1 / 18

On the analysis of a multi-threshold Markovian risk model

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.

bonita
Download Presentation

On the analysis of a multi-threshold Markovian risk model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)

More Related