On the Impulse Control of Jump Diffusions

Bayraktar, Erhan; Emmerling, Thomas; Menaldi, José Luis

doi:10.1137/120863836

Cited by 25 publications

(20 citation statements)

References 13 publications

(26 reference statements)

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“…Remark 6. Recall that the functions (w ρ ) ρ>0 appearing in the nonlocal consistency inequalities (12) are an arbitrary family of uniformly bounded real-valued maps. However, a close inspection of the proof of Theorem 4 reveals that we only use these inequalities with w ρ = u ρ where u ρ is a solution of the scheme.…”

Section: Convergence Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

Azimzadeh¹,

Bayraktar²,

Labahn³

2018

SIAM J. Control Optim.

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In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.

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Section: Convergence Resultsmentioning

confidence: 99%

“…The interested reader can review[15,25,33,29,32,20,16,19,12,17,23,6,13,26,1] for information and recent developments on impulse control. We do not claim this list to be exhaustive.…”

mentioning

confidence: 99%

Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

Azimzadeh¹,

Bayraktar²,

Labahn³

2018

SIAM J. Control Optim.

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“…4,Sec. 4.2] in the context of regular stochastic control; [14] in the context of impulse control). We use this assumpton since, as outlined in the introduction, in our approach the proof of the semiconvexity of the value function will be a crucial step towards the proof of the C 1 regularity.…”

Section: Remark 22mentioning

confidence: 99%

“…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.…”

mentioning

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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“…We are aware of only a few works [46,20] that study zero-sum SDGs with impulse control (along with some related studies on switching controls; e.g., [42,43,7]). This is in spite of the fact that impulse control problems have enjoyed a resurgence (see, e.g., [6,17]) due to a demand for more realistic financial models (e.g., fixed transaction costs and liquidity risk) [34,13,39,18] and their link to backwards stochastic differential equations [33].…”

Section: Context and Literaturementioning

confidence: 99%

A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

Azimzadeh

2017

Appl Math Optim

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We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.

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On the Impulse Control of Jump Diffusions

Cited by 25 publications

References 13 publications

Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

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

A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

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