I am searching for a reference I think I saw in a book by either Shreve or Oskendahl. I am struggling with a theoretical question.
As I recall how it was posed, the idea of no prior information (or a filtration) being a necessity for an SDE could be confused. If we have a stock and are trying to solve an SDE for its price diffusion, at time t we assume only prior information, which is sensible.
But the passage I am recalling talked about an SDE that, if I remember, included both interest rates and stock prices. The distinction made was that - even if a bond had random interest payments - one could treat it less strictly than the solution for the stock's price.
In essence, if I recall, the notion was that we know the par value of the bond at all times, and even if it has random payoffs at any time t, that is very different than the stock having a continuously evolving stock price diffusion which is itself the sum of prior prices and infinitessimal returns patterns. Thus, the (as I recall the words) stock price model had everything being endogenous, while the varying interest rate was what it was at time t, and thus more exogenous if you will, and one could be looser in treating it formally.