Three-dimensional Brownian motion and the golden ratio rule

We show that the first time at which the excursion of the radial part of three-dimensional Brownian motion away from its running minimum and the running minimum itself form the golden ratio is as close as possible to the time of the ultimate minimum in a normalised mean deviation sense. Among other things this offers a rigourous optimality argument for the choice of the golden retracement in technical analysis of asset prices.

Joint work with K. Glover and H. Hulley