Constructing maximal germ couplings of Brownian motions with drift (Q6597239)

In [\textit{P. A. Ernst} et al., Stochastic Processes Appl. 129, No. 2, 355--380 (2019; Zbl 1403.60062)] (see also [\textit{F. Völlering}, ``On maximal agreement couplings'', Preprint, \url{arXiv:1608.01511}]) the notion of a ``maximal agreement coupling'' of two processes is considered.\N\NRoughly, given two Markov processes, they consider the problem of coupling them so that the trajectories coincide for ``as long as possible''.\N\NIn that paper the problem is shown to be well-defined (there is a unique distribution for the random time up until which the two processes will coincide) and some constructions of maximal coupling are given, for example for two Brownian motions with different drifts (see section 5.2).\N\NIn the paper under review, another maximal coupling is given for standard Brownian motion and a Brownian motion with constant drift. This is done by a clever choice of a reflection.\N\NGiven a standard Brownian motion \((B_t:t \ge 0)\) in the real line and a real parameter \(\theta\) the authors consider the process defined by \N\[\NH^\theta_t = \left\lbrace\begin{array}{ll}B_t &\text{ for }t \le \tau = \sup\lbrace s \ge 0: B_s = \frac{1}{2}\theta s\rbrace, \\\N\theta t - B_t &\text{ otherwise.} \end{array}\right.\N\]\NThey show that \(H^\theta_t\) is a Brownian motion with constant drift equal to \(\theta\) starting at \(0\). This is not immediate because the random time \(\tau\) is not a stopping time. They furthermore show that it is a maximal coupling.\N\NThey use this to deduce some properties of the distribution of the time \(\tau\), and also propose a sort of algorithm for simulation of the coupling (here the difficulty is that the time \(\tau\) depends on the entire trajectory of \(B_{\bullet}\)).