Three-Dimensional Brownian Motion and the Golden Ratio Rule

The golden ratio has fascinated people of diverse interests for at least 2,400 years (see e.g. [24]). In mathematics (and the arts) two quantities a and b are in the golden ratio if the ratio of the sum of the quantities a+b to the larger quantity a is equal to the ratio of the larger quantity a to the smaller quantity b . This amounts to setting (a+b)/a = a/b =: φ and solving φ2−φ−1 = 0 which yields φ = (1+√5)/2 = 1.61 . . . Apart from being abundant in nature, and finding diverse applications ranging from architecture to music, the golden ratio has also found more recent uses in technical analysis of asset prices (in strategies such as Fibonacci retracement representing an ad-hoc method for determining support and resistance levels). Despite its universal presence and canonical role in diverse applied areas, we