【AP】Robust multi-period portfolio selection(2)

Navigator

Pre Blog Link

Robust multi-period portfolio selection(1)

Robust optimization and asymmetric distribution

考虑一般形式的随机规划
min?x{c′x:G(x,ξ~)≥0,x∈X}(3.1)\min_x\{\pmb{c}'\pmb{x}:G(\pmb{x}, \tilde{\xi})\geq 0, \pmb{x}\in\mathcal{X}\}\tag{3.1} xmin?{ ccc′xxx:G(xxx,ξ~?)≥0,xxx∈X}(3.1)
其中$\widetilde{\xi \text{ξ}\xi }\in {\mathbb{R}}^m$为一个随机向量，支撑集(support set)${\mathbb{S}}_{\xi }?{\mathbb{R}}^m$. 模型$(3.1)$的解$x\text{x}x$满足随机约束条件，即对于任意$\widetilde{\xi \text{ξ}\xi }\in {\mathbb{S}}_{\xi }$均要求约束满足，但是实际上这种情况一般是不可能的.
对于鲁棒优化问题的求解步骤一般是构建一个紧的, compact不确定集合$U_{\Omega }$，并且集合的大小取决于常数$\Omega$. 然后在集合$U_{\Omega }$中求解问题

1. $x\text{x}x$是方程$(3.1)$的最优解对于$\xi \text{ξ}\xi \in U_{\Omega }$.
2. 解$x\text{x}x$满足随机约束的概率是一个$\Omega$中可以求出的函数(tractable function)，模型$(3.1)$中的鲁棒部分为
   min?x{c′x:G(x,ξ)≥0,ξ∈UΩ,x∈X}(3.2)\min_x\{\pmb{c}'\pmb{x}: G(\pmb{x}, \pmb{\xi})\geq 0, \pmb{\xi}\in U_\Omega, \pmb{x}\in \mathcal{X}\}\tag{3.2} xmin?{ ccc′xxx:G(xxx,ξ?ξ??ξ)≥0,ξ?ξ??ξ∈UΩ?,xxx∈X}(3.2)
   即转换为一个非随机优化问题. 如果可以找到$u_{\Omega }$的上界(upper bound)，即满足
   $$\mathbb{P}(G(x\text{x}x,\widetilde{\xi \text{ξ}\xi })\ge 0)\ge u_{\Omega }$$
   可以找出一个常数$\Omega$使得$u_{\Omega }$接近于1，即$x\text{x}x$在一定概率下满足$(3.2)$的鲁棒部分.

This gives a probability guarantee of the solution $x\text{x}x$ of robust counterpart.

常用刻画不确定集合的方法为椭圆集合(Ben-Tal & Nemirovski, 1998¹; Ghaoui et al., 2003²; Gulpinar et al., 2013³, 2016⁴).
UΩ={ξ:∥ξ∥2≤Ω}(3.3)U_\Omega=\{\pmb{\xi}:\lVert \pmb{\xi}\rVert_2\leq \Omega\}\tag{3.3} UΩ?={ ξ?ξ??ξ:∥ξ?ξ??ξ∥2?≤Ω}(3.3)

It can be proved that if $G(x\text{x}x,\xi \text{ξ}\xi )$ is linear in $\xi \text{ξ}\xi$, the robust counterpart in $(3.2)$ with $U_{\Omega }$ given by $(3.3)$ can be converted intp a convex cone constraint that can be solved using interior point algorithms, where the probability guarantee is not less than $u_{\Omega }=1?\exp (?{\Omega }^2/2)$. Hence, robust optimization ideas can also be used to deal with the following probability constraint.

