Consider the following model.
$X_{n+1}$ given $X_n, X_{n-1},...,X_0$ has a Poisson distribution with mean $\lambda=a+bX_n$ where $a>0,b\geq{0}$. If $b\geq 1$, then $E[X_{n+1}|X_n]= a+bX_n > X_n$. Hence $(X_n)$ is a submartingale. Can we transform it into a nonnegative martingale/supermartingale?
Thanks in advance!
The meaning of "transform" in "transform it into a nonnegative martingale/supermartingale" is not completely clear but in the supercritical case, that is, if $b\gt1$, $M_n=b^{-n}((b-1)X_n+a)$ defines a positive martingale $(M_n)_{n\geqslant0}$.
In the critical case $b=1$, it is known (but not obvious) that the process $(X_n)_{n\geqslant0}$ is recurrent for every $a$ small enough and transient for every $a$ large enough (in the specific Poisson setting, the former means $a\leqslant\frac12$ and the latter means $a\gt\frac12$).
These are valid for every reproducing and immigration distributions. Using the specific Poisson setting, one can show that $$ M_n=\frac1{1+X_n} $$ is a supermartingale if the parameters $(a,b)$ are such that $s(a)\geqslant 1/b$, where $$s(a)=\inf\limits_{0\leqslant t\leqslant1-1/\mathrm e}\mathrm e^{at}(1-t). $$ This covers the case when $b=1$ and $a\geqslant\mathrm e/(\mathrm e-1)$.