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Part6-Notes

# Part6-Notes - Review Notes for Loss Models 1 ACTSC 431/831...

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Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2008 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 0 } is said to be a Poisson process with rate λ > 0 if the following three conditions hold: (a) N 0 = 0 . (b) The number of claims in any time interval of length t has a Poisson distribution with mean λt , namely, for all s 0 and t > 0, Pr { there are n claims in ( s, s + t ] } = Pr { N t + s - N s = n } = ( λt ) n e - λt n ! . (c) The process { N t , t 0 } has stationary and independent increments. Stationary increments mean that for all n = 1 , 2 , ..., 0 t 0 < t 1 < · · · < t n and h 0, the distribution of the random vector ( N t 1 + h - N t 0 + 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 , ..., 0 t 0 < t 1 < · · · < t n , random variables N t 1 - N t 0 , N t 2 - N t 1 , · · · , N t n - N t n - 1 are independent.

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Part6-Notes - Review Notes for Loss Models 1 ACTSC 431/831...

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