I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff).
Suppose that at time t we hold the portfolio $\left(a_t,b_t,c_t\right)$ where $a_t,b_t$ and $c_t$ represent the number of units held at time t of securities with respective price processes $A_t,B_t$ and $C_t$. Assume $\left(a_t,b_t,c_t\right)$ are previsible. Letting $V_t$ be the value of this portfolio at time t.
The instantaneous change in the value of the portfolio, including cash inflows and outflows is therefore:
I understand that for $a_tdA_t$, it means the original holdings of $a_t$ multiplied by the change in value. For $da_tA_t$, it means changes in number of units held multiplied by the value at time t.
However, I'm struggling to understand intuitively why there is a $da_tdA_t$. Would greatly appreciate if anyone can help me understand!