Terminal wealth floor ($w_T\ge \xi$)

A terminal portfolio wealth floor is a prescribed lower bound, possibly a random variable, on terminal portfolio portfolio wealth. In a complete market, its state-price density cost cannot exceed initial portfolio wealth. Nonnegative admissibility replaces a possibly negative floor by its positive part.

For a nonnegative terminal wealth floor whose state-price density cost is strictly below the available budget, pointwise maximization of $U(y)-\lambda {\zeta }_{T}y$ over $y\ge \xi$ gives the displayed payoff. The positive multiplier exhausts the state-price budget constraint. This requires the usual utility and integrability hypotheses ensuring that the multiplier exists and the payoff is replicable.