I have a very fundamental question regarding simulation of DRIFTED geometric brownian motion.
We have the standard Blackos Scholes model:
$dS(t)=r S(t)dt+\sigma S(t) dW^{\mathbb{P}}(t)$, where $W^{\mathbb{P}}(t)$ is the standard Wiener process under probability measure $\mathbb{P}$.
If we want to simulate this, using constant $\Delta t$, we use the recursive formula:
$S_{t+1}=S_te^{(r-\frac{\sigma^2}{2})\Delta t+\sigma \sqrt{\Delta t} Z_t }$, where $Z_t \sim N(0,1)$.
Now assume that we want to change the drift such that:
$W^{\mathbb{Q}}(t) = W^{\mathbb{P}}(t) - \int_0^t \theta_s ds$ is a brownian motion under $\mathbb{Q}$ such that:
$dS(t)=(r + \sigma \theta )S(t)dt+\sigma S(t) dW^{\mathbb{Q}}(t)$
Now this is where I become unsure. If I want to simulate the drifted process, is it just fine to use the similar method as:
$S_{t+1}=S_te^{(r+\sigma \theta-\frac{\sigma^2}{2})\Delta t+\sigma \sqrt{\Delta t} Z_t }$, where $Z_t \sim N(0,1)$.
OR is it that I have to use $Z_t \sim N(\theta, 1)$?
Im not that strong in the change of measure part, so thats why I'm a bit unsure. Would appreciate for help.
Thanks