# Milstein discretization of the CIR process

Given the CIR process $$\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$$ - I want to show that its Milstein scheme is $$\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\Delta\epsilon_i}+ 0.25\sigma^2\Delta\epsilon^2 d$$ - so that the innovation term can be written as $$\ X_{i+1} = A_i(\epsilon_{i+1} + \sqrt{C_i}^2) + B_i = A_iZ_{i+1} + B_i$$, where A_i, B_i, C_i are functions of of X_i and I want to give A_i, B_i, C_i for the CIR.

• What is your question, do you want help deriving the step? What did you try? – Bob Jansen Apr 11 at 13:22
• yes exactly I need help to rewrite the innovation term with the parameters A_i, B_i and C_i as functions of X_i. I was able to derive the solution of the process, but don't really know where to start with the rewriting process. – 303 Waters Please Apr 12 at 8:17