The structure of martingales generated by restrictions on stochastically continuous fields with independent increments to curves. I. (English. Ukrainian original) Zbl 0986.60041

Mishura, Yu. S.; Zolota, A. V.

Theory Probab. Math. Stat. 61, 137-152 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 131-144 (2000).

This paper is devoted to the following problem. Let a stochastically continuous random field \(x\) with independent increments and parametrized curve \(\Gamma\) be given in a nonnegative quadrant of the plane. Restriction of \(x\) to \(\Gamma\) generates some flow of \(\sigma\)-algebras. The problem is to find the structure of the martingale, adapted with the indicated flow. It is proved that the restriction of \(x\) to \(\Gamma\) is a semimartingale, and a martingale, generated by the restriction of \(x\) to \(\Gamma,\) is constructed by martingale components of \(x.\) In all the cases this martingale is given as a sum of “stochastic” line integrals with respect to martingale components of the field \(x.\) The cases of increasing, decreasing and mixed curves are considered. The formula of transition from stochastic line integrals to planar integrals are given.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G60	Random fields
60G51	Processes with independent increments; Lévy processes
60G44	Martingales with continuous parameter

Keywords:

martingales; stochastically continuous fields; curves on the plane