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ξ~∈Rm\tilde{\pmb{\xi}}\in\mathbb{R}^mξξξ~∈Rm为一个随机向量，支撑集(support set)Sξ⊂Rm\mathbb{S}_\xi\subset\mathbb{R}^mSξ⊂Rm. 模型(3.1)(3.1)(3.1)的解x\pmb{x}xxx满足随机约束条件，即对于任意ξ~∈Sξ\tilde{\pmb{\xi}}\in\mathbb{S}_\xiξξξ~∈Sξ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UΩU_\OmegaUΩ，并且集合的大小取决于常数Ω\OmegaΩ. 然后在集合UΩU_\OmegaUΩ中求解问题

x\pmb{x}xxx是方程(3.1)(3.1)(3.1)的最优解对于ξ∈UΩ\pmb{\xi}\in U_\Omegaξξξ∈UΩ.
解x\pmb{x}xxx满足随机约束的概率是一个Ω\OmegaΩ中可以求出的函数(tractable function)，模型(3.1)(3.1)(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,ξξξ)≥0,ξξξ∈UΩ,xxx∈X}(3.2)
即转换为一个非随机优化问题. 如果可以找到uΩu_\OmegauΩ的上界(upper bound)，即满足
P(G(x,ξ~)≥0)≥uΩ\mathbb{P}(G(\pmb{x}, \tilde{\pmb{\xi}})\geq 0)\geq u_\Omega P(G(xxx,ξξξ~)≥0)≥uΩ
可以找出一个常数Ω\OmegaΩ使得uΩu_\OmegauΩ接近于1，即x\pmb{x}xxx在一定概率下满足(3.2)(3.2)(3.2)的鲁棒部分.

This gives a probability guarantee of the solution x\pmb{x}xxx 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={ξξξ:∥ξξξ∥2≤Ω}(3.3)

It can be proved that if G(x,ξ)G(\pmb{x}, \pmb{\xi})G(xxx,ξξξ) is linear in ξ\pmb{\xi}ξξξ, the robust counterpart in (3.2)(3.2)(3.2) with UΩU_\OmegaUΩ given by (3.3)(3.3)(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 uΩ=1−exp⁡(−Ω2/2)u_\Omega=1-\exp(-\Omega^2/2)uΩ=1−exp(−Ω2/2). 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,ξξξ~)≥0}≥1−ε(3.4)
选择合适的常数Ω\OmegaΩ，使得uΩ=1−εu_\Omega=1-\varepsilonuΩ=1−ε.
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,ξ~)G(\pmb{x}, \tilde{\pmb{\xi}})G(xxx,ξξξ~)具有如下双线性(bilinear form)形式
G(x,ξ~)=h0(x)+∑i=1mhi(x)ξ~i(3.5)G(\pmb{x}, \tilde{\pmb{\xi}})=h_0(\pmb{x})+\sum_{i=1}^mh_i(\pmb{x})\tilde{\xi}_i\tag{3.5} G(xxx,ξξξ~)=h0(xxx)+i=1∑mhi(xxx)ξ~i(3.5)
其中hi(i=0,1,…,m)h_i(i=0,1,\dots, m)hi(i=0,1,…,m)是关于x\pmb{x}xxx的线性方程，ξ~i\tilde{\xi}_iξ~i表示原始不确定性(primitive uncertainty). 根据计量经济学理论，可以对ξ\pmb{\xi}ξξξ设置如下标准条件⁶
{E[ξ~]=0E[ξ~ξ~′]=I(3.6)\left\{ \begin{aligned} &\mathbb{E}[\tilde{\pmb{\xi}}]=\pmb{0}\\ &\mathbb{E}[\tilde{\pmb{\xi}}\tilde{\pmb{\xi}}']=\pmb{I} \end{aligned}\tag{3.6} \right. ⎩⎨⎧E[ξξξ~]=000E[ξξξ~ξξξ~′]=III(3.6)
支撑集
S=[−l,u]\mathbb{S}=[-\pmb{l}, \pmb{u}] S=[−lll,uuu]
其中l=(l1,l2,…,lm)′,u=(u1,u2,…,um)′,−∞≤li≤ui≤∞\pmb{l}=(l_1, l_2, \dots, l_m)', \pmb{u}=(u_1, u_2, \dots, u_m)', -\infty\leq l_i\leq u_i\leq \inftylll=(l1,l2,…,lm)′,uuu=(u1,u2,…,um)′,−∞≤li≤ui≤∞, 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={ξξξ:∃uuu,uuu∈R+m,ξξξ=uuu−uuu,∥P−1uuu+Q−1uuu∥2≤Ω,ξξξ∈[−lll,uuu]}(3.7)
其中矩阵P\mathbf{P}P和Q\mathbf{Q}Q满足如下性质
{P=diag(p1,…,pm)Q=diag(q1,…,qm)pi=σf(ξ~i)>0forward devationsqi=σb(ξ~i)>0backward devations\left\{ \begin{aligned} &\mathbf{P}=diag(p_1, \dots, p_m)\\ &\mathbf{Q}=diag(q_1, \dots, q_m)\\ &p_i=\sigma_f(\tilde{\xi}_i)>0\quad&\text{forward devations}\\ &q_i=\sigma_b(\tilde{\xi}_i)>0\quad&\text{backward devations} \end{aligned} \right. ⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧P=diag(p1,…,pm)Q=diag(q1,…,qm)pi=σf(ξ~i)>0qi=σb(ξ~i)>0forward devationsbackward devations
不确定集合FΩ\mathcal{F}_\OmegaFΩ为紧的凸集合，参数Ω\OmegaΩ控制集合的大小. 可以发现，当P=Q=I\mathbf{P}=\mathbf{Q}=\mathbf{I}P=Q=I且l=u=∞\mathbf{l}=\mathbf{u}=\inftyl=u=∞时，FΩ\mathcal{F}_\OmegaFΩ退化为UΩU_\OmegaUΩ.

