How to show that $Z(t)=e^{σ W(t)-\frac{1}{2}σ^2t}$ is a martingale

Question

How would I show that the stochastic process $Z(t)=e^{σ W(t)-\frac{1}{2}σ^2t}$ where $σ>0$ is a martingale?

I'm not sure how to do this, can someone please explain?

Answer 1

Hint: There are at least two ways to prove this.

1) Show $\mathbf E[Z(t)|\mathcal F(s)]=\mathbf E\big[e^{σ (W(t)-W(s))-\frac{1}{2}σ^2(t-s)}Z(s)\big|W(s)\big]=Z(s), \,\forall s<t.$

To do this, ask yourself what the distribution function of $W(t)-W(s)$ is, then set up the appropriate integrate and compute.

2) Use Ito's lemma to show that the term involving $dt$ of $dZ(t)$ vanishes.

Answer 2

You may apply Novikov's (Kazamaki's) condition on the adapted process $X_t = \sigma$.