The scalars $x_{t}, y_{t}, u_{t}$, are related by the equations

$$x_{t}=x_{t-1}+u_{t-1}, \quad y_{t}=x_{t-1}+\eta_{t-1}, \quad t=1, \ldots, T,$$

where $\left\{\eta_{t}\right\}$ is a sequence of uncorrelated random variables with means of 0 and variances of 1. Given that $\hat{x}_{0}$ is an unbiased estimate of $x_{0}$ of variance 1 , the control variable $u_{t}$ is to be chosen at time $t$ on the basis of the information $W_{t}$, where $W_{0}=\left(\hat{x}_{0}\right)$ and $W_{t}=\left(\hat{x}_{0}, u_{0}, \ldots, u_{t-1}, y_{1}, \ldots, y_{t}\right), t=1,2, \ldots, T-1$. Let $\hat{x}_{1}, \ldots, \hat{x}_{T}$ be the Kalman filter estimates of $x_{1}, \ldots, x_{T}$ computed from

$$\hat{x}_{t}=\hat{x}_{t-1}+u_{t-1}+h_{t}\left(y_{t}-\hat{x}_{t-1}\right)$$

by appropriate choices of $h_{1}, \ldots, h_{T}$. Show that the variance of $\hat{x}_{t}$ is $V_{t}=1 /(1+t)$.

Define $F\left(W_{T}\right)=E\left[x_{T}^{2} \mid W_{T}\right]$ and

$$F\left(W_{t}\right)=\inf _{u_{t}, \ldots, u_{T-1}} E\left[\sum_{\tau=t}^{T-1} u_{\tau}^{2}+x_{T}^{2} \mid W_{t}\right], \quad t=0, \ldots, T-1$$

Show that $F\left(W_{t}\right)=\hat{x}_{t}^{2} P_{t}+d_{t}$, where $P_{t}=1 /(T-t+1), d_{T}=1 /(1+T)$ and $d_{t-1}=V_{t-1} V_{t} P_{t}+d_{t} .$

How would the expression for $F\left(W_{0}\right)$ differ if $\hat{x}_{0}$ had a variance different from $1 ?$