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Uniform Markov Renewal Theory and Ruin Probabilities in Markov Random Walks  [PDF]
Cheng-Der Fuh
Mathematics , 2004, DOI: 10.1214/105051604000000260
Abstract: Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time \tau=\tau(b)=inf{n:S_n>b}. When a drift \mu of the random walk S_n is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities P_{\pi}{\tau
Asymptotic Estimate for the Probability of an Exceedance over Negatively Associated Renewal Thresholds and the Ruin Probability in Dividend Barrier Models

数学物理学报(A辑) , 2009,
Abstract: In this paper, we obtain an asymptotic estimate for the probability of an exceedance over negatively associated renewal thresholds, which extends the corresponding result of Robert (2005)12]. Furthermore, using a new method, we also derive a more rigorous proof of the asymptotics for the ruin probability in dividend barrier models.
The Distribution of the Time of Ruin, the Surplus Immediately before Ruin and Deficit at Ruin under Two Sided Risk Renewal Process  [PDF]
Joseph Justin Rebello, K. K. Thampi
Journal of Mathematical Finance (JMF) , 2017, DOI: 10.4236/jmf.2017.73032
Abstract: In the renewal risk theory, the study of two sided jumps has been attracted by many researchers since its introduction. After the development of the distribution of modified inter time claim occurrence, the explicit expressions for ruin theory components in the literature under some assumptions, in this work, we examine probability density of the time of ruin, surplus immediately before ruin and deficit at ruin respectively under two sided risk process using some basic assumptions. Explicit expressions for distribution of interest are being derived.
On the ruin problem in the renewal risk processes perturbed by diffusion  [PDF]
Min Song
Mathematics , 2008,
Abstract: In this paper, we consider the perturbed renewal risk process. Systems of integro-differential equations for the Gerber-Shiu functions at ruin caused by a claim and oscillation are established, respectively. The explicit Laplase transforms of Gerber-Shiu functions are obtained, while the closed form expressions for the Gerber-Shiu functions are derived when the claim amount distribution is from the rational family. Finally, we present numerical examples intended to illustrate the main results.
Ruin Probability in Compound Poisson Process with Investment
Yong Wu,Xiang Hu
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/286792
Abstract: We consider that the surplus of an insurer follows compound Poisson process and the insurer would invest its surplus in risky assets, whose prices satisfy the Black-Scholes model. In the risk process, we decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively. We derive the integro-differential equations for these ruin probabilities these ruin probabilities. When the claim sizes are exponentially distributed, third-order differential equations of the ruin probabilities are derived from the integro-differential equations and a lower bound is obtained.
The probability of exceeding a piecewise deterministic barrier by the heavy-tailed renewal compound process  [PDF]
Zbigniew Palmowski,Martijn Pistorius
Mathematics , 2008,
Abstract: We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity.
Ruin Probability in a Risk Model under Unexpected Random Heavy-tailed Claims

BI Xiu-Chun,ZHANG Shu-Guang,

数学物理学报(A辑) , 2012,
Abstract: This paper sets up a generalized renewal risk model based on zhe reality of existing unexpected heavy-tailed claims and gives tail equivalence relationship for the ruin probability, which indicates that unexpected heavy-tailed claims can result in ruin.
Comparison of Ruin Probabilities in Compound Poisson Risk Model  [PDF]
Dol Nath Khanal
Open Journal of Statistics (OJS) , 2019, DOI: 10.4236/ojs.2019.91004
Abstract: Compound Poisson risk model has been simulated. It has started with exponential claim sizes. The simulations have checked for infinite ruin probabilities. An appropriate time window has been chosen to estimate and compare ruin probabilities. The infinite ruin probabilities of two-compound Poisson risk process have estimated and compared them with standard theoretical results.
From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes  [PDF]
Boris Tsirelson
Mathematics , 2012,
Abstract: A uniform key renewal theorem is deduced from the uniform Blackwell's renewal theorem. A uniform LDP (large deviations principle) for renewal-reward processes is obtained, and MDP (moderate deviations principle) is deduced under conditions much weaker than existence of exponential moments.
Ruin Probabilities of Double Compound Poisson Risk Model under Proportional Reinsurance and Interest Force
Jia Niannian,Xia Yunfan
Journal of Modern Mathematics and Statistics , 2012, DOI: 10.3923/jmmstat.2012.10.13
Abstract: The researchers introduced interest force and reduced the risk of the insurance company with the proportional reinsurance under double compound Poisson risk model. Differential-Integral equations of ruin probabilities in finite and infinite time were provided. These conclusions have theoretical significance for the insurance company measuring ruin risk.
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