Three-dimensional Brownian motion and the golden ratio rule

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Let X=(Xₜ)ₜ≥₀ be a transient diffusion process in (0,∞) with the diffusion coefficient σ>0 and the scale function L such that Xₜ→∞ as t→ ∞, let Iₜ denote its running minimum for t≥0, and let θ denote the time of its ultimate minimum I∞. Setting c(i,x)=1-2L(x)/L(i) we show that the stopping time τ*={t≥0 Xₜ≥ f*(Iₜ)} minimizes E(θ-τ-θ) over all stopping times τ of X (with finite mean) where the optimal boundary f* can be characterized as the minimal solution to f'(i)=-σ²(f(i))L'(f(i))/c(i,f(i))[L(f(i))-L(i)]∫ᵢᶠ⁽ⁱ⁾cᵢ'(i,y)[L(y) -L(i)]/σ²(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 τ *={t≥0ₜ-Iₜ/Iₜ≥φ}, where φ=(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.