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In Öksendal he defines the Ito integral for adapted processes, and everything works out smoothly. However, my professor tells me that in much of the standard litterature, e.g. Williams: Diffusions, Markov Processes and Martingales vol 2, the integral is only defined for predictable processes, a class strictly smaller than that of adapted processes. My question is then, why? What is the practical difference?

Severin
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    If I remember well, Oksendal book is integration with respect to continuous semimartingale, you need predictable condition when you integrate over jump process like Lévy process and stuff check George Lowther blog "almost sure" blog to get the full story – TheBridge Jan 03 '22 at 21:43
  • Ah yes, this seems to be true. In Karatzas Shreve's Brownian Motion and Stochastic Calculus he writes:

    "In order to define the stochastic integral with respect to general martingales [...] (possibly discontinuous, such as the compensated Poisson process), one has to select an even narrower class of integrands among the so-called predictable processes. "

    Thanks!

     – Severin Jan 06 '22 at 21:40

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