scholarly journals Some State-Specific Exit Probabilities in a Markov-Modulated Risk Model

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jingchao Li ◽  
Shuanming Li
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Target Level ◽  
Scale Functions ◽  
Exit Probabilities ◽  
Markov Modulated

In this paper, we study some state-specific one-sided exit probabilities in a Markov-modulated risk process including the probability that ruin occurs without or with the surplus visiting certain states; the probability that ruin occurs without or with a claim occurring in certain states; the probability that the surplus attains a target level without or with visiting certain states; and the probability that the surplus attains a target level without or with a claim occurring in certain states. We also investigate the corresponding two-sided first exit probabilities without (or with) the surplus visiting certain states or without (or with) claims occurring in certain states. All these probabilities can be expressed elegantly in terms of some modified matrix scale functions which are easily computable.

scholarly journals A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

2017 ◽  
Vol 54 (1) ◽  
pp. 267-285 ◽  
Author(s):  
Onno J. Boxma ◽  
Esther Frostig ◽  
David Perry
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Insurance Risk ◽  
Deficit At Ruin ◽  
Scale Functions ◽  
The Deficit At Ruin ◽  
The Laplace Transform ◽  
Surplus Before Ruin ◽  
Set Up ◽  
Spectrally Negative Lévy Processes

AbstractWe consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

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.

scholarly journals The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

2008 ◽  
Vol 38 (1) ◽  
pp. 53-71 ◽  
Author(s):  
Shuanming Li ◽  
Yi Lu
Keyword(s):  
Risk Model ◽  
Jump Process ◽  
Risk Process ◽  
Penalty Functions ◽  
Arrival Process ◽  
Claim Amount ◽  
Markov Jump ◽  
Expected Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

scholarly journals The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

2008 ◽  
Vol 38 (01) ◽  
pp. 53-71 ◽  
Author(s):  
Shuanming Li ◽  
Yi Lu
Keyword(s):  
Risk Model ◽  
Jump Process ◽  
Risk Process ◽  
Penalty Functions ◽  
Arrival Process ◽  
Claim Amount ◽  
Markov Jump ◽  
Expected Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

Expansions for Markov-modulated systems and approximations of ruin probability

1996 ◽  
Vol 33 (01) ◽  
pp. 57-70
Author(s):  
Bartłomiej Błaszczyszyn ◽  
Tomasz Rolski
Keyword(s):  
Ruin Probability ◽  
Marked Point ◽  
Factorial Moment ◽  
Risk Process ◽  
Rate Function ◽  
Marked Point Process ◽  
Probability Of Ruin ◽  
Actual Risk ◽  
Moment Measures ◽  
Markov Modulated

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β ∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a 0 + β ∗ a 1 + ·· ·+ (β∗ ) nan + o((β ∗) n ) for β ∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β ∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

scholarly journals Ruin problems in Markov-modulated risk models

2017 ◽  
Vol 12 (1) ◽  
pp. 23-48 ◽  
Author(s):  
David C.M. Dickson ◽  
Marjan Qazvini
Keyword(s):  
Continuous Time ◽  
Risk Model ◽  
Numerical Algorithms ◽  
Binomial Model ◽  
Risk Models ◽  
Time Of Ruin ◽  
Compound Binomial ◽  
Compound Binomial Model ◽  
Markov Modulated ◽  
The Time Of Ruin

AbstractChen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

scholarly journals On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

1984 ◽  
Vol 14 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Jean-Marie Reinhard
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Risk Models ◽  
Markovian Environment ◽  
Necessary And Sufficient ◽  
Natural Way ◽  
Quantities Of Interest

AbstractWe consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Constant barrier strategies in a two-state Markov-modulated dual risk model

2011 ◽  
Vol 27 (4) ◽  
pp. 679-690 ◽  
Author(s):  
Xue-min Ma ◽  
Kui Luo ◽  
Guang-ming Wang ◽  
Yi-jun Hu
Keyword(s):  
Risk Model ◽  
Dual Risk Model ◽  
Markov Modulated
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