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![Page 1: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/1.jpg)
Return probabilities for stochastic fluid flows and their use in collective risk theory
Andrei Badescu
Department of Statistics
University of Toronto
![Page 2: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/2.jpg)
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
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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)
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• 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.
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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).
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Stochastic Fluid Queues• Extensively used in telecommunications to model the traffic as a continuous
fluid flow.
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The fluid flow analysis
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Risk processes analyzed as fluid flows
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![Page 9: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/9.jpg)
Return times to initial level
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Return times to initial level
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![Page 11: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/11.jpg)
Return times to initial level
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![Page 12: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/12.jpg)
Return times to initial level
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![Page 13: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/13.jpg)
Return times to initial level
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Return times to initial level
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![Page 15: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/15.jpg)
Return times to initial level B)
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![Page 16: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/16.jpg)
Return times to initial level C)
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![Page 17: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/17.jpg)
The roots of the generalized Lundberg equation
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![Page 18: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/18.jpg)
Several other first passages:• First passages in a finite buffer fluid flow:
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![Page 19: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/19.jpg)
“Basic” Insurance Risk Model:
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(2006) Ramaswami -
(2005) Stanford Latouche, Drekic, Breuer, Badescu, -
(2005) Remiche Stanford, Latouche, Breuer, Badescu, -
![Page 20: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/20.jpg)
Insurance Risk Models – Gerber-Shiu discounted penalty function
• The Gerber-Shiu discounted penalty function:
• The vector based Gerber-Shiu discounted penalty function:
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1],)0(|)(|))(|),(([)]([ SiuRIRReEu siis
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(2007)Badescu andAhn -
![Page 21: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/21.jpg)
Perturbed Insurance Risk Model
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![Page 22: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/22.jpg)
Barrier/Threshold Risk Models
(2007) Ramaswami Badescu, Ahn, -
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![Page 23: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/23.jpg)
Multi-threshold Risk Models
00 b
(2008) Landriault Badescu, -
(2007b) Landriault Drekic, Badescu, -
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![Page 24: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/24.jpg)
Future research – Taxation models
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![Page 25: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/25.jpg)
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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![Page 26: Return probabilities for stochastic fluid flows and their use in collective risk theory Andrei Badescu Department of Statistics University of Toronto.](https://reader035.fdocuments.us/reader035/viewer/2022070412/56649ea25503460f94ba5ede/html5/thumbnails/26.jpg)
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