We have already defined a self financing strategy as one in which the portfolio gain comes from the gains in the assets and nothing else.

Complete Market & First Fundamental Theorem of Asset Pricing

A market is complete if every contingent claim is attainable i.e if and only if any contingent claim has at least one replicating portfolio that is self-financing

Portfolio Replication Interpretation of Market Completeness.
For a risky asset $X$ a portfolio is replicating if for every contingent claim is there are primitive securities $S_0, \cdots, S_n$ with constant coefficients $a_0, \cdots, a_n$ such that at time $T$, the payoff of the replicating portfolio is equal to the payoff of the contingent claim
\begin{align*}
\forall \omega \in \Omega, \qquad X(\omega) = \sum_{j=0}^n a_j S_{n,T}(\omega)
\end{align*}and by no arbitrage, the price $X(0)$ at time $0$ of receiving $X$ at $T$ is therefore
\begin{align*}
X(0) = \sum_{j=0}^n a_j S_{n,0} = \sum_{j=0}^n a_j e^{-rT}\mathbb{E_Q} (S_{n,T}) = e^{-rT}\mathbb{E_Q} (X)
\end{align*}
Arbitrage provides a beautiful way of pricing derivatives as long as we are in a complete market.

THEOREM: First Fundamental Theorem of Asset Pricing.

An arbitrage-free market is complete if and only if there exists a unique probability measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ under which discounted asset prices are Q-martingales.[ref]Bingham & Kiesel. Risk Neutral Valuation – Pricing and Hedging of Financial Derivatives[/ref]

A continuous time financial market consisting of traded securities and trading strategies is said to be arbitrage free and complete if for every choice of numeraire there exists a unique equivalent martingale measure such that all security prices relative to that numeraire are martingales under that measure[ref]Yue-Kuen Kwok: Mathematical Models of Financial Derivatives[/ref]

It is sometimes said that the model satisfies the No Free Lunch with Vanishing Risk (NFLVR) condition.

Self-financing strategy implies that $$X = V_T = V_t (\phi, \psi) + \int_t^T \phi_u dS_t$$