A restriction to nonnegative wealth is suﬀicient to preclude all arbitrage opportunities in ﬁnancial models that have risk neutral probabilities that are valid for all simple strategies. Imposing nonnegative wealth does not constrain agents from making the choice they would make under the standard integrability condition. This conclusion does not depend on whether the markets are complete.
Dybvig, Philip H. and Huang, Chi-fu, "Nonnegative Wealth, Absence of Arbitrage, and Feasible Consumption Plans" (1988). Cowles Foundation Discussion Papers. 1103.