The differential form of a stochastic variable can be expressed as $$dx=a(x)dt+b(x)dw(t)$$, here w(t) presents the Wiener process and satisfies ##(dw)^2=dt##.(adsbygoogle = window.adsbygoogle || []).push({});

For the function f(x), the derivation of its differential form in the book by Gardiner is

$$df(x)=f'(x)dx+(1/2)f''(x)dx^2=f'(x)[a(x)dt+b(x)dw(t)]+(1/2)f''(x)[a(x)dt+b(x)dw(t)]^2$$

taking into account ##(dw)^2=dt## and only take the first order of dt, we get the Ito formula

$$df(x)=[f'(x)a(x)+(1/2)(b(x))^2]dt+f'(x)b(x)dw(t)$$

Here is my questions:

1, In the above derivation, which step shows the Ito rule?

2, How to derive the "Stratonovich formula" by the same way? That's which step I should change when I use Stratonovich rule to get the differential form of the function f(x).

Thank you!

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# The derivation of Ito formula and Stratonovich formula

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