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

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Nicolas Perkowski, Johannes Ruf, others




The Annals of Probability, (43)6, 3133–3176.


Complex Systems, Computation and Language, Computational Finance, and Computers and Society