P{G(x,ξ~)≥0}≥1?ε(3.4)\mathbb{P}\{G(\pmb{x}, \tilde{\pmb{\xi}})\geq 0\}\geq 1-\varepsilon\tag{3.4} P{ G(xxx,ξ?ξ??ξ~?)≥0}≥1?ε(3.4)
选择合适的常数$\Omega$，使得$u_{\Omega }=1?\epsilon$.
本文引入非对称分布刻画不确定集合(Chen et al. 2007⁵)，设置函数$G(x\text{x}x,\widetilde{\xi \text{ξ}\xi })$具有如下双线性(bilinear form)形式
$$G(x\text{x}x,\widetilde{\xi \text{ξ}\xi })=h_0(x\text{x}x)+\sum _{i=1}^m{h}_i(x\text{x}x){\tilde{\xi }}_i\text{\hspace{1em}(3.5)}$$
其中$h_i(i=0,1,\dots ,m)$是关于$x\text{x}x$的线性方程，${\tilde{\xi }}_i$表示原始不确定性(primitive uncertainty). 根据计量经济学理论，可以对$\xi \text{ξ}\xi$设置如下标准条件⁶
$$?\begin{array}{rl}& \mathbb{E}[\widetilde{\xi \text{ξ}\xi }]=0\text{0}0\\ & \mathbb{E}[\widetilde{\xi \text{ξ}\xi }{\widetilde{\xi \text{ξ}\xi }}^{\prime }]=I\text{I}I\end{array}\text{\hspace{1em}(3.6)}$$
支撑集
$$\mathbb{S}=[?l\text{l}l,u\text{u}u]$$
其中$l\text{l}l=(l_1,l_2,\dots ,l_m{)}^{\prime },u\text{u}u=(u_1,u_2,\dots ,u_m{)}^{\prime },?\infty \le l_i\le u_i\le \infty$, Chen et al提出的不确定集合如下
FΩ={ξ:?u?,u?∈R+m,ξ=u??u?,∥P?1u?+Q?1u?∥2≤Ω,ξ∈[?l,u]}(3.7)\mathcal{F}_\Omega=\{\pmb{\xi}:\exist \underline{\pmb{u}}, \overline{\pmb{u}}\in\mathbb{R}_+^m, \pmb{\xi}=\overline{\pmb{u}}-\underline{\pmb{u}}, \lVert \mathbf{P}^{-1}\overline{\pmb{u}}+\mathbf{Q}^{-1}\underline{\pmb{u}}\rVert_2\leq \Omega, \pmb{\xi}\in[-\pmb{l}, \pmb{u}]\}\tag{3.7} FΩ?={ ξ?ξ??ξ:?uuu?,uuu∈R+m?,ξ?ξ??ξ=uuu?uuu?,∥P?1uuu+Q?1uuu?∥2?≤Ω,ξ?ξ??ξ∈[?lll,uuu]}(3.7)
其中矩阵$\mathbf{P}$和$\mathbf{Q}$满足如下性质
$$?\begin{array}{rlr}& \mathbf{P}=diag(p_1,\dots ,p_m)& \\ & \mathbf{Q}=diag(q_1,\dots ,q_m)& \\ & p_i={\sigma }_f({\tilde{\xi }}_i)>0& \text{forward devations}\\ & q_i={\sigma }_b({\tilde{\xi }}_i)>0& \text{backward devations}\end{array}$$
不确定集合${\mathcal{F}}_{\Omega }$为紧的凸集合，参数$\Omega$控制集合的大小. 可以发现，当$\mathbf{P}=\mathbf{Q}=\mathbf{I}$且$\mathbf{l}=\mathbf{u}=\infty$时，${\mathcal{F}}_{\Omega }$退化为$U_{\Omega }$.

Intuitively, to capture distributional asymmetries, we decompose the primitive data uncertainty into two random variables.

{u?~=max?{ξ~,0}u?~=max?{?ξ~,0}ξ~=u?~?u?~\left\{ \begin{aligned} &\tilde{\overline{\pmb{u}}}=\max\{\tilde{\pmb{\xi}}, 0\}\\ &\tilde{\underline{\pmb{u}}}=\max\{-\tilde{\pmb{\xi}}, 0\}\\ &\tilde{\pmb{\xi}}=\tilde{\overline{\pmb{u}}}-\tilde{\underline{\pmb{u}}} \end{aligned} \right. ?????????uuu~=max{ ξ?ξ??ξ~?,0}uuu?~?=max{ ?ξ?ξ??ξ~?,0}ξ?ξ??ξ~?=uuu~?uuu?~??

The multipliers ${\mathbf{P}}^{?1}$ and ${\mathbf{Q}}^{?1}$ normalize the effective peturbation contributed by both $\widetilde{\overline{u\text{u}u}}$ and $\widetilde{\underline{u\text{u}u}}$ such that the norm of the aggregated values falls within the budget of uncertainty.

