Martingale selection theorem for a stochastic sequence with relatively open convex values

Dmitry B. Rokhlin

paper

Martingale selection theorem for a stochastic sequence with relatively open convex values

arXiv:math/0602587

Abstract

For a set-valued stochastic sequence $(G_{n})_{n = 0}^{N}$ with relatively open convex values $G_{n} (ω)$ we give a criterion for the existence of an adapted sequence $(x_{n})_{n = 0}^{N}$ of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular conditional upper distributions of the elements $G_{n}$ . This result is a refinement of the main result of author's previous paper (Teor. Veroyatnost. i Primen., 2005, 50:3, 480--500), where the sets $G_{n} (ω)$ were assumed to be open and where were asked if the openness condition can be relaxed.

7 pages