Let $Z_n$ be an $(F_n)$ adapted sequence of random variables satisfying
$$
E[Z_{n+1}|\mathscr{F}_n] \le (1+b_n)Z_n + c_n – d_n
$$
where $b_n,c_n,d_n$ are all non-negative sequences satisfying $\sum_{n=0}^\infty (b_n+c_n) < \infty$.
Therefore, $E[Z_{n+1}|\mathscr{F}_n] \le e^{b_n}Z_n + c_n – d_n \le e^{b_n}Z_n + c_n$ and as a result $X_n = e^{-B_{n-1}}Z_n – C_n$ is a supermartingale where $B_n = \sum_{i=0}^nb_i$ and $C_n = \sum_{i = 0}^{n-1}e^{-B_i}c_i$.
Since $(X_n)^- \le C_n < \sum_{j=0}^\infty c_j < \infty$ for every $n$ by the supermartingale convergence theorem, $\lim_{n \to \infty} X_n = X_\infty$ exists almost surely.
Question: Now, the author concludes that since $\sum_{j=0}^\infty (b_j+c_j) < \infty$ we have that $\lim_{n \to \infty}Z_n = Z_\infty$ almost surely. I don't quite see how this is obvious? I know $Z_n$ is a function of $X_n$, so if $X_n$ converges then $e^{-B_{n-1}}Z_n – C_n$ also converges. However, here we have $e^{-\sum_{i=0}^{n-1} b_i}Z_n – \sum_{i = 0}^{n-1}e^{-B_i}c_i$ – and I'm not sure how $\sum_{j=0}^\infty (b_j+c_j) < \infty$ is sufficient.
Best Answer
$Z_n=(X_n+C_n)e^{B_{n-1}} \to (X_{\infty}+C)e^{B}$ where $C=\sum e^{-b_n}c_n$ and $B=\sum b_n$. Note that $\sum c_n <\infty$ implies that $\sum c_ne^{-b_n} <\infty$.