Abstract. We show that for a quantum L-martingale (X(t)), p > 2, there exists a Doob-Meyer decomposition of the submartingale (|X(t)|). A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an L-martingale, p > 2, is constructed. Using this construction, the uniqueness of the Doob-Me...

A general set of distribution-free conditions is described under which an i.i.d. sequence of random variables is preserved under optional skipping. 1. Introduction and motivation. This paper discusses a general set of conditions under which an i.i.d. sequence of random variables ξ 1 , ξ 2 ,. .. , taking values in a measurable space (X, B), with common distribution F , is preserved under " optio...

Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently – well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov’s Grundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as ...

We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory. AMS Classification: 46L53, 46L51

4 Martingales 35 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Doob Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Optional Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Inequalities and Convergence . . . . . . . . . . . ...