Definition (Portfolio)

A portfolio is a pair $\phi=((a_{t})_{t\in[0,T]},(b_{t})_{t\in[0,T]})$ of $(\mathcal{F}_{t})_{t\ge 0}$-adapted processes. The value of the portfolio at time $t$ is $$V_{t}(\phi)=a_{t}S_{t}+b_{t}A_{t}$$Hence, at expiration, the value of the portfolio is given by $$V_{T}(\phi)=a_{T}S_{T}+b_{T}A_{T}$$

Definition (Self-financing portfolio)

A portfolio is self-financing if $$dV_{t}(\phi)=a_{t}dS_{t}+b_{t}dA_{t}\quad\forall t$$i.e. no money is injected or removed

Definition (Admissible portfolio)

A portfolio is admissible if $$V_{t}(\phi)\ge 0\quad\forall t\in[0,T]$$

Definition (Replicating portfolio)

Let $H$ be a $\mathcal{F}_{t}$-measurable RV, the portfolio $\phi$ is said to replicate $H$ if $$V_{T}(\phi)=H$$