Part6-Notes-431-2011-F

Part6-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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Unformatted text preview: Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2011 Part 6 The Classical Continuous Time Ruin Model In this part, times are measured in years, unless stated otherwise . 1. Poisson Process: Let N t be the number of claims up to time t or the number of claims occurring in the time interval (0 , t ] , t > 0. The process { N t , t } is said to be a Poisson process with rate > 0 if the following three conditions hold: (a) N = 0 . (b) The number of claims in any time interval of length t has a Poisson distribution with mean t , namely, for all s 0 and t > 0, Pr { there are n claims in ( s, s + t ] } = Pr { N t + s- N s = n } = ( t ) n e- t n ! . (c) The process { N t , t } has stationary and independent increments. Stationary increments mean that for all n = 1 , 2 ,..., t < t 1 < < t n and h 0, the distribution of the random vector ( N t 1 + h- N t + h , N t 2 + h- N t 1 + h , , N t n + h- N t n- 1 + h ) does not depend on h . In particular, N t + h- N h has the same distribution as N t . Independent increments mean that for all n = 1 , 2 ,..., t < t 1 < < t n , random variables N t 1- N t , N t 2- N t 1 , , N t n- N t n- 1 are independent. For the Poisson process { N t , t } , (a) For any t > 0, N t has a Poisson distribution with mean t ....
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This note was uploaded on 02/08/2012 for the course ACTSC 431 taught by Professor Laundriualt during the Fall '09 term at Waterloo.

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Part6-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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