1
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ This question and its answer by yourself are as absurd, mathematically speaking, as on the other page mathoverflow.net/questions/82274/…. Re the question, I would be curious to see any "textbooks result" "given for integer $\alpha$" (even for $\alpha=1$, the integral as written above does not make sense). $\endgroup$
    – Did
    Jan 26, 2012 at 20:42
  • $\begingroup$ Just put the right answer, taking into account the discussion with George. You will see from this that my answer here is obviously correct and does not worth a downvote. Thanks. $\endgroup$
    – Jon
    Jan 26, 2012 at 20:45
  • $\begingroup$ You seem to be in a delusional state about the status of the discussion with George. // Unsurprisingly, no answer to my request for (at least) one example of the textbooks mentioned in your question. $\endgroup$
    – Did
    Jan 26, 2012 at 20:49
  • $\begingroup$ @Didier: Sorry Didier but I am not in a delusional state. I have just accepted your answer. I agree with you that there is a problem. What else? Just George declared that $(dW)^\alpha=0$ with $\alpha>2$ and this is what I obtain below. So, why downvote? I live mathematics like a pleasure and I may be wrong. It occurred to me sometime in my twenty years long career. I just learn from my errors and go ahead. Of course, you have been helpful and I gave you thanks accepting your answer and deleting wrong statements. Period. $\endgroup$
    – Jon
    Jan 26, 2012 at 21:00

1 Answer 1

0
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.