hi, my question is -- if we differentiate a function of an Ito process X(t) and t with Ito's lemma, can that function involve stochastic terms? can these stochastic terms be correlated to X(t)?specifically, i want to differentiate f:f{X(t),t} = X(t) + integral[0,t]{a(t)dW} + b(t)where a(t) and b(t) are deterministic functions of t; and W is a Brownian motion driving X(t), i.e. X(t) = m(X,t)dt + s(X,t)dWis that ok? or do we run into troubles?? to check, i was thinking to represent the Ito integral as Ito process and then use 2D Ito, but is such representation even possible??

For this case, I'd write f as f = g + h, apply Ito's lemma to g and h, and add: df = dg + dh. The functions on t and x would be g(t,x) = x + bt and h(t, x) = x. For g, use dX_t = m dt + s dW_t, and for h dY_t = a_t dW_t. Hope this helps!

For the case when f and X admit stochastic differentials there exists formula Ito-Ventzel representing differential of their composition.

thank you manolom and list,@manolom : so you rkn the differential for the ito integral "Integral[0,t]{a(s)dW(s)}" is a(t)dW(t)? sorry im like a total noob in stochastic calculus..@list: thanks! im still looking for the formula, might give my librray a call tomorow for that Lazrieva and Toronjadze paper

Quote@manolom : so you rkn the differential for the ito integral "Integral[0,t]{a(s)dW(s)}" is a(t)dW(t)? sorry im like a total noob in stochastic calculus..Yes.

QuoteOriginally posted by: Shtraussthank you manolom and list,@manolom : so you rkn the differential for the ito integral "Integral[0,t]{a(s)dW(s)}" is a(t)dW(t)? sorry im like a total noob in stochastic calculus..@list: thanks! im still looking for the formula, might give my librray a call tomorow for that Lazrieva and Toronjadze paperyou can also check the formula in Rozovskii book