I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

Question

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$

I solved and got $$X\left(t\right)= a\left(\frac{T-t}{T}\right)+\frac{bt}{T}+\left(T-t\right)\int_{0}^{t}\frac{dB\left(s\right)}{T-s}$$

I want to show $$\operatorname{Cov}\left(X\left(t\right),X\left(s\right)\right)=\min\left(s,t\right)- \frac{st}{T}$$

I start $$\operatorname{Cov}\left(X\left(t\right),X\left(s\right)\right)=E\left[\left(X\left(t\right)-E\left[X\left(t\right)\right]\right)\left(X\left(s\right)-E\left[X\left(s\right)\right]\right)\right]$$

and we know that $$E\left(X\left(t\right)\right)=a\left(\frac{T-t}{T}\right)+\frac{bt}{T}$$

I got $$\operatorname{Cov}\left(X\left(t\right),X\left(s\right)\right)=E\left[\left(\left(T-t\right)\int_{0}^{t}\frac{dB\left(s\right)}{T-s}\right)\left(\left(T-s\right)\int_{0}^{t}\frac{dB\left(t\right)}{T-t}\right)\right]$$

and I can not Continue. thanks for help.

Answer 1

As usual, this is Itô's isometry: the factors $(T-t)$ and $(T-s)$ are deterministic hence, assuming without loss of generality that $s\leqslant t$, one is left with $$ E\left[\int_{0}^{t}\frac{dB\left(u\right)}{T-u}\cdot\int_{0}^{s}\frac{dB\left(u\right)}{T-u}\right]=\int_0^{s}\frac{du}{(T-u)^2}=\frac{s}{T(T-s)}. $$ The RHS times $(T-t)\cdot(T-s)$ is $$ \frac{s\cdot(T-t)}{T}=s-\frac{st}T. $$