Yes you can, but you probably want to at least be predictable, and you need to first develop some theory in order to give meaning to the integral, and impose sufficient conditions on the sample paths of W. The point of this post is that we do not require any prior knowledge of stochastic integration nor that we have well-behaved versions (such as right-continuous) of the process W.

]]>Do you know whether the Theorem 5 result , for a martingale, can be extended to a more general class of stochastic processes ? Maybe something like being integrable, adapted to the natural filtration of , a.s. cadlag and a.s. uniformly bounded on ?

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