Question

Use ito's formula to prove thaat hte stochastic process is martingale

Answer 1

Answer\\

Suppose you define an arbitrary stochastic process, for example

Xt:=W8t−8t where Wt is a Brownian motion.

The question is, how could I deduce that this stochastic process is a martingale or not using Itô's formula?

The only thing I know is:

Looking at the stochastic integral ∫KdM where M={Mt} is a martingale, which is right continuous with left limit, null at 0 and satisfies suptE[Mt]<∞ and K a stochastic process bounded and predictable, then ∫KdM is a martingale too.

But I'm not sure if this is helpful in this situation. An example of how to solve such types of problems would be appreciated.

Just to be sure, I state Itô's formula which I know so far.

Let {Xt} a general Rn valued semimartingale and f:Rn→R such that f∈C2. Then {f(Xt)} is again a semimartingale and we get Itô's formula (in differential form):

df(Xt)=∑i=1nfxi(Xt)dXt,i+12∑i,j=1nfxi,xj(Xt)d⟨Xi,Xj⟩t