我如何計算 $Covleft(int_{0}^{s}W_u,du,,,,,int_{0}^{t}W_v,dvright)$
我該如何計算？\begin{align} Cov\left(\int_{0}^{s}W_u,du,,,,,\int_{0}^{t}W_v,dv\right) \end{對齊}
感謝您的關注。
你知道$E\left[\int_{0}^{s}W_udu\right]=E\left[\int_{0}^{t}W_vdv\right]=0$。根據定義 \begin{align} & Cov\left(\int_{0}^{s}W_u,du,,,,\int_{0}^{t}W_v,dv\right)=E \left[\int_{0}^{s}W_u,du\int_{0}^{t}W_v,dv\right]-0 \end{align} 然後 \begin{align} & Cov\left( \int_{0}^{s}W_u,du,,,,\int_{0}^{t}W_v,dv\right)=\int_{0}^{s}\int_{0 }^{t}E,[W_uW_v],,du,dv \end{align} 由於 $E,[W_uW_v]=min {,u,,v }$ 因此 \begin{對齊} & Cov\left(\int_{0}^{s}W_u,du,,,,\int_{0}^{t}W_v,dv\right)=\int_{0}^ {s}\int_{0}^{t}min {,u,,v },,du,dv \end{align} 對於 $s<t$ 的情況
\begin{align} \int_{0}^{s}\int_{0}^{t}min {,u,,v },,du,dv=\int_{0}^ {s}\int_{0}^{s}min {,u,,v },,du,dv+\int_{0}^{s}\int_{s}^{t} min {,u,,v },,du,dv \end{align} 我們立即有 \begin{align} \int_{0}^{s}\int_{0}^{t }min {,u,,v },,du,dv=\frac{1}{3}s^3+\frac{1}{2}(ts)s^2 \end {align} 按照與上述相同的步驟，對於 $s > t$ 的情況，我們還可以顯示 \begin{align} \int_{0}^{s}\int_{0}^{t}min {\ ,u,,v },,du,dv=\frac{1}{3}t^3+\frac{1}{2}(st)s^2 \end{align} 因此， \begin{align} Cov\left(\int_{0}^{s}W_u,du,,,,\int_{0}^{t}W_v,dv\right)=\frac{1 }{3}min{s^3,,,t^3}+\frac{1}{2}|ts|min{s^2,,,t^2} \end {對齊}