Recursive methods for two-dimensional risk processes with common shocks

Lan Gong – University of Toronto(joint work with Andrei L. Badescu and Eric C.K. Cheung)

Summary

•Introduction•A recursive approach•A Gerber Shiu function at claim instants•Numerical illustrations•Conclusions

Introduction• Chan et al. (2003) , Dang et al. (2009)

• - the initial capital of the i-th class of business;• - the premium rate of the i-th class of business;• - the k-th claim amount in the i-th risk process, with common cdf

and pdf ; • - the counting process for the i-th risk process. are common shock correlated Poisson processes occurring at rates respectively.

where are independent Poisson

processes with rates ;

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Introduction

•

•

•

•

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References• Chan et al. (2003)

• Cai and Li (2005)

• Yuen et al. (2006)

• Li et al. (2007)

• Dang et al. (2009)

Introduction• Chan et al. (2003) for

• Dang et al. (2009)

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Phase-type claims – survival probability

• Let follow independent PH distributions with parameters (α, T) and (β, Q).

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Gerber Shiu function•

• is a penalty function that depends on the surplus

levels at time Tor in both processes.

• Here are few choices of the penalty functions1. 2. 3.

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Gerber Shiu function for ruin at n-th claim instant

Expected discounted deficit• Considering the first case when ruin occurs at the first claim

instant in {U1(t)} only and using a conditional argument gives

• By similar method, one immediately has . Hence by adding

, we obtain the starting point of recursion.

• If , and , the three integrals reduce to

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Exponential claims-Expected discounted deficit•

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•

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Mixture of Erlangs claims - Survival probability

•

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• We denote the survival probability associated to the time of ruin Tand, by .

•

•

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Numerical illustrations

• u1=2 ,u2=10, c1=3.2, c2=30, 1/µ1=1 and 1/µ2=10.

• Case 1: Independent model — λ11=λ22=2; λ12=0. Case 2: Three-states common shock model — λ11=λ22=1.5;

λ12=0.5. Case 3: Three-states common shock model — λ11=λ22=0.5;

λ12=1.5. Case 4: One-state common shock model — λ11 = λ22 = 0;

λ12 = 2.

• Note that λ1 = λ2 = 2, and θ1 = 0.6 and θ2 = 0.5.

Numerical illustrations

• In Case 1, after 100 iterations we obtain a ruin probability of 0.6306428 that is very close to the exact value of 0.6318894.

•

)()(

)()(

),(} )(),(max{

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11

22

11

21

22

11

uu

uu

uuuu

or

Numerical illustrations

• Cai and Li (2005, 2007) provided simple bounds for Ψand(u1, u2) given by

)}(),(min{

),()()(

22

11

21

22

11

uu

uuuu

and

Numerical illustrations

• δ = 0.05

Numerical illustrations

• This quantity is achieved by letting w1(y, z) = y+z and w2(.,. ) = w12(.,.) =0

Numerical illustrations

• This quantity is achieved by letting w2(y, z) = y+z and w1(.,. ) = w12(.,.) =0

Conclusions

• Several extensions:

1. Correlated claims

2. Correlated inter-arrival times and the resulting claims

3. Renewal type risk models