# Local martingale

W is standard Brownian motion; $Y_t = \delta_{1-t}(W_t),\text{ for } 0\le t < 1; Y_t = 0 \ \text{ otherwise};$ where $\delta_s(x) = \frac1{\sqrt{2\pi s}}e^{-\frac {x^2}{2s}}$ How to show that $Y_t$ is a local martingale? This is my first time to touch a local martingale, really have no idea how to proceed with this one.

THANKS A LOT!!

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How come the standard computation does not work? Consider $0\leqslant s\lt t\lt1$ and compute $E(Y_t\mid\mathcal Y_s)$ where $\mathcal Y_s=\sigma(Y_u;u\leqslant s)$. A first step might be to show that the sigma-algebra $\mathcal Y_s$ is $\sigma(W_u;u\leqslant s)$. –  Did Feb 11 '12 at 2:45
@DidierPiau I got it. Thanks! –  BVFanZ Feb 11 '12 at 3:16
Good. You might wish to write your solution as an answer. –  Did Feb 11 '12 at 6:57