Consider the following $f_{t+1}(z)=p_{12} f_{t}(z/A)+ p_{21} f_{t}(z/B)+p_{22} f_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f_t$ is a probability distribution. Are there any nice distribution families that are preserved under the transformation? Fail that, are there $f_t$ such that $f_{t+1}$ has a closed form?

It's motivated by the following problem:

Let there be two simple bonds that either default or pay off a return on investment (That may or may not be correlated), denote the bonds as random variables $Z_1$ and $Z_2$.

Now throw in a population of investors, with wealth following a distribution $W$, investing some fixed percentage of their income in the two bonds(investing a fixed percentage is a Nash equilibrium under the model I'm working with). The resulting after-investment wealth distribution will be a mixture of dilations of the original distribution, I'm trying to find a distribution to work with that will make things simple when studying the behavior of the system over time.

Any ideas?