General drawdown-based de Finetti optimization for spectrally negative Lévy risk processes

2018 ◽  
Vol 55 (2) ◽  
pp. 513-542 ◽  
Author(s):  
Wenyuan Wang ◽  
Xiaowen Zhou
Keyword(s):  
Value Function ◽  
Small Class ◽  
General Version ◽  
Ruin Time ◽  
Underlying Process ◽  
Optimal Dividend ◽  
Running Maximum ◽  
Admissible Strategies ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

Abstract For spectrally negative Lévy risk processes we consider a general version of de Finetti's optimal dividend problem in which the ruin time is replaced with a general drawdown time from the running maximum in its value function. We identify a condition under which a barrier dividend strategy is optimal among all admissible strategies if the underlying process does not belong to a small class of compound Poisson processes with drift, for which the take-the-money-and-run dividend strategy is optimal. It generalizes the previous results on dividend optimization from ruin time based to drawdown time based. The associated drawdown functions are discussed in detail for examples of spectrally negative Lévy processes.

OPTIMAL DIVIDEND–REINSURANCE WITH TWO TYPES OF PREMIUM PRINCIPLES

2015 ◽  
Vol 30 (2) ◽  
pp. 224-243 ◽  
Author(s):  
Hui Meng ◽  
Ming Zhou ◽  
Tak Kuen Siu
Keyword(s):  
Numerical Analysis ◽  
Impulse Control ◽  
Value Function ◽  
Proportional Transaction Costs ◽  
Ruin Time ◽  
Closed Form Solutions ◽  
Impulse Control Problem ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
The Value Function

A combined optimal dividend/reinsurance problem with two types of insurance claims, namely the expected premium principle and the variance premium principle, is discussed. Dividend payments are considered with both fixed and proportional transaction costs. The objective of an insurer is to determine an optimal dividend–reinsurance policy so as to maximize the expected total value of discounted dividend payments to shareholders up to ruin time. The problem is formulated as an optimal regular-impulse control problem. Closed-form solutions for the value function and optimal dividend–reinsurance strategy are obtained in some particular cases. Finally, some numerical analysis is given to illustrate the effects of safety loading on optimal reinsurance strategy.

Spectrally negative Lévy processes with applications in risk theory

2001 ◽  
Vol 33 (1) ◽  
pp. 281-291 ◽  
Author(s):  
Hailiang Yang ◽  
Lianzeng Zhang
Keyword(s):  
Lévy Processes ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Levy Processes ◽  
Point Of View ◽  
Ruin Time ◽  
Classical Risk Process ◽  
Risk Processes ◽  
First Time ◽  
Spectrally Negative Lévy Processes

In this paper, results on spectrally negative Lévy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, the first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Lévy process theory's point of view and in a unified and simple way.

Draw-down Parisian ruin for spectrally negative Lévy processes

2020 ◽  
Vol 52 (4) ◽  
pp. 1164-1196
Author(s):  
Wenyuan Wang ◽  
Xiaowen Zhou
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Fixed Amount ◽  
Ruin Time ◽  
Surplus Process ◽  
Potential Measure ◽  
Parisian Ruin ◽  
Exit Problems ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

AbstractDraw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin.

scholarly journals Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information

Symmetry ◽  
2018 ◽  
Vol 10 (7) ◽  
pp. 276 ◽  
Author(s):  
Qingyou Yan ◽  
Le Yang ◽  
Tomas Baležentis ◽  
Dalia Streimikiene ◽  
Chao Qin
Keyword(s):  
Transaction Cost ◽  
Risk Information ◽  
Insurance Company ◽  
Optimal Control Strategy ◽  
Ruin Time ◽  
Discounted Cost ◽  
Capital Injection ◽  
Surplus Process ◽  
Optimal Dividend ◽  
A Company

This paper considers the optimal dividend and capital injection problem for an insurance company, which controls the risk exposure by both the excess-of-loss reinsurance and capital injection based on the symmetry of risk information. Besides the proportional transaction cost, we also incorporate the fixed transaction cost incurred by capital injection and the salvage value of a company at the ruin time in order to make the surplus process more realistic. The main goal is to maximize the expected sum of the discounted salvage value and the discounted cumulative dividends except for the discounted cost of capital injection until the ruin time. By considering whether there is capital injection in the surplus process, we construct two instances of suboptimal models and then solve for the corresponding solution in each model. Lastly, we consider the optimal control strategy for the general model without any restriction on the capital injection or the surplus process.

