On a Risk Model with Surplus-dependent Premium and Tax Rates

In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007). In the compound Poisson risk model, Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007) showed that a simple relationship between the ruin probabilities in the risk model with and without tax exists. This so-called tax identity was later generalized to a surplus-dependent tax rate by Albrecher et al. (Insur Math Econ 44(2):304–306, 2009). The present paper further generalizes these results to the Gerber–Shiu function with a generalized penalty function involving the maximum surplus prior to ruin. We show that this generalized Gerber–Shiu function in the risk model with tax is closely related to the ‘original’ Gerber–Shiu function in the risk model without tax defined in a dividend barrier framework. The moments of the discounted tax payments before ruin and the optimal threshold level for the tax authority to start collecting tax payments are also examined.     

The Tax Identity For Markov Additive Risk Processes

Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Lévy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally-negative MAPs. We use the Sparre Andersen risk processes with phase-type interarrivals to illustrate the ideas in their simplest form.     

On the Markov-modulated insurance risk model with tax

In this paper, we consider the Markov-modulated insurance risk model with tax. We assume that the claim inter-arrivals, claim sizes and premium process are influenced by an external Markovian environment process. The considered tax rule, which is the same as the one considered by Albrecher and Hipp [Blätter DGVFM 28(1):13–28, 2007], is to pay a certain proportion of the premium income, whenever the insurer is in a profitable situation. A system of differential equations of the non-ruin probabilities, given the initial environment state, are established in terms of the ruin probabilities under the Markov-modulated insurance risk model without tax. Furthermore, given the initial state, the differential equations satisfied by the expected accumulated discounted tax until ruin are also derived. We also give the analytical expressions for them by iteration methods.     

Omega diffusion risk model with surplus-dependent tax and capital injections

In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). The study of this new risk model is closely related to the Azéma–Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma–Yor process below a constant level until the first passage time of another Azéma–Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of Li and Zhou (2013) and Zhang (2015). We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma–Yor process.     

Analysis of a drawdown-based regime-switching Lévy insurance model

In this paper, we propose a new drawdown-based regime-switching (DBRS) Lévy insurance model in which the underlying drawdown process is used to model an insurer’s level of financial distress over time, and to trigger regime-switching transitions. By some analytical arguments, we derive explicit formulas for a generalized two-sided exit problem. We specifically state conditions under which the survival probability is not trivially zero (which corresponds to the positive security loading conditions of the proposed model). The regime-dependent occupation time until ruin is later studied. As a special case of the general DBRS model, a regime-switching premium model is given further consideration. Connections with other existing risk models (such as the loss-carry-forward tax model of Albrecher and Hipp, 2007) are established.     

Ruin probability in the presence of interest earnings and tax payments

In this paper we investigate the ruin probability in a general risk model driven by a compound Poisson process. We derive a formula for the ruin probability from which the Albrecher–Hipp tax identity follows as a corollary. Then we study, as an important special case, the classical risk model with a constant force of interest and loss-carried-forward tax payments. For this case we derive an exact formula for the ruin probability when the claims are exponential and an explicit asymptotic formula when the claims are subexponential.     

On taxed spectrally negative Lévy processes with draw-down stopping

In this paper we consider a spectrally negative Lévy risk model with tax. With the ruin time replaced by a draw-down time with a linear draw-down function and for a constant tax rate, we find expressions for the present values of tax payments. They generalize previous results in Albrecher et al. (2008). Alternative proofs are given for the special case of Cramér–Lundberg risk models. Optimal barrier taxation policies are discussed.     

The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure

We study the distribution of tax payments in the model of Kyprianou and Zhou [Kyprianou, A.E., Zhou, X., 2009. General tax structures and the Lévy insurance risk model. J. Appl. Probab. (in press)], that is a Lévy insurance risk model with a surplus-dependent tax rate. More precisely, after a short discussion on the so-called tax identity, we derive a recursive formula for arbitrary moments of the discounted tax payments until ruin and we identify the distribution of the tax payments when there is no force of interest.     

On the dual risk model with tax payments

In this paper, we study the dual risk process in ruin theory (see e.g. Cramér, H. 1955. Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes. Ab Nordiska Bokhandeln, Stockholm, Takacs, L. 1967. Combinatorial methods in the Theory of Stochastic Processes. Wiley, New York and Avanzi, B., Gerber, H.U., Shiu, E.S.W., 2007. Optimal dividends in the dual model. Insurance: Math. Econom. 41, 111–123) in the presence of tax payments according to a loss-carry forward system. For arbitrary inter-innovation time distributions and exponentially distributed innovation sizes, an expression for the ruin probability with tax is obtained in terms of the ruin probability without taxation. Furthermore, expressions for the Laplace transform of the time to ruin and arbitrary moments of discounted tax payments in terms of passage times of the risk process are determined. Under the assumption that the inter-innovation times are (mixtures of) exponentials, explicit expressions are obtained. Finally, we determine the critical surplus level at which it is optimal for the tax authority to start collecting tax payments.     

