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Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE \begin{equation} dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1} \end{equation} ?

Notice that $\{r_t\}$ is our process of interest, $k_r$ and $\theta_r$ are constant parameters and $dW_t$ denotes Wiener increment.

I would need a closed-form solution of $(1)$ (or even a good analytical proxy) so as to compare it with solution of my numerical simulation of $(1)$. I am aiming at determining the order of accuracy of my numerical solution, so I would need an analytical solution (or something like that) of $(1)$ so as to compute the average error of my simulation code, that is $\mathbb{E}\left(r_t^{TRUE}-r_t^{SIMULATED}\right)$.

Any suggestion or good source?

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    $\begingroup$ There's no closed form solution for the square root diffusion $r_t$ that I know of. You can however write down its (conditional) moments in closed-form. Similarly, you can write down closed-form solutions for the prices of zero coupon bonds and bond options under the CIR model. You can compare both the moments and the prices with your simulation results. $\endgroup$ – Kevin Aug 6 '20 at 20:51
  • $\begingroup$ Also asked here: math.stackexchange.com/questions/3782360/… $\endgroup$ – fesman Aug 7 '20 at 5:34

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