Three-dimensional Brownian motion and the golden ratio rule

Let X=(Xt)t.0 be a transient diffusion process in (0,.) with the diffusion coefficient .>0 and the scale function L such that Xt.. as t.., let It denote its running minimum for t.0, and let . denote the time of its ultimate minimum I.. Setting c(i,x)=1.2L(x)/L(i) we show that the stopping time ..=inf{t.0|Xt.f.(It)} minimizes E(|...|..) over all stopping times . of X (with finite mean) where the optimal boundary f. can be characterized as the minimal solution to f.(i)=..2(f(i))L.(f(i))c(i,f(i))[L(f(i)).L(i)].f(i)ic.i(i,y)[L(y). L(i)].2(y)L.(y)dy staying strictly above the curve h(i)=L.1(L(i)/2) for i>0. In particular, when X is the radial part of three-dimensional Brownian motion, we find that ..=inf{t.0..Xt.ItIt..}, where .=(1+.5)/2=1.61. is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigorous optimality argument for the choice of the well-known golden retracement in technical analysis of asset prices.