1. Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto

2. The insurance risk model Insurer’s surplus: - the initial capital. - the premium rate - the number of claims at time t. Quantities of interest: - time to ruin - surplus prior to ruin - deficit at ruin - dividends - taxation

3. References 1) Poisson arrivals with independent inter-claim times and claim sizes: Gerber and Shiu (1998) Lin an Willmot (1999, 2000) 2) Renewal arrivals with independent inter-claim times and claim sizes: Willmot and Dickson (2003) Li and Garrido (2004, 2005) Gerber and Shiu (2005) 3) Dependent inter-claim times and claim sizes: Albrecher and Boxma (2004, 2005) Marceau, Cossette and Landriault (2006)

4. Goal: - To construct and analyze a completely dependent structure between the inter-claim times and the claim sizes (not of a renewal type). Model assumptions: –Claims are Phase-type distributed (PH, Neuts 1975) –Claim arrival process follows a versatile point process - Markovian Arrival Process (MAP, Neuts 1979) Methodology: –Using the connections between fluid flows and risk processes (Asmussen 1995). –Using Matrix Analytic Methods (MAMs) – the main tool of analyzing fluid models.

5. Matrix Analytic Methods popular modeling tools giving the ability to construct and analyze in an algorithmically tractable manner a wide class of stochastic models (e.g. telecommunication). MAMs involve a constant interplay between formal algebraic manipulation of mathematical expressions and probabilistic interpretation of these expressions. MAMs avoid the use of theory of eigen-values (algorithmic tractability). PH distributions represent the simplest introduction to MAMs (the distribution of a random variable is defined through a matrix).

6. Stochastic Fluid Queues Extensively used in telecommunications to model the traffic as a continuous fluid flow.

7. The fluid flow analysis 1 2 3 mm+1 m+2 m+3 m+n

8. Risk processes analyzed as fluid flows

9. Return times to initial level

17. The roots of the generalized Lundberg equation

18. Several other first passages: First passages in a finite buffer fluid flow:

19. “Basic” Insurance Risk Model:

20. Insurance Risk Models – Gerber-Shiu discounted penalty function The Gerber-Shiu discounted penalty function: The vector based Gerber-Shiu discounted penalty function:

21. Perturbed Insurance Risk Model

22. Barrier/Threshold Risk Models t t

23. Multi-threshold Risk Models

24. Future research – Taxation models

25. Future research – premium level dependent risk model Insurance model: Fluid model: We need to determine the LST of the busy period.

26. “ To do work in computational mathematics is… a commitment to a more demanding definition of what constitutes the solution of a mathematical problem. When done properly, it conforms to the highest standard of scientific research. ” [NEUTS]