Let
- $T>0$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\in[0,\:T]}$ be a right-continuous filtration on $(\Omega,\mathcal A)$
- $B$ be a Brownian motion on $(\Omega,\mathcal A,\mathcal F,\operatorname P)$
Note that $$\mathcal E_0\left\{\Phi\in\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}):\Phi\text{ is }\mathcal F\text{-progressively measurable}\right\}$$$$\mathcal E_0:=\left\{\Phi\in\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}):\Phi\text{ is }\mathcal F\text{-progressively measurable}\right\}$$ is a closed subspace of $\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]})$, $$\mathcal E:=\left\{\Phi\in\mathcal E_0:\Phi=\sum_{i=1}^n\zeta_{i-1}1_{(t_{i-1},\:t_i]}\text{ for some }n\in\mathbb N\text{, }0\le t_0<\cdots<t_n\le T\text{ and }\zeta_0,\ldots,\zeta_{n-1}\in\mathcal L^0(\operatorname P)\right\}$$ is a dense closed subspace of $\mathcal E_0$, $$\mathcal M^2:=\left\{X\subseteq\mathcal L^2(\operatorname P):X\text{ is an almost surely right-continuous }\mathcal F\text{-martingale with }\operatorname P\left[X_0=0\right]=1\right\}$$ equipped with $$\left\|X\right\|_{\mathcal M^2}^2:=\sup_{t\in[0,\:T]}\left\|X_t\right\|_{\mathcal L^2(\operatorname P)}^2=\left\|X_T\right\|_{\mathcal L^2(\operatorname P)}^2$$ is a complete semi-normed space and $$\mathcal M^2_c:=\left\{X\in\mathcal M^2:X\text{ is almost surely continuous}\right\}$$ is a closed subspace of $\mathcal M^2$. Let $$(\Phi\cdot B)_t:=\sum_{i=1}^n\zeta_{i-1}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)\;\;\;\text{for }t\in[0,\:T]$$ for $\Phi\in\mathcal E$ of the form as in the definition of $\mathcal E$. Then $\Phi\cdot B\in\mathcal M^2_c$ and $$\mathcal E\to\mathcal M^2_c\;,\;\;\;\Phi\mapsto\Phi\cdot B\tag 1$$ is a linear isometry. Thus, $(1)$ can be uniquely extended to a linear isometry $\mathcal E_0\to\mathcal M^2_c$. Let $$\mathfrak T:=\left\{\left\{t_0,\ldots,t_n\right\}:n\in\mathbb N\text{ and }0=t_0<\cdots<t_n=T\right\}$$ and $$\left|\varsigma\right|:=\max_{1\le i\le n}(t_i-t_{i-1})\;\;\;\text{for }\varsigma=\left\{t_0,\ldots,t_n\right\}\in\mathfrak T\;.$$
Question:$\;\;\;$Let $\Phi\in\mathcal E_0$, $\varsigma=\left\{t_0,\ldots,t_n\right\}\in\mathfrak T$, $\lambda\in[0,1]$, $$\tau_i:=(1-\lambda)t_{i-1}+\lambda t_i\;\;\;\text{for }i\in\left\{1,\ldots,n\right\}$$ and $$\Phi_t^{\varsigma,\:\lambda}:=\sum_{i=1}^n\Phi_{\tau_i}1_{(t_{i-1},\:t_i]}(t)\;\;\;\text{for }t\in[0,T]\;.$$ If $$\Phi_t\in\mathcal L^2(\operatorname P)\;\;\;\text{for all }t\in[0,T]\tag 2$$ and $$\left(t\mapsto\Phi_t\right)\in C^0\left([0,T],\mathcal L^2(\operatorname P)\right)\;,\tag 3$$ then $$\left\|\Phi^{\varsigma,\:\lambda}-\Phi\right\|_{\mathcal E}\xrightarrow{\left|\varsigma\right|\to 0+}0\;.\tag 4$$ If $\lambda=0$, then $(4)$ implies $$\left\|\left(\Phi^{\varsigma,\:0}\cdot B\right)_T-\left(\Phi\cdot B\right)_T\right\|_{\mathcal L^2(\operatorname P)}=\left\|\Phi^{\varsigma,\:0}\cdot B-\Phi\cdot B\right\|_{\mathcal M^2}\xrightarrow{\left|\varsigma\right|\to 0+}0\tag 5$$ by definition of $\Phi\cdot B$. $(5)$ is the reason why many books define the Itō integral as the "mean-square limit" of $$\sum_{i=1}^n\Phi_{t_{i-1}}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)\;,$$ i.e. the $\mathcal L^2(\operatorname P)$-limit of $(\Phi^{\varsigma,\:0}\cdot B)_T$, as $|\varsigma|\to0+$ in the first place. I don't like this definition, cause it obscures what's actually happening and unnecessarily restricts the class of integrands.
I want to study the relation between the Itō (which corresponds to the choice $\lambda=0$) and the Stratonovich integral (which corresponds to $\lambda=1/2)$ and the way we can convert one into the other. My problem is, that I can't find any book which introduces the Stratonovich integral rigorously. Motivated by the last paragraph, all authors simply define the Stratonovich integral to be the mean-square limit of $$\sum_{i=1}^n\Phi_{\tau_i}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)$$ as $|\varsigma|\to0+$ with $\lambda=1/2$ (and hence $\tau_i=(t_{i-1}+t_i)/2$) and they don't even state under which assumptions on $\Phi$ this limit even exists!
So, the question is: How can we define the Stratonovich integral as rigorously as in the construction of the Itō integral that I've described above? Since I want to compare both integrals and derive a correction term for the conversion, it's clear that the corresponding classes of integrands must be somehow compatible (otherwise, we couldn't compare their integrals).