Itô calculus for the rest of us

One of the areas of my research is stochastic differential equations (SDE). I posted about it several times before. One of the things students and collaborators keep asking me about SDEs is the weird stochastic Itô Calculus. Itô Calculus is different from what you learn in 101 calculus. In particular, the chain rule is not longer valid. Let me explain it with an example. Suppose you have the following equation

dX = a dW(t)

where W(t) is the Wiener stochastic process: it is a process in which W(0) = 0 and with Gaussian independent increments W(t)-W(s) \sim {\cal N}(0,t-s), where {\cal N}(\mu,\sigma^2) are Gaussian random numbers with mean \mu and variance \sigma^2. This equation is the simplest SDE and its solution is (obviously),

X(t) = a W(t)

But there is a different way to look at SDEs. Since they describe the trayectory of X(t) subject to W(t) we can instead ask ourselves what is the probability to find that X(t) = x in average, i.e. after many different runs of W(t). Let’s call P(x,t) that probability which is known to satisfy the Fokker-Planck equation

\displaystyle{\frac{dP}{dt}=a\frac{d^2P}{dx^2}}

which is the simple diffusion equation. The following figure shows the relationship between X(t) and P(x,t). While X(t) is a simple trayectory (for a given W(t) realization), P(x,t) is the probability to find X(t) = x over all possible realizations of W(t).

pdf1

The relationship between X(t) and P(x,t) is more general than for the example used here. In general, if we have the SDE

dX(t) = f(X,t)dt + g(X,t) dW(t)

in the Itô sense, then P(x,t) is the solution of

\frac{\partial P}{\partial t}= - \frac{\partial}{\partial x}\left[f(x,t)P(x,t)\right] + \frac{\partial^2}{\partial x^2}\left[g^2(x,t)P(x,t)\right]

The term f(x,t) is usually know as the drift and g(x,t) is the diffusion term.

Once the relationship between the Fokker-Planck equation and its SDE is given, it is very easy to understand Itô Calculus. For example, given the above SDE dX = a dW(t), what is the SDE that will satisfy Y(t) = X^2(t)?

Instead of using Itô’s lemma,  we will use the Fokker Planck equation and traditional calculus. Thus \partial/\partial x = 2 \sqrt{y} \partial/\partial y. We also have that P(x,t) dx = P(y,t) dy or P(x,t) = 2 \sqrt{y} P(y,t). Putting all these things together we get

\displaystyle{\frac{\partial}{\partial t}2\sqrt{y}P(y,t) = \frac{a^2}{2}\left\{2\frac{\partial}{\partial y}+4y\frac{\partial^2}{\partial y^2}\right\}2\sqrt{y} P(y,t)}

and with a little bit of algebra we get

\displaystyle{\frac{\partial P}{\partial t}=-\frac{\partial}{\partial y}[a^2 P(y,t)] + \frac{a^2}{2}\frac{\partial^2}{\partial y^2}[4yP(y,t)]}

which, according to the general relationship between Fokker-Planck equation and SDE, corresponds to the following SDE

dY = a^2 dt + 2a \sqrt{Y} dW(t)

as the Itô’s lemma says. Obviously, this way to get the Itô lemma is painful in general, so you’d better use it directly on the SDE instead of the method shown here which goes through the Fokker-Planck equation. However, it is interesting to see that you can recover it from the Fokker-Planck equation.

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6 Responses

  1. Minos Niu says:

    Great post. My research is using SDE to explain brain functions. We model the brain activity as a diffusion process across the neuronal network. I’ve been looking for tutorials to link Fokker-Planck equations together with Ito’s intergral. This post just does that. Thanks!

  2. Riz says:

    I am grad student of maths & currently I am studying Ito, the article written is so easy to understand as mostly books use alien language

  3. Jean-Francois Bido says:

    Great post! I am currently working on building a Libor Market Model and this post helped me having a more intuitive understanding of Itô’s lemma.
    NB: I think there are small typos in both Fokker-Planck equations

  4. admin says:

    Thanks! Could you send me the typos?

  5. Jean-Francois Bido says:

    Imho it should be

    \frac{dP}{dt}=\frac{a^2}{2}\frac{d^2P}{dx^2}

    and

    \frac{\partial P}{\partial t}= – \frac{\partial}{\partial x}\left[f(x,t)P(x,t)\right] + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[g^2(x,t)P(x,t)\right]

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