Equilibrium Dividend Strategy with Non-exponential Discounting in a Dual Model

ABSTRACT. We study the optimal dividend problem in the dual model where dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al. [6] solved the case with i.i.d. hyperexponential jumps; they showed the optimality of a (periodic) barrier strategy where dividends are paid at dividend-decision times if and only if the surplus is above some level. In this paper, we generalize the results for a general spectrally positive Lévy process with additional terminal payoff/penalty at ruin, and also solve the case with classical bail-outs so that the surplus is restricted to be nonnegative.The optimal strategies as well as the value functions are concisely written in terms of the scale function.Numerical results are also given.AMS 2010 Subject Classifications: 60G51, 93E20, 91B30 JEL Classifications: C44, C61, G24, G32, G35

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“…20) and(4.21), we have (L − q)Z (q,r) (b * − x) = 0. Applying this and (4.18) in (4.7), we have the result for 0 < x < b * .…”

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confidence: 90%

On the optimality of periodic barrier strategies for a spectrally positive Lévy process

Pérez

Yamazaki

2017

Insurance: Mathematics and Economics

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“…Remark 2. (i) In a certain sense, the extended HJB system of equations in Definition 3.2 is consistent with Definition 3.1 in Li et al [21] and Definition 3.1 in Zhao et al [29]. (ii) If terminal time is fixed, the Definition 3.1 is equivalent to the Definition 3.2, however, in the case of considering bankruptcy, it is not a simple equivalent.…”

Section: Remarkmentioning

confidence: 71%

“…Applying the operator d dx − β to (21) and after some calculations, we obtain 9) and Remark 6, we have…”

Section: Remarkmentioning

confidence: 96%

“…Many researchers have studied the control problem in the dual model. See, for example, Pérez and Yamazaki [24], Li et al [21], Avanzi and Gerber [5], Foucart et al [17], Avanzi et al [4,6], Zhao et al [31,32] and Yin et al [28] and so on.…”

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confidence: 99%

“…Under the above discount function, Chen et al [11,12] discuss the optimal dividend problem in the dual model and a general diffusion model, respectively. In a compound Poisson dual model, Li et al [21] studies the absolutely continuous dividend problem with the mixture of two exponential discount functions. In a diffusion model, Zhao et al [29] solves the same dividend problem with a mixture of exponential discount functions and a pseudo-exponential discount function.…”

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confidence: 99%

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Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes

Zhong¹,

Zhao²,

Chen³

2021

JIMO

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In the dual risk model, we study the periodic dividend problem with a non-exponential discount function which results in a time-inconsistent control problem. Viewing it within the game theoretic framework, we extend the Hamilton-Jacobi-Bellman (HJB) system of equations from the fixed terminal to the time of ruin and derive the verification theorem, and we generalize the theory of classical optimal periodic dividend. Under two special non-exponential discount functions, we obtain the closed-form expressions of equilibrium strategy and the corresponding equilibrium value function in a compound Poisson dual model. Finally, some numerical examples are presented to illustrate the impact of some parameters.

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