We work on a filtered probability space. Let $H$ be predictable, locally bounded, and non-negative, and let $X$ be a local sub-martingale. The goal is to show that the stochastic integral $H\cdot X$ is a local sub-martingale.
My progess so far: By the Doob-Meyer decomposition we know that there exists a unique local martingale $M$ and a unique predictable, increasing process $A$ started at $0$ such that $X=M+A$. We know that $H\cdot M$ is a local martingale (since $H$ is predictable and locally bounded, and $M$ is a local martingale). Moreover, $\int H \,dA$ (a pathwise Stieltjes integral) is predictable; this is discussed in Stochastic integral with respect to a predictable process again predictable?.
My question: Why is $\int H \,dA$ increasing?
Because $H$ is non-negative and $A$ is increasing.
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