scholarly journals A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Aili Zhang ◽  
Zhang Liu
Keyword(s):  
Risk Model ◽  
Poisson Model ◽  
Risk Process ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ruin Time ◽  
Scale Functions ◽  
Surplus Process ◽  
Dividend Payments ◽  
Special Cases

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

2010 ◽  
Vol 108-111 ◽  
pp. 1103-1108
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Process ◽  
Closed Form Expression ◽  
Standard Wiener Process ◽  
Form Expression ◽  
Expected Discounted Penalty Function ◽  
Premium Rate ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

scholarly journals ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS

2014 ◽  
Vol 44 (3) ◽  
pp. 635-651 ◽  
Author(s):  
Chuancun Yin ◽  
Yuzhen Wen ◽  
Yongxia Zhao
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Levy Process ◽  
Dual Model ◽  
Threshold Strategy ◽  
Classical Risk Model ◽  
Surplus Process ◽  
Optimal Dividend ◽  
Special Cases ◽  
The Classical Risk Model

AbstractIn this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Lévy process before dividends are deducted. This model includes the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to an admissible strategy whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that the optimal dividend strategy is formed by a threshold strategy.

scholarly journals Some Optimal Dividends Problems

2004 ◽  
Vol 34 (1) ◽  
pp. 49-74 ◽  
Author(s):  
David C.M. Dickson ◽  
Howard R. Waters
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Present Value ◽  
Classical Risk Model ◽  
Optimal Dividends ◽  
Surplus Process ◽  
Dividend Payments ◽  
Constant Dividend Barrier ◽  
De Finetti ◽  
The Classical Risk Model

We consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin.

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 RUIN PROBABILITY UNDER COMPOUND POISSON MODELS WITH RANDOM DISCOUNT FACTOR

2004 ◽  
Vol 18 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Kai W. Ng ◽  
Hailiang Yang ◽  
Lihong Zhang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Poisson Model ◽  
Convergence Result ◽  
Discount Factor ◽  
Insurance Risk ◽  
Compound Poisson ◽  
Analysis Techniques ◽  
Surplus Process ◽  
Poisson Models

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

scholarly journals On Risk Model with Dividends Payments Perturbed by a Brownian Motion – An Algorithmic Approach

2008 ◽  
Vol 38 (1) ◽  
pp. 183-206 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Brownian Motion ◽  
Risk Model ◽  
Risk Process ◽  
Insurance Company ◽  
Algorithmic Approach ◽  
Compound Poisson ◽  
Expected Time ◽  
Premium Rate ◽  
Surplus Process ◽  
Phase Type

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given level b as dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the level b. The dividends are paid until the surplus down-crosses the level a, a < b . We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

Shear compliance of two-dimensional pores possessing N -fold axis of rotational symmetry

2008 ◽  
Vol 464 (2091) ◽  
pp. 759-775 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Rotational Symmetry ◽  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Special Cases ◽  
Fourfold Symmetry ◽  
Shear Compliance

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N -fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 θ . An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

scholarly journals Solution of Hamilton-Jacobi-Bellman Equation in Optimal Reinsurance Strategy under Dynamic VaR Constraint

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Bellman Equation ◽  
Closed Form Expression ◽  
Proportional Reinsurance ◽  
Form Expression ◽  
Optimal Reinsurance ◽  
Surplus Process ◽  
Hamilton Jacobi Bellman Equation ◽  
Generalized Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

This paper analyzes the optimal reinsurance strategy for insurers with a generalized mean-variance premium principle. The surplus process of the insurer is described by the diffusion model which is an approximation of the classical Cramér-Lunderberg model. We assume the dynamic VaR constraints for proportional reinsurance. We obtain the closed form expression of the optimal reinsurance strategy and corresponding survival probability under proportional reinsurance.

PORTFOLIO SELECTION BY MINIMIZING THE PRESENT VALUE OF CAPITAL INJECTION COSTS

2014 ◽  
Vol 45 (1) ◽  
pp. 207-238 ◽  
Author(s):  
Ming Zhou ◽  
Kam C. Yuen
Keyword(s):  
Portfolio Selection ◽  
Risk Model ◽  
Numerical Study ◽  
Optimal Investment ◽  
Investment Policy ◽  
Model Parameters ◽  
Capital Injection ◽  
Surplus Process ◽  
Insurance Portfolio ◽  
Special Cases

AbstractThis paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

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