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However something needs to be clarified about B_t and dc

B_t follows a Brownian motion in time t

You can cut equal time intervals and each interval B changes by a normal random variable with 0 mean and time length variance; each B change in a time interval has nothing to do with another B change in a different interval.

And

hence

As Δt gets small, mathematicians argue the square of ΔB has a quite steady mean Δt with a negligible (higher order) variance 2*Δt². Hence here comes

Ito's lemma

dB*dB=dt;
dt*dt=o(dt);
dB*dt=o(dt)


Thus, to approach a difference of a function which involves a brownian motion B in it directly or indirectly, you have to use Taylor expansion with order 2 to capture the innegligible dB².

thus here come the dc

where

o(dt) is higher order term of dt, negligible the same way we do our normal calculus. And the final formual for dc is:

Last edited by George,Y (2008-04-20 03:09:29)