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I'm don't really understand what's the meaning of fractional derivative, neither where it apply in the nature. Nevertheless, I often see that formally for a Brownian motion, we use the notation $dB_t=(dt)^{1/2}$.

  • Q1) Does it mean that despite the fact that the Brownian motion has no derivative, it has a $\frac{1}{2}-$derivative ?

  • Q2) More generally, I know that if $F$ has bounded variation, then it's derivable a.e. So if $f$ has $p$-bounded variation (i.e. $$\lim_{n\to \infty }\sum_{a\leq t_0<t_1<...< t_n\leq b}|f(t_{i+1})-f(t_i)|^p<\infty,$$ but not $q-$bounded variation for all $q<p$, would it make sense to say that $df$ repreent the $\frac{1}{p}-$derivative of $f$ ? (i.e. $df=(dt)^{\frac{1}{p}}$).

  • Q3) Do you have an example of function that has quadric variation but is not of bounded variation on a compact set ? (in determinist case, i.e. not the Brownian motion).

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  • $\begingroup$ What have you tried? $\endgroup$ – user526015 Feb 28 '19 at 13:16
  • $\begingroup$ @Surb If you check out the help pages of MSE, e.g. here math.stackexchange.com/help/how-to-ask you'll find that you should first try to solve the problem yourself and let us know what you tried, why it didn't work out etc. Just asking questions without showing any attempt to solve it yourself makes it unlikely that people are willing to answer. $\endgroup$ – user526015 Feb 28 '19 at 13:26
  • $\begingroup$ @Surb Also if these are more complicated questions, it is always better to show some effort in solving them. I am however also interested in the answer (and thus gave your answer an upvote) and had hoped that you have some first thoughts on it that I could get my hands on.. However, good look with the question :-) $\endgroup$ – user526015 Feb 28 '19 at 13:45
  • $\begingroup$ I wrote an answer a while ago about fractional calculus, which I think might be relevant here. $\endgroup$ – polfosol Mar 9 '19 at 10:01
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I guess that in the definition of a function $f$ of bounded $p$-variation should be the condition $$\sup \sum_{a\leq t_0<t_1<...< t_n\leq b}|f(t_{i+1})-f(t_i)|^p<\infty.$$ At least so is for $p=1$. Also $f(t_i)$ should be defined for each $t_i$ in the sum. Below I’ll use this definition.

Now about Q3. There exists a function $f$ continuous of $[0,1]$ such that variation of $f$ on $[0,1]$ is unbounded [LYB, p.338]. Namely, put $f(x)=x\cos\tfrac \pi{2x}$ for $0<x\le 1$ and $f(0)=0$. Given a natural $n$, put $$(t_0,t_1,\dots, t_k)=(0, \tfrac 1{2n},\tfrac 1{2n-1},\dots,\tfrac 12,1).$$ Then

$$\sum|f(t_{i+1})-f(t_i)|=1+\frac 12+\dots+\frac 1n\to\infty.$$

On the other hand, I conjectured that $2$-variation of $f$ on $[0,1]$ is bounded. I tried to prove this, but I found no easy way to do this. Maybe somebody will find it for this or an other similar function with unbounded variation.

References

[LYB] I. Lyashko, V. Yemel’yanov, O. Boyarchuk, Mathematical analysis, vol. 1, Kyiv, Vyshcha shkola, 1992 (in Ukrainian).

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  • $\begingroup$ My most important question is Q2)... I'm very surprised that at this no body gave an answer to this one... May be it's more difficult than what I thougt). By the way, Q1) is just an introduction of Q2), but no answer as well.... Is my question that complicate ? (I guess that with your reputation you must be a master !) $\endgroup$ – user649261 Mar 5 '19 at 22:35
  • $\begingroup$ @user649261 :-) I think the reputation shows popularity, not masterliness. Also I’m not a specialist in analysis. Maybe your question recieved no answer because a notion of a fractional derivative does not belong to a common analysis background, so you chances to recive it may be better if you provide its definition, and also a definition of a Brownian motion. $\endgroup$ – Alex Ravsky Mar 6 '19 at 1:07

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