Three-dimensional Brownian motion and the golden ratio rule

Abstract

Let $X=(X_t{)}_{t\ge 0}$ be a transient diffusion process in $(0,\infty )$ with the diffusion coefficient $\sigma >0$ and the scale function $L$ such that $X_t\to \infty$ as $t\to \infty$, let $I_t$ denote its running minimum for $t\ge 0$, and let $\theta$ 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 $\mathsf{E}(|\theta -\tau |-\theta )$ over all stopping times $\tau$ of $X$ (with finite mean) where the optimal boundary $f_{\ast }$ 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 $\phi =(1+\sqrt{5})/2=1.61\dots$ 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.

Published in at http://dx.doi.org/10.1214/12-AAP859 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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