## 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