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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

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Welcome to Math.SE. You'll get much better answers to your question if you edit it to make it clearer. Explain what you mean by "clear drift", and if possible use LaTeX to format your equation. Also, I added some more tags, which will make it easier for people to find your question. –  Nate Eldredge Mar 5 '13 at 23:39
    
If the process has not drift, it means is martingale.. and by applying ito's I think we can see this. –  user65229 Mar 6 '13 at 0:09

1 Answer 1

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.

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Excellent; many thanks, I'll work it out from here ! –  user65229 Mar 6 '13 at 17:08
    
@user65229 You are welcome. Good luck! –  saz Mar 6 '13 at 18:42

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