# How to design back-testing (validation) for such modified Vasicek model?

Consider a classical Black Scholes model ,

$$\frac{dS}{S} = \mu dt + \sigma dW$$ , where $dW$ is a Brownian motion, that $W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$.

The back-testing strategy is straight-forward: Once $\mu$ and $\sigma$ is recognized from the samples, $dS / S_t \sim N(\mu \cdot dt, \sigma^2 dt)$. So back-testing of the model becomes hypothesis testing of a normal distribution's mean and standard deviation.

However, I'm coming out with such a modified Vasicek model:

$$dr_t = a(b-r_t)dt + (c + d \cdot r_t) dW$$

This modifies the original Vasicek model: $dr_t = a(b-r_t)dt + \sigma dW$ as I notice the samples shows time-variant $\sigma(t)$ which has a strong linear correlation with $r_t$.

Now, how could I design the back-testing to validate my model?

I thought to back-test the $a$ and $b$ first, at least, $$E[dr_t + a \cdot r_t dt] = a \cdot b \cdot dt$$ , this is a constant.

But $dr_t + a \cdot dt$ 's standard deviation is not constant, I'm lost how to set the Hypothesis testing's criterion! Leaving alone how to back-test the $c$ and $d$ part?

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