Exponential representations of continuous two-parametric martingales. (Russian) Zbl 0663.60038

Mishura, Yu. S.

Teor. Veroyatn. Mat. Stat., Kiev 38, 88-96 (1988).

The paper is devoted to the investigation of the structure of two- parameter continuous positive martingales. It is proved, for example, that a positive, continuous, weak semimartingale \(X_ t\) is a local strong martingale if and only if \(X_ t\) has a representation: \[ X_ t=\exp \{m_ t+[m]_ t-(n^ 1*n^ 2+n^ 1*<n^ 2>^ 2+<n^ 1>^ 1*n^ 2-(3/2)<<n^ 1>^ 1*<n^ 2>^ 2)_ t\}, \] where \(m_ t\) is a local strong martingale, and \(n^ i_ t\), \(i=1,2\), are local martingales which satisfy some system of stochastic equations.

Reviewer: Yu.S.Mishura

MSC:

60G44	Martingales with continuous parameter
60G60	Random fields

Keywords:

positive two-parameter martingale; exponential representation; strong martingale; local strong martingale