One can rewrite the problem in terms of products of i.i.d. random variables as follows.

Assume that $X_t$ has distribution density $f_t$. Then, the relation between $f_t$ and $f_{t+1}$ means that one can choose $X_{t+1}=X_tZ_{t+1}$, where the $Z_t$ are i.i.d. and $Z_t=A$ or $B$ or $A+B$, with probabilities $p_{12}A$, $p_{21}B$ and $p_{22}(A+B)$, respectively.

Hence one must assume that the three numbers $p_{12}A$, $p_{21}B$ and $p_{22}(A+B)$ sum to $1$ for the relation between $f_t$ and $f_{t+1}$ to make sense, and when this is so, $X_t=X_0Z_1Z_2\cdots Z_t$.

This tells you that $E(X_t)=E(X_0)m^t$ for every $t$, with $m=E(Z_1)$, that is, $m=p_{12}A^2+p_{21}B^2+p_{22}(A+B)^2$; that $t^{-1}\log X_t$ converges almost surely to $\mu=E(\log Z_t)$, that is, $\mu=p_{12}A\log A+p_{21}B\log B+p_{22}(A+B)\log(A+B)$; and that $\log X_t$ follows the multinomial distribution with parameters $t$ and $(p_{12}A,p_{21}B,p_{22}(A+B))$ (or more precisely, the convolution of this with the distribution of $\log X_0$).

Unfortunately, these remarks do not help much if one is interested in closed form formulas. Sorry.

One can rewrite the problem in terms of products of i.i.d. random variables as follows.

Assume that $X_t$ has distribution density $f_t$. Then, the relation between $f_t$ and $f_{t+1}$ means that one can choose $X_{t+1}=X_tZ_{t+1}$, where the $Z_t$ are i.i.d. and $Z_t=A$ or $B$ or $A+B$, with probabilities $p_{12}A$, $p_{21}B$ and $p_{22}(A+B)$, respectively.

Hence, for the relation between $f_t$ and $f_{t+1}$ to make sense, one must assume that the three nonnegative numbers $p_{12}A$, $p_{21}B$ and $p_{22}(A+B)$ sum to $1$, and when this is so, $X_t=X_0Z_1Z_2\cdots Z_t$.

This tells you that:

• $E(X_t)=E(X_0)m^t$ for every $t$, with $m=E(Z_1)$, that is, $$ m=p_{12}A^2+p_{21}B^2+p_{22}(A+B)^2. $$
• $t^{-1}\log X_t$ converges almost surely to $\mu=E(\log Z_1)$, that is, $$ \mu=p_{12}A\log A+p_{21}B\log B+p_{22}(A+B)\log(A+B). $$
• $\log X_t$ follows the multinomial distribution with parameters $t$ and $(p_{12}A,p_{21}B,p_{22}(A+B))$, or, more precisely, the convolution of this multinomial with the distribution of $\log X_0$.