The Annals of Applied Probability

Three-dimensional Brownian motion and the golden ratio rule

Kristoffer Glover, Hardy Hulley, and Goran Peskir

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Let $X=(X_{t})_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $\sigma>0$ 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 $\theta$ denote the time of its ultimate minimum $I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time

\[\tau_{*}=\inf\{t\ge0\vert X_{t}\ge f_{*}(I_{t})\}\]

minimizes $\mathsf{E}(\vert\theta-\tau\vert-\theta)$ over all stopping times $\tau$ of $X$ (with finite mean) where the optimal boundary $f_{*}$ can be characterized as the minimal solution to


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 $\varphi=(1+\sqrt{5})/2=1.61\ldots$ 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.

Article information

Ann. Appl. Probab., Volume 23, Number 3 (2013), 895-922.

First available in Project Euclid: 7 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]
Secondary: 34A34: Nonlinear equations and systems, general 49J40: Variational methods including variational inequalities [See also 47J20] 60G44: Martingales with continuous parameter

Optimal prediction transient diffusion Bessel process Brownian motion the golden ratio the maximality principle Fibonacci retracement support and resistance levels constant elasticity of variance model strict local martingale bubbles


Glover, Kristoffer; Hulley, Hardy; Peskir, Goran. Three-dimensional Brownian motion and the golden ratio rule. Ann. Appl. Probab. 23 (2013), no. 3, 895--922. doi:10.1214/12-AAP859.

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