Regularity and B-S equation

Question

Let $t_{0}>0,$ $S_{t_{0}}=S_{0}$ and $\forall t>0\quad \frac{dS_t}{S_t}=\mu dt+\sigma dW_t$.
Are there any regularity results for $(\mu, \sigma)\mapsto S(\mu,\sigma)$?

Answer 1

As mentioned in https://en.wikipedia.org/wiki/Geometric_Brownian_motion we have

$$S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right).$$

So the mapping $(\mu, \sigma)\mapsto S(\mu,\sigma)$ is smooth.