# How do I create error bounds after parameter calibration?

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.