Open Access
June 2013 Three-dimensional Brownian motion and the golden ratio rule
Kristoffer Glover, Hardy Hulley, Goran Peskir
Ann. Appl. Probab. 23(3): 895-922 (June 2013). DOI: 10.1214/12-AAP859

Abstract

Let X=(Xt)t0 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 t0, and let θ denote the time of its ultimate minimum I. Setting c(i,x)=12L(x)/L(i) we show that the stopping time

τ=inf{t0|Xtf(It)}

minimizes E(|θτ|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)=L1(L(i)/2) for i>0. In particular, when X is the radial part of three-dimensional Brownian motion, we find that

τ=inf{t0|XtItItφ},

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.

Citation

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Kristoffer Glover. Hardy Hulley. Goran Peskir. "Three-dimensional Brownian motion and the golden ratio rule." Ann. Appl. Probab. 23 (3) 895 - 922, June 2013. https://doi.org/10.1214/12-AAP859

Information

Published: June 2013
First available in Project Euclid: 7 March 2013

zbMATH: 06162080
MathSciNet: MR3076673
Digital Object Identifier: 10.1214/12-AAP859

Subjects:
Primary: 60G40 , 60J60 , 60J65
Secondary: 34A34 , 49J40 , 60G44

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

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 3 • June 2013
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