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The derivation of Ito formula and Stratonovich formula

  1. Sep 1, 2014 #1
    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!
     
  2. jcsd
  3. Sep 8, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
     
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