How to estimate the variance of this stochastic process?

Question

I have an unobservable stochastic quantity $\lambda(t)$, which I analytically know the variance of, that is

$$\text{Var}(\lambda(t))= \frac{\theta \sigma^2}{2\kappa}$$

My goal is to get an estimate of $\sigma^2$.

I can observe S and K historically at times $t=1,2,..$, and know that the following approximately holds

$$ \lambda(t) \approx S(t)+K(t)$$

Does it make sense to simply take the sample variance of the time series $(S(t)+K(t))$, let's call it $\hat{\sigma}^2_S$, and say that a reasonable estimate of $\sigma^2$ is $2 \kappa\frac{\hat{\sigma}^2_S}{\theta}$?

It should be mentioned that the distribution of $\lambda$ is not pretty, but I have the mean and variance.

Answer 1

If $$ \Lambda = S + K $$ then you can look at samples of $S+K$ and estimate the variance of $\Lambda$ by the variance of $S+K$. If $\theta$ and $\kappa$ are known constants then you can do the algebra to derive $\sigma^2$.

The question is how the above equality holds. If it holds almost surely, then you are done. If it holds in probability then you are done too.

Maybe it helps to look at

$$ \Lambda_t = S_t + K_t + \epsilon_t $$ and model that resiudal $\epsilon$.

Answer 2

$$ \text{Var}(dL) = \text{Var}(dK) + \text{Var}(dS) + \text{cov}(dS, dK) $$ I hope u have some simultaneous observations on $dK$ and $dS$ to estimate the covariance or a good assumption about it.