(Every strictly positive asset is a generalized geometric Brownian motion). Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which is defined a Brownian motion $W(t), 0 \leq t \leq T$. Let $\mathcal{F}(t), 0 \leq t \leq T$, be the filtration generated by this Brownian motion. Assume there is a unique risk-neutral measure $\widetilde{\mathbb{P}}$, and let $\widetilde{W}(t), 0 \leq t \leq T$, be the Brownian motion under $\widetilde{\mathbb{P}}$ obtained by an application of Girsanov's Theorem, Theorem 5.2.3.
Corollary 5.3.2 of the Martingale Representation Theorem asserts that every martingale $\widetilde{M}(t), 0 \leq t \leq T$, under $\widetilde{\mathbb{P}}$ can be written as a stochastic integral with respect to $\widetilde{W}(t), 0 \leq t \leq T$. In other words, there exists an adapted process $\widetilde{\Gamma}(t), 0 \leq t \leq T$, such that
$$
\widetilde{M}(t)=\widetilde{M}(0)+\int_0^t \widetilde{\Gamma}(u) d \widetilde{B}(u), \quad 0 \leq t \leq T .
$$
Now let $V(T)$ be an almost surely positive ("almost surely" means with probability one under both $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ since these two measures are equivalent), $\mathcal{F}(T)$-measurable random variable. According to the risk-neutral pricing formula (5.2.31), the price at time $t$ of a security paying $V(T)$ at time $T$ is
$$
V(t)=\tilde{\mathbb{E}}\left[e^{-\int_t^T R(u) d u} V(T) \mid \mathcal{F}(t)\right], \quad 0 \leq t \leq T .
$$
(i) Show that there exists an adapted process $\tilde{\Gamma}(t), 0 \leq t \leq T$, such that
$$
d V(t)=R(t) V(t) d t+\frac{\widetilde{\Gamma}(t)}{D(t)} d \widetilde{W}(t), \quad 0 \leq t \leq T .
$$