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CIR process

The CIR process or the Cox-Ingersoll-Ross process was actually treated by W. Feller long before Cox, Ingersoll and Ross. The name Feller process has however, been taken for a wider class of diffusion processes. The CIR process satisfies the SDE
dX(t)=κ(θ-X(t))dt+X(t)^1/2σdW(t),
X(0)=x₀.
This model is used as a model for short rates and for stochastic volatility e.g. in the Heston model. In order for the model to proper model non-negative entities such as volatility and interest rates we nedd to ensure that the process stays stricly positive. If 2κ&theta>σ² then X will not hit zero.
It is unfortunately not possible to find an explicit expression for the solution. But we can simulate using the Euler scheme. Below you can see a simuation where you can change the parameters.