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A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.

If a nonnegative value $V$ dominates the payoff and $a=rV-{\mathcal{L}}_{r}V\ge 0$, delta holdings $V\prime (S)$ combined with a reserve satisfying $dD=(rD+a(S))dt$ give a self-financing portfolio of wealth $V(S)+D$. The reserve retains the surplus instead of consuming it, so the wealth dominates the payoff at every exercise time.

An arbitrage is a zero-cost self-financing portfolio whose terminal payoff is nonnegative in every state and strictly positive with positive probability.

A contingent claim is replicated when an admissible self-financing portfolio has exactly its terminal payoff. In the absence of arbitrage, two replicating strategies for the same payoff have the same initial cost.

A complete market is a financial market in which every contingent claim in the chosen class can be replicated by an admissible self-financing portfolio. In an arbitrage-free finite market, completeness is equivalent to uniqueness of the equivalent martingale measure.

Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio. With no initial capital, a one-period position of nonpositive initial cost and nonnegative terminal payoff permits nonnegative consumption at both dates.

In a one-factor market whose filtration is the usual augmentation of the natural Brownian filtration, with nonzero spot volatility and local martingale deflator $Y$, the Brownian martingale representation theorem constructs a nonnegative replicating strategy for a bounded nonnegative contingent claim. The minimal initial cost among nonnegative self-financing portfolios is $\mathbb{E}[Y_T{\xi }_T]$.

Buy one maturity-$(T-1)$ bond and short $1+f$ maturity-$T$ bonds. Reinvest the first bond's unit maturity payoff in $1/P_{T-1}(T)$ maturity-$T$ bonds. This self-financing portfolio pays $r_T-f$ at $T$. The replication uses only the adjacent maturities and does not require completeness of the whole bond market.

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.

A collection of holdings in traded assets. Its value is the dot product of the holdings with the asset prices. A self-financing portfolio is an investment portfolio whose trading requires no external funding.

A numéraire strategy is a self-financing portfolio with zero consumption whose wealth is strictly positive at every date. Its wealth process can carry consumption received at an earlier date forward to a later date.