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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?

L. Johnson
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1 Answers1

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Just learned that the $\star$-integral is called the Hänggi-Klimontovich integral which has interesting properties in connection with the relativistic Langevin equation.

When the quadratic covariaton of $G$ and $W$ exists then
\begin{align} \langle G,W\rangle_s&=\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}\Big(G(t_{k+1})-G(t_k)\Big)\Big(W(t_{k+1})-W(t_{k})\Big)\\[3mm] &=\int_0^sG(t)\star\,dW(t)-\int_0^sG(t)\,dW(t)\,. \end{align} In contrast, the Stratonovich integral $\int_0^sG(t)\circ\,dW(t)$ is known to satisfy \begin{align} \color{red}{\frac{1}{2}}\langle G,W\rangle_s&=\frac{1}{2}\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}\Big(G(t_{k+1})-G(t_k)\Big)\Big(W(t_{k+1})-W(t_{k})\Big)\\ &=\lim_{\Delta t\rightarrow 0}\sum_{k=1}^{N-1}\left(\frac{G(t_{k+1})+G(t_k)}{2}-G(t_k)\right)\Big(W(t_{k+1})-W(t_{k})\Big)\\[3mm] &=\int_0^sG(t)\circ\,dW(t)-\int_0^sG(t)\,dW(t)\,. \end{align} From both these equations it follows that \begin{align} \underbrace{\int_0^sG(t)\star\,dW(t)}_{\text{Hänggi-Klimontovich}}=2\underbrace{\int_0^sG(t)\circ\,dW(t)}_{\text{Stratonovich}}-\underbrace{\int_0^sG(t)\,dW(t)}_{\text{Ito}}\,. \end{align}

JwJJJJ
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Kurt G.
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