Let's assume that \begin{equation} Y_t -Y_{t-1} = \beta(X_t - X_{t-1}) + \alpha + \varepsilon_t \end{equation}
where $Y_t -Y_{t-1}$, $X_t-X_{t-1}$, $\varepsilon_t$ fulfill all assumptions of linear regression model.
Does such relation always imply that $Y_t$ has a trend, i.e. $Y_t = \beta X_t + \alpha t + \gamma + \varepsilon_t'$?
All I came to is that the opposite relation always holds.