scholarly journals Ruin Probability in a Generalized Risk Process under Interest Force with Homogenous Markov Chain Premiums

2013 ◽  
Vol 2 (4) ◽  
Author(s):  
Phung Duy Quang
Keyword(s):  
Markov Chain ◽  
Ruin Probability ◽  
Risk Process ◽  
Interest Force ◽  
Homogenous Markov Chain

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 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.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (03) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (3) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

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.

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