# How to estimate the variance of this stochastic process?

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.

• yes it is correct if your $\approx$ is a "reasonable" estimate. – MJ73550 Aug 5 '16 at 11:56
• it is correct unless S(t) and K(t) are correlated because in this case var(S+K) is not equal to var(S) + var(K). – Malick Aug 5 '16 at 22:19
• Does it really matter, if S and K are correlated? If you have a sample of S(t) + K(t) you can estimate the variance of that sample regardless of the distribution of S and K, right? – Ami44 Aug 7 '16 at 14:58
• What is the process in question? Where did you get the formula for the variance? This is interesting for the background of the question. – SRKX Jan 5 '17 at 6:44

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 $\sigma^2$.
$$\Lambda_t = S_t + K_t + \epsilon_t$$ and model that resiudal $\epsilon$.
$$\text{Var}(dL) = \text{Var}(dK) + \text{Var}(dS) + \text{cov}(dS, dK)$$ I hope u have some simultaneous observations on $dK$ and $dS$ to estimate the covariance or a good assumption about it.
• If he has a simultaneous sample of $S$ and $K$, them he can compute a sample of $\lambda$ and does not need to care about the covariance... – SRKX Jan 5 '17 at 6:46