Two-parameter martingales associated with an integer-valued random measure. (English. Russian original) Zbl 0738.60038

Rojtgarts, A. D.

Theory Probab. Math. Stat. 43, 153-160 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 135-142 (1990).

Summary: Let \(\mu\) be an integer-valued random measure on the plane with continuous compensator \(\nu\) with respect to the two-parameter flow of \(\sigma\)-algebras generated by it; the jumps of the measure \(\mu\) do not have finite-accumulation points, and the flow generated by it satisfies the Cairoli-Walsh conditions. It is proved that every square-integrable martingale or martingale of integrable variation associated with this flow is the sum of two terms, of which the first admits a representation as a stochastic integral of the first kind with respect to the measure \(\mu-\nu\), and the second a representation as a stochastic integral of the second kind with respect to the measure \((\mu-\nu)(\mu-\nu)\). An explicit form is found for the integrands. The results obtained are used to solve a filtering problem for two-parameter martingales from observations of the measure \(\mu\).

MSC:

60G44	Martingales with continuous parameter
60G57	Random measures

Keywords:

integer-valued random measure; Cairoli-Walsh conditions; representation as a stochastic integral; two-parameter martingales