Black Scholes model with Hull White interest rates
Mainly for my own reference, but also responding to popular demand: Here is a quick computation regarding risk-neutral modelling equity type assets following a geometric Brownian motion with interest rates following the Hull-White model in its exact simulation. It complements the Hull-White notes.
The black scholes model for a total return index $S$ (e.g. a not-dividend-paying stock) is $$\frac{dS_t}{S} = x_t dS + \sigma_S dW_t^S = x_t dS + \sigma_S \rho dW_t^x + \sigma_S \sqrt{1-\rho^2} dW_t^y.$$
Taking the logarithm and using Itô's formula we have $$d(\ln S_t) = \left( x_t - \frac{1}{2} \sigma_S^2 \right) dt + \sigma_s \rho dW_t^x + \sigma_s \sqrt{1-\rho^2} dW_t^y.$$
Thus $$ \ln S_t = \ln S_0 + \int_0^t x_u du - \frac{1}{2} \sigma_S^2 t + \sigma_s \rho \int_0^t dW_s^x + \sigma_s \sqrt{1-\rho^2} \int_0^t dW_s^y. $$
If $x_t$ is an OU process, i.e. $dx_t = - \alpha x_t + \sigma_x dW^x$, it has the form $$ x_t = \exp(-\alpha t) x_0 + \sigma_x \int_0^t \exp(\alpha (t-s)) dW^x_s $$ and its integral is $$ \int_0^t x_s ds = x_0 \frac{1}{\alpha} (\exp(-\alpha t)-1) + \frac{\sigma_x}{\alpha} \int_0^t (\exp(\alpha (t-s))-1) dW^x_s. $$
We plug this in to see $$ \begin{aligned} \ln S_t = &\ln S_0 - x_0 \frac{1}{\alpha} (\exp(-\alpha t)-1) - \frac{1}{2} \sigma_S^2 t + \frac{\sigma_x}{\alpha} \int_0^t (\exp(\alpha (t-s))-1) dW^x_s \\ & + \sigma_s \rho \int_0^t dW_s^x + \sigma_s \sqrt{1-\rho^2} \int_0^t dW_s^y. \end{aligned} $$
For the covariances, we have $$ \begin{aligned} Cov\left[\ln S_t, x_t \right] &= CoV\left[\int_0^t x_s ds, x_t\right] + \sigma_s \rho Cov\left[ \int_0^t dW_s^x, x_t \right] \\ & = CoV\left[\int_0^t x_s ds, x_t\right] + \sigma_s \sigma_x \rho Cov\left[ \int_0^t dW_s^x, \int_0^t \exp(\alpha (t-s)) dW^x_s \right] \\ & = CoV\left[\int_0^t x_s ds, x_t\right] + \sigma_s \sigma_x \rho E\left[ \left(\int_0^t dW_u^x \right) \left(\int_0^t \exp(\alpha (t-s)) dW^x_s \right)\right] \\ & = CoV\left[\int_0^t x_s ds, x_t\right] + \sigma_s \sigma_x \rho \int_0^t \exp(\alpha (t-s)) ds \\ & = CoV\left[\int_0^t x_s ds, x_t\right] + \sigma_s \sigma_x \rho \frac{1}{\alpha} ( \exp(\alpha t) - 1) ds \end{aligned} $$ and $$ \begin{aligned} Cov\left[\ln S_t, \int_0^t x_s ds \right] &= V\left[\int_0^t x_s ds\right] + \sigma_s \rho Cov\left[ \int_0^t dW_s^x, \int_0^t x_s ds \right] \\ & = V\left[\int_0^t x_s ds\right] + \frac{\sigma_x \sigma_s \rho}{\alpha} E\left[ \left(\int_0^t dW_u^x \right) \left( \int_0^t (\exp(\alpha (t-s))-1) dW^x_s \right)\right] \\ & = V\left[\int_0^t x_s ds\right] + \frac{\sigma_x \sigma_s \rho}{\alpha} \left ( \int_0^t \exp(\alpha (t-s)) ds - t \right) \\ & = V\left[\int_0^t x_s ds\right] + \frac{\sigma_x \sigma_s \rho}{\alpha} \left ( \frac{1}{\alpha} (exp(\alpha t)-1) - t \right). \end{aligned} $$