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On the joint distribution of the supremum and terminal value of a uniformly integrable martingale. (English) Zbl 0828.60030

Engelbert, Hans J. (ed.) et al., Stochastic processes and optimal control. Lectures presented at the 9th winterschool held in Friedrichroda, Germany, March 1-7, 1992. Montreux: Gordon and Breach Science Publishers. Stochastics Monogr. 7, 183-199 (1993).
This paper provides characterization results of the following kind: “Given a probability measure \(\nu\) on \(\mathbb{R}^+\), when (and only when) does there exist a uniformly integrable and continuous martingale such that the probability measure of the terminal value of the maximal function coincides with \(\nu\)?” This result is due to the author. A new proof is given for the following result due to Rogers: “Given a probability measure \(\eta\) on \(\{(x,y) \in \mathbb{R}^+ \times \mathbb{R}; x \geq y\}\), when (and only when) does there exist a uniformly integrable and continuous martingale such that the joint probability measure of the terminal value of the maximal function and the terminal value of the martingale itself coincides with \(\eta\)”?
For the entire collection see [Zbl 0816.00029].
Reviewer: A.Gut (Uppsala)

MSC:

60G44 Martingales with continuous parameter
60G70 Extreme value theory; extremal stochastic processes
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