I am currently working on stochastic processes and I have met a stumbling block in the Ito integral


$$\int\_{t\_0}^tdt'G(t')[dW(t')]^\alpha$$


with $\alpha\in\mathbb{R}$ and $\alpha>0$. Textbooks result is given for integer $\alpha$ but not in the more general case that could not exist. Of course, also some good references are welcome.



A class of Ito integrals
It can be shown that $[dW(t)]^\alpha=0$ with $\alpha\in\mathbb{R}$ and $\alpha\ge 3$ generalizing the integer case.


Let us consider the stochastic differential equation $dX(t)=[dW(t)]^\alpha$ with $\alpha>0$. We can write the solution in the form $X(t)=X(t\_0)+\int\_{t\_0}^t[dW(t)]^\alpha$ with the integral in the Ito sense. Then, we have to evaluate this integral with the sum
\begin{equation}
 S\_n=\sum\_{k=1}^n[W(t\_k)-W(t\_{k-1})]^\alpha.
\end{equation}
The power of the Brownian process can be evaluated in the following way
\begin{equation}
 [W(t\_k)-W(t\_{k-1})]^\alpha = [(1+W(t\_k)+W(t\_{k-1}))-1]^\alpha=
\end{equation}
\begin{equation}
 (-1)^\alpha\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)(-1)^{l\_1}(1+W(t\_k)+W(t\_{k-1}))^{l\_1}=
\end{equation} 
\begin{equation}
 (-1)^\alpha\sum\_{l\_1=0}^\infty\sum\_{l\_2=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)\left(\begin{array}{c}
 l\_1 \\
 l\_2
\end{array}\right)(-1)^{l\_1}
[W(t\_k)-W(t\_{k-1})]^{l\_2}
\end{equation}
provided $|W(t\_k)-W(t\_{k-1})|<1$. Now, we can use stochastic calculus to remove powers higher than 2 and it is easy to see that
\begin{equation}
 S\_n=(-1)^\alpha\sum\_{k=1}^n\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)(-1)^{l\_1}-
 (-1)^\alpha\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)l\_1(-1)^{l\_1}\sum\_{k=1}^n[W(t\_k)-W(t\_{k-1})]+
\end{equation}
\begin{equation}
 (-1)^\alpha\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)\frac{l\_1(l\_1-1)}{2}(-1)^{l\_1}
 \sum\_{k=1}^n[W(t\_k)-W(t\_{k-1})]^2.
\end{equation}
So, we have the required expansion with coefficients
\begin{eqnarray}
 \mu\_0&=&\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)(-1)^{l\_1} \nonumber \\
 \mu\_1&=&\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)l\_1(-1)^{l\_1} \nonumber \\
 \mu\_2&=&\sum\_{l\_1=0}^\infty\left(\begin{array}{c}
 \alpha \\
 l\_1
 \end{array}\right)\frac{l\_1(l\_1-1)}{2}(-1)^{l\_1}
\end{eqnarray}
Now we see immediately that $\mu\_0=\left.(1-x)^\alpha\right|\_{x=1}=0$. Besides, we get immediately the result that, for any real $\alpha\ge 3$, we have again $[dW(t)]^\alpha=0$ as in this case the coefficients are all zero when $\mu\_1$ and $\mu\_2$ are evaluated thorugh Abel summation. Finally, when $0<\alpha<1$ both the coefficients $\mu\_1$ and $\mu\_2$ are divergent and maybe no meaning can be attached to them (I have in mind summable divergent series here, any suggestion is greatly appreciated).


