# Incorporating explanatory variables in a Ornstein-Uhlenbeck model

In my research im dealing with a longitudinal data set, which consists of a single response variable and a set of several predictor variables, which comprises a mixture of individual specific constants and other time-varying variables. As my data is longitudinal, I have data on several individuals with a varying amount of (unevenly spaced) observations per individual.

For the response variable that I am modelling, it would be very untuitive to assume a mean-reverting process, where the mean is individual specific. I am familiar with the standard Ornstein-Uhlenbeck model and its parameters. However, I was wondering what the best way would be to incorporate the additional predictor variables into my model. The Ornstein-Uhlenbeck process can be summarized by:

$dy_{it} = \kappa(\mu_i-y_{it})dt + \sigma dW_t$,

where $y_{it}$ refers to the measured response variable of individual $i$ at time $t$. I was thinking I could estimate a specific mean $\mu_i$ for every individual in my dataset. However, I am not sure how I can inporate the additional predictor variables into my model. I could set $\mu_i$=$\mathbf{x}_{it}^\intercal\mathbf{\beta} + \alpha_i$, so that I am estimating a contstant per individual, as well as a general parameter vector that links the predictors to the mean of the Ornstein-Uhlenbeck process. However, I think the non-constant predictors mess up this approach, as I dont think that the mean $\mu_i$ is supposed to be time-varying for a specific individual.

What are your thoughts on this? Does anyone has a suggestion how to best incorporate additional explanatory variables in a Ornstein-Uhlenback model (related to longitudinal or panel data)?