Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

Angelis, Tiziano De; Federico, Salvatore; Ferrari, Giorgio

doi:10.1287/moor.2016.0841

Cited by 32 publications

(36 citation statements)

References 48 publications

(70 reference statements)

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“…The proof is organised in several steps. It follows arguments similar to those recently employed in [15] for a two-dimensional elliptic problem, and it extends to our two-dimensional setting the arguments presented in [38], Section 25.…”

Section: The Auxiliary Optimal Stopping Problemmentioning

confidence: 70%

“…It is worth noticing that the number of papers characterising the optimal policy in multidimensional singular stochastic control problems is still limited (see [15] for a recent contribution). In some early papers (see, e.g., [9], [36], [43] and [45]) fine analytical methods based on the dynamic programming principle and the theory of variational inequalities are employed to study the regularity of the value function of multi-dimensional singular stochastic control problems, and of the related free boundary.…”

Section: Introductionmentioning

confidence: 99%

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On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.Public Debt Control 2 of a crisis. This negative effect on economic growth from high debt levels has been observed in 18 different advanced economies (see [8]).In this paper we propose a continuous-time stochastic model for the control of the debtto-GDP ratio. The problem we have in mind is that of a government aiming to answer the question: How much is too much? 1 Following classical macroeconomic theory (see, e.g., [5]), in any given period the debt ratio stock grows by the existing debt stock multiplied by the difference between real interest rate and GDP growth, less the primary budget balance 2 . We assume that the government can reduce the level of the debt-to-GDP ratio by adjusting the primary budget balance, e.g. through fiscal interventions like raising taxes or reducing expenses. We therefore interpret the cumulative interventions on the debt ratio as the government's control variable, and we model it as a nonnegative and nondecreasing stochastic process. Uncertainty in our model comes through the GDP growth rate of the country, and its inflation and nominal interest rate which, by Fisher law [21], directly affect the growth rate of the debt ratio. We first assume that they are general not necessarily Markovian stochastic processes whose dynamics is not under government control. Indeed, the level of these macroeconomic variables is usually regulated by an autonomous Central Bank, whose action, however, is not modelled in this paper (see, e.g., [10] and [25] for problems related to the optimal control of inflation).Since high debt-to-GDP ratios can constrain economic growth making it more difficult to break the burden of the debt, we assume that debt ratio generates an instantaneous cost/penalty. This is a general nonnegative convex function of the debt ratio level, that the government would like to k...

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Section: The Auxiliary Optimal Stopping Problemmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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“…This is motivated by the fact that in recent years several specific problems of this kind have appeared in the literature (cf. [6], [8], [10], [12], [16], [17]) and further ones are on their way. Often these papers are motivated by real-world applications where dimension two (or higher) plays a crucial role (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the known uniqueness arguments [22,, originally established in [19] and further refined in [11], extend to this fully two-dimensional case as well (cf. [6], [8], [12], [16], [17]) and this enables one to characterise the optimal stopping boundary as the unique solution to a nonlinear integral equation in the class of continuous functions (with the value function being expressed in terms of the optimal stopping boundary itself). This characterisation of the optimal stopping boundary is known to be sufficient for practical purposes of optimal stopping (using Picard iteration for numerics for instance) and higher degrees of regularity can then be studied subsequently as/if needed.…”

Section: Introductionmentioning

confidence: 99%

Continuity of the optimal stopping boundary for two-dimensional diffusions

Peškir¹

2019

Ann. Appl. Probab.

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We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary's discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Hölder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.

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“…More relevant to this paper models have been studied by several authors in the economics literature: see Dixit and Pindyck [17,Chapter 11] and references therein. 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. 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].…”

Section: Introductionmentioning

confidence: 99%

Irreversible Capital Accumulation with Economic Impact

Motairi

Zervos

2016

Appl Math Optim

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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.

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Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

Cited by 32 publications

References 48 publications

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Continuity of the optimal stopping boundary for two-dimensional diffusions

Irreversible Capital Accumulation with Economic Impact

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