# The derivation of Ito formula and Stratonovich formula

1. Sep 1, 2014

### chern

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. Sep 8, 2014