Itō’s Insight

Consider the stochastic process

f(B_t)

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

Ito’s lemma states that

df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt

The first term is recognizable from the chain rule in classical calculus, but why the second term? If dB_t is truly infinitesimal, it doesn’t even seem possible that df \neq f'(B_t) dB_t .

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

dB_t = B_t - B_{t+dt} \sim N(0,dt)

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

The picture at the start considers an example f(B_t) where f'(0)=0 , thereby suppressing the “intuitive” or classical term in Ito’s lemma. The reason why df>0 in that picture is the reason why that “non-intuitive” term is needed.

Loosely speaking, wherever f has curvature, dB_t will diffuse around that curvature sufficiently to influence the expected result on the order of dt . (The specific example in the picture makes this trivial, as f(\sqrt{t}) = t .)

Why doesn’t dB_t dominate away (dB_t)^2 (i.e. dt )? Because E[(dB_t)^2] does not come from (E[dB_t])^2 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.

 

3 Responses to “Itō’s Insight”

  • randy says:

    Hey super cool content here!
    Is there any way from someone that has knowledge of calculus 3 and mathematical statistics 2 or less to understand that? I would love to share Ito’s lemma with my Financial Technology club.

    Thanks!

  • Ito says:

    Hello,
    why is the derivative f’ in the considered example of f(B_t) = (B_t)^2 equal to zero?

    • admin says:

      In that sentence we are considering only the (example) case where B_t = 0.
      I will clarify the text; thank you for pointing this out.


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