On the time to reach maximum for a variety of constrained Brownian motions

We derive, the joint probability density of the maximum), ( mtMP M and the time at which this maximum is achieved, for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over mt M, the marginal density is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain ‘agreement formulae ’ that are encountered more generally in probabilistic studies of Brownian motion processes.) ( mtP) ( mtP