This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
2026-03-27 17:36:20.1774632980
Extensions of the Ito integral
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It all depends on what you want the set $\mathcal{V}^{m\times n}$ to be used for. In Oksendal's monograph, what it denotes is the following. Each set $\mathcal{V}^{m\times n}(0,T)$ denotes the set of $m\times n$ matrix processes with entries integrable over $(0,T]$. With Oksensdal's definition of $\mathcal{V}^{m\times n}$, this set becomes the set of $m\times n$ matrix processes with entries integrable over any interval (S,T] for $0 \le S \le T$. It's just a convenience, really, not any profound "extension" of the integral.