Three-dimensional Brownian motion and the golden ratio rule

Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $σ>0$ and the scale function $L$ such that $X_t\rightarrow\infty$ as $t\rightarrow \infty$, let $I_t$ denote its running minimum for $t\ge0$, and let $θ$ denote the time of its ultimate minimum $I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time $τ_*=\inf\{t\ge0\vert X_t\ge f_*(I_t)\}$ minimizes $\mathsf{E}(\vertθ-τ\vert-θ)$ over all stopping times $τ$ of $X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as the minimal solution to $f'(i)=-\frac{σ^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y) -L(i)]}{σ^2(y)L'(y)}\,dy$ 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 $τ_ *=\inf\biggl\{t\ge0\Big\vert\frac{X_t-I_t}{I_t}\geφ\biggr\},$ where $φ=(1+\sqrt{5})/2=1.61\ldots$ 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.

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