Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \end{align} Moreover, suppose that there is a stock $S^{(1)}$ such that \begin{align} d S^{(1)}_{t} = \alpha S^{(1)}_{t} \, dt + \sigma S^{(1)}_{t}dW_{t}^{(1)}. \end{align} Assume that $a,b,\sigma_{r},\alpha,\sigma>0$ and that the initial values of the processes are known constants and that $W^{(0)}$ and $W^{(1)}$ are independent.

First question: Is $r_{t}$ adapted to the (augmented) natural filtration of $S^{(0)}$ and $S^{(1)}$?

Second question: Given $T>0$, does the Novikov condition hold, i.e. \begin{align} \mathbb{E}\left[\text{e}^{\frac{1}{2}\int_{0}^{T}\left(\frac{\alpha-r_{t}}{\sigma}\right)^{2}dt}\right]<\infty? \end{align}



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