Calculation of finite time ruin probabilities for some risk models

Cardoso, Rui M. R.; Waters, Howard R.

doi:10.1016/j.insmatheco.2004.11.005

Cited by 6 publications

(9 citation statements)

References 14 publications

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“…However, we can choose a suitable function for

{\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j})

; when x j ≥ b we have

H_{h} (x MathClass-punc; x_{j}) MathClass-rel= F_{h} ((), x MathClass-bin+ {MathClass-op\int}_{τ}^{τ MathClass-bin+ h} I_{II} (U_{II} (s)) ds MathClass-bin- (c MathClass-bin- ĉ) h) \leq F_{h} (x MathClass-bin+ ĉh) \leq {\overset{F}{falsemml-overline}}_{d} (x MathClass-bin+ ĉh) MathClass-punc.

Hence, defining

{\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\overset{F}{falsemml-overline}}_{d} (x MathClass-bin+ ĉh) MathClass-punc, MathClass-rel\forall x_{j}

, we obtain a proper upper bound for V ( u , b , t ) and also a proper lower bound for the finite time ruin probability. This development is an improvement to the results from Cardoso and Waters .…”

Section: Markov Chains and Algorithms For Model IImentioning

confidence: 68%

“…Cardoso and Waters , in their Section 4.1, consider that the lower and upper bounds for H h ( x ; x j ) are, respectively,

{\underset{H}{falsemml-underline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\underset{F}{falsemml-underline}}_{d} (x) and {\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\overset{F}{falsemml-overline}}_{d} (x) MathClass-punc.

…”

Section: Markov Chains and Algorithms For Model IImentioning

confidence: 99%

“…To derive our numerical algorithms, we follow the methodology used by Cardoso and Waters , Section 2 and by Cardoso and Waters , Section 4.1. We construct discrete time Markov chains, denoted

{Z_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

{{\underset{Z}{falsemml-underline}}_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

and

{{\overset{Z}{falsemml-overline}}_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

, which approximate the surplus process or bound it, below and above, up to the time of ruin.…”

Section: Introductionmentioning

confidence: 99%

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Dividends in finite time horizon

Cardoso

2012

Appl Stoch Models Bus & Ind

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In this paper, we consider the classical risk model modified in two different ways by the inclusion of a dividend barrier. For Model I, we present numerical algorithms, which can be used to approximate or bound the expected discounted value of dividends up to a finite time horizon, t, or ruin if this occurs earlier. We extend this by requiring the shareholders to provide the initial capital and to pay the deficit at ruin each time it occurs so that the process then continues after ruin up to time t. For Model I, we assume the full premium income is paid as dividends whenever the surplus exceeds a set level.In our Model II, we assume dividends are paid at a rate less than the rate of premium income. Copyright © 2012 John Wiley & Sons, Ltd.

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“…However, we can choose a suitable function for

{\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j})

; when x j ≥ b we have

H_{h} (x MathClass-punc; x_{j}) MathClass-rel= F_{h} ((), x MathClass-bin+ {MathClass-op\int}_{τ}^{τ MathClass-bin+ h} I_{II} (U_{II} (s)) ds MathClass-bin- (c MathClass-bin- ĉ) h) \leq F_{h} (x MathClass-bin+ ĉh) \leq {\overset{F}{falsemml-overline}}_{d} (x MathClass-bin+ ĉh) MathClass-punc.

Hence, defining

{\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\overset{F}{falsemml-overline}}_{d} (x MathClass-bin+ ĉh) MathClass-punc, MathClass-rel\forall x_{j}

Section: Markov Chains and Algorithms For Model IImentioning

confidence: 68%

“…Cardoso and Waters , in their Section 4.1, consider that the lower and upper bounds for H h ( x ; x j ) are, respectively,

{\underset{H}{falsemml-underline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\underset{F}{falsemml-underline}}_{d} (x) and {\overset{H}{falsemml-overline}}_{h} (x MathClass-punc; x_{j}) MathClass-rel= {\overset{F}{falsemml-overline}}_{d} (x) MathClass-punc.

…”

Section: Markov Chains and Algorithms For Model IImentioning

confidence: 99%

“…To derive our numerical algorithms, we follow the methodology used by Cardoso and Waters , Section 2 and by Cardoso and Waters , Section 4.1. We construct discrete time Markov chains, denoted

{Z_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

{{\underset{Z}{falsemml-underline}}_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

and

{{\overset{Z}{falsemml-overline}}_{n}}_{n MathClass-rel= 0}^{MathClass-rel\infty}

, which approximate the surplus process or bound it, below and above, up to the time of ruin.…”

Section: Introductionmentioning

confidence: 99%

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Dividends in finite time horizon

Cardoso

2012

Appl Stoch Models Bus & Ind

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“…Dickson (1991) considered the case where the premium rate changes when the surplus crosses an upper barrier. More recently, Cardoso and Waters (2005) presented a numerical method for calculating finite time ruin probabilities for the same problem. In a recent paper Malinovskii (2008) provides an analytical solution to a problem similar to ours, but only in the case where claim amounts are exponentially distributed.…”

Section: Introductionmentioning

confidence: 99%

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

Afonso¹,

Reis²,

Waters³

2009

ASTIN Bull.

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In 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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“…A risk map of natural disaster is a regionalization map that identifies the places that might be adversely affected in the event of natural disasters. When we employ probabilities to represent risks [2], the map is called a probability-risk map. For generating this kind of map, it is necessary to estimate the probability distribution p(x) about an index x measuring adverse event, such as earthquake and flood.…”

Section: Introductionmentioning

confidence: 99%