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{ξξξ~,0}uuu~=max{−ξξξ~,0}ξξξ~=uuu~−uuu~

The multipliers P−1\mathbf{P}^{-1}P−1 and Q−1\mathbf{Q}^{-1}Q−1 normalize the effective peturbation contributed by both u‾~\tilde{\overline{\pmb{u}}}uuu~ and u‾~\tilde{\underline{\pmb{u}}}uuu~ such that the norm of the aggregated values falls within the budget of uncertainty.

使用p(ξ~)p(\tilde{\xi})p(ξ~)和q(ξ~)q(\tilde{\xi})q(ξ~)描述均值为0的forward deviations和backward deviations
p(ξ~)=inf⁡{αp:αp≥0,E[exp⁡(ϕαpξ~)]≤exp⁡(ϕ22),∀ϕ>0}(3.8)p(\tilde{\xi})=\inf\bigg\{ \alpha_p:\alpha_p\geq 0, \mathbb{E}\bigg[\exp\bigg(\frac{\phi}{\alpha_p}\tilde{\xi}\bigg)\bigg]\leq \exp\bigg(\frac{\phi^2}{2}\bigg),\forall \phi>0\bigg\}\tag{3.8} p(ξ~)=inf{αp:αp≥0,E[exp(αpϕξ~)]≤exp(2ϕ2),∀ϕ>0}(3.8)
和
q(ξ~)=inf⁡{βq:βq≥0,E[exp⁡(−ϕβqξ~)]≤exp⁡(ϕ22),∀ϕ>0}(3.9)q(\tilde{\xi})=\inf\bigg\{ \beta_q:\beta_q\geq 0, \mathbb{E}\bigg[\exp\bigg(-\frac{\phi}{\beta_q}\tilde{\xi}\bigg)\bigg]\leq \exp\bigg(\frac{\phi^2}{2}\bigg),\forall \phi>0\bigg\}\tag{3.9} q(ξ~)=inf{βq:βq≥0,E[exp(−βqϕξ~)]≤exp(2ϕ2),∀ϕ>0}(3.9)
且当ξ~\tilde{\xi}ξ~的支撑集[−l,u][-l, u][−l,u]有限时，ppp和qqq为有限；而当支撑集无穷时，ppp和qqq未必有限. 但是当随机变量ξ~\tilde{\xi}ξ~服从正态分布时，ppp和qqq是有界的并且等于标准偏离(Chen et al. 2007⁵)实际上，随机变量ξ~\tilde{\xi}ξ~的精确分布无从得知，所以p(ξ~)p(\tilde{\xi})p(ξ~)和q(ξ~)q(\tilde{\xi})q(ξ~)需要从数据中估计出来.

