A Remarkable σ-finite Measure Associated with Last Passage Times and Penalisation Problems

Abstract

In this paper, we give a global view of the results we have obtained in relation with a remarkable class of submartingales, called (Σ), and which are stated in (Najnudel and Nikeghbali, 0906.1782 (2009), 0910.4959 (2009), 0911.2571 (2009) and 0911.4365 (2009)). More precisely, we associate to a given submartingale in this class (Σ), defined on a filtered probability space Open image in new window , satisfying some technical conditions, a σ-finite measure Open image in new window on Open image in new window , such that for all t≥0, and for all events Open image in new window : Open image in new window where g is the last hitting time of zero of the process X. This measure Open image in new window has already been defined in several particular cases, some of them are involved in the study of Brownian penalisation, and others are related with problems in mathematical finance. More precisely, the existence of Open image in new window in the general case solves a problem stated by D. Madan, B. Roynette and M. Yor, in a paper studying the link between Black-Scholes formula and last passage times of certain submartingales. Once the measure Open image in new window is constructed, we define a family of nonnegative martingales, corresponding to the local densities (with respect to ℙ) of the finite measures which are absolutely continuous with respect to Open image in new window . We study in detail the relation between Open image in new window and this class of martingales, and we deduce a decomposition of any nonnegative martingale into three parts, corresponding to the decomposition of finite measures on Open image in new window as the sum of three measures, such that the first one is absolutely continuous with respect to ℙ, the second one is absolutely continuous with respect to Open image in new window and the third one is singular with respect to ℙ and Open image in new window . This decomposition can be generalized to supermartingales. Moreover, if under ℙ, the process (X t ) t≥0 is a diffusion satisfying some technical conditions, one can state a penalisation result involving the measure Open image in new window , and generalizing a theorem given in (Najnudel et al., A Global View of Brownian Penalisations, 2009). Now, in the construction of the measure Open image in new window , we encounter the following problem: if Open image in new window is a filtered probability space satisfying the usual assumptions, then it is usually not possible to extend to Open image in new window (the σ-algebra generated by Open image in new window ) a coherent family of probability measures (ℚ t ) indexed by t≥0, each of them being defined on Open image in new window . That is why we must not assume the usual assumptions in our case. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths (typically adapted and continuous or adapted and cadlag versions) for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. In order to fix this problem, we introduce another augmentation of filtrations, intermediate between the right continuity and the usual conditions, and call it N-augmentation in this paper. This augmentation has also been considered by Bichteler (Stochastic integration and stochastic differential equations, 2002). Most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation still hold under the N-augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration.