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 ($\tilde{E}(.|\mathcal{F}_t)$)?


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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.