Is there a way to derive an expression for the density of the transformation of a continuous random variable $X$ (with prob. density $p(X)$ supported on $\mathbb{R}^d$) as follows?
$$Y = X + f(X)$$
What properties must $f$ satisfy for this to be possible?
I know that we can derive an expression for the density of a transformation $Z = f(X)$ when $f$ is a smooth, invertible function.
We have $$ \mathbb P(Y\le t) = \mathbb P(X + f(X)\le t) $$ so if we have a formula for $f$, gathering the conditions for which $X + f(X)\le t$, we can retrieve the density probability function $P_{X+f(X)}$ by differentiating $F_Y(t) = \mathbb P(Y\le t)$.
Example: $f=\rm Id$ gives $F_Y(t) = F_X(\frac t 2 )$.