Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes

Guo, Xin; Wu, Guorong

doi:10.1137/080716001

Cited by 39 publications

(60 citation statements)

References 35 publications

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“…6]. For a viscosity approach to the quasi-variational inequality characterizing the value function of impulse problems, see Davis et al [15], Guo and Wu [19]. Øksendal [28], Øksendal et al [30] study the robustness of a class of impulse problems with respect to the intervention costs.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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

Basei

2019

Math Meth Oper Res

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“…On the theoretical side, starting from the classical book [17], several works investigated QVIs associated to stochastic impulse optimal control in R n . Among them, we mention the recent [43] in a diffusion setting and [14,29] in a jump-diffusion setting. In particular, [17, Ch.…”

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confidence: 99%

“…More recently, these methods have been extended to Markov processes valued in metric spaces (see [25]); again a complete description of the solution is shown in one dimensional examples.Contribution. From the methodological side our work is close to [43]. As in the latter, we follow a direct analytical method based on viscosity solutions and we do not employ a guess-and-verify approach( 4 ).…”

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confidence: 99%

Irreversible investment with fixed adjustment costs: a stochastic impulse control approach

2019

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We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with respect to the relevant parameters of the problem.A.M.S. Subject Classification: 93E20 (Optimal stochastic control); 35Q93 (PDEs in connecton woth control and optimization); 35D40 (Viscosity solution); 35B65 (Smoothness and regularity of solutions).Related literature. First of all, it is worth noticing that the stochastic impulse control setting has been widely employed in several applied fields: e.g., exchange and interest rates [21,51,56], portfolio optimization with transaction costs [34,49,57], inventory and cash management [12,20,27,30,31,44,45,58,62,67,68,71], real options [47,54], reliability theory [7]. More recently, games of stochastic impulse control have been investigated with application to pollution [39].From a modeling point of view, the closest works to ours can be considered [3,6,26,35,49]. On the theoretical side, starting from the classical book [17], several works investigated QVIs associated to stochastic impulse optimal control in R n . Among them, we mention the recent [43] in a diffusion setting and [14,29] in a jump-diffusion setting. In particular, [17, Ch. 4] deals with Sobolev type solutions, whereas [43] deals with viscosity solutions. These two works prove a W 2,pregularity, with p < ∞, for the solution of QVI, which, by classical Sobolev embeddings, yields a C 1 -regularity. However, it is typically not easy to obtain by such regularity information on the structure of the so called continuation and action regions, hence on the candidate optimal control. If this structure is established, then one can try to prove a verificiation theorem to prove that the candidate optimal control is actually optimal. In a stylized one dimensional example, [43, Sec. 5] successfully employs this method by exploiting the regularity result proved in [43, Sec. 4] to depict the structure of the continuation and action region for the problem at hand. Concerning verification, we need to mention the recent paper [15], which provides a non-smooth verification theorem in a quite general setting based on the stochastic Perron method to construct a viscosity solution to QVI; also this paper, in the last section, provides and application of the results to a one dimensional problem with an implementable solution. In dimension one other approaches, based on excessive mappings and iterated optimal stopping schemes, have been successfully employed in the context of stochastic impulse control (see [3,6,35,46]). More recently, these ...

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“…The focus of this paper is on identifying the regularity of the value function for impulse control under a general jump diffusion setting on the whole space and with unbounded controls. Regularity in various relevent contexts has been examined by many in the literature; see, e.g., [1], [2], [3], [4], [6], [7], [8], [11], [12]. Recently, [8] (resp., [4]) identified W 2,p loc (R n ) regularity of the value function of impulse control for a pure diffusion (resp., jump diffusion) with unbounded controls.…”

Section: Introductionmentioning

confidence: 99%

“…Regularity in various relevent contexts has been examined by many in the literature; see, e.g., [1], [2], [3], [4], [6], [7], [8], [11], [12]. Recently, [8] (resp., [4]) identified W 2,p loc (R n ) regularity of the value function of impulse control for a pure diffusion (resp., jump diffusion) with unbounded controls. In both of these papers, the authors utilized classical PDE arguments along with recent viscosity results for impulse control [17] to establish regularity.…”

Section: Introductionmentioning

confidence: 99%

Impulse Control of Jump Diffusions

Bayraktar

Emmerling

Menaldi

Applied Stochastic Control of Jump Diffusions

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Abstract. Regularity of the impulse control problem for a nondegenerate n-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in [M. H. A. Davis, X. Guo, and G. Wu, SIAM J. Control Optim., 48 (2010), pp. 5276-5293]. Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function u of the impulse control problem satisfies u ∈ W 2,p loc (R n ).

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Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes

Cited by 39 publications

References 35 publications

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

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

Irreversible investment with fixed adjustment costs: a stochastic impulse control approach

Impulse Control of Jump Diffusions

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