scholarly journals Bounds for the Ruin Probability of a Discrete-Time Risk Process

2009 ◽  
Vol 46 (01) ◽  
pp. 99-112 ◽  
Author(s):  
Maikol A. Diasparra ◽  
Rosario Romera
Keyword(s):  
Markov Chain ◽  
Interest Rate ◽  
Discrete Time ◽  
Ruin Probability ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Proportional Reinsurance ◽  
Rate Process ◽  
Stationary Policy ◽  
The Interest Rate

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

scholarly journals Bounds for the Ruin Probability of a Discrete-Time Risk Process

2009 ◽  
Vol 46 (1) ◽  
pp. 99-112 ◽  
Author(s):  
Maikol A. Diasparra ◽  
Rosario Romera
Keyword(s):  
Markov Chain ◽  
Interest Rate ◽  
Discrete Time ◽  
Ruin Probability ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Proportional Reinsurance ◽  
Rate Process ◽  
Stationary Policy ◽  
The Interest Rate

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

Ruin Probability in a Generalised Risk Process under Rates of Interest with Homogenous Markov Chains

2014 ◽  
Vol 4 (3) ◽  
pp. 283-300
Author(s):  
Phung Duy Quang
Keyword(s):  
Markov Chain ◽  
Ruin Probability ◽  
Risk Process ◽  
Upper Bounds ◽  
Ruin Probabilities ◽  
Countable Number ◽  
Probability Inequalities ◽  
Homogenous Markov Chain ◽  
Recursive Equations ◽  
Risk Processes

AbstractThis article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.

scholarly journals RUIN PROBABILITIES UNDER AN OPTIMAL INVESTMENT AND PROPORTIONAL REINSURANCE POLICY IN A JUMP DIFFUSION RISK PROCESS

2009 ◽  
Vol 51 (1) ◽  
pp. 34-48 ◽  
Author(s):  
YIPING QIAN ◽  
XIANG LIN
Keyword(s):  
Ruin Probability ◽  
Market Price ◽  
Optimal Investment ◽  
Jump Diffusion ◽  
Risk Process ◽  
Closed Form Expression ◽  
Proportional Reinsurance ◽  
Insurance Company ◽  
Risky Assets ◽  
Price Of Risk

AbstractIn this paper, we consider an insurance company whose surplus (reserve) is modeled by a jump diffusion risk process. The insurance company can invest part of its surplus in n risky assets and purchase proportional reinsurance for claims. Our main goal is to find an optimal investment and proportional reinsurance policy which minimizes the ruin probability. We apply stochastic control theory to solve this problem. We obtain the closed form expression for the minimal ruin probability, optimal investment and proportional reinsurance policy. We find that the minimal ruin probability satisfies the Lundberg equality. We also investigate the effects of the diffusion volatility parameter, the market price of risk and the correlation coefficient on the minimal ruin probability, optimal investment and proportional reinsurance policy through numerical calculations.

scholarly journals Ruin Probabilities of a Discrete-time Multi-risk Model

2015 ◽  
Vol 44 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Andrius Grigutis ◽  
Agneška Korvel ◽  
Jonas Šiaulys
Keyword(s):  
Discrete Time ◽  
Ruin Probability ◽  
Arrival Time ◽  
Risk Model ◽  
Recursive Formula ◽  
Ruin Probabilities ◽  
Numerical Examples ◽  
Premium Rate ◽  
Insurance Business ◽  
Independent Series

In this work,  we investigate a  multi-risk model describing insurance business with  two or more independent series of claim amounts. Each series of claim amounts consists of independent nonnegative random variables. Claims of each series occur periodically with some fixed   inter-arrival time. Claim amounts occur until they   can be compensated by a common premium rate and the initial insurer's surplus.  In this article, wederive a recursive formula for calculation of finite-time ruin probabilities. In the case of bi-risk model, we present a procedure to calculate the ultimate ruin probability. We add several numerical examples illustrating application  of the derived formulas.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8635

scholarly journals Ruin Probability in Compound Poisson Process with Investment

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Yong Wu ◽  
Xiang Hu
Keyword(s):  
Differential Equations ◽  
Poisson Process ◽  
Ruin Probability ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Risky Assets ◽  
Black Scholes Model ◽  
Black Scholes

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.

scholarly journals The Present Value of a Series of Cashflows: Convergence in a Random Environment

1995 ◽  
Vol 25 (2) ◽  
pp. 81-94 ◽  
Author(s):  
Andrew J. G. Cairns
Keyword(s):  
Interest Rate ◽  
Random Environment ◽  
General Class ◽  
Almost Sure Convergence ◽  
Sufficient Condition ◽  
Rate Process ◽  
Present Value ◽  
The Interest Rate

AbstractThe present paper considers the present value, Z(t), of a series of cashflows up to some time t. More specifically, the cashflows and the interest rate process will often be stochastic and not necessarily independent of one another or through time. We discuss under what circumstances Z(t) will converge almost surely to some finite value as t→∞. This problem has previously been considered by Dufresne (1990) who provided a sufficient condition for almost sure convergence of Z(t) (the Root Test) and then proceeded to consider some specific examples of such processes. Here, we develop Dufresne's work and show that the sufficient condition for convergence can be proved to hold for quite a general class of model which includes the growing number of Office Models with stochastic cashflows.

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