On the analysis of a multi-threshold Markovian risk model

1. On the analysis of a multi-threshold Markovian risk model Andrei Badescu – University of Toronto Steve Drekic – University of Waterloo David Landriault – University of Waterloo IME 2007, University of Piraeus, Piraeus, Greece The authors gratefully acknowledge the support provided by NSERC

2. Outline • Introduction : Fluid flow process vs surplus process • A multi-level threshold-type risk model with Markovian claim arrivals (MAP) • Analysis of the expected discounted dividend payments • Numerical illustration

3. Connection between fluid flow process and surplus process • Asmussen (1995) • Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford (2005a) • Badescu, Breuer, Drekic, Latouche and Stanford (2005b) • Ahn, Badescu and Ramaswami (2006) • Ahn and Ramaswami (2004, 2005) • Ramaswami (2007)

4. A fluid flow process • A bivariate Markov process: where - : the level of the fluid buffer - : a CTMC that describes the states of the environmental process • The fluid level is such that • For , the fluid level increases at rate c(i) > 0 • For , the fluid level decreases at rate c(i) > 0 • The finite state space • The infinitesimal generator

5. A surplus process • An insurer’s surplus where - : initial capital - : premium rate - : number of claims by time t - : claim sizes

6. Fluid flow process vs surplus process

7. A risk model with Markovian arrivals • Claim number process : Markov Arrival Process of order m • : the initial state probability vector • : transition rates among states without an arrival • : transition rates among states at the time of an arrival • Claim sizes : a transition from to at the time of a claim yields a claim size of distribution of order n

8. A risk model with Markovian arrivals Equivalent fluid flow representation • - the ascending phases - of order • - the descending phases - of order • the infinitesimal generator of such a process: with

9. A threshold-type risk model with MAP • Idea: Use the connection between fluid flow processes and risk processes to analyze threshold-type risk models defined in a Markovian environment • Generalizes the class of risk models studied in the context of a threshold-type dividend strategy by • Lin and Sendova (2007) • Albrecher and Hartinger (2007) • Zhou (2006) Cramer-Lundberg risk model

10. A multi-threshold risk model with MAP • Insurer’s surplus:

11. A multi-threshold risk model with MAP

12. Expected discounted dividend payments • Objective : analysis of the expected discounted dividend payments • Methodology • sample path analysis • recursive calculation : adding a surplus layer at each iteration • Idea • starting point : barrier-free risk model (known) • proceed recursively by adding the next top layer • : expected discounted dividends (with initial surplus u)for the risk model

13. A multi-threshold risk model with MAP Risk process constructed by ignoring the first (i-1) layers

14. Expected discounted dividend payments • First term : expected discounted dividend from time 0 to the time that the surplus process reaches level bi or any ruin level for the first time • Second term : expected discounted dividend received thereafter

15. Expected discounted dividend payments • First term : expected discounted dividend from time 0 to the time that the surplus level is less than bi for the first time • Second term : expected discounted dividend received during the first sojourn of the surplus level in the bottom layer • Third term : expected discounted dividend received thereafter

16. Expected discounted dividend payments or equivalently

17. A numerical illustration MAP contagion model example * see Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford (2005) • Dependence structure between the claim sizes and the interclaim times • Two environments: • First environment (i.e. standard environment) – only small claims • Second environment (i.e. infectious environment) – small and large claims

18. A numerical illustration With a gross premium rate of c = 2.5 and B = (20, 40, 60), consider the following 4 dividend strategies: Expected discounted dividend payments prior to ruin (δ = 0.001)