I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For instance:
$$Y(t)= \exp(\sigma X(t)−0.5\sigma^2t)$$ where $X(t)$ is S.B.M.
How to approach this problem?
Many thanks
It's not necessary to apply Itô's formula in this case. Instead you can use the independence of increments of the Brownian motion $(X_t)_t$ and the knowledge about exponential moments of normal distributed random variables to check whether it's a martingale.
$$\mathbb{E}(Y_t \mid \mathcal{F}_s) = e^{-\frac{1}{2}\sigma^2 \cdot t} \cdot \mathbb{E} \big(e^{\sigma \cdot (X_t-X_s) + \sigma \cdot X_s} \mid \mathcal{F}_s \big) = e^{-\frac{1}{2}\sigma^2 \cdot t + \sigma \cdot X_s} \cdot \mathbb{E} \big(e^{\sigma \cdot (X_t-X_s)} \mid \mathcal{F}_s \big) = \ldots$$
Alternatively, you can try to find a suitable function $g$ such that $$Y_t = Y_0 + \int_0^t g(s) \, dX_s$$ because stochastic integrals with respect to a Brownian motion are martingales right from the definition. To find such a function $g$, apply Itô's formula.