Potentials and local times associated with two-parameter purely discontinuous strong martingales. (English. Ukrainian original) Zbl 0923.60057

Mishura, Yu. S.; Ol’tsik, Ya. O.

Theory Probab. Math. Stat. 56, 137-149 (1998); translation from Teor. Jmovirn. Mat. Stat. 56, 133-144 (1997).

Existence and properties of local times for continuous multi-parameter strong martingales with respect to their quadratic variation in the case when Tanaka theory is applied have been obtained by P. Imkeller [J. Multivariate Anal. 19, 348-365 (1986; Zbl 0601.60042)]. R. F. Bass [Z. Wahrscheinlichkeitstheorie Verw. Geb. 67, 433-459 (1984; Zbl 0551.60047)] gave sufficient conditions for existence of local time for purely discontinuous one-parameter semi-martingales. This paper is devoted to prove the existence of a local time for purely discontinuous strong martingales on the plane which are random perturbations of stable fields and to which Tanaka’s formula is not applied. This result generalizes the analogous Bass sufficient conditions of existence of a local time.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G60	Random fields
60G44	Martingales with continuous parameter
60J55	Local time and additive functionals

Keywords:

two-parameter martingale; local time; potential

Citations:

Zbl 0601.60042; Zbl 0551.60047