hull-white modelas an extension of the vasicek

Question

Hull-White model

As an extension of the Vasicek model, Hull-White model (1990) assumed that the short interest rate process follows the mean-reverting stochastic differential equation SDE:

dr_t = k(Θ_t - r_t)dt +σdW_t²,

where k and σ are positive constants,  Θ_t  is time-dependent function which will be used to currently fit the term structure of interest rate in the market.

while the interest rates are considerable to be stochastic, the tradeable asset evolves according to a geometric Brownian motion. So under the risk-natural measure Q, the dynamic of the instantaneous asset price is given by

dS_t =rS_tdt + σS_t dW_t¹,

where the two increments W_t¹ and W_t²are independent Brownian motion process with

dW_t¹ dW_t² = ρ dt

 ρ representing the instantaneous correlation parameter between the asset price and the short interest rate.

Applying Ito lemma to (e^kt) to find the differential of a time-dependent function of a stochastic interest rate process

D(r e^kt) = e^kt dr + k r e^kt dt = kΘ_te^kt dt + σe^kt dWt²

Then integrate both sides over [s,t] and simplify the equation,

1. Complete what I did before to prove  it is Gaussian and then  is following the normal distribution in clear and specific points?