CIR Process: Exact Distribution and Simulation Error Measurement

Summary

The document asks whether the Cox–Ingersoll–Ross short-rate stochastic differential equation has a closed-form solution or a useful analytical proxy. The motivation is to assess a numerical simulation by comparing simulated values with a true process and estimating average error, with the broader aim of determining the method’s accuracy order.

It provides the mean-reverting square-root diffusion equation and identifies its rate process, constant mean-reversion parameters, volatility parameter, and Wiener increment. It does not include an answer, a candidate solution, simulation results, or a discussion of how to define error when the exact path is not available. Thus it frames a numerical-methods question rather than presenting a validated benchmark; any comparison would need to distinguish pathwise error from distributional or moment-based checks.

Key ideas

• The CIR model describes a mean-reverting process with volatility proportional to the square root of its level.
• The document seeks an analytical benchmark for evaluating numerical simulation accuracy.
• It proposes comparing simulated and true values through an expected error measure.
• No solution, proxy, or numerical evidence is supplied in the document.

Tags

# CIR model. Is there a closed-form solution or even a good proxy of analytical solution?


# CIR model. Is there a closed-form solution or even a good proxy of analytical solution?












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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