Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications

Aïd, René; Basei, Matteo; Callegaro, Giorgia; Campi, Luciano; Vargiolu, Tiziano

doi:10.1287/moor.2019.0989

Cited by 38 publications

(99 citation statements)

References 29 publications

(39 reference statements)

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“…That is, independently of the action of agent j, agent i will always force the process X to stay in her inaction region I i . A rigorous formalization of (φ 1 ,φ 2 ) can be obtained by the arguments employed in Definition 2.2 of Aïd et al (2018). We now show that the policies (φ 1 ,φ 2 ) previously defined are in fact admissible.…”

Section: Construction Of a Candidate Solutionmentioning

confidence: 76%

“…Our methodology is different with respect to that of Aïd et al (2018). Here we obtain candi-4 For other works modeling the pollution control problem as a dynamic game one can refer, among others, to the example in Section 4 of De Angelis and Ferrari (2016), Long (1992) and van der Ploeg and de Zeeuw (1991).…”

Section: Introductionmentioning

confidence: 92%

“…On the other hand, from a mathematical point of view, ours is one of the first papers dealing with a two-player nonzero-sum stochastic impulse game. It is worth noticing that a verification theorem for two-player nonzero-sum stochastic impulse games, in which the uncontrolled process is a multi-dimensional Itô-diffusion, has been recently proved in Aïd et al (2018). There the authors give a set of sufficient conditions under which the solutions (in an appropriate sense) of a system of coupled constrained PDE problems (the so-called quasi-variational inequalities, QVIs 5 ) identify equilibrium values of the game.…”

Section: Introductionmentioning

confidence: 99%

“…This is in contrast to the eight equations one would obtain by imposing C 0 and C 1 -regularity of the solutions to the system of QVIs (cf. Aïd et al (2018)). The rest of the paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a strategic model of pollution control

Ferrari

Koch

2018

Ann Oper Res

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This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants' emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions' level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Itôdiffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm's operating profit and the government's running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government.

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Section: Construction Of a Candidate Solutionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a strategic model of pollution control

Ferrari

Koch

2018

Ann Oper Res

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“…The latter inequality contradicts that v (2) ≤ v (1) on I, and therefore shows that a * 2 ≥ a * 1 . Analogous arguments can be employed to obtain b * 1 ≥ b * 2 .…”

Section: 1mentioning

confidence: 91%

On the singular control of exchange rates

Ferrari

Vargiolu

2019

Ann Oper Res

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Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current level of the exchange rate. The central bank aims at minimizing the total expected costs of interventions on the exchange market, plus a total expected holding cost. We formulate this problem as an infinite time-horizon stochastic control problem with controls that have paths which are locally of bounded variation. The exchange rate evolves as a general linearly controlled one-dimensional diffusion, and the two nondecreasing processes giving the minimal decomposition of a bounded-variation control model the cumulative amount of foreign currency that has been purchased and sold by the central bank. We provide a complete solution to this problem by finding the explicit expression of the value function and a complete characterization of the optimal control. At each instant of time, the optimally controlled exchange rate is kept within a band whose size is endogenously determined as part of the solution to the problem. We also study the expected exit time from the band, and the sensitivity of the width of the band with respect to the model's parameters in the case when the exchange rate evolves (in absence of any intervention) as an Ornstein-Uhlenbeck process, and the marginal costs of controls are constant. The techniques employed in the paper are those of the theory of singular stochastic control and of one-dimensional diffusions.

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Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates

Basei

2019

Math Meth Oper Res

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We consider a retailer who buys energy in the wholesale market and resells it to final consumers. The retailer has to decide when to intervene to change the price he asks to his customers, in order to maximize his income. We model the problem as an infinite-horizon stochastic impulse control problem. We characterize an optimal price strategy and provide analytical existence results for the equations involved. We then investigate the dependence on the intervention cost. In particular, we prove that the measure of the continuation region is asymptotic to the fourth root of the cost. Finally, we provide some numerical results and consider a suitable extension of the model. MSC Classification: 93E20, 91B70, 91B24.

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Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications

Cited by 38 publications

References 29 publications

On a strategic model of pollution control

On a strategic model of pollution control

On the singular control of exchange rates

Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates

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