This is my first time asking a question here, so please tell me if there is something I should ask differently.
We are in the setting of $(\mathbb R^n, \mathcal F, \mathbb Q)$ being a probability space, and $(S_t)_{t=1}^n$ where $S_t \colon \mathbb R^n \rightarrow R, (x_1,\ldots ,x_n) \mapsto x_t$ being a martingale in its own filtration denoted $(\mathcal F_t)_{t=1}^n$.
Now I am finding two definitions for a trading strategy:
- A trading strategy $ (\xi_t)_{t=1}^n $ is a predictable process, meaning that $\xi_t $ is $\mathcal F_{t-1}$-measurable,
- A trading strategy $ \Delta = ( \Delta_{t} )_{t=0}^{n-1} $ consists of bounded Borel measurable functions $\Delta_t\colon \mathbb R^t \rightarrow \mathbb R$ where $0\leq t < T$.
The explanation given for equivalence was that for a discrete process $(\xi_t)_{t=1}^n$ it holds that $(\xi_t)_t$ is predictable in the first sense if and only if $\xi_t \overset{\mathbb Q\text{-a.s.}}{=} \Delta_{t-1} (x_1,\ldots ,x_{t-1}) $ for some bounded measurable functions $\Delta_t\colon \mathbb R^t\rightarrow \mathbb R$. Again the filtration is the natural filtration of the coordinate process.
The backwards implication is clear (at least I hope I'm not making a mistake by thinking it is) and the forward implication is supposed to follow from some basic transformation lemma, however the only one I'm finding is the martingale transformation, with which I don't see a solution.
My question is why are the two definitions equivalent, and if it is indeed by the explanation given above, which transformation lemma is used and how?