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,


1 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
  • $\begingroup$ Changed the typo $\frac{\partial\Sigma(t,X_t)}{\partial t}$ to $\frac{\partial\Sigma(t,X_t)}{\partial X_t}$. $\endgroup$
    – Gordon
    Jul 3, 2015 at 15:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.