Recursive methods for two-dimensional risk processes with common shocks

1. 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)

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

3. 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 ;

4. Introduction

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

6. Introduction • Chan et al. (2003) for • Dang et al. (2009)

7. An alternative recursive approach

8. Phase-typeclaims – survival probability • Let follow independent PH distributions with parameters (α, T) and (β, Q).

9. 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 functions

10. Gerber Shiu function for ruin at n-th claim instant Where correspond to the cases {τ1<τ2}, {τ2<τ1} and {τ1=τ2} respectively.

11. 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

12. Exponential claims-Expected discounted deficit • 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

13. Expected discounted deficit

14. Mixture of Erlangs claims - Survival probability • Using equation (4) for n=0 and λ11=λ22=0 along with the trivial condition , we obtain

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

17. 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.

18. 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.

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

20. Numerical illustrations • δ = 0.05

21. Numerical illustrations • This quantity is achieved by letting w1(y, z) = y+zand w2(.,. ) = w12(.,.) =0

22. Numerical illustrations • This quantity is achieved by letting w2(y, z) = y+zand w1(.,. ) = w12(.,.) =0

23. Conclusions • Several extensions: • Correlated claims • Correlated inter-arrival times and the resulting claims • Renewal type risk models