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_{t}^{2},
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_{t}dt + σS_{t} dW_{t}^{1},
where the two increments W_{t}^{1} and W_{t}^{2}are independent Brownian motion process with
dW_{t}^{1} dW_{t}^{2 }= 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_{t}e^{kt} dt + σe^{kt} dWt^{2}
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?