带干扰广义Elang(n)风险过程的破产前最大盈余

本文考虑了索赔时间间距为广义Erlang(n)分布的带干扰更新(Sparre Andersen)风险过程.所用的方法类似于Albrecher,et al.(2005),即将广义Erlang(n)随机变量分解成n个独立的指数随机变量的和.建立了破产前最大盈余所满足的积分-微分方程,讨论了索赔量分布为K<,m>分布时的特殊情形.     

The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang(n) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.     

常利率风险模型下T-A目标函数的最大化

本文研究了带常利率扩散风险模型,考虑了下面的目标函数V(x,L)=E(integral from n=0 toτe~(-βt)dL_t+integral from n=0 toτe~(-βt)∧dt|R_0~L=x),这里常数∧≥0.我们称上面的表达式为T-A(Thonhauser and Albrecher)目标函数.对于常利率下的扩散模型,通过随机控制理论(HJB方程),T-A目标函数的最大化问题得以解决:对于有界分红率,最优策略是门槛策略;对于无界分红率,最优策略是边界策略.     

保费收入服从泊松过程的一类相依更新风险模型研究

本文研究了保费收入过程是泊松过程和聚合理赔过程中理赔间隔时间和个别理赔额之间具有Boudreault et al.(2006)中所描述的相依结构的一类更新风险模型.运用生成函数、离散形式的Dickson-Hipp算子和反Z变换等一系列方法,推导出了该模型的Gerber-Shiu函数的生成函数的精确表达式,以及它所满足的瑕疵更新方程.     

The complete parsimony haplotype inference problem and algorithms based on integer programming,branch-and-bound and Boolean satisfiability

Haplotype inference by pure parsimony (Hipp) is a well-known paradigm for haplotype inference. In order to assess the biological significance of this paradigm, we generalize the problem of Hipp to the problem of finding all optimal solutions, which we call Chipp. We study intrinsic haplotype features, such as backbone haplotypes and fat genotypes as well as equal columns and decomposability. We explicitly exploit these features in three computational approaches that are based on integer linear programming, depth-first branch-and-bound, and Boolean satisfiability. Further we introduce two hybrid algorithms that draw upon the diverse strengths of the approaches. Our experimental analysis shows that our optimized algorithms are significantly superior to the baseline algorithms, often with orders of magnitude faster running time. Finally, our experiments provide some useful insights into the intrinsic features of this important problem.     

Archimedean copulas in finite and infinite dimensions—with application to ruin problems

In this paper we discuss the link between Archimedean copulas and L₁ Dirichlet distributions for both finite and infinite dimensions. With motivation from the recent papers Weng et al. (2009) and Albrecher et al. (2011) we apply our results to certain ruin problems.     

Switching tax structure and payouts in endogenous bankruptcy models

In the spirit of Leland (H.E. Leland, Corporate Debt Value, Bond Covenant, and Optimal Capital Structure, J. Finance 49 (1994), pp. 1213–1252), we consider a structural credit risk model with tax provisions under the assumption of a positive payout rate. By defining a more general tax structure than in (Leland, 1994), we introduce a general switching corporate tax rate function and analytically derive the value of the tax benefits claim, the whole capital structure and the smooth pasting condition. In this set-up, the endogenous failure level is derived and both the singular and joint effect of the two introduced risk factors (payouts and tax asymmetry) on optimal managerial financing decisions are studied. Results show a quantitatively significant impact on optimal debt issuance and leverage ratios, bringing them to values more in line with historical norms and providing a way to explain differences in observed leverage across firms.     

Convergence rates of the strong law for stationary mixing sequences

Summary In this note we estimate the rate of convergence in Marcinkiewicz-Zygmung strong law, for partial sumsS _n of strong stationary mixing sequences of random variables. The results improve the corresponding ones obtained by Tze Leung Lai (1977) and Christian Hipp (1979).     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

基于跳扩散过程的Omega模型

文章通过在Omega模型中加入布朗运动扰动项,提出了一种跳扩散Omega破产模型.在索赔额为指数分布的情形下,给出了破产率函数是常数时的破产概率函数表达式.文章进一步研究了破产概率和盈余过程的“负占有时”之间的关系,并给出了破产概率函数的第二种推导过程.最后通过两个数值试验,将我们的模型与Albreeher和Lautscham (2013)的Omega模型的破产概率进行了比较分析.     

On the Markov-dependent risk model with tax

In this paper we consider the Markov-dependent risk model with tax payments in which the claim occurrence, the claim amount as well as the tax rate are controlled by an irreducible discrete-time Markov chainSystems of integro-differential equations satisfied by the expected discounted tax payments and the non-ruin probability in terms of the ruin probabilities under the Markov-dependent risk model without tax are establishedThe analytical solutions of the systems of integro-differential equations are also obtained by the iteration method.