A dilemma about the definition of the stochastic integral $\int_a^b\Phi\:{\rm d}W$

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $T>0$
• $(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration of $\mathcal A$
• $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$

$\Phi:\Omega\times[0,T]\to\mathbb R$ is called elementary $\mathcal F$-predictable $:\Leftrightarrow$ $$\Phi=\sum_{i=1}^k1_{(t_{i-1},\:t_i]}^{[0,\:T]}\eta_i\tag1$$ for some $k\in\mathbb N$, $0\le t_0<\cdots<t_k\le T$ and $\mathcal F_{t_{i-1}}$-measurable $\eta_i:\Omega\to\mathbb R$ with $|\eta_i(\Omega)|\in\mathbb N$ for $i\in\left\{1,\ldots,k\right\}$. Let $$\mathcal E:=\left\{\Phi:\Omega\times[0,T]\to\mathbb R\mid\Phi\text{ is elementary }\mathcal F\text{-predictable}\right\}$$ be equipped with the norm inherited from $L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}\right)$ and $$\mathcal I^2:=\left\{\Phi\in\mathcal L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}\right):\Phi\text{ is }\mathcal F\text{-predictable}\right\}\;.$$ If $\Phi\in\mathcal E$, then $$\int\Phi\:{\rm d}W:=\sum_{i=1}^k\eta_i\left(W_{t_i}-W_{t_{i-1}}\right)$$ and $$\int_a^b\Phi\:{\rm d}W:=\int 1_{(a,\:b]}^{[0,\:T]}\Phi\:{\rm d}W\tag2$$ for $0\le a\le b\le T$. Moreover, $$(\Phi\cdot W)_t:=\int_0^t\Phi\:{\rm d}W\;\;\;\text{for }t\in[0,T]\;.$$ Now, $$\mathcal E\ni\Phi\mapsto\Phi\cdot W\tag3$$ is a linear isometry into the space of square-integrable continuous $\mathcal F$-martingales $M_c^2(\mathcal F,\operatorname P)$ equipped with the usual norm. $\mathcal E$ is a dense subset of $$\mathcal I^2:=\left\{\Phi\in\mathcal L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}\right):\Phi\text{ is }\mathcal F\text{-predictable}\right\}$$ and hence there is a unique isometric and linear extension $\Xi$ of $(3)$ to $\mathcal I^2$. If $\Phi\in\mathcal I^2$, then $$\Phi\cdot W:=\Xi(\Phi)$$ and $$\int\Phi\:{\rm d}W:=(\Phi\cdot W)_T\;.$$

If $0\le a\le b\le T$, how should we define $$\int_a^b\Phi\:{\rm d}W\tag4$$ for $\Phi\in\mathcal I^2$?

The definition found in the most books is, $$\int_a^b\Phi\:{\rm d}W:=(\Phi\cdot W)_b-(\Phi\cdot W)_a\;.\tag5$$ But there is an obvious problem with $(4)$: If $X:\Omega\to\mathbb R$ is bounded and $\mathcal F_a$-measurable, shouldn't the object $$\int_a^bX\Phi\:{\rm d}W$$ be well-defined? However, with $(4)$ this is not the case, since $X\Phi$ is obviously not $\mathcal F$-predictable and hence $X\Phi\cdot W$ is undefined. So, $(4)$ is a bad choice. Another option would be $$\int_a^b\Phi\:{\rm d}W:=\int1_{(a,\:b]}\Phi\:{\rm d}W\;.\tag6$$ But with $(5)$ we encounter an other problem: The processes $\Phi\cdot W$ and $$\left(\int_0^t\Phi\:{\rm d}W\right)_{t\in[0,\:T]}$$ should be indistinguishable; but with $(5)$ they are only modifications.

So, do we need to mimic the described construction of the stochastic integral for any possible initial value $a\in[0,T]$ and define $(4)$ in terms of a restriction, i.e. $$\int_b^T\Phi\:{\rm d}W:=\int_b^T\left.1_{(b,\:T]}^{[a,\:T]}\Phi\right|_{\Omega\times[b,\:T]}\:{\rm d}W$$ for $\Phi:\Omega\times[a,T]\to\mathbb R$ suitable in the obvious manner?

Answer 1

Conventionally, the stochastic integral $\Phi\cdot W$ is defined for $\Phi\in\mathcal I^2$ as a continuous martingale via the Ito isometry, and then $\int_a^b \Phi\,dW$ is just notation for $(\Phi\cdot W)_b-(\Phi\cdot W)_a$. The subtlety you raise is basically our sloppy identification of $(\Phi\cdot W)_{t\wedge b}$ with $\int_0^t 1_{(0,b]}\Phi\,dW$. For a fixed $b\in[0,T]$ these two (viewed as martingales in $t$) are indistinguishable. For fixed $t$, the former is a.s. continuous in $b$ while the latter is merely continuous in probability (and in $L^2$) in $b$. Of course, $(b,t)\mapsto (\Phi\cdot W)_{t\wedge b}$ provides an a.s. jointly continuous modification of $\int_0^t 1_{(0,b]}\Phi\,dW$. (The paper "Calcul stochastique dépendant d'un paramètre" by C. Stricker and M. Yor [Z. Wahrsch. Verw. Gebiete 45 (1978) 109–133] addresses such issues in a broader context; see https://link.springer.com/article/10.1007%2FBF00715187.)