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).
> 
> 
> 



How can we define the Stratonovich integral rigorously?
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!


