How to use Itô's formula to deduce that a stochastic process is a martingale?

Question

I'm working through different books about financial mathematics and solving some problems I get stuck.

Suppose you define an arbitrary stochastic process, for example

$ X_t := W_t^8-8t $ where $ W_t $ 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 $ \int K dM $ where $ M=\{M_t\} $ is a martingale, which is right continuous with left limit, null at $0$ and satisfies $ sup_t E[M_t] < \infty$ and $ K $ a stochastic process bounded and predictable, then $ \int K dM $ 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 $\{X_t\}$ a general $ \mathbb{R}^n $ valued semimartingale and $f: \mathbb{R}^n \to \mathbb{R}$ such that $ f\in C^2 $. Then $ \{f(X_t)\} $ is again a semimartingale and we get Itô's formula (in differential form):

$$ df(X_t) = \sum_{i=1}^n f_{x_i}(X_t)dX_{t,i} + \frac{1}{2}\sum_{i,j=1}^n f_{x_i,x_j}(X_t)d\langle X_i,X_j\rangle_t$$

Answer 1

In general, if you have a process that you can write under the form $F(B_t,t)$ where $F$ is $\mathcal{C}^{2,1}$ then Itô's lemma gives you the drift term and diffusion term of $dF$. Then if the resulting SDE has a null drift (that's where Black Scholes PDE comes from), and you get a only local martingale. For it to be a proper martingale you can look at theorem 1.

But you have easier sufficient conditions, in particular if you only need martingale property over finite time intervals. Those conditions are about the integrability of the quadratic variation process, but as I can't remember them exactly, I won't try to derive them here. They must appear in any book over stochastic integration with respect to Brownian motion.

Best regards

Answer 2

For Itô Processes $dX(t) = \mu(t) \mathrm{d}t + \sigma(t) \mathrm{d}W(t)$ you have the result that (under appropriate assumptions which ensure that the local martingale is a martingale, e.g. $E( (\int \sigma(t)^2 \mathrm{d}t )^{1/2} ) < \infty$, etc.): $X$ is a martingale $\Leftrightarrow$ $\mu(t) = 0$.

So in order to check if a process $X$ is a martingale use Itô to get its "$\mathrm{d}X = \ldots$-representation" and check the coefficient of $dt$ on zero.

(I believe the exact result can be found in Øksendal, Bernt K.: Stochastic Differential Equations: An Introduction with Applications)

Answer 3

Hint

Let $\,H_0(x,t)=1$ , $H_1(x,t)=x$ and for every $n\ge 2$ set $${{H}_{n}}(x,t)=x {{H}_{n -1}}(x,t)-(n-1)\,t\,{{H}_{n-2}}(x,t)$$ then ${{H}_{n }}(W_t ,t)$ is a Martingale. For exapmple $$H_1(W_t,t)=W_t$$ $$\qquad H_2(W_t,t)=W_t^2-t$$ $$\qquad\qquad H_3(W_t,t)=W_t^3-3tW_t$$ $$\vdots $$

Answer 4

Rather simply and generally when you take the stochastic differential of a process and get no drift term but simply an ito integral, then this process is a martingale. From memory that's how you retrieve some pde equations whose solutions lead to martingale (take the differential, look at the dt partial differentials term, then look for solution that would yield a vanishing dt term)