XXXI: 16, 176-189, LNM 1655 (1997)
ÉMERY, Michel
Closed sets supporting a continuous divergent martingale (
Martingale theory)
This note gives a characterization of all closed subsets $F$ of $
R^d$ such that, for every $F$-valued continuous martingale $X$, the limit $X_\infty$ exists in $F$ (or $
R^d$) with non-zero probability. The criterion is as follows: To each $F$ is associated a smaller closed set $F'$ obtained, roughly speaking, by chopping off all prominent parts of $F$; this map $F\mapsto F'$ is iterated, giving a decreasing sequence $(F^n)$ with limit $F^\infty$; the condition is that $F^\infty$ is empty. (If $d=2$, $F^\infty$ is also the largest closed subset of $F$ such that all connected components of its complementary are convex)
Comment: Two similar problems are discussed in
1485Keywords: Continuous martingales,
Asymptotic behaviour of processesNature: Original
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