2013/03/07 by Kristoffer Glover, Hardy Hulley, Goran Peškir +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Brownian motion #Combinatorics #Mathematical physics #Mathematics #Physics #Quantum mechanics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.1214/12-aap859
published as Annals of Applied Probability 2013, Vol. 23, No. 3, 895-922 · 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)
openalex publication_date 2013/03/07 · arxiv created 2013/03/12 · arxiv updated 2013/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Let X=(Xt)t≥0 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 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 τ*=inf\t≥0\vert Xt≥ f*(It)\ minimizes 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)=-\fracσ2(f(i))L'(f(i))c(i,f(i))[L(f(i))-L(i)]∫if(i)\fracci'(i,y)[L(y)-L(i)]σ2(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≥0\vert\fracXt-ItIt≥φ\biggr\, 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.