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To shorten the notation, let's write $T_t = T(D_t,y_t)$ and $\delta_t = \delta(D_t,y_t)$. There are two ways to show that, in fact, the dynamics of $$ \xi_t = \xi(D_t, y_t,t) = e^{-\int_0^t \delta_s ds}\, T_t $$ is given by $$ \frac{d\xi_t}{\xi_t} = \left( -\delta_t + \frac{\mathscr{L} T_t}{T_t} \right)dt \quad+\quad \text{diffusion terms}. $$ First way ...


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if we take a digital option and price under BS then you can do the whole thing by direct verification. i.e. $N(d_2)$ solves the PDE and converges to the final pay-off pointwise. So if the final pay-off has a finite number of jump discontinuities then subtract a linear combination of digitals to reduce to the continuous case.


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Number one, the central limit theorem means a lot of things that may not be normal end up looking normal when lots of little 'experiments' or impacts are added up. Number 2, when dealing with finance you need a model that seems plausible. An arithmetic Brownian motion could go negative, but stock prices can't. On the other hand, it seems quite plausible ...



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