I am reading the part of constructing B&S price for stock paying dividends. The simplest model used continuous yield dividend. But I can not see that rigorous in term of formulations.
Firstly, in case of non-paying dividend. We start by $$ \frac{dS_t}{S_t} = \mu dt + \sigma d\hat{W}_t \hspace{1cm} \frac{dB_t}{B_t}=rdt $$ Then by applying Girsanov, transforming $\hat{W}_t \rightarrow W_t $ with prime of risk $\frac{r-\mu}{\sigma}$, we have $$ \frac{dS_t}{S_t} = r dt + \sigma dW_t \hspace{1cm} \frac{dS^*_t}{S^*_t}=\sigma dW_t $$ Under this martingale measure and associated Brownian, the self-financing porfolio is defined by $$ V_t = H^1 S_t + H^2 B_t \hspace{1cm} dV_t = H^1 dS_t + H^2 dB_t $$ The important consequence is that the discounted self-financing and discounted Stock process, $V^*_t$ and $S^*_t$ are martingale, thus one can build replicating portfolio for any continent claim at time $T$ by applying "Martingale representation theorem".
Now going into case with continuous dividend paying, after what I've seen in the Musiela's book[1], he supposed $$ dQ_t = q S_t dt $$ Then define an auxiliary "stock process" $\tilde{S}_t = e^{qt} S_t$ and start the formulation by this process $$ \frac{d\tilde{S}_t}{\tilde{S}_t} = (\mu+q) dt + \sigma d\hat{W}_t \hspace{1cm} \frac{dB_t}{B_t}=rdt $$ Then by applying Girsanov, transforming $\hat{W}_t \rightarrow \tilde{W}_t $ with prime of risk $\frac{r-\mu - q}{\sigma}$, we have $$ \frac{d\tilde{S}_t}{\tilde{S}_t} = r dt + \sigma d\tilde{W}_t \hspace{1cm} \frac{d\tilde{S}^*_t}{\tilde{S}^*_t}=\sigma d\tilde{W}_t $$ The self-financing portfolio become $$ V_t = H^1 e^{-qt} \tilde{S}_t + H^2 B_t \hspace{1cm} dV^*_t = H^1 e^{-qt} d\tilde{S}^*_t $$ Things sounds good until now, and after. But the point that I do not understand is that the discounted portfolio is neither a martingale (nor local martin gal ??). In this case, we can not apply the "martingale representation theorem", and the build of replicating portfolio will fail. However in the book, he do not mention this point, and continue to apply the pricing formula, without proving $V^*_t$ is a marginal under the transformed measure.
Can any one help me this point ? Or this is not that simple and there need a more sophisticated formulation?
[1] "Martingale Methods in Financial Modelling" _ Marek Musiela, Marek Rutkowski 2004, §3.2.1, p.148
