Example

Take

$$ f(x)=\frac12 x^2. $$

Then

$$ f'(x)=x, \qquad f''(x)=1. $$

Applying Ito formula gives

$$ \frac12 B_T^2

\int_0^T B_t,dB_t + \frac12\int_0^T 1,dt. $$

So

$$ \int_0^T B_t,dB_t

\frac12 B_T^2-\frac T2

\frac12(B_T^2-T). $$

Significance

This matches the martingale from Phase 6 and shows how Ito formula turns stochastic integrals into explicit expressions.

Source Links

• STA447
• Phase 8 Atomic Reading Order
• Phase 8 — Itô’s Formula 与应用
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