Thank you for your notes. However, I have a question. Let X_t be a continuous martingale. What should be the requirements for the process b in the stochastic integral

\int_0^t dX_s/b(X_s)

For some reason in Liptser, Shiryaev (2001) on page 200, the requirement is b^2 >= c > 0. Is it really necessary for the integrand to be bounded?

Thank you.

Best,

Alex. ]]>

Thanks a lot for your great blog. I learn a lot from it.

Here is my question. Sorry if it is stupid. You write above:

“Also, the stochastic integral should satisfy bounded convergence in probability. That is, if is a sequence of predictable processes converging to a limit …”

You have probably explained somewhere in what sense the sequence of predictable processes does converge, but I am unable to find where. Or is it obvious ? Is it UCP convergence or almost sure uniform convergence on compact sets ?

Thanks and regards ]]>

“Then, you can take which converges uniformly and, hence, is continuous.”

Here, you define the integral to be 0, outside the set A which actually yields \omega-wise continuity (not P-a.s. continuity). This way of defining the integral is fine. However, if we do not make this assumption and the only thing that we know is that it is possible to find a sequence of simple processes (continuous and adapted) that converge uniformly P-a.s. is it still enough to add all zero sets from F_oo (not complete) to F_t to claim that the limit process is adapted?

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When defining the stochastic integral, it is possible to find simple processes such that their integrals are continuous and converge uniformly with probability 1, and thus, the continuity is preserved almost surely. Since the convergence is only in almost sure sense, the limit does not have to be adapted unless the filtration is complete.

Note quite. If a sequence of (simple) integrals converges uniformly with probability 1, to a limit , then there exists a set with probability 1 such that convergence is uniform on A. We know that A can be taken to be in because each of the simple integrals is -measurable. Then, you can take which converges uniformly and, hence, is continuous. So, we need to know that A is in for all t, which is guaranteed by the requirement that all zero probability sets in are in . Completeness is a stronger condition than is necessary.

]]>We want the integral wrt a BM to be continuous and adapted. When defining the stochastic integral, it is possible to find simple processes such that their integrals are continuous and converge uniformly with probability 1, and thus, the continuity is preserved almost surely. Since the convergence is only in almost sure sense, the limit does not have to be adapted unless the filtration is complete.

SImilar construction in Liptser, Shiryaev “Statistics of random processes”, V.2, p.102 also requires the filtration to be augmented by all zero sets from the original sigma-algebra which is assumed to be complete.

Could you please explain these deferences in construction of stochastic integrals (for example, wrt Brownian motion)? For what cases we have to require the filtration (and or the original probability space) to be complete? ]]>

I have a question on that completeness issue. You say that instead of completeness of filtrations we can add zero probability sets from F_oo to F_t. Does F_oo have to be complete?

Thank you.

Alex. ]]>