Itō’s Insight

Consider the stochastic process

where is standard Brownian motion (or the Wiener process) and is a twice-differentiable function.

Ito’s lemma states that

The first term is recognizable from the chain rule in classical calculus, but why the second term? If is truly infinitesimal, it doesn’t even seem possible that .

To understand Ito’s lemma intuitively, think of as a little stochastic variable, specifying ‘s change during the next .

This models Brownian motion (or the Wiener process) completely. Now should be a little stochastic variable too, modeling the stochastic process .

The picture at the start considers an example where , thereby suppressing the “intuitive” or classical term in Ito’s lemma. The reason why in that picture is the reason why that “non-intuitive” term is needed.

Loosely speaking, wherever has curvature, will diffuse around that curvature sufficiently to influence the expected result on the order of . (The specific example in the picture makes this trivial, as .)

Why doesn’t dominate away (i.e. )? Because does not come from as happens with classical differentials. The strong law of large numbers implies that a stochastic differential’s expected value pushes its integral on a faster order than its deviation does.

 

brownian motion, ito calculus, ito's lemma, stochastic calculus, stochastic differential, stochastic differential equation, wiener process

Quantitative Finance, System Modelling