Before revision: We consider a notion of weak no arbitrage condition commonly known as Robust No Unbounded Profit with Bounded Risk (RNUPBR) in the context of continuous time markets with small proportional transaction costs . We show that the RNUPBR condition on terminal liquidation value holds if and only if there exists a strictly consistent local martingale system (SCLMS) . Moreover, we show that RNUPBR condition implies the existence of optimal solution of the utility maximization problem defined on the terminal liquidation value .
After revision: We consider a notion of weak no arbitrage condition commonly known as Robust No Unbounded Profit with Bounded Risk (RNUPBR) in the context of continuous time markets with small proportional transaction costs in the sense that the utility maximization problems defined on terminal liquidation value holds if and only if there exists a strictly consistent local martingale system (SCLMS) . Moreover, we show that RNUPBR condition implies the existence of optimal solution of the utility maximization problem defined on the terminal liquidation value .