How can we define the Stratonovich integral rigorously? 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 $$\mathcal E_0:=\left\{\Phi\in\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}):\Phi\text{ is }\mathcal F\text{-progressively measurable}\right\}$$ 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).

 A: I think there might be a typo in your write up. Take a look at either chapter 8 from the book http://press.princeton.edu/titles/7566.html (http://www-math.mit.edu/~dws/ito/ito8.pdf). I will try to give a brief explanation, hope it helps. 
Essentially the reason for defining Ito's integral of the process adapted process $h(s)$ with respect to Brownian Motion $W$ as the limit 
$$\label{eqn1} \tag 1
\int_0^t h(s)dW(s) := \lim_{n \to \infty } \sum_{i=0}^n h(t_{i}\wedge t)( W( t_{i+1}\wedge t ) - W( t_{i}\wedge t ) )
$$ for any sequence of random partitions tending towards the identity ucp(uniformly over compacts in probability) is the ability to use Ito's isometry. In other words, the fact that for a simple process, $\xi(t) = \sum_i \xi_i 1_{(t_{i+1},t_i]}(t)$ with $\xi_i$ $\mathcal{F}_{t_i}-$measurable, the identity 
\begin{equation}
\mathbf{E} \left[ \int_0^t\xi(s)dW(s) \right]^2= \mathbf{E}\int_0^t\xi(s)^2ds
\end{equation}
holds, allows us to extend the class of processes we can integrate to a large class of processes: adapted with $\mathbf{P}\left( \int_0^t\xi(s)^2ds < \infty\right)=1.$ Having said this if you change the point in \eqref{eqn1} from $t_i$ to anything else bigger that $t_i$, Ito's isometry is not valid anymore. Stochastic integrals can still be defined through the Skorohod integral (https://en.wikipedia.org/wiki/Skorokhod_integral) for which I don't think there is a straightforward explanation. 
Formally speaking when people refer to Stratonovich integral, I don't think the mean the limit in (1) with $h$ evaluated at the point $(t_i + t_{i+1})/2$, I think they mean the limit of sums of the form 
$$
\sum_{i=0}^n \left( \frac{ h(t_{i}\wedge t) + h(t_{i+1}\wedge t)}{2} \right )( W( t_{i+1}\wedge t ) - W( t_{i}\wedge t ) ),
$$
see Section 8.1.2 from Strook's book attached, 
Of course Ito isometry does not apply straightforwardly in this case either. But not simple algebra implies 
$$ \tag 2
\left( h(t_{i}) + h(t_{i+1}) \right )( W( t_{i+1} ) - W( t_{i} ) )/2 = h(t_{i}) ( W( t_{i+1} ) - W( t_{i} ) )+\left( h(t_{i+1})-h(t_{i}) \right )( W( t_{i+1} ) - W( t_{i} ) )/2,
$$
here I substitute $t_i \mapsto t_i\wedge t$ to avoid writing minimums everywhere. Now upon taking the sum and then the limit, the first term converged to the Ito integral while the second converges to the quadratic covariation $[h,W]$. With this intuition in hand, you arrive to the definition of the Stratonovich integral:
$$
\int_0^t h(s)\circ dW(s) = \int_0^t h(s)dW(s) + \frac{1}{2} \left[ h,W \right](t).
$$
And the appropriate set of $h$ are the ones for which both the Ito integral and the quadratic covariation are defined. This is how is defined in Protter's book that @NawafBou-Rabee posted on the comments.
Also @pavel mentioned rough paths which is a further (algebraic flavored) way of defining all sorts of integrals, but I let someone else with better expertise on that area comment.
Hope it helps!
