It's Lemma

Statement of the lemma

$dx(t)\; =\; a(x,t)\backslash ,dt\; +\; b(x,t)\backslash ,dW\_t$

$f(x(t))$ is also an It process.

$df(x(t),t)\; =\; \backslash left(\; a(x,t)\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +\; \backslash frac\{b(x,t)*b(x,t)*\; \backslash frac\{\backslash partial^2f\}\{\backslash partial\; x^2\}\}\{2\}\; \backslash right)\; dt\; +\; b(x,t)\backslash frac\{\backslash partial\; f\}\{dx\}\backslash ,dW\_t$

Informal proof

$df\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\backslash ,dx\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\backslash ,dt\; +\; \backslash frac\{1\}\{2\}\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x^2\}\backslash ,dx^2+\; \backslash cdots$

$df\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}(a\backslash ,dt\; +\; b\backslash ,dW\_t)\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\backslash ,dt\; +\; \backslash frac\{1\}\{2\}\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x^2\}(a^2(dt)^2\; +\; 2.a.b\backslash ,dt\backslash ,dW\; +\; b^2(dW^2))+\; \backslash cdots$

$dW^2\; =\; E\backslash left(dW^2\backslash right)$, as $E\backslash left(dW^2\backslash right)\; =\; dt.$

$df\; =\; \backslash left(\; a\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +\; \backslash frac\{b*b*\backslash frac\{\backslash partial^2f\}\{\backslash partial\; x^2\}\}\{2\}\backslash right)\; dt\; +\; b\backslash frac\{\backslash partial\; f\}\{dx\}\backslash ,dW\_t$

See also

• Wiener process
• It calculus

External links

• Derivation, Prof. Thayer Watkins
• Discussion, quantnotes.com
• informal proof, optiontutor

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