Use Theorem 15.15.c) to show that the stochastic integrals for the right and left continuous simple processes \(f(t, \omega):=\sum_{j=1}^{n} \phi_{j-1}(\omega) \mathbb{1}_{\left[s_{j-1}, s_{j}\right)}(t)\) and \(g(t, \omega):=\) \(\sum_{j=1}^{n} \phi_{j-1}(\omega) \mathbb{1}_{\left(s_{j-1}, s_{j}\right]}(t)\) coincide.

Data From Theorem 15.15

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15.15 Theorem. Let (B+)to be a BM, (F)to be an admissible filtration and f = . Then we have for all t