Intuition and/or visualisation of Ito integral/Ito's lemma - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:09:56Z http://mathoverflow.net/feeds/question/29750 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29750/intuition-and-or-visualisation-of-ito-integral-itos-lemma Intuition and/or visualisation of Ito integral/Ito's lemma vonjd 2010-06-28T06:32:45Z 2012-06-23T23:59:29Z <p>Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. <a href="http://en.wikipedia.org/wiki/Riemann_sum" rel="nofollow">Wikipedia:Riemann sum</a></p> <p>The <a href="http://en.wikipedia.org/wiki/Ito_integral" rel="nofollow">Ito integral</a> has due to the unbounded total variation but bounded quadratic variation an extra term (sometimes called Ito correction term). The standard intuition for this is a <a href="http://en.wikipedia.org/wiki/Taylor_series" rel="nofollow">Taylor expansion</a>, sometimes <a href="http://en.wikipedia.org/wiki/Jensen_inequality#Proofs" rel="nofollow">Jensen's inequality</a>.</p> <p>But normally there is more than one intuition for a mathematical phenomenon, e.g. in Thurston's paper, <a href="http://arxiv.org/abs/math/9404236" rel="nofollow">"On Proof and Progress in Mathematics"</a>, he gives seven different elementary ways of thinking about the derivative.</p> <p><strong>My question</strong><br/> Could you give me some other intuitions for the Ito integral (and/or Ito's lemma as the so called "chain rule of stochastic calculus"). The more the better and from different fields of mathematics to see the big picture and connections. I am esp. interested in new intuitions and intuitions that are not so well known.</p> http://mathoverflow.net/questions/29750/intuition-and-or-visualisation-of-ito-integral-itos-lemma/29758#29758 Answer by Michael Greinecker for Intuition and/or visualisation of Ito integral/Ito's lemma Michael Greinecker 2010-06-28T09:54:22Z 2010-06-28T09:54:22Z <p>Robert Anderson used nonstandard analysis to generate Brownian motion from a finite random walk obtained from coin tosses, where "finite" means indexed by an infinite, non-standard natural number. The corresponding random walk has bounded variation under a non-standard bound. One can then do everything in terms such an random walk, as has been done without rigorous justification before. The Itô-integral can be obtained from a Stiltjes-integral on the random walk, they differ only by an infinitesimal. An outline of the arguments can be found <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183537617" rel="nofollow">here</a>. For the details, see:</p> <p>MR0464380 (57 #4311) Anderson, Robert M. A non-standard representation for Brownian motion and Itô integration. Israel J. Math. 25 (1976), no. 1-2, 15--46.</p> http://mathoverflow.net/questions/29750/intuition-and-or-visualisation-of-ito-integral-itos-lemma/29800#29800 Answer by Andrey Rekalo for Intuition and/or visualisation of Ito integral/Ito's lemma Andrey Rekalo 2010-06-28T15:00:58Z 2010-06-29T20:28:04Z <p>I find the intuitive explanation by <a href="http://www.amazon.co.uk/Paul-Wilmott-Quantitative-Finance-2nd/dp/0470018704" rel="nofollow">Paul Wilmott</a> particularly appealing.</p> <p>Fix a small $h>0$. The stochastic integral $$\int_0^{h} f(W(t))\ dW(t)=\lim\limits_{N\to\infty}\sum\limits_{j=1}^{N} f\left(W(t_{j-1})\right)\left(W(t_{j})-W({t_{j-1}})\right),\quad t_j= h\frac{j}{N},$$ involves adding up an infinite number of random variables. Let's substitute every term $f\left(W(t_{j-1})\right)$ with its formal Taylor expansion. Then there are several contributions to the sum: those that are <em>a sum of random variables</em> and those that are <em>a sum of the squares of random variables</em>, and then there are <em>higher-order terms</em>.</p> <p>Add up a large number of independent random variables and the Central Limit Theorem kicks in, the end result being a normally distributed random variable. Let's calculate its mean and standard deviation.</p> <p>When we add up $N$ terms that are normal, each with a mean of $0$ and a standard deviation of $\sqrt{h/N}$, we end up with another normal, with a mean of $0$ and a standard deviation of $\sqrt{h}$. This is our $dW$. Notice how the $N$ disappears in the limit.</p> <p>Now, if we add up the $N$ <em>squares</em> of the same normal terms then we get something which is normally distributed with a mean of $$N\left(\sqrt{\frac{h}{N}}\right)^2=h$$ and a standard deviation which is $h\sqrt{2/N}.$ This tends to zero as $N$ gets larger. In this limit we end up with, in a sense, our $dW^2(t)=dt$, because <em>the randomness as measured by the standard deviation disappears leaving us just with the mean $dt$</em>. </p> <p>The higher-order terms have means and standard deviations that are too small, disappearing rapidly in the limit as $N\to\infty$. </p> http://mathoverflow.net/questions/29750/intuition-and-or-visualisation-of-ito-integral-itos-lemma/100445#100445 Answer by Pavel Bažant for Intuition and/or visualisation of Ito integral/Ito's lemma Pavel Bažant 2012-06-23T11:38:20Z 2012-06-23T23:59:29Z <p>I know this thread is already two years old, but, while preparing for a path integration exam, I arrived at an intuitive picture that sheds some light on the origin of the extra term. The picture represents an integral of a smooth function with respect to a concrete realization of Brownian motion. The sum of the areas of the green rectangles represents the difference between Ito (using the left point of each interval) and "anti-Ito" (using the right point of each interval) for sampling of the Brownian motion represented by the red line. Finer sampling leads to smaller rectangles, but they <strong>overlap more and more</strong> (because Brownian motion is not monotonic), so even if the area occupied by them tends to zero, the sum of their areas does not. This suggests (only suggests -- it is an upper bound on the difference, not a lower bound) that there is a "room" for Ito and "anti-Ito" to differ in their values. Stratonovich can be expected to lie somewhere in between.</p> <p>Look at the following image:</p> <p><a href="https://lh6.googleusercontent.com/-bEPzm01WyGk/T-WplGQAc3I/AAAAAAAAACQ/mZr-5p0VUrw/s317/integral-wrt-brownian-motion.png" rel="nofollow">https://lh6.googleusercontent.com/-bEPzm01WyGk/T-WplGQAc3I/AAAAAAAAACQ/mZr-5p0VUrw/s317/integral-wrt-brownian-motion.png</a></p> <p><br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="https://lh6.googleusercontent.com/-bEPzm01WyGk/T-WplGQAc3I/AAAAAAAAACQ/mZr-5p0VUrw/s317/integral-wrt-brownian-motion.png" alt="Brownian Motion"></p>