  We derive P(M,t_m), the joint probability density of the maximum M and the
time t_m 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 M, the marginal density P(t_m) 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 P(t_m) 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.
