suppose I have a jump-drift process $y_t$ such that $y_t$ drifts at a rate $-\beta y_t dt$ and at some poisson rate $\lambda$ $y_t$ is drawn from a normal distribution with mean zero and variance $\sigma^2$. What is the distribution of $y_{t+\Delta}$?
Is the following correct: after $\Delta$ units of time, either $y_t$ didn't switch and $y_{t+\Delta} = e^{-\beta\Delta}y_t$, with complementary probability there has been a switch and hence $t_{t+\Delta}$ is a normal with mean zero and variance $\sigma^2$. Hence
$$y_{t+\Delta}\vert y_t \sim \mathcal{N}\left(e^{-(\beta + \lambda)\Delta}y_t,\quad (1-e^{-\lambda\Delta})^2\sigma^2\right)$$
