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Ito integral is generally defined through the sums

$$S_n=\sum_{i=1}^nG(\tau_i)(W(t_i)-W(t_{i-1}))$$

then the limit $\lim_{n\rightarrow\infty}S_n$ must exist in the rms sense. This definition can be easily generalized to cases like

$$S_n=\sum_{i=1}^nG(\tau_i)(W(t_i)-W(t_{i-1}))^\alpha$$

with $\alpha\in\mathbb{R}$ and $\alpha>0$ or

$$S_n=\sum_{i=1}^nG(\tau_i)|(W(t_i)-W(t_{i-1})|$$

where different classes of Wiener processes are considered. From these one can formally define, on the lines of stochastic calculus, $[dW]^\alpha$ or $|dW|$ and the corresponding Ito integral. It is not difficult to show that, at least for some cases, these reduce to the standard form $adW+bdt$ and so are computable.

Given this, I have done the following formal manipulations for stochastic calculus. I consider the square root of a Wiener process and I put

$$dX=[dW]^\frac{1}{2}=(\mu_0+\mu_1|dW|+\mu_2dt)\left(\frac{1-i}{2}{\rm sign}(dW)+\frac{1+i}{2}\right)$$

then I do the square and I get

$$(dX)^2=\mu_0^2{\rm sign}(dW)+dW$$

rather than $(dX)^2=dW$ ($\mu_1=1/2\mu_0$, $\mu_2=-1/8\mu_0^3$): There is a residual Bernoulli process with mean 0 and variance 1 that should be eventually interpreted in the sense of a Ito integral. This can be accomplished by noting that ${\rm sign}(x)=x/|x|$ but this integral would be eventually zero as

$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n{\rm sign}(W(t_i)-W(t_{i-1}))=0.$$

My question is this: Is this manipulation correct? Have I got the square root of a Wiener process? If yes, what meaning should I assign to the residual Bernoulli process?

Thanks.

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When one uses notations (and maybe concepts) as unconventional as $[dW]^{1/2}$ and $|dW|$, the least one can do is to define precisely their meaning. –  Did Jan 24 '12 at 9:01
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Jon: You probably do not realize the irony of the last sentence of your comment... But let us stick to the content of your post: (1) If $[dW]^{1/2}$ means $dt$, then write $dt$. (1') If $dX=[dW]^{1/2}$ and $[dW]^{1/2}$ is $dt$, then $X=t+$constant? (2) What is $|dW|$? (3) What process does sign$(dW)$ refer to? // Curious to know which textbooks on stochastic calculus you are relying on. –  Did Jan 24 '12 at 9:37
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Maybe it is inappropriate to intervene the discussion: I agree with @Didier that the symbols (like $[dW]^\frac{1}{2}$)in your (Jon's) question are undefined. In the last comment, you came up with stuff (like $(W(t)-W(s))^\alpha$) which is on the other hand pretty well defined. Maybe you want to restate your question in integral form? Are you interested in $\langle |W(t)-W(s)|^{1/2} \rangle$??? –  Fabian Jan 24 '12 at 12:42
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@Jon: let me intervene a tiny bit. (a) Since you goal is to get an answer to your question, perhaps it would be a good idea to make sure that your question is written in a way that, at the very least, experts in the field can understand your notation. While I am by no means an expert in SDE, Didier does have pretty good credentials there. (b) BTW, in regards to $|dW|$, you may want to recall that almost surely a Wiener process has unbounded variation in any interval, this explains why Didier says you won't see formulae like $\int|dW|$. –  Willie Wong Jan 24 '12 at 15:08
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... which may be (but I am not sure) the root of your problems: your formula for $dX$ ends up setting two "infinite" values to be equal. (BTW, formally speaking, are you sure $\left([dW]^{1/2}\right)^2$ by your definition equals $dW$ and not $|dW|$?) –  Willie Wong Jan 24 '12 at 15:12

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