In this paper we review some old and new results about the enlargement of filtrations problem, as well as their applications to credit risk and insider trading problems. The enlargement of filtrations problem consists in the study of conditions under which a semimartingale remains a semimartingale when the filtration is enlarged, and, in such a case, how to find the Doob-Meyer decomposition. Fi...

Rivest & Wells (2001) proposed estimators of the marginal survival functions in a right-censored model that assumes an Archimedean copula between the survival time and the censoring time. We study the extension of these estimators to the context of rightcensored semi-competing risks data with an independent second level censoring time. We intensively use martingale techniques to derive their la...

We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob–Khas’minskii theorem. The second par...

In this paper we extend the notion of g-evaluation, in particular g-expectation, of Peng [10, 11] to the case where the generator g is allowed to have a quadratic growth. We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation, and consequently the reverse comparison theorem of quadratic BSDEs as ...

We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of nonMarkovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob-Khas’minskii theorem. The second part...

In the theory of progressive enlargements of filtrations, the supermartingale Zt = P (g > t | Ft) associated with an honest time g, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale Zt, based on Doob’s maximal identity. We thus give new examples of progressive enla...

Motivated by the Kyle-Back model of \insider trading", we consider certain classes of linear transformations of two independent Brownian motions and study their canonical decomposition, i.e., their Doob-Meyer decomposition as semimartingales in their own ltration. In particular we characterize those transformations which generate again a Brownian motion. 1 Support of the EU through the TMR cont...

The main result announced here is a negative solution of the Kakutani-Doob problem [3 ] on measurability of stochastic processes, assuming the continuum hypothesis. Thus the positive solution proposed earlier by M. Mahowald [ó] is incorrect (the last step in the argument applies the Fubini theorem to sets in a product space which need not be measurable). Complete proofs will appear in the Proce...

Schrr odinger processes are deened as mixtures of Brownian bridges which preserve the Markov property. In nite dimensions, they can be characterized as h-transforms in the sense of Doob for some space-time harmonic function h of Brownian motion, and also as solutions to a large deviation problem introduced by Schrr odinger which involves minimization of relative entropy with given marginals. As...