of the singular control term Xl tdηt (understood in the sense like [2]), the decrease in the algae population rather than by controlling the discharge....

    [...]

  • ...Combining Equations (26) and (27) leads to [2] (x2) ≥ J(x2; q, η̃) = J(x1; q, η̄)− β ∫ 0 −0 dηs...

    [...]

  • ...Applying the dynamic programming principle as in some of the literatures ([2], Chapter 2....

    [...]

Journal ArticleDOI
TL;DR: Meyer-$\sigma$-fields is proposed to use as a flexible tool to model information flow in stochastic control problems, illustrating in a first case study how different signals on exogenous jumps lead to different optimal controls.
Abstract: In stochastic control problems delicate issues arise when the controlled system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. We propose to use Meyer-$\sigma $-fields as a flexible tool to model information flow in such situations. The possibilities of this approach are illustrated first in a very simple linear stochastic control problem and then in a fairly general formulation for the singular stochastic control problem of irreversible investment with inventory risk. For the latter, we illustrate in a first case study how different signals on exogenous jumps lead to different optimal controls, interpolating between the predictable and the optional case in a systematic manner.

6 citations


Cites background from "Irreversible Capital Accumulation w..."

  • ...Federico and Pham [2014], Ferrari [2015], Al Motairi and Zervos [2017], De Angelis et al....

    [...]

  • ...…control problem which has been of considerable interest in the literature; see e.g. Arrow (1966), Dixit and Pindyck (1994), Bertola (1998), Merhi and Zervos (2007), Riedel and Su (2011), Federico and Pham (2014), Ferrari (2015), Al Motairi and Zervos (2017), De Angelis, Federico and Ferrari (2017)....

    [...]

  • ...Federico and Pham [2014], Ferrari [2015], Al Motairi and Zervos [2017], De Angelis et al. [2017])....

    [...]

  • ...Arrow [1966], Dixit and Pindyck [1994], Bertola [1998], Merhi and Zervos [2007], Riedel and Su [2011], Federico and Pham [2014], Ferrari [2015], Al Motairi and Zervos [2017], De Angelis et al. [2017]. Let us consider a controller who can choose her actions based on the information flow conveyed by a Meyer-σ-field Λ satisfying P(F ) ⊂ Λ ⊂ O(F ), where F is a complete, right-continuous filtration generated, e....

    [...]

  • ...Arrow [1966], Dixit and Pindyck [1994], Bertola [1998], Merhi and Zervos [2007], Riedel and Su [2011], Federico and Pham [2014], Ferrari [2015], Al Motairi and Zervos [2017], De Angelis et al....

    [...]

01 Oct 2007
TL;DR: This correspondence provides a novel method for solving high-dimensional singular control problems, and enables the theory of reversible investment to be extended, as sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions.
Abstract: This paper builds a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence provides a novel method for solving high-dimensional singular control problems, and enables us to extend the theory of reversible investment: sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions. Consequently, our regularity result links singular controls and Dynkin games through sequential optimal stopping problems.

6 citations

References
More filters
Book
01 Jan 1994
TL;DR: In this article, Dixit and Pindyck provide the first detailed exposition of a new theoretical approach to the capital investment decisions of firms, stressing the irreversibility of most investment decisions, and the ongoing uncertainty of the economic environment in which these decisions are made.
Abstract: How should firms decide whether and when to invest in new capital equipment, additions to their workforce, or the development of new products? Why have traditional economic models of investment failed to explain the behavior of investment spending in the United States and other countries? In this book, Avinash Dixit and Robert Pindyck provide the first detailed exposition of a new theoretical approach to the capital investment decisions of firms, stressing the irreversibility of most investment decisions, and the ongoing uncertainty of the economic environment in which these decisions are made. In so doing, they answer important questions about investment decisions and the behavior of investment spending.This new approach to investment recognizes the option value of waiting for better (but never complete) information. It exploits an analogy with the theory of options in financial markets, which permits a much richer dynamic framework than was possible with the traditional theory of investment. The authors present the new theory in a clear and systematic way, and consolidate, synthesize, and extend the various strands of research that have come out of the theory. Their book shows the importance of the theory for understanding investment behavior of firms; develops the implications of this theory for industry dynamics and for government policy concerning investment; and shows how the theory can be applied to specific industries and to a wide variety of business problems.

10,879 citations

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
18 Dec 1992
TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.
Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

3,885 citations

Journal ArticleDOI
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


Additional excerpts

  • ...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]....

    [...]

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


"Irreversible Capital Accumulation w..." refers background in this paper

  • ...Irreversible capacity expansion models have attracted considerable interest and can be traced back to Manne [38] (see Van Mieghem [47] for a survey)....

    [...]