On page 56 of Baxter and Rennie (Financial Calculus), we have
- The definition of a continuous stochastic process, in terms of the drift $\mu_s$ and volatality $\sigma_s$. Its important to keep in mind that the drift and volatality can be processes themselves.
- The uniquness result, a part of which tells us that for a given $\mu_s$ , $\sigma_s$ and $X_0$, we can construct only a single process $X_t$.
However, the bottom part of the page introduces SDEs, which may have multiple solutions. The definition of an SDE is given in terms of drift and volatality which may depend upon the current value of the process - the notation being $\mu(X_t,t)$ and $\sigma(X_t,t)$.
I think $\mu(X_t,t)$ and $\sigma(X_t,t)$ are special cases of $\mu_s$ and $\sigma_s$, which would imply that an SDE cannot have multiple solutions. This is obviously wrong, and I would like to know why - hopefully in terms of how those two notations differ. Thank you in advance!