To test whether a process is a Martingale (Stochastic calculus)?

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If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to calculate partial differentials of $Y_t$ with respect to $t$ and $W_t$. Any suggestions will be appreciated !