Three-dimensional brownian motion and the golden ratio rule

Let X = (X_t)_t≥0 be a transient diffusion process in (0,∞) with the diffusion coefficient σ < 0 and the scale function L such that X_t → ∞ 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 [equation presented] minimizes E(|θ-ρ|-θ) over all stopping times τ of X (with finite mean) where the optimal boundary f. can be characterized as the minimal solution to [equation presented] 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 where ψ = (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.