# How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?

My concern arises from the fact that in the following system:

$x_k = (\mu, \sigma)^T = x_{k-1}$

$Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$

that I cannot separate the states I want to estimate (X = $(\mu, \sigma)^T$) from the noise $Z_k$ in my measurement equation. I tried taking logs in the measurement equation but then now I don't know how to deal with the variance of $log(Z_k)$ and also $log(Z_k)$ is not a zero mean process. Can the system be specified as a linear discrete system?

• If you're going to put $\mu$ and $\sigma$ as states, then no, this is not a linear system. Typically you would estimate those as parameters, by maximum likelihood. The MLE is even available in closed form here. Is there some reason you're not doing that instead? – Chris Haug Apr 29 '18 at 16:39
• It was supposed to be a simple exercise in using the Kalman filter. How would maximum likelihood estimation be used in conjunction with a linear Kalman filter? What would be the states for example? – Vykta Wakandigara Apr 29 '18 at 19:27
• For a fixed set of parameters, the Kalman filter gives you the likelihood as a byproduct of its computation. You then optimize over that. The state would just be $Y_k$ itself (i.e. with no observation error). – Chris Haug Apr 29 '18 at 21:10
• But it seems like estimating $\mu$ need $\sigma$ and vice versa. Are they done simultaneously? What is the algorithm for incorporating MLE in a Kalman filter? – Vykta Wakandigara Apr 29 '18 at 22:12
• You can estimate by optimizing over all parameters jointly. If your goal is to just get an estimate, you can use the closed form for the MLE conditional on the first observation (i.e. take the mean and variance of $\Delta Y_k$, it will work fine). If you have other specific initial conditions for the first time point (e.g. some fixed normal distribution, or exact diffuse conditions), the Kalman filter will output the parameters of the filtering distributions which you need to compute the likelihood. You could then use something like BFGS to optimize that. What initial conditions did you want? – Chris Haug Apr 29 '18 at 23:50