Abstract
1. Introduction. It is a simple but very useful observation that a probabil- ity measure Q which is not absolutely continuous with respect to some reference measure P has a nonnegative P -supermartingale as its “Radon–Nikodym deriva- tive.” For instance, such supermartingales appear naturally in the generalization of Girsanov’s theorem to measures without absolute continuity relation as in Yoeurp (1985), or when working with killed diffusions. Conversely, given a nonnegative supermartingale, under suitable assumptions on the probability space, it is possible to reconstruct a measure associated to it, the so-called Föllmer measure. The behavior of the Föllmer measure characterizes the most important properties of the supermartingale; see Föllmer (1972, 1973); see also Ruf (2013a, 2013b), Cui (2013) and Larsson and Ruf (2014) for appli- cations in the detection of strict local martingales. Further applications include, among others, potential theory [Airault and Föllmer (1974), Föllmer (1972)], sim- ple proofs of the main semimartingale decomposition theorems [Föllmer (1973)], filtration enlargements [Kardaras (2012), Yoeurp (1985)], filtration shrinkage [Föllmer and Protter (2011), Larsson (2014)] and a simple approach to the study of conditioned measures [Perkowski and Ruf (2012)]. Measures associated to nonnegative supermartingales have also appeared natu- rally in the duality approach to stochastic control. The dual formulation has been developed for several important applications, such as utility maximization [see, among many others, Föllmer and Gundel (2006), Karatzas et al. (1991), Kramkov
Authors
Nicolas Perkowski, Johannes Ruf, others
Year
2015
Journal
The Annals of Probability, (43)6, 3133–3176.
Keywords
Complex Systems, Computation and Language, Computational Finance, and Computers and Society