The Hull White model is [d X_{t}=a(t)left(b(t)-X_{t} ight) d t+sigma(t) d W_{t}] In this problem take (a(t)=theta_{1}

The Hull White model is

\[d X_{t}=a(t)\left(b(t)-X_{t}\right) d t+\sigma(t) d W_{t}\]

In this problem take \(a(t)=\theta_{1} t, b(t)=\theta_{2} \sqrt{t}\) and \(\sigma(t)=\theta_{3} t\) where \(\theta_{1}=2, \theta_{2}=0,7, \theta_{3}=0.8\), and \(\Delta t=0.001\).

(a) Generate a single path of the process by choosing a \(\Delta t=0.001\) from \(t=0\) to \(t=1\).

(b) Use the Sim.DiffProc package in \(\mathrm{R}\) to estimate the parameters \(\theta_{1}, \theta_{2}\), \(\theta_{3}\). Use the Euler method and compare with the known values of the parameters.

(c) Repeat part (b) with all the methods available in the package. Write a conclusion based on the results obtained.