In my recent work I've become interested in working with the minimizer of $$ \mathbb{E}[(Y-Z)^2] + \lambda P(Z), $$ $Y$ is an observed random variable, $P$ is a positive-convex penalty function, $Z$ is a measurable random variable with respect to the $\sigma$-algebra generated by $Y$ and $\lambda\geq 0$.

If $\lambda=0$ this is the MSE and is minimized by the conditional expectation. My question is, is there a well developed theory of "penalized conditional expectation"?
That is a theory studying the above equation's minimizer? So far I have found nothing really. All help is greatly appreciated.