I'm currently looking at page 20 of chapter 6 of the SP6 ActEd notes. They have defined the processes $$ E_t := B_t^{-1} V_t \\ D_t := B_t^{-1} S_t $$ where B_t is the accumulated value of 1 unit of capital invested at the risk-free rate, V_t is the value of the derivative price (which we are trying to replicate), and S_t is the price of the underlying at time t. However, it then goes on to say that E_t and D_t are previsible. I'm fairly sure that this is not true - am I wrong?
You're right; there’s no technical requirement on \(D_t\) and \(E_t\) to be previsible. However, \(\phi(t)\) is only previsible in the sense that it’s the number of shares held at time t so that at time t+dt the portfolio equals the derivative value. In the same sense \(\psi(t)\) is also previsible because its value can be determined by the data available at time t only (without looking into the future at t+dt). The main point is that over every small time period we can know what our holdings of stock and cash will be at the beginning of that time period.
