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 \[f'(i)=-\frac{ \sigma^2(f(i))L'(f(i))}{ c(i,f(i))[L(f(i))-L(i)]}\int_i^{ f(i)}\frac{ c_i'(i,y)[L(y) -L(i)]}{ \sigma^2(y)L'(y)}\,dy\] 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 \[\tau_ *=\inf\biggl\{ t\ge0\Big\vert\frac{ X_t-I_t}{ I_t}\ge\varphi\biggr\},\] 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.
Categories: math.PR