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.

Ruin probabilities for competing claim processes

2004 ◽  
Vol 41 (03) ◽  
pp. 679-690 ◽  
Author(s):  
Miljenko Huzak ◽  
Mihael Perman ◽  
Hrvoje Šikić ◽  
Zoran Vondraček
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Ruin Probabilities ◽  
Classical Risk Process ◽  
Risk Processes

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Ruin probabilities via local adjustment coefficients

1995 ◽  
Vol 32 (03) ◽  
pp. 736-755 ◽  
Author(s):  
Søren Asmussen ◽  
Hanne Mandrup Nielsen
Keyword(s):  
Ruin Probability ◽  
Arrival Rate ◽  
Zero Solution ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Poisson Arrival ◽  
Claim Size Distribution ◽  
Exponential Change ◽  
Risk Processes ◽  
Standard Risk

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.

Ruin probabilities via local adjustment coefficients

1995 ◽  
Vol 32 (3) ◽  
pp. 736-755 ◽  
Author(s):  
Søren Asmussen ◽  
Hanne Mandrup Nielsen
Keyword(s):  
Ruin Probability ◽  
Arrival Rate ◽  
Zero Solution ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Poisson Arrival ◽  
Claim Size Distribution ◽  
Exponential Change ◽  
Risk Processes ◽  
Standard Risk

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.

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 probabilities for competing claim processes

2004 ◽  
Vol 41 (3) ◽  
pp. 679-690 ◽  
Author(s):  
Miljenko Huzak ◽  
Mihael Perman ◽  
Hrvoje Šikić ◽  
Zoran Vondraček
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Ruin Probabilities ◽  
Classical Risk Process ◽  
Risk Processes

LetC1,C2,…,Cmbe independent subordinators with finite expectations and denote their sum byC. Consider the classical risk processX(t) =x+ct-C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinatorsCi. Formulae for the probability that ruin is caused byCiare derived. These formulae can be extended to perturbed risk processes of the typeX(t) =x+ct-C(t) +Z(t), whereZis a Lévy process with mean 0 and no positive jumps.

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