Increasing process is natural if and only if it is predictable

Question

Let $\{A_t\}_{t \geqslant 0}$ be a progressively measurable stochastic process defined on a filtered space $(\Omega, F, \{F_t\}_{t \geqslant 0},P)$ such that every sample path is right continuous and left hand limit exists (RCLL).$\{A_t\}_{t \geqslant 0}$ is said to be increasing if $A_0 \equiv 0$ and $A_s \leqslant A_t$ almost surely whenever $0 \leqslant s \leqslant t$ and predictable if $A_t$ is measurable with respect to $F_{t-}= \sigma(\bigcup_{s<t}F_s)$.

Let $\{A_t\}_{t \geqslant 0}$ be an increasing process defined on a filtered space $(\Omega, F, \{F_t\}_{t \geqslant 0},P)$. $\{A_t\}_{t \geqslant 0}$ is said to be natural if, for every bounded martingale $\{M_t\}_{t \geqslant 0}$ of RCLL samples paths and every $t \in (0,\infty) $, we have:

\begin{equation*} \mathbb{E}[\int_0^tM_sdA_s] = \mathbb{E}[\int_0^tM_{s-}dA_s] \end{equation*}

The integrals are treated as Lebesgue-Stieljes integrals and $M_{s-}(\omega)$ is the limit of $M_t(\omega)$ as $t \to s$ while $t<s$.

We want to prove that an increasing process is natural if and only if it is predictable. I have many references to this result but none of which is direct. Is there a direct reference for its proof?

Answer 1

Regarding the natural and predictable increasing process, the following books are helpful:

P. A. Meyer, Probability and Potentials, Blaisdell, Waltham, 1966. p.111--, \S VII.3 D.18, Th.19-20.

C. Dellacherie & P. Meyer, Probabilities and Potential B, volume 72 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 1982. p.126--, Th.VI 61--.

Answer 2

P. Protter's book Stochastic Integration and Differential Equations discusses natural vis-a-vis predictable. He relies more on natural in the first edition of the book, less so in the second edition; it might be helpful to look at both.