The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is:
$$\frac{dP}{dN}(T|\mathcal{F}_t)=\frac{1}{N(T)P(t,T)}$$
Suppose I now want to price an option on stock $S_t$, but under non-deterministic rates, i.e. $N(t)=e^{\int_0^tr(t)dt}$. I am interested in $C(t_0,T)=\mathbb{E}_{t_0}^N\left[e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]$. Suppose I want to use the Radon-Nikodym from above, I could do the following:
$$C(t_0,T)=P(t_0,T)\mathbb{E}_{t_0}^N\left[P(t_0,T)^{-1}e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]=P(t_0,T)\mathbb{E}_{t_0}^P\left[(S_T-K)^+\right]$$
But now I'd need to figure out the process $S_t$ under $P(t,T)$. Supposing that under $N(t)$, $S_t$ follows the process:
$$S_t=S_0e^{\int_0^tr(t)dt-0.5 \sigma^2 t +\sigma W_t}$$
How can I use the Radon-Nikodym:
$$\frac{1}{N(t) P(t,T)}=\frac{1}{e^{\int_0^tr(t)dt} \mathbb{E}^N[e^{\int_0^tr(t)dt}]}$$
To apply the Cameron-Martin-Girsanov Theorem to the process for $S_t$ under $N_t$?
Here is my attempt:
We want Radon-Nikodym that looks like $exp\left\{\int_0^T \mu (t) dW_t-0.5 \int_0^T \mu^2 (t) dt\right\}$. By CMG Theroem, applying this Radon-Nikodym to a Brownian $W_t$ will result in a new measure under which the same Brownian motion will acquire a drift $\int_0^T \mu(t) dt$.
Instead, we have:
$$exp\left\{-\int_0^Tr(t)dt\right\}*constant$$
Is there a way to proceed further?
