{"id":8049,"date":"2026-05-26T17:49:56","date_gmt":"2026-05-26T16:49:56","guid":{"rendered":"https:\/\/sinatootoonian.com\/?p=8049"},"modified":"2026-05-26T17:52:49","modified_gmt":"2026-05-26T16:52:49","slug":"crash-course-on-ito-calculus","status":"publish","type":"post","link":"https:\/\/sinatootoonian.com\/index.php\/2026\/05\/26\/crash-course-on-ito-calculus\/","title":{"rendered":"Crash Course on Ito Calculus"},"content":{"rendered":"\n

These are my notes on the Ito calculus as presented in Chapter 8 of Potter’s “A First Course in Random Matrix Theory.” I follow that chapter pretty closely, filling in some of the gaps in the presentation as I go.<\/em><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Brownian Motion<\/h2>\n\n\n\n

We start with Brownian motion process, $X_t$. This has drift $\\mu$, and diffusion variance $\\sigma^2$, so, $$ X_t \\sim \\mathcal{N}(\\mu t, \\sigma^2 t).$$<\/p>\n\n\n\n

The Normal distribution is infinitely divisible<\/em>. This means we can break a variable distributed that way into a sum of independent contributions. <\/p>\n\n\n\n

This is intuitive for our variable above. We can think of its overall position $X_t$ as the sum of infinitesimal motions up to that point. In fact, that’s how the Brownian motion<\/a> was originally observed, the result of countless random, independent perturbations, of e.g. dust particles under a microscope.<\/p>\n\n\n\n

To formalize this, we split the time interval $[0,t]$ into $N$ sections. We can now think of $X_t$ as the sum of perturbations in each interval: $$X_t = \\sum_{k=0}^{N-1} \\delta X_k.$$<\/p>\n\n\n\n

Each perturbation will contribute its own portion of drift and diffusion, $$ \\delta X_k \\sim \\mathcal{N}(\\mu \\delta t, \\sigma^2 \\delta t).$$ We can write this more suggestively as $$ \\delta X_k = \\mu \\delta t + \\sigma \\mathcal{N}(0, \\delta t).$$ This equation makes it clear that each contribution has a non-random part, the drift, and a random diffusive part.<\/p>\n\n\n\n

If we take the limit as $\\delta t \\to 0$, we get $$ dX_t = \\mu dt + \\sigma dB_t,$$ where $dB_t$ is a centered, infinitesimal Gaussian perturbation<\/p>\n\n\n\n

One subtlety that’s crucial to Ito’s approach is exactly when the perturbations are supposed to arrive. Ito assumes that the perturbations arrive at the beginning of the interval. This then means that the the perturbation at $t$ will be independent of the value $X_t$ before it. This is called Ito’s prescription.<\/strong><\/p>\n\n\n\n

Transforming by a function<\/h3>\n\n\n\n

Now what happens when we transform through a function? Let’s apply a Taylor expansion to second order:<\/p>\n\n\n\n

$$dF = F(X + dX) – F(X) = F'(X) dX + {F^{”}(X) \\over 2} dX^2.$$<\/p>\n\n\n\n

$$F'(X) dX = F'(X) (\\mu\\, dt + \\sigma dB_t).$$<\/p>\n\n\n\n

$$ dX^2 = \\mu^2 dt^2 + \\sigma^2 dB_t^2 + 2 \\mu \\sigma^2 dB_t dt.$$<\/p>\n\n\n\n

If we expand this out as <\/p>\n\n\n\n

$$ dX^2 = \\mu^2 dt^2 + \\left(\\sigma^2 dB_t^2 – \\sigma^2 dt\\right) + \\sigma^2 dt + 2 \\mu \\sigma^2 dB_t dt.$$<\/p>\n\n\n\n

The first term is $O(dt^2)$, the last term is $O(dt^{3\/2})$. The second term has mean 0, but turns out to be of order $dt$, so we can’t ignore it in isolation. But we’ll be integrating such terms, and they’re all uncorrelated, so Ito showed that we can ignore them in the integration. <\/p>\n\n\n\n

So, as long as we agree that we’re going to be integrating things, we end up with $$ dX^2 \\to \\sigma^2 dt,$$ and our differential becomes $$ dF = F'(X) dX + \\underbrace{{\\sigma^2 \\over 2} F^{\\prime\\prime}(X) dt}_{\\text{Ito correction}}.$$ Notice the Ito<\/em> correction <\/em> relative to the ordinary (non-stochastic) case.<\/p>\n\n\n\n

Example: Variance of Brownian Noise<\/h4>\n\n\n\n

Suppose we have a driftless Brownian noise process, $$ dX_t = \\sigma dB_t.$$<\/p>\n\n\n\n

We expect the variances of $X_t$ to be $\\sigma^2 t$. Let’s see if we can derive that.<\/p>\n\n\n\n

The variance of $X_t$ is $$\\var(X_t) = \\EE X_t^2 – (\\EE X_t)^2 = \\EE X_t^2.$$<\/p>\n\n\n\n

So if we take $F(X) = X^2$, then $$ \\var(X_t) = \\EE F(X_t).$$<\/p>\n\n\n\n

Using Ito’s formula above, $$ dF_t = F'(X_t) dX_t + {\\sigma^2 \\over 2}F^{”}(X_t) dt = 2X_t dX_t + \\sigma^2 dt = 2\\sigma X_t dB_t + \\sigma^2 dt.$$<\/p>\n\n\n\n

We then get $$F_T = \\int_0^T 2\\sigma X_t dB_t + \\sigma^2 dt.$$<\/p>\n\n\n\n

By Ito’s prescription, $dB_t$ is independent of $X_t$, so the first term integrates to zero, though it presumably has some fluctuations.<\/p>\n\n\n\n

The second term gives $\\sigma^2 T.$<\/p>\n\n\n\n

So, $ \\EE F_T = \\sigma^2 T,$ as expected.<\/p>\n\n\n\n

The Langevin Equation<\/h2>\n\n\n\n

To motivate the Langevin equation, let’s start with a process with increments are $$ dX_t = dB_t + F(X_t) dt.$$ This is a combination of Brownian motion with a perturbing force. Brownian motion will increase variance with time. This gives us variability that we can sculpt to produce various effects. <\/p>\n\n\n\n

We might be interested in:<\/p>\n\n\n\n