Extended CIR and discretization

Question

Did someone know how to discretize this process efficiently :

$dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$

I am looking for something more sophisticated than the trivial Euler Schema :

$X(t_{k+1}) =X(t_{k}) + \kappa[\theta(t_k)-X(t_{k})]\Delta t + \sigma \sqrt{X(t_{k})}\Delta_k W$

Thanks in advance,

Answer 1

Milstein Scheme

This scheme is described in Glasserman (2003) and in Kloeden and Platen (1992) for general processes.Hence, for simplicity, we can assume that the Stochastic Process is driven by the SDE \begin{align} &dX_t=\Xi(t,X_t)dt+\Sigma(t,X_t)dW_t\\ \end{align} Milstein discretization is, \begin{align} dX_{t+\Delta t}=X_t+\Xi(t,X_t)dt+\Sigma(t,X_t)\sqrt{\Delta t}\,Z+\frac{1}{2}\Sigma(t,X_t)\frac{\partial\Sigma(t,X_t)}{\partial X_t}\Delta t(Z^2-1) \ \end{align} where Z is is a standard normal variable.The coefficients of C.I.R process are \begin{align} \Xi(t,X_t) = \kappa(\theta(t) − X_t) \end{align} and \begin{align} \Sigma(t,X_t) = \sigma\sqrt X_t \end{align} by application of Milstein scheme,we have

\begin{align} dX_{t+\Delta t}=X_t+\kappa(\theta(t) − X_t)\Delta t+\sigma\sqrt{\Delta t \,X_t}\,Z+\frac{1}{4}\sigma^2\Delta t(Z^2-1) \end{align} Other Methods as follow

1. Transformed volatility scheme
2. IJK Scheme
3. Quadratic exponential method