return probabilities for stochastic fluid flows and their use in collective risk theory andrei...
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Return probabilities for stochastic fluid flows and their use in collective risk theory
Andrei Badescu
Department of Statistics
University of Toronto
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
)(
1
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uc
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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)
• 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.
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).
Stochastic Fluid Queues• Extensively used in telecommunications to model the traffic as a continuous
fluid flow.
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“Basic” Insurance Risk Model:
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(2006) Ramaswami -
(2005) Stanford Latouche, Drekic, Breuer, Badescu, -
(2005) Remiche Stanford, Latouche, Breuer, Badescu, -
Insurance Risk Models – Gerber-Shiu discounted penalty function
• The Gerber-Shiu discounted penalty function:
• The vector based Gerber-Shiu discounted penalty function:
])0(|)(|))(|),(([)( uRIRReEu ss
1],)0(|)(|))(|),(([)]([ SiuRIRReEu siis
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(2007)Badescu andAhn -
Perturbed Insurance Risk Model
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(2008)Breuer andBadescu -
Barrier/Threshold Risk Models
(2007) Ramaswami Badescu, Ahn, -
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Multi-threshold Risk Models
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(2008) Landriault Badescu, -
(2007b) Landriault Drekic, Badescu, -
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Future research – Taxation models
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Future research – premium level dependent risk model
• Insurance model:
• Fluid model:
• We need to determine the LST of the busy period .
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“ 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. ”
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