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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… (More)

- Marcel Nutz
- 2012

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Pathby-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that… (More)

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √ T in time T and depends only on the initial value of the… (More)

We study the existence of optimal actions in a zero-sum game infτ supP E P [Xτ ] between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem infτ E(Xτ ) for a class of sublinear expectations E(·) such as the G-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear… (More)

We establish the duality formula for the superreplication price in a setting of volatility uncertainty which includes the example of random G-expectation. In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.

- Marcel Nutz
- ArXiv
- 2009

We study utility maximization for power utility random elds with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. We show that any optimal strategy leads to a solution of the corresponding Bellman equation. The optimal strategies are described pointwise in terms of the opportunity process, which is… (More)

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to in nity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models.… (More)

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… (More)

We study the utility maximization problem for power utility random elds in a semimartingale nancial market, with and without intermediate consumption. The notion of an opportunity process is introduced as a reduced form of the value process of the resulting stochastic control problem. We show how the opportunity process describes the key objects: optimal… (More)

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our… (More)