Characterization of max-continuous local martingales vanishing at infinity. (English) Zbl 1354.60045

Acciaio, Beatrice; Penner, Irina

Electron. Commun. Probab. 21, Paper No. 71, 10 p. (2016).

Summary: We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by A. Nikeghbali and M. Yor [Ill. J. Math. 50, No. 1–4, 791–814 (2006; Zbl 1101.60059)] for continuous filtrations. Our generalization is complementary to the one presented by C. Kardaras [Stochastic Processes Appl. 124, No. 1, 373–384 (2014; Zbl 1300.60044)], and is obtained by means of similar tools.

MSC:

60G44	Martingales with continuous parameter
60G48	Generalizations of martingales
60G40	Stopping times; optimal stopping problems; gambling theory
60G07	General theory of stochastic processes

Keywords:

local martingales; predictable stopping time; predictable set; time of maximum

Citations:

Zbl 1101.60059; Zbl 1300.60044