An investment-consumption arbitrage is an admissible trading plan that starts with zero capital, withdraws nonnegative consumption at every date, and ends with no unpaid debt, with positive consumption occurring with positive probability. One may include final liquidation as the last consumption payment; equivalently require nonnegative terminal wealth and count its positive liquidation value.
The budget must be self-financing apart from these withdrawals: each withdrawal is funded by asset holdings, not by outside money. The terminal solvency condition is essential. Borrowing money, consuming it immediately and leaving a negative terminal balance is not an arbitrage. In a model with a positive numéraire, unused positive terminal wealth can be paid out as final consumption, connecting this definition to pure-investment arbitrage.
A pure-investment arbitrage has zero initial cost, no intervening consumption, and a nonnegative terminal payoff that is strictly positive with positive probability.
Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio.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.
A terminal-consumption arbitrage has zero initial cost and a nonnegative terminal payoff that is strictly positive with positive probability. Allowing negative initial cost gives the broader one-period notion in which some consumption may occur immediately.
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