We prove that non-commutative martingale transforms are of weak type (1, 1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1 is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x = (xn) ∞ n=1 in L 1(M), adapted to (Mn)n=1, and any sequence of signs (εn) ∞ n=1, ∥∥∥∥ε1x1 + N ∑ n=2 εn(xn...

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a generic framework for introducing stochasticity into variational principles through concept of semi-martingale driven principle and constraining component variables be compatible with driving semi-martingale. Within this corresponding choice constraints, Euler-Poincare equation can easily dedu...

In this paper and its companion [J. Opt. Soc. Am. A.23, 2251 (2006)], the problem of ray propagation in nonuniform random half-plane lattices is considered. Cells can be independently occupied according to a density profile that depends on the lattice depth. An electromagnetic source external to the lattice radiates a monochromatic plane wave that undergoes specular reflections on the occupied ...

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X1 = Z1. We call X a dynamic bridge, because its terminal value Z1 is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration FX and the filtration FX,Z jointly generated by X and Z. Our construction is heavily based on parabo...

We characterize those vector-valued stochastic processes (with a finite index set and defined on an arbitrary stochastic base) which can become a martingale under an equivalent change of measure. This solves a problem which arises in the study of finite period securities markets with one riskless bond and a finite number of risky stocks. In this setting, our characterization provides necessary ...