三维布朗运动与黄金比例法则
Three-dimensional Brownian motion and the golden ratio rule
摘要：
令$X=(X_t)_{t\ge0}$是$(0,\infty)$中的瞬态扩散过程，其扩散系数为$σ>0$，标度函数为$L$，使得$X_t\rightarrow\infty$为$t\rightarrow \infty$，令$I_t$表示$t\ge0$的运行最小值，令$θ$表示其最终最小值$I_{\infty}$的时间。设置$c(i,x)=1-2L(x)/L(i)$，我们表明，停止时间\[τ_*=\inf\{t\ge0\vert X_t\ge f_*(I_t)\}\]最大限度地减少$\mathsf{E}(\vertθ-τ\vert-θ)$在所有停止时间$X$ $τ$（有限均值），其中最佳边界$f_*$可以被描述为最小的解决方案，\[f'(i)=-\frac{σ^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y) -L(i)]}{σ^2(y)L'(y)}\,dy\]严格停留在曲线$h(i)=L^{-1}(L(i)/2)$以上的$i>0$。特别地，当$X$是三维布朗运动的径向部分时，我们发现\[τ_ *=\inf\biggl\{t\ge0\Big\vert\frac{X_t-I_t}{I_t}\geφ\biggr\},\]，其中$φ=(1+\sqrt{5})/2=1.61\ldots$是黄金比例。推导出的结果适用于问题的最优交易中存在的泡沫，我们表明，黄金比例规则提供了一个严格的最优性参数的选择著名的黄金回撤在技术分析的资产价格。
英文摘要：
Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $σ>0$ 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 $θ$ 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 $\mathsf{E}(\vertθ-τ\vert-θ)$ over all stopping times $τ$ of $X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as the minimal solution to \[f'(i)=-\frac{σ^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y) -L(i)]}{σ^2(y)L'(y)}\,dy\] 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 $φ=(1+\sqrt{5})/2=1.61\ldots$ 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.
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