# Calibration for CIR Model Discretization for Predictor Corrector and Milstein method

I'm new to Quantitative Finance. I've data which I need to fit a CIR model and estimate its parameters.

$$dX_{t+1} = a(b-X_{t})dt + \sigma \sqrt{X_t}dW_{t}$$

While I can fit and obtain parameparameterates using Euler-maurayana discretization and then linear regression closed form solution, I'd like to explore better approaches. This method was quite easy as after discretizing it, it became:

$$\frac{S_{t+1}-S{t}}{\sqrt S_{t}} = \frac{ab\Delta t}{\sqrt{S_{t}}} - a\sqrt{S_{t}}\Delta t + \sigma \sqrt{\Delta t} \epsilon_{t}$$

where $$\epsilon_{t}$$ is Normal(0,1)

I estimated a and b through OLS and got sigma estimate through error variance. I'd like to estimate parameters of this CIR model through better discretizing approaches. I tried discretizing the model using Milstein and Prediction and Error Correction Method. But both aren't in traditional linear regression form (either there are nonlinear terms or error terms aren't just normal). I'm not aware of how to proceed from here. Since there's stochastic component here, I don't know how to estimate these parameters (including sigma) with approaches (gradient descent or something).

• Use the transformation $y =\sqrt{x}$ and obtain the transformed SDE for which Milstein and Euler schemes are the same.