This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems (SCPS) as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABP). In particular, we show that the NUPBR and NLABP conditions in the robust sense for the smaller bid-ask spreads is the equivalent characterization of the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.Definition 1.1. Given the stock price (S t ) t∈[0,T ] with transaction cost (λ t ) t∈[0,T ] such that 0 < λ t < 1 a.s. for all t ∈ [0, T ], we call the pair (S, Q) a CPS ifwhere (S t ) t∈[0,T ] is a local martingale under Q and Q ∼ P. Moreover, if we have inf t∈[0,T ] λ t S t − |S t −S t | > 0, P-a.s., the pair (S, Q) is said to be a strictly consistent price system (SCPS).We should note that whetherS is required to be a local martingale or a true martingale in the above definition depends on the numéraire and numéraire-based admissibility of self-financing portfolios; see section 5 of [23] and [26] for details. Sufficient conditions for the existence of CPS for stock price processes with strictly positive and continuous paths have been extensively studied in the literature. One well-known example is the conditional full support condition proposed by [14]. Other related Date: August 12, 2018.

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This paper studies the continuous time utility maximization problem on consumption with addictive habit formation in incomplete semimartingale markets. Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into a time-separable utility maximization problem with a shadow random endowment on the product spaceExistence and uniqueness of the optimal solution are established using convex duality approach, where the primal value function is defined on two variables, that is, the initial wealth and the initial standard of living. We also provide sufficient conditions on the stochastic discounting processes and on the utility function for the well-posedness of the original optimization problem. Under the same assumptions, classical proofs in the approach of convex duality analysis can be modified when the auxiliary dual process is not necessarily integrable.

An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity‐averse an agent is. This inclusion of ambiguity attitude, via an α‐maxmin nonlinear expectation, renders the stopping problem time‐inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one‐dimensional diffusion with drift and volatility uncertainty, we show that any initial stopping policy will converge to an equilibrium through a fixed‐point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst‐case (or best‐case) analysis. In a concrete example of real options valuation under model ambiguity, all equilibrium stopping policies, as well as the best one among them, are fully characterized under appropriate conditions. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity‐averse, the more eager to stop—so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.

Abstract. This paper studies the utility maximization problem on the terminal wealth with both random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of the super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the convex duality analysis. As an important application of the duality theory, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form similar to [5] as well as in the usual sense using acceptable portfolios.