I've been trying to get a grasp on some of the basics of interest rate modeling, and am looking to simulate rates using the 2 factor Hull White model, which I am aware offers a more realistic model of rates which allows for imperfect correlation between instantaneous forward rates.
I've found resources on the web which reduce the model to an additive Gaussian one, where one has for the short rate $r(t)$:
$r(t) = \varphi(t) + x_1(t) + x_2(t)$
where $x_1, x_2$ are mean reverting processes governed by:
$dx_1(t) = -a_1x_1(t)\cdot dx + \sigma_1\cdot dW_1(t)$
$dx_2(t) = -a_1x_2(t)\cdot dx + \sigma_2\cdot dW_2(t)$
with $dW_1(t)dW_2(t) = \rho$, and $\varphi(t)$ is deterministic and chosen to fit the initial forward rate curve $f(0,t)$:
$\varphi(t) = f(0,t) + \frac{\sigma_1^2}{2a_1^2}\left(1-e^{-a_1t}\right)^2+\frac{\sigma_2^2}{2a_2^2}\left(1-e^{-a_2t}\right)^2+\rho\frac{\sigma_1\sigma_2}{a_1a_2}\left(1-e^{-a_1t}\right)\left(1-e^{-a_2t}\right)$
This tells me how to simulate the short rate (by updating $x_1,x_2$ at each time increment and adding to $\varphi$), but my question is, how could one simulate the evolution of the whole forward curve? I have also found (unwieldy) closed-form expressions for $P(t,T)$ the price of a term $T$ zero coupon bond at time $t$, from which you can obtain the forward curve, but is there a way to generate the forward curve at time $t+\Delta t$ by updating the curve at time $t$, akin to the way we can do it for the short rate $r(t)$?
