An optimization of a continuous time risk process

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

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Continuous-time monotone stochastic recursions and duality

Advances in Applied Probability ◽

10.1239/aap/1013540172 ◽

2000 ◽

Vol 32 (2) ◽

pp. 426-445 ◽

Cited By ~ 14

Author(s):

Karl Sigman ◽

Reade Ryan

Keyword(s):

Continuous Time ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Risk Process ◽

Stationary Increments ◽

The One ◽

And Diffusion ◽

Steady State Probabilities ◽

Two Barriers

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

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RECURSIVE CALCULATION OF RUIN PROBABILITIES AT OR BEFORE CLAIM INSTANTS FOR NON-IDENTICALLY DISTRIBUTED CLAIMS

Astin Bulletin ◽

10.1017/asb.2014.30 ◽

2014 ◽

Vol 45 (2) ◽

pp. 421-443 ◽

Cited By ~ 13

Author(s):

Anisoara Maria Raducan ◽

Raluca Vernic ◽

Gheorghita Zbaganu

Keyword(s):

Continuous Time ◽

Ruin Probability ◽

Risk Model ◽

Recursive Algorithm ◽

Risk Process ◽

Ruin Probabilities ◽

Numerical Examples ◽

Claim Number ◽

Recursive Calculation ◽

The Impact

AbstractIn this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

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Numerical Evaluation of Continuous Time Ruin Probabilities for a Portfolio with Credibility Updated Premiums

Astin Bulletin ◽

10.2143/ast.40.1.2049236 ◽

2010 ◽

Vol 40 (1) ◽

pp. 399-414 ◽

Cited By ~ 5

Author(s):

Lourdes B. Afonso ◽

Alfredo D. Egídio dos Reis ◽

Howard R. Waters

Keyword(s):

Continuous Time ◽

Finite Time ◽

Numerical Evaluation ◽

Risk Process ◽

Ruin Probabilities ◽

Probability Of Ruin ◽

Experience Rating ◽

Classical Risk Process ◽

Major Consideration

AbstractThe probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.

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Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums

Astin Bulletin ◽

10.2143/ast.39.1.2038059 ◽

2009 ◽

Vol 39 (1) ◽

pp. 117-136 ◽

Cited By ~ 6

Author(s):

Lourdes B. Afonso ◽

Alfredo D. Egídio dos Reis ◽

Howard R. Waters

Keyword(s):

Gamma Distribution ◽

Continuous Time ◽

Finite Time ◽

Ruin Probability ◽

Numerical Evaluation ◽

Risk Process ◽

Ruin Probabilities ◽

Aggregate Claims ◽

Classical Risk Process ◽

Major Consideration

AbstractIn this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

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Ruin probability for discrete risk processes

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.4.1441 ◽

2019 ◽

Vol 56 (4) ◽

pp. 420-439

Author(s):

Ivana Geček Tuden

Keyword(s):

Random Walk ◽

Discrete Time ◽

Continuous Time ◽

Ruin Probability ◽

Risk Process ◽

Dual Process ◽

Type Formula ◽

Fixed Level ◽

Risk Processes

Abstract We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

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On a bivariate risk process with a dividend barrier strategy

Annals of Actuarial Science ◽

10.1017/s1748499514000165 ◽

2014 ◽

Vol 9 (1) ◽

pp. 3-35 ◽

Cited By ~ 1

Author(s):

Luyin Liu ◽

Eric C. K. Cheung

Keyword(s):

Continuous Time ◽

Risk Process ◽

Capital Allocation ◽

Proportional Reinsurance ◽

Barrier Strategy ◽

Ruin Time ◽

Optimal Pair ◽

Numerical Examples ◽

Common Shocks ◽

Individual Line

AbstractIn this paper, we study a continuous-time bivariate risk process in which each individual line of business implements a dividend barrier strategy. The insurance portfolios of the two insurers are correlated as they are subject to common shocks that induce dependent claims. To analyse the expected discounted dividends until the joint ruin time of the bivariate process (i.e. exit from the positive quadrant), we propose a discrete-time counterpart of the model and apply a bivariate extension of the Dickson−Waters discretisation with the use of a bivariate Panjer-type recursion. Detailed numerical examples under different dependencies via common shocks, copulas and proportional reinsurance are discussed, and applications to optimal problems in reinsurance, capital allocation and dividends are given. It is also illustrated that the optimal pair of dividend barriers maximising the dividend function is dependent on the initial surplus levels. A modified type of dividend barrier strategy is proposed towards the end.

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Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

Journal of Applied Probability ◽

10.1017/s0021900200002965 ◽

2007 ◽

Vol 44 (02) ◽

pp. 285-294 ◽

Cited By ~ 2

Author(s):

Qihe Tang

Keyword(s):

Asymptotic Formula ◽

Continuous Time ◽

Heavy Tails ◽

Tail Behavior ◽

Renewal Model ◽

Discounted Aggregate Claims ◽

Aggregate Claims ◽

Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

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