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Following Shreve's notation,

$d(e^{-\int_{0}^tR(s)ds}X(t))=e^{-\int_0^tR(s)ds}\sigma(t)\Delta(t) S(t)(\frac{\alpha(t)-R(t)}{\sigma(t)}dt+dW_t)$. In order to make $d(e^{-\int_{0}^tR(s)ds}X(t))$ a martingale, he defines $d\tilde{W}_t=\frac{\alpha(t)-R(t)}{\sigma(t)}dt+dW_t$, and the new probability measure $\tilde{P}_t$ under which $d\tilde{W}_t$ is a brownian motion. But $\tilde{P}_t$ relies on $R(t),\alpha(t),\sigma(t)$. My question is, since $\tilde{p}_t$ is not determinant, what do you mean by the expectation under risk-neutral measure ($\tilde{E}(.|\mathcal{F}_t)$)?

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closed as unclear what you're asking by Quantuple, LocalVolatility, SmallChess, muffin1974, Gordon Feb 13 '17 at 18:08

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