使用$p(\tilde{\xi })$和$q(\tilde{\xi })$描述均值为0的forward deviations和backward deviations
$$p(\tilde{\xi })=\inf \{{\alpha }_p:{\alpha }_p\ge 0,\mathbb{E}[\exp \left(\frac{?}{{\alpha }_p}\tilde{\xi }\right)]\le \exp (\frac{{?}^2}{2}),??>0\}\text{\hspace{1em}(3.8)}$$
和
$$q(\tilde{\xi })=\inf \{{\beta }_q:{\beta }_q\ge 0,\mathbb{E}[\exp (?\frac{?}{{\beta }_q}\tilde{\xi })]\le \exp (\frac{{?}^2}{2}),??>0\}\text{\hspace{1em}(3.8)}$$
且当$\tilde{\xi }$的支撑集$[?l,u]$有限时，$p$和$q$为有限；而当支撑集无穷时，$p$和$q$未必有限. 但是当随机变量$\tilde{\xi }$服从正态分布时，$p$和$q$是有界的并且等于标准偏离(Chen et al. 2007⁵)实际上，随机变量$\tilde{\xi }$的精确分布无从得知，所以$p(\tilde{\xi })$和$q(\tilde{\xi })$需要从数据中估计出来.

Fig-1给出了一个关于不确定集合${\mathcal{F}}_{\Omega }$的二维的例子，可以发现$F_{\Omega }$是关于$(0,0)$的不对称集合. 在${\mathcal{F}}_{\Omega }$下随机约束条件可以表示为
$$G(x\text{x}x,\xi \text{ξ}\xi )\ge 0,?\xi \text{ξ}\xi \in {\mathcal{F}}_{\Omega }\text{\hspace{1em}(3.10)}$$
下面两个定理在后续分析中是很关键的，定理的证明可以在Chen et al. (2007)⁵中找到.
定理 3.1：如果分布支撑集$\mathbb{S}$是有限的，$(3.10)$中和以下结果等价
$$\begin{array}{rl}& ?\zeta \text{ζ}\zeta \in {\mathbb{R}}^m,s\text{s}s,v\text{v}v\in {\mathbb{R}}_{+}^m\\ & ?\begin{array}{l}{h}_0(x\text{x}x)\ge \Omega \Vert \zeta \text{ζ}\zeta {\Vert }_2+{s\text{s}s}^{\prime }u\text{u}u+{v\text{v}v}^{\prime }l\text{l}l\\ {\zeta }_i\ge ?p_i(h_i(x\text{x}x)+s_i?v_i)\\ {\zeta }_i\ge q_i(h_i(x\text{x}x)+s_i?v_i)\end{array}\end{array}\text{\hspace{1em}(3.11)}$$
定理3.2：令随机向量$\widetilde{\xi \text{ξ}\xi }$满足标准条件$(3.6)$，$x\text{x}x$为鲁棒可行集中的向量，可以得到
P{G(x,ξ~)≥0}≥1?exp?(?Ω2/2)(3.12)\mathbb{P}\{G(\pmb{x}, \tilde{\xi})\geq 0\}\geq 1-\exp(-\Omega^2/2)\tag{3.12} P{ G(xxx,ξ~?)≥0}≥1?exp(?Ω2/2)(3.12)

如果我们通过历史数据得到了关于$\tilde{\xi }$的分布信息，可以得到关于$p(\tilde{\xi })$和$q(\tilde{\xi })$的估计结论如下
定理3.3[Natarajan et al, 2008⁷]： 如果关于随机向量$\tilde{\xi }$的分布信息或者历史数据已知，可以通过以下公式估计出$p(\tilde{\xi })$和$q(\tilde{\xi })$
$$?\begin{array}{rl}& p(\tilde{\xi })=\underset{\pi >0}{\sup }\left\{\sqrt{2\frac{\ln (\mathbb{E}(\exp (\pi \tilde{\xi })))}{{\pi }^2}}\right\}\\ & q(\tilde{\xi })=\underset{\pi >0}{\sup }\left\{\sqrt{2\frac{\ln (\mathbb{E}(\exp (?\pi \tilde{\xi })))}{{\pi }^2}}\right\}\end{array}$$