chern
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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##.
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!
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!