The structure of martingales generated by restrictions on stochastically continuous fields with independent increments to curves. I (Q2740456)

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.