# How can the increments of a CIR process be derived?

For a CIR process, which has SDE $$dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t$$ how can I derive the increments over the discrete time-interval from $$r_t$$ to $$r_{t+1}$$?

• $dr_t$ is replaced by $\Delta r_t$, $dt$ is replaced by $\Delta t$ and $dW_t$ is replaced by $\xi_i \sqrt{\Delta t}$ where $\xi_i$ is random $N(0,1)$ Oct 21 '20 at 10:52
• An alternative, "exact" simulation would be via the non-central Chi-Squared distribution, for example explained here: deriscope.com/docs/Andersen_Jaeckel_Kahl_2010.pdf Oct 21 '20 at 11:27
• Oct 21 '20 at 11:28
• @noob2, would you be able to explain why that is the case? Oct 21 '20 at 12:04
• It is the Euler Maruyama method, the simplest (but not the only, as Kermitfrog pointed out) way to simulate an SDE in discrete time. $dW_t$ is random zero-mean and has a variance proportional to $dt$, so the standard deviation is proportional to $\sqrt{dt}$ Oct 21 '20 at 12:21

I am not totally sure I understand what you want to achieve. It seems like you are interested in discretizing CIR SDE. This can be done using the Euler-Murayama scheme for an equidistant decomposition of the time interval $$[0, T]$$, $$\{0=t_0<\dots.
First of all, let us write the model dynamics: $$r_t=r_0+\alpha\int_0^t(\mu-r_s)ds+\sigma\int_0^t\sqrt{r_s}dW_s$$
We need to discretize this process: $$r_{t+\Delta t}=r_t+\alpha(\mu-r_t)\Delta t+\sigma\sqrt{r_t}W_{\Delta t}$$ with $$\Delta t=\frac{T}{n}$$ and $$W_{\Delta t}\sim\mathcal N\left(0,\frac{T}{n}\right)\Rightarrow W_{\Delta t}=\sqrt{\frac{T}{n}}\varepsilon,$$ with $$\varepsilon$$ being a standard normal random variable.