Fig-1给出了一个关于不确定集合FΩ\mathcal{F}_\OmegaFΩ的二维的例子，可以发现FΩF_\OmegaFΩ是关于(0,0)(0, 0)(0,0)的不对称集合. 在FΩ\mathcal{F}_\OmegaFΩ下随机约束条件可以表示为
G(x,ξ)≥0,∀ξ∈FΩ(3.10)G(\pmb{x}, \pmb{\xi})\geq 0, \forall \pmb{\xi}\in\mathcal{F}_\Omega\tag{3.10} G(xxx,ξξξ)≥0,∀ξξξ∈FΩ(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S\mathbb{S}S是有限的，(3.10)(3.10)(3.10)中和以下结果等价
∀ζ∈Rm,s,v∈R+m{h0(x)≥Ω∥ζ∥2+s′u+v′lζi≥−pi(hi(x)+si−vi)ζi≥qi(hi(x)+si−vi)(3.11)\begin{aligned} &\forall\pmb{\zeta}\in\mathbb{R}^m, \pmb{s}, \pmb{v}\in\mathbb{R}_+^m\\ & \begin{cases} h_0(\pmb{x})\geq \Omega\lVert\pmb{\zeta}\rVert_2+\pmb{s}'\pmb{u}+\pmb{v}'\pmb{l}\\ \zeta_i\geq -p_i(h_i(\pmb{x})+s_i-v_i)\\ \zeta_i\geq q_i(h_i(\pmb{x})+s_i-v_i) \end{cases} \end{aligned}\tag{3.11} ∀ζζζ∈Rm,sss,vvv∈R+m⎩⎪⎨⎪⎧h0(xxx)≥Ω∥ζζζ∥2+sss′uuu+vvv′lllζi≥−pi(hi(xxx)+si−vi)ζi≥qi(hi(xxx)+si−vi)(3.11)
定理3.2：令随机向量ξ~\tilde{\pmb{\xi}}ξξξ~满足标准条件(3.6)(3.6)(3.6)，x\pmb{x}xxx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ξ~\tilde{\xi}ξ~的分布信息，可以得到关于p(ξ~)p(\tilde{\xi})p(ξ~)和q(ξ~)q(\tilde{\xi})q(ξ~)的估计结论如下
定理3.3[Natarajan et al, 2008⁷]：如果关于随机向量ξ~\tilde{\xi}ξ~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(ξ~)p(\tilde{\xi})p(ξ~)和q(ξ~)q(\tilde{\xi})q(ξ~)
{p(ξ~)=sup⁡π>0{2ln⁡(E(exp⁡(πξ~)))π2}q(ξ~)=sup⁡π>0{2ln⁡(E(exp⁡(−πξ~)))π2}\left\{ \begin{aligned} &p(\tilde{\xi})=\sup_{\pi>0}\bigg\{\sqrt{2\frac{\ln(\mathbb{E}(\exp(\pi\tilde{\xi})))}{\pi^2}}\bigg\}\\ &q(\tilde{\xi})=\sup_{\pi>0}\bigg\{\sqrt{2\frac{\ln(\mathbb{E(\exp(-\pi\tilde{\xi}))})}{\pi^2}}\bigg\} \end{aligned} \right. ⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧p(ξ~)=π>0sup{2π2ln(E(exp(πξ~)))}q(ξ~)=π>0sup{2π2ln(E(exp(−πξ~)))}

Robust Convex Optimization ↩︎
worst case VaR and Robust portfolio_A conic programming approach ↩︎
a robust optimization approach to asset-la under time-varying inv opp ↩︎
A robust asset-liability man- agement framework ↩︎
a robust optimization perspective on stochastic programming ↩︎ ↩︎ ↩︎
如果ξ~\tilde{\xi}ξ~具有均值μξ\mu_\xiμξ以及非对角方差协方差矩阵Σξ\Sigma_\xiΣξ，令f~=Σξ−12(ξ~−μξ)\tilde{f}=\Sigma_\xi^{-\frac{1}{2}}(\tilde{\xi}-\mu_\xi)f~=Σξ−21(ξ~−μξ)，那么f~\tilde{f}f~满足标准条件(3.6)(3.6)(3.6). ↩︎
Incorporating asymmetric distri- butional information in robust value-at-risk optimization ↩︎

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

【AP】Robust multi-period portfolio selection(3)
Navigator Pre Blog Link Robust conterparts of multi-period portfolio problems The relationship betwe ...
【AP】Least-squares approach to risk parity in portfolio selection(1)
导航 Paper Link Authors Pub Date Abstract & Introduction Risk parity problem Minimum variance opti ...
【AP】a pratical guide to robust optimization（2）
Navigator Pre Link Choosing the uncertainty set Linearly adjustable robust counterpart: linear in wh ...
【RO】Robust formulation controls higher moments
Navigator Robust formulation Skew normal distribution Empirical Tests Matlab Code Robust formulation ...
【STL】rb_tree (multi)set (multi)map
rb_tree rb_tree源码实现 G2.9版本的rb_tree源码 rb_tree底层实现红黑树,其示意图和代码如下: 4个需要注意的模板参数:Key,Value,KeyOfValue,Comp ...
【cpu处理器分析】【AP】【bp】【cp】
BP基带处理器: 其实很多玩家都只听过基带这个东西,但不知道这个到底是什么.基带处理器管理的是手机一切无线信号(除了wifi,蓝牙,NFC等等),一款手机支持多少种网络模式,支持4G还是3G,都是 ...
【AP_EJOR】Robust solutions to multi-objective linear programs with uncertain data(1)
Navigator Paper Link Introduction main contributions Preliminaries Paper Link Robust solutions to mu ...
【翻译】Robust High-Resolution Video Matting with Temporal Guidance
Robust High-Resolution Video Matting with Temporal Guidance 论文地址 RobustVideoMatting 代码地址论文阅读笔记版权声明 ...
【AP】On the Bayesian interpretation of Black-Litterman(2)
导航 Pre Link An empirical example for APT Simultaneous views on drift and volatility Pre Link On the ...

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

Pre Blog Link

Robust optimization and asymmetric distribution

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

最新文章

热门文章