I have a power transform $f$ I am applying to an Ornstein-Uhlenbeck stochastic process $\{X(t), t\geq 0\}$:

$$dX(t) = \kappa (\mu - X(t)) dt + \sigma dW_{t}.$$

From here, I was able to plug in my sample data I am trying to fit into the toy model, write the transition density from $t$ to $t+1$ and write down my likelihood function,

$$L(\hat{X(1)}, \dots \hat{X(n)} | \hat{X(0)}) = \prod_{i=1}^{n}p(X(i-1), X(i))$$

take the logarithm, take the derivatives, and solve a system of linear equations for optimal $\hat{\kappa}, \hat{\mu}, \hat{\sigma}$.

However, at this point, I noticed that some papers I've looked at include error bounds in the form of a constant band say, $\hat{\kappa} \pm C$.

At this point, I feel like I cross the barrier from what I consider probability theory and calculus into the realm of statistics and fall flat on my face.

I have -no idea- how one might make an error bound of some form for these parameters and I'm having trouble even finding notes on how this is usually done.

When I read about confidence intervals, I think I understand on some superficial level that I want to treat $\kappa, \mu, \sigma$ as a random variable so I can write down something like $\mathbb{P}[-\alpha/2 < \kappa < \alpha/2] = 0.95$ but I don't know how I would do that since I only have 1 set of "real" data and presumably I would need to know something about the distribution of said data to even write down a probability.


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