Prove Corollary 5.3 .2 by the following steps.
(i) Compute the differential of $\frac{1}{Z(t)}$, where $Z(t)$ is given in Corollary 5.3.2.
(ii) Let $\widetilde{M}(t), 0 \leq t \leq T$, be a martingale under $\tilde{\mathbb{P}}$. Show that $M(t)=$ $Z(t) \widetilde{M}(t)$ is a martingale under $\mathbb{P}$.
(iii) According to Theorem 5.3.1, there is an adapted process $\Gamma(u), 0 \leq u \leq T$, such that
$$
M(t)=M(0)+\int_0^T \Gamma(u) d W(u), 0 \leq t \leq T .
$$
Write $\widetilde{M}(t)=M(t) \cdot \frac{1}{Z(t)}$ and take its differential using Itô's product rule.
(iv) Show that the differential of $\widetilde{M}(t)$ is the sum of an adapted process, which we call $\widetilde{\Gamma}(t)$, times $d \widetilde{W}(t)$, and zero times $d t$. Integrate to obtain (5.3.2).
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5 Risk-Neutral Pricing