Martingales in Homogeneous spaces

Details Martingales in Homogeneous spaces Martingales in Homogeneous spaces

2011-06-30

arxiv.org/abs/1106.6246v1

Mathematics

Probability

10 pages

Scientific paper

Let $G/H$ be a reductive homogeneous space and $\nabla^{G/H}$ a $G$-invariant connection. Our interesse is to study $\nabla^{G/H}$-martingales in $G/H$. In fact, we yields a correspondence between $\nabla^{G/H}$-martingales and local martingales $\m$, where $\m$ is the subspace of Lie algebra $\g$ such that $\g = \h \oplus \m$ such that $Ad(H)(\m)\subset \m$. Here $\h$ is the Lie subalgebra of $H$. As application we show that martingales in the sphere $S^{n}$ are in 1-1 correspondence with local martingales in $\mathbb{R}^{n}$.

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