Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$.
Now say we have a equation $P_{t+1}=u\cdot (A_{m}-Y_{t+1})^{+}$ where $(Y_{m}-Y_{t+1})^{+}$ = $ \max \{(A_{m}-Y_{t+1}),0\}$ and $u$ is a constant.
So $P_{t+1}=u\cdot(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})^{+}$.
Can we write $E[P_{t+1} \mid Y_{t}]= u\cdot \int_{-\infty}^{(A_{m}-a - \rho Y_{t})}(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})\phi(\epsilon_{t+1})d\epsilon_{t+1}$ ?