Exact and Asymptotic Solutions for the Time-Dependent Problem of Collective Ruin I

Knessl, Charles; Peters, Craig S.

doi:10.1137/s0036139993250579

Cited by 9 publications

(5 citation statements)

References 5 publications

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“…λ/(µp). More generally, it was shown by Asmussen (1984), Knessl and Peters (1994) (with p = 1) and Pervozvansky (1998) that the finite time ruin probability ψ(x, t) is given by…”

Section: 2mentioning

confidence: 99%

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Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Avram¹,

Palmowski²,

Pistorius³

2008

Ann. Appl. Probab.

125

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Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution.

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“…λ/(µp). More generally, it was shown by Asmussen (1984), Knessl and Peters (1994) (with p = 1) and Pervozvansky (1998) that the finite time ruin probability ψ(x, t) is given by…”

Section: 2mentioning

confidence: 99%

“…λ/(µp). More generally, it was shown by Asmussen (1984), Knessl and Peters (1994) (with p = 1) and Pervozvansky (1998) that the finite time ruin probability ψ(x, t) is given by ψ(x, t) = 1 − ψ(x, t) = [1 − Ce −γx ]1 (γ>0) + w(x, t), (62) where w(x, t) = 1 π λ µp…”

Section: Introductionmentioning

confidence: 99%

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Avram¹,

Palmowski²,

Pistorius³

2008

Ann. Appl. Probab.

125

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show abstract

“…Using (2.5), in [36] the Laplace transform of the survival probability φ(t, y) = 1−ψ(t, y) is found when σ P = σ R = 0 and claims are exponentially distributed. However, this transform involves a ratio of confluent hypergeometric functions, and is therefore difficult to invert, the exception is when λ = r in which case the solution is rather simple.…”

Section: Analytical and Numerical Solutionsmentioning

confidence: 99%

Ruin Models with Investment Income

Paulsen

2010

Encyclopedia of Quantitative Finance

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This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integrodifferential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].

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“…In the finite time horizon case analytical solutions are hard to come by. Using (2.5), in [36] the Laplace transform of the survival probability φ(t, y) = 1−ψ(t, y) is found when σ P = σ R = 0 and claims are exponentially distributed. However, this transform involves a ratio of confluent hypergeometric functions, and is therefore difficult to invert, the exception is when λ = r in which case the solution is rather simple.…”

Section: Analytical and Numerical Solutionsmentioning

confidence: 99%

Ruin models with investment income

Paulsen¹

2008

Probab. Surveys

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This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].Comment: Published in at http://dx.doi.org/10.1214/08-PS134 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Exact and Asymptotic Solutions for the Time-Dependent Problem of Collective Ruin I

Cited by 9 publications

References 5 publications

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Ruin Models with Investment Income

Ruin models with investment income

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