We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1011 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random Gexpectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set Θ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in Θ. The nonlinear Lévy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-differential equation.Let X = (X t ) t∈R + be an R d -valued process with càdlàg paths and X 0 = 0, defined on a measurable space (Ω, F) which is equipped with a nonlinear expectation E(·). For our purposes, this will be a sublinear operatorwhere P is a set of probability measures on (Ω, F) and E P [ · ] is the usual expectation, or integral, under the measure P . In this setting, if Y andfor all bounded Borel functions ϕ, and if Y and Z are of the same dimension, they are said to be identically distributed iffor all bounded Borel functions ϕ. We note that both definitions coincide with the classical probabilistic notions if P is a singleton. Following [8, Definition 19], the process X is a nonlinear Lévy process under E(·) if it has stationary and independent increments; that is, X t − X s and X t−s are identically distributed for all 0 ≤ s ≤ t, and X t − X s is independent of (X s 1 , . . . , X sn ) for all 0 ≤ s 1 ≤ · · · ≤ s n ≤ s ≤ t. The particular case of a classical Lévy process is recovered when P is a singleton. Let Θ be a set of Lévy triplets (b, c, F ); here b is a vector, c is a symmetric nonnegative matrix, and F is a Lévy measure. We recall that each Lévy triplet characterizes the distributional properties and in particular the infinitesimal generator of a classical Lévy process. More precisely, the asso-where, e.g., h(z) = z1 |z|≤1 . Our goal is to construct a nonlinear Lévy process whose local characteristics are described by the set Θ, in the sense that the analogue of the Kolmogorov equation will be the fully nonlinear (and somewhat nonstandard) partial integro-differential equation v t (t, x) − sup (b,c,F )∈Θ bv x (t, x) + 1 2 tr[cv xx (t, x)] (1.2) + [v(t, x + z) − v(t, x) − v x (t, x)h(z)] F (dz) = 0.

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family P of possible physical measures. A robust notion NA 1 (P) of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: NA 1 (P) holds if and only if every P ∈ P admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.