# Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are interchangeable, but going from one interpretation to the other will change the form of the equations). The situation is more subtle than I put it, as there are systems where Ito is in fact the more natural one. Even some very recent papers addressing the dilemma have come up with claims of cases where neither approach seems to work particularly well.

I am wondering why in finance it seems that people are only using the Ito approach. I've heard claims that the Stratonovich definition might open up arbitrage in some cases, but I would be interested in hearing about the generality of this statement and if it should always be true. Finally, and more importantly, I would like to know the intuition behind picking Ito calculus in finance in the first place.

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Hi amlrg! Welcome to quant.SE and thank you for your question. –  Bob Jansen Jul 13 '14 at 22:23

In fact Ito and Stratonovich calculus are both mathematically equivalent. In the following paper you can e.g. see that both derivations lead to the same result, i.e. the Black-Scholes equation:

Black-Scholes option pricing within Ito and Stratonovich conventions by J. Perello, J. M. Porra, M. Montero and J. Masoliver

From the abstract:

Options financial instruments designed to protect investors from the stock market randomness. In 1973, Fisher Black, Myron Scholes and Robert Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation. Herein, we derive the Black-Scholes equation for the option price using the Stratonovich calculus along with a comprehensive review, aimed to physicists, of the classical option pricing method based on the Ito calculus. We show, as can be expected, that the Black-Scholes equation is independent of the interpretation chosen. We nonetheless point out the many subtleties underlying Black-Scholes option pricing method.

The main fact that in finance the Ito calculus is chosen over Stratonovich is that it has a natural interpretation and intuition: Because the left endpoints of the intervals in the limiting process are being chosen this could be interpreted as the fact that in finance you don't know any future stock prices. In the Stratonovich calculus you choose the midpoints which would lie in the future (and are therefore, strictly speaking, unknown).

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While I think both answers are correct and useful, I personally chose the one that I felt better answered the "most important" part of the question, i.e. what the intuition behind choosing Ito is. –  alarge Jul 15 '14 at 0:06
@amlrg: Thank you :-) –  vonjd Jul 15 '14 at 6:37

My understanding is because the Ito's integration definition keeps the martingale property.

With Brownian motion $W(t, \omega)$ defined, to define stochastic integration in a Riemann–Stieltjes style: $$\int_0^t f(t, \omega) d W(t, \omega) = \lim_{\| \Delta_n\| \to 0 } \sum_{i=1}^{n} f(\tau_i,\omega) \left ( W(t_i, \omega) - W(t_{i-1}, \omega) \right )$$ , the choice to made is that which point from $[t_{i-1}, t_i]$ shall $\tau_i$ pick?

If $\tau_i = (t_{i-1}+t_i) /2$, this is the Stratonovich integration.

But look at the $f(t,\omega)$, if it's determined, or denoted as $f(t)$, we have $\int_0^t f(t) d W(t, \omega)$ is a martingale. This is natural (by intuition) as $W(t, \omega)$ is a martingale, as a special case.

So we hope the intuition goes on, that for a general $f(t,\omega)$, the integration is also a martingale.

This lead to $\tau_i = t_{i-1}$, the Ito's integration.

On @amlrg 's comment, about choosing martingale or non-arbitrage while defining stochastic calculus, it's a bit long so I'm appending the answer here.

Well, I'm not a math guy so below is just my guessing.

My guess is that when Ito etc were building up the theory, comparing to "non-arbitrage", "martingale" might be more interesting as it has more applications, a simpler concept, and more potential to extend.

"Non-arbitrage" is about pricing in quant finance, besides that there were multiple kinds of problems related to stochastic calculus, for reference, pls consult examples given in Oksendal's book "Stochastic Differential Equations" chapter 1.

Also, martingale is simpler. To define non-arbitrage other concepts such as self-financing, conditional expectation are needed.

Martingale is also a more fundamental concept. This makes it easier to extend the Ito calculus. If the $f(t,\omega)$ does not have bounded squared variation, so long as the exceptions has $0$ Lebesgue measure, the Ito integral still works: it's no longer a martingale, but still has "local martingale" property. It's said that it can be further extend to Malliavin calculus, applying to the calculus of variations, defined on Hilbert space etc.. but that is already beyond my poor limited math knowledge.

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+1: The martingale property of the Ito integral is another important point indeed. –  vonjd Jul 13 '14 at 15:23
Could you perhaps add some text as to why martingales should be the intuitively correct way to go? Is this, like @vonjd explained in his answer, because then you guarantee that you can't be "looking up ahead"? Does it really matter, though, if you don't run into arbitrage problems (the Black-Scholes turned out to be the same regardless of the definition)? –  alarge Jul 13 '14 at 16:06
@amlrg pls see my answer updated. –  athos Jul 14 '14 at 2:34
"My guess is that when Ito etc were building up the theory, comparing to "non-arbitrage", "martingale" might be more interesting as it has more applications, a simpler concept, and more potential to extend." I doubt this because if I'm well informed, Itō wasn't creating the theory of stochastic differential equation for finance issues. –  NikolajK Apr 10 at 21:30