scholarly journals Optimal loss-carry-forward taxation for Lévy risk processes stopped at general draw-down time

2019 ◽  
Vol 51 (03) ◽  
pp. 865-897 ◽  
Author(s):  
Wenyuan Wang ◽  
Zhimin Zhang
Keyword(s):  
Objective Function ◽  
Insurance Company ◽  
Tax System ◽  
Ruin Time ◽  
Return Function ◽  
Numerical Examples ◽  
Optimal Tax ◽  
Spectrally Negative Lévy Process ◽  
Tax Payments ◽  
Risk Processes

AbstractMotivated by Avram, Vu and Zhou (2017), Kyprianou and Zhou (2009), Li, Vu and Zhou (2017), Wang and Hu (2012), and Wang and Zhou (2018), we consider in this paper the problem of maximizing the expected accumulated discounted tax payments of an insurance company, whose reserve process (before taxes are deducted) evolves as a spectrally negative Lévy process with the usual exclusion of negative subordinator or deterministic drift. Tax payments are collected according to the very general loss-carry-forward tax system introduced in Kyprianou and Zhou (2009). To achieve a balance between taxation optimization and solvency, we consider an interesting modified objective function by considering the expected accumulated discounted tax payments of the company until the general draw-down time, instead of until the classical ruin time. The optimal tax return function and the optimal tax strategy are derived, and some numerical examples are also provided.

scholarly journals Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

2016 ◽  
Vol 53 (2) ◽  
pp. 572-584 ◽  
Author(s):  
Erik J. Baurdoux ◽  
Juan Carlos Pardo ◽  
José Luis Pérez ◽  
Jean-François Renaud
Keyword(s):  
Risk Process ◽  
Levy Processes ◽  
Insurance Company ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Scale Functions ◽  
Surplus Process ◽  
Parisian Ruin ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

Abstract Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

scholarly journals The Optimal Dividend Payout Model with Terminal Values and Its Application

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Xiankang Luo ◽  
Peimin Chen ◽  
Jiangming Ma
Keyword(s):  
Value Function ◽  
Analytic Solution ◽  
Important Application ◽  
Stochastic Dynamic ◽  
Optimal Dividend ◽  
Optimal Value ◽  
The Status ◽  
Optimal Dividend Policy ◽  
The Relationship ◽  
Theoretical Results

For some firms with large nonliquid assets, preferred shareholders can still get back a little bit of money when the firms finish disbursement of loans at the status of bankruptcy. For such a situation, to investigate the optimal dividend policy, a stochastic dynamic dividend model with nonzero terminal bankruptcy values is put forward in this paper. Moreover, an analytic solution for the optimal objective function of the discounted dividends is provided and verified. An important application of this result is that it can be employed to construct the solution for the optimal value function on the dividend problem with bailouts at bankruptcy. Further, the relationship for the solutions of these two different problems is demonstrated. In the end, some numerical examples are provided to support our theoretical results and the corresponding economic interpretations are illustrated.

scholarly journals On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

2012 ◽  
Vol 44 (3) ◽  
pp. 886-906 ◽  
Author(s):  
Jiaqin Wei ◽  
Rongming Wang ◽  
Hailiang Yang
Keyword(s):  
Diffusion Model ◽  
Regime Switching ◽  
Value Function ◽  
Risk Theory ◽  
System Of Differential Equations ◽  
Arrival Times ◽  
Switching Diffusion ◽  
Optimal Dividend ◽  
Optimal Dividend Strategy ◽  
The Value Function

In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

scholarly journals Parisian ruin of self-similar Gaussian risk processes

2015 ◽  
Vol 52 (3) ◽  
pp. 688-702 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Lanpeng Ji
Keyword(s):  
Normal Approximation ◽  
Asymptotic Relation ◽  
Exact Asymptotics ◽  
Ruin Time ◽  
Self Similar ◽  
Parisian Ruin ◽  
Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

scholarly journals Threshold strategies for risk processes and their relation to queueing theory

2011 ◽  
Vol 48 (A) ◽  
pp. 29-38 ◽  
Author(s):  
Onno J. Boxma ◽  
Andreas Löpker ◽  
David Perry
Keyword(s):  
Queueing Theory ◽  
Ruin Probability ◽  
Risk Model ◽  
Insurance Company ◽  
Threshold Strategy ◽  
Ruin Time ◽  
Risk Processes

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability, and the total dividend, using methods and results from queueing theory.

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