WtW_tWt 与 ∫0tWsds\int_0^t W_s ds∫0tWsds 的相关系数 What is the correlation 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 What is the correlation between WtW_{t}Wt and ∫0tWsds\int_{0}^{t}W_{s}ds∫0tWsds ? 解析 令 It=∫0tWsdsI_t=\int_0^t W_s dsIt=∫0tWsds。 协方差： Cov(Wt,It)=∫0tE[WtWs]ds=∫0ts ds=t22.\operatorname{Cov}(W_t,I_t)=\int_0^t \mathbb{E}[W_tW_s]ds=\int_0^t s\,ds=\frac{t^2}{2}.Cov(Wt,It)=∫0tE[WtWs]ds=∫0tsds=2t2. 方差：Var(Wt)=t\operatorname{Var}(W_t)=tVar(Wt)=t，且 Var(It)=∫0t∫0tmin(s,u) ds du=t33.\operatorname{Var}(I_t)=\int_0^t\int_0^t \min(s,u)\,ds\,du=\frac{t^3}{3}.Var(It)=∫0t∫0tmin(s,u)dsdu=3t3. 因此相关系数 Corr(Wt,It)=t2/2t⋅t3/3=32.\boxed{\operatorname{Corr}(W_t,I_t)=\frac{t^2/2}{\sqrt{t\cdot t^3/3}}=\frac{\sqrt{3}}{2}}.Corr(Wt,It)=t⋅t3/3t2/2=23.