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Marcel Nutz

Publications and source records attributed to Marcel Nutz.

At least 73 records · Page 4Linked to original sources
arXiv2015

Robust Fundamental Theorem for Continuous Processes

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family $\mathcal{P}$ of possible physical measures. A robust notion ${\rm NA}_{1}(\mathcal{P})$ 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: ${\rm NA}_{1}(\mathcal{P})$ holds if and only if every $P\in\mathcal{P}$ 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.

q-fin.MF↗
arXiv2015

Robust Superhedging with Jumps and Diffusion

We establish a nondominated version of the optional decomposition theorem in a setting that includes jump processes with nonvanishing diffusion as well as general continuous processes. This result is used to derive a robust superhedging duality and the existence of an optimal superhedging strategy for general contingent claims. We illustrate the main results in the framework of nonlinear Lévy processes.

q-fin.MF↗
arXiv2015

Arbitrage and duality in nondominated discrete-time models

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.

q-fin.GN↗
arXiv2015

Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract mea-surable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in ver-ification arguments for classical solutions of Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed by an exten-sion of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending.

math.PR↗
arXiv2015

Nonlinear Lévy Processes and their Characteristics

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.

math.PR↗
arXiv2014

Martingale Inequalities and Deterministic Counterparts

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 results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging.

math.PR↗
arXiv2014

Consistent Price Systems under Model Uncertainty

We develop a version of the fundamental theorem of asset pricing for discrete-time markets with proportional transaction costs and model uncertainty. A robust notion of no-arbitrage of the second kind is defined and shown to be equivalent to the existence of a collection of strictly consistent price systems.

q-fin.MF↗
arXiv2014

Measurability of Semimartingale Characteristics with Respect to the Probability Law

Given a càdlàg process $X$ on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let $\mathfrak{P}_{sem}$ be the set of all probability measures $P$ under which $X$ is a semimartingale. We construct processes $(B^P,C,ν^P)$ which are jointly measurable in time, space, and the probability law $P$, and are versions of the semimartingale characteristics of $X$ under $P$ for each $P\in\mathfrak{P}_{sem}$. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.

math.PR↗
arXiv2014

Stochastic target games with controlled loss

We study a stochastic game where one player tries to find a strategy such that the state process reaches a target of controlled-loss-type, no matter which action is chosen by the other player. We provide, in a general setup, a relaxed geometric dynamic programming principle for this problem and derive, for the case of a controlled SDE, the corresponding dynamic programming equation in the sense of viscosity solutions. As an example, we consider a problem of partial hedging under Knightian uncertainty.

math.OC↗
arXiv2014

Superreplication under Model Uncertainty in Discrete Time

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a suitable Banach space. The main ingredient is a closedness result for the set of claims which can be superreplicated from zero capital; its proof relies on medial limits.

q-fin.PR↗
arXiv2013

Random G-expectations

We construct a time-consistent sublinear expectation in the setting of volatility uncertainty. This mapping extends Peng's G-expectation by allowing the range of the volatility uncertainty to be stochastic. Our construction is purely probabilistic and based on an optimal control formulation with path-dependent control sets.

math.PR↗
arXiv2013

Utility Maximization under Model Uncertainty in Discrete Time

We give a general formulation of the utility maximization problem under nondominated model uncertainty in discrete time and show that an optimal portfolio exists for any utility function that is bounded from above. In the unbounded case, integrability conditions are needed as nonexistence may arise even if the value function is finite.

q-fin.PM↗
arXiv2013

Constructing Sublinear Expectations on Path Space

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 G-expectation, 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.

math.PR↗
arXiv2012

Weak Dynamic Programming for Generalized State Constraints

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.

math.OC↗
arXiv2012

Pathwise Construction of Stochastic Integrals

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-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 the integrator is a semimartingale. This method applies to any predictable integrand.

math.PR↗
arXiv2012

Superhedging and Dynamic Risk Measures under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

q-fin.RM↗
arXiv2012

A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations

We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This value process is characterized by a second order backward SDE, which can be seen as a non-Markovian analogue of the Hamilton-Jacobi-Bellman partial differential equation. Moreover, our value process yields a generalization of the G-expectation to the context of SDEs.

math.PR↗