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

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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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Chan et al. (2003) Cai and Li (2005) Yuen et al. (2006) Li et al. (2007) Dang et al. (2009)

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Chan et al. (2003) for Dang et al. (2009)

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

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is a penalty function that depends on the surplus levels at time T or in both processes. Here are few choices of the penalty functions 1. 2. 3. 4.

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Where correspond to the cases {τ 1 <τ 2 }, {τ 2 <τ 1 } and {τ 1 =τ 2 } respectively.

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Considering the first case when ruin occurs at the first claim instant in {U 1 (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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The idea that we use to find a computational tractable solution of (16) is based on mathematical induction. Therefore, the expected discounted deficit when ruin happens at the instant of the first claim is given by

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Using equation (4) for n=0 and λ 11 =λ 22 =0 along with the trivial condition, we obtain

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We denote the survival probability associated to the time of ruin T and, by.

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u 1 =2,u 2 =10, c 1 =3.2, c 2 =30, 1/µ 1 =1 and 1/µ 2 =10. Case 1: Independent model — λ 11 =λ 22 =2; λ 12 =0. Case 2: Three-states common shock model — λ 11 =λ 22 =1.5; λ 12 =0.5. Case 3: Three-states common shock model — λ 11 =λ 22 =0.5; λ 12 =1.5. Case 4: One-state common shock model — λ 11 = λ 22 = 0; λ 12 = 2. Note that λ 1 = λ 2 = 2, and θ 1 = 0.6 and θ 2 = 0.5.

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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.

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Cai and Li (2005, 2007) provided simple bounds for Ψ and (u 1, u 2 ) given by

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δ = 0.05

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This quantity is achieved by letting w 1 (y, z) = y+z and w 2 (.,. ) = w 12 (.,.) =0

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This quantity is achieved by letting w 2 (y, z) = y+z and w 1 (.,. ) = w 12 (.,.) =0

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Several extensions: 1.Correlated claims 2.Correlated inter-arrival times and the resulting claims 3.Renewal type risk models

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