Solving a BK equation

[Maelstro]

Hello everyone

I'm trying to solve this equation I got by using Itô lemma on some Black-Karasinski model.

[math]d r(t)=r(t)\left(k \ln (\theta)+\frac{\sigma^2}{2}-k \ln (r(t))\right) d t+\sigma r(t) d W(t)[/math]
but I'm having a bit of trouble expressing the short rate, and then generating a few trajectories in python.

Thanks in advance to anyone who takes the time to help me

 

[ngnguyen]

Hint: Denote [imath]x(t) = \ln r(t)[/imath] and apply the Ito's lemma
[math]dx(t) = k(\ln(\theta) - x(t))dt + \sigma dW_t \tag{1}[/math]From [imath](1)[/imath], we find that [imath]x(t)[/imath] follows the OU process and can be calculated analytically.