Nonnegative predictable transforms preserve submartingale gains

Set $Z=H\cdot X$ and $d_k=E[X_k-X_{k-1}\mid {\mathcal{F}}_{k-1}]$. By the transform domain $Z$ is adapted and integrable with $Z_0=0$. Linearity and the known-variable identity give $d_k=E[X_k\mid {\mathcal{F}}_{k-1}]-X_{k-1}\ge 0$ a.s. by the submartingale hypothesis Martingale submartingale and supermartingale.

Apply [F2] with input $X_k-X_{k-1}$ and finite known factor $H_k$. The input and its product are integrable, so [F2] also guarantees $H_k{d}_k\in L^1$ and gives $E[Z_k-Z_{k-1}\mid {\mathcal{F}}_{k-1}]=H_k{d}_k\ge 0$ a.s. Adding the known $Z_{k-1}$ yields $E[Z_k\mid {\mathcal{F}}_{k-1}]\ge Z_{k-1}$ for every $k\ge 1$. For timewise bounds, [F1] verifies the product hypothesis. AC is inherited from the conditional classes; no positivity of $X$ or boundedness of an unbounded $H$ is inferred.