6
A Fully Consistent Dynamical Model: Generalized-Poisson Loss Model
The Generalized-Poisson Loss (GPL) model can be formulated as follows. Consider a probability space supporting a number n of independent Poisson processes N1, . . . , Nn with time-varying, and possibly stochastic, intensities λ1, . . . , λn under the risk-neutral measure 216 The risk-neutral expectation conditional on the market information up to time t , including the pool loss evolution up to t , is denoted by 217. Intensities, if stochastic, are assumed to be adapted to such information.
Define the stochastic process
(6.1)
218
for positive integers α1, . . . , αn. In the following we refer to the Zt process simply as the GPL process. We will use this process as a driving process for the cumulated portfolio loss 219 , the relevant quantity for our payoffs. In Brigo et al. (2006a, 2006b) the possible use of the GPL process as a driving tool for the default counting process 220 is illustrated instead.
The characteristic function of the ...

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