Is there a name for the stochastic integral using the right end of each interval?

I am studying the Ito integral which is defined such that:

$\int_{0}^{s}G(t)dW(t)=\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}G(t_k)[W(t_{k+1})-W(t_{k})]$

Now, I know the Stratonovich form uses the midpoint of each interval. But I was wondering if there is also a name for the version that uses the right end point of each interval? i.e.:

$\int_{0}^{s}G(t)\star dW(t)=\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}G(t_{k+1})[W(t_{k+1})-W(t_{k})]$

Where $\star$ is some new notation used for this new type of integral. If there is, is there then some easy way to rewrite:

$dX=b(X(t),t)dt+\sigma(X(t),t)dW(t)$

In this new format?

Solution 1:

The correct definition of the Stratonovich integral uses the average of $G(t_k)$ and $G(t_{k+1})$ rather than $t_k$ and $t_{k+1}$ (which is the midpoint) : $$ \int_{0}^{s}G(t)\circ dW(t)=\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}\frac{G(t_k)+G(t_{k+1})}{2}[W(t_{k+1})-W(t_{k})] $$ see this Q&A and references therein. Clearly, this implies $$ \underbrace{\int_{0}^{s}G(t)\circ dW(t)}_{\text{Stratonovich}}=\frac{1}{2}\underbrace{\int_{0}^{s}G(t)\,dW(t)}_{\text {Ito}}+\frac{1}{2}\int_{0}^{s}G(t)\star dW(t)\,. $$ Just learned that the $\star$-integral is called the Hänggi-Klimontovich integral which has interesting properties in connection with the relativistic Langevin equation.