We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a semi-martingale X does not transfer from the reference filtration F to a larger filtration G if the information starts growing up to a positive time. Then we study the case G = F ∨ H when there exists a second special semi-martingale Y enjoying the predictable representation property with respect to H. We establish conditions under which the triplet (X, Y, [X, Y ]) enjoys the predictable representation property with respect to G.

In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.2000 Mathematics Subject Classification: 92D25 (35R09, 60G55)

Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale respectively on the same probability space (Ω, F, P ), both enjoying the strong predictable representation property. We propose a martingale representation result for the square-integrable (P, G)-martingales, where G = F ∨ H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of d i=1 F i , where, fixed i ∈ (1, . . . , d), F i is the reference filtration of a real martingale M i , which enjoys the (P, F i ) predictable representation property. A second application falls into the framework of credit risk modeling and in particular into the study of the progressive enlargement of the market filtration by a default time. More precisely, when the risky asset price is a multidimensional semi-martingale enjoying the strong predictable representation property and the default time satisfies the density hypothesis, we present a new proof of the analogous of the classical Kusuoka's theorem.

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We consider a probabilistic model of a heterogeneous population P subdivided into homogeneous sub-cohorts. A main assumption is that the frailties give rise to a discrete, exchangeable random vector. We put ourselves in the framework of stochastic filtering to derive the conditional distribution of residual lifetimes of surviving individuals, given an observed history of failures and survivals. As a main feature of our approach, this study is based on the analysis of behaviour of the vector of ‘occupation numbers’.