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This question can be seen as an inverse of Why predictable processes?

I'm currently trying to get a basic understanding of stochastic integration (in particular Itô integrals), and I'm a bit confused about the introduction of different kinds of measurability.

I'm reading Kallenberg, and when they defined the integral wrt to the Brownian motion we in particular assumed the integrands to be progressive and when generalizing to semi-martingales with possible jump discontinuities we assume the integrands to be predictable.

I'm aware that predictability

  1. implies progressiveness
  2. is needed to ensure that the integrals are semi-martingales when the integrators might not be Brownian motions, so I understand why predictable processes are necessary to define.

My question is why do we define progressive processes then? Is it just to have a larger class of integrable processes or is there some other deeper reason?

Any help is greatly appreciated.

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    $\begingroup$ The comment here is related. I think it boils down to the integrator being BM allows for a larger class of integrands. There could also be historical reasons. Older books knew only the Ito integral w.r.t. BM. Progressive seems from those old days. $\endgroup$
    – Kurt G.
    Mar 12, 2022 at 11:42
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    $\begingroup$ Another interesting result is Bichteler's characterisation of semimartingales as the largest class of stochastic integrators. $\endgroup$
    – Kurt G.
    Mar 12, 2022 at 11:46
  • $\begingroup$ Thank you! This helps a lot! $\endgroup$ Mar 13, 2022 at 17:22

1 Answer 1

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"Progressive" is a natural notion in many ways. If $B$ is a progressive process then $\omega\mapsto\int_0^t B_s(\omega)\, ds$ is $\mathcal F_t$-measurable (provided the integral converges). If also $T$ is a stopping time then $B_T:\omega\to B_{T(\omega)}(\omega)$ is a random variable—more precisely, $\mathcal F_T$-measurable.

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