The joint law of a~max-continuous local submartingale and its maximum
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 693-709
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We consider the family of converging max-continuous local submartingales
starting from zero. An equivalence relation for processes of this family is
introduced so that two processes are called equivalent if their joint laws of
the terminal value and the maximum are the same. We single out a subfamily of
processes of simple structure that have a unique (in the sense of the
distribution) representative in each equivalence class. Next, using an
extension of Monroe's theorem, we embed a process of this subfamily in
a Brownian motion using a minimal time-change, and from this embedding
construct a continuous local martingale from the same equivalence class.
Moreover, it is found that whether a process from this family belongs to the
class of uniformly integrable martingales depends only on its equivalence
class. So, these results provide an alternative approach to the problems of
characterization of the distribution of a continuous local martingale and its
maximum, which were considered by L. C. G. Rogers and P. Vallois in the early 1990s.
Keywords:
Skorokhod embedding problem, time-change, local max-level martingale, local submartingale, max-continuous random process,
single jump processes, running maximum of a process.
@article{TVP_2020_65_4_a2,
author = {A. A. Gushchin},
title = {The joint law of a~max-continuous local submartingale and its maximum},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {693--709},
publisher = {mathdoc},
volume = {65},
number = {4},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a2/}
}
A. A. Gushchin. The joint law of a~max-continuous local submartingale and its maximum. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 693-709. http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a2/