Suppose I have the following process
$$y_{t+1} = y_t \exp (\varepsilon_{t+1})$$
where $\varepsilon_{t+1}\sim _{iid} N(\frac{\sigma v^2 }{2},v^2 )$, $v> 0$. I would like to calculate $E[y_{t+1}|y_t]$
For this, I apply conditional expectation in the equation above. So
$$E[y_{t+1} | y_t ] =E [y_t \exp (\varepsilon_{t+1})| y_t] = y_t E [ \exp (\varepsilon_{t+1})| y_t].$$
Now I’m unsure of saying that $\exp (\varepsilon_{t+1})$ and $y_t$ are independent. I suspect this to be true, since $y_t = y_{t-1} \varepsilon_{t} $ and $\{\varepsilon_t\}_{t=0}^{\infty}$ is a iid process. If the independence of $\exp (\varepsilon_{t+1})$ and $y_t$ is true, I can conclude that
$$E[y_{t+1} | y_t ] = y_t E [ \exp (\varepsilon_{t+1})| y_t] = y_t E [ \exp (\varepsilon_{t+1})] = y_t \exp( \frac{\sigma v^2 }{2} + \frac{v^2}{2}).$$
Is my argument correct?