Optimality of an $(s, S)$ Policy with Compound Poisson and Diffusion Demands: A Quasi-variational Inequalities Approach

Bensoussan, Alain; Liu, R. H.; Sethi, Suresh P.

doi:10.1137/s0363012904443737

Cited by 100 publications

(89 citation statements)

References 19 publications

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“…with strategy L* given by (23). This then establishes V (x) ≥ V L (x) for any other strategy L and V (x) = V L * (x).…”

Section: There Is a Unique Solutionmentioning

confidence: 72%

“…As an alternative to the above full characterization of the optimization problem (the "analytic way"), another quite common (Bensoussan et al [23] call it "probabilistic") approach in the literature is to maximize a certain value function over a (small) restricted class of admissible strategies, say barrier type strategies or simple impulse controls (cf. Avram et al [14], Loeffen [90,88], Gerber & Shiu [54,55]).…”

Section: Discussion Of the Hjb Equation -Verification Argumentsmentioning

confidence: 99%

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Optimality results for dividend problems in insurance

Albrecher

Thonhauser

2009

Rev. R. Acad. Cien. Serie A. Mat.

145

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“…with strategy L* given by (23). This then establishes V (x) ≥ V L (x) for any other strategy L and V (x) = V L * (x).…”

Section: There Is a Unique Solutionmentioning

confidence: 72%

Section: Discussion Of the Hjb Equation -Verification Argumentsmentioning

confidence: 99%

Optimality results for dividend problems in insurance

Albrecher

Thonhauser

2009

Rev. R. Acad. Cien. Serie A. Mat.

145

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show abstract

“…The QVIs are then solved explicitly for the minimal cost functions and the associated optimal reinsurance-injection controls are found. For other applications of QVI methods and regular-impulse control theory, readers are referred to Bensoussan et al [2] and Candenillas et al [3] and [4]. In our paper we consider two types of reinsurance -proportional reinsurance and excess-of-loss reinsurance separately.…”

Section: Introductionmentioning

confidence: 99%

Minimal cost of a Brownian risk without ruin

Luo

Taksar

2012

Insurance: Mathematics and Economics

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Abstract. In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The objective is to find the optimal reinsurance and cash injection strategy that minimizes the total cost to keep the company's surplus process non-negative, i.e. without ruin, where the cost function is defined as the total discounted value of the injections. The optimal solution is found explicitly by solving the according quasi-variational inequalities (QVIs).

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“…Singular control with discretionary stopping has been studied in [35,61]. In addition to the economic papers cited in this introduction, impulse Brownian control has been used to study manufacturing systems [53], paper plants for paper making and cutting [54], natural resource economics such as forest harvesting problems [2,3,4,5], inventory control problems [12,18,21,28,65,75], and financial problems such as dividend problems [6,27,49,67], inflation and pricing problems [78], portfolio optimization problems with transaction costs [24,47,51,63,77], and exchange rate problems [26,48,62]. See [52] for a survey on using impulse control to solve these financial problems.…”

mentioning

confidence: 99%

“…The QVI problem is a pure analytical problem that relates closely to the free boundary problem. Many authors start with the QVI problems, relying on the verification theorem developed in [20]; see for example, [17,18,21,75]. The potential drawback of this approach is that a slight difference in the formulation of a Brownian or diffusion control problem from the setting in [20] may require developing a new verification theorem, presumably mimicking the development in reference.…”

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confidence: 99%

Brownian Inventory Models with Convex Holding Cost, Part 1: Average-Optimal Controls

Dai

Yao

2013

Stochastic Systems

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We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed cost and a proportional cost. The challenge is to find an adjustment policy that balances the holding cost and adjustment cost to minimize the long-run average cost. When both upward and downward fixed costs are positive, our problem is an impulse control problem. When both fixed costs are zero, our problem is a singular or instantaneous control problem. For the impulse control problem, we prove that a four-parameter control band policy is optimal among all feasible policies. For the singular control problem, we prove that a two-parameter control band policy is optimal.We use a lower-bound approach, widely known as the "verification theorem", to prove the optimality of a control band policy for both the impulse and singular control problems. Our major contribution is to prove the existence of a "smooth" solution to the free boundary problem under some mild assumptions on the holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. We demonstrate that the lower-bound approach also works for a Brownian inventory model which prohibits an inventory backlog. A companion paper ("Brownian inventory models with convex holding cost, part 2: discountoptimal controls", Stochastic Systems, Vol. 3, No. 2, pp. 500-573) explains how to adapt the lower-bound approach to study a Brownian inventory model under a discounted cost criterion. 1. Introduction. This paper concerns the optimal control of Brownian inventory models under a long-run average cost criterion. It serves two purposes. First, it provides a tutorial on the powerful lower-bound approach, known as the "the verification theorem", for proving the optimality of a control band policy among all feasible policies. The tutorial is rigorous and self-contained with the exception of the standard Itô's formula. Second, it contributes to the literature by proving the existence of a "smooth" solution to the free boundary problem with a general convex holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. The companion paper [33] studies the optimal control of Brownian inventory models under a discounted cost criterion.

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Optimality of an $(s, S)$ Policy with Compound Poisson and Diffusion Demands: A Quasi-variational Inequalities Approach

Cited by 100 publications

References 19 publications

Optimality results for dividend problems in insurance

Optimality results for dividend problems in insurance

Minimal cost of a Brownian risk without ruin

Brownian Inventory Models with Convex Holding Cost, Part 1: Average-Optimal Controls

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