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Convex Analysisの二,三の進展について

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

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.

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Arbitrage and Duality in Nondominated Discrete-Time Models

The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization. This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs.

Pricing, Hedging and Optimally Designing Derivatives via Minimization of Risk Measures

In this paper, we study a class of quadratic backward stochastic differential equations (BSDEs), which arises naturally in the utility maximization problem with portfolio constraints. We first establish the existence and uniqueness of solutions for such BSDEs and then give applications to the utility maximization problem. Three cases of utility functions, the exponential, power, and logarithmic ones, are discussed.

Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem

This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property.

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On the Extension of the Namioka-Klee Theorem and on the Fatou Property for Risk Measures

We consider a stochastic …nancial incomplete market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of the utility maximization from terminal wealth, with utility functions that are …nite valued over (a;1), a 2 [1 ;1), and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a uni…ed way for both the cases: a 2 R or a = 1 . By duality methods we prove the existence of the solutions to the primal and dual problems and show that a singular component in the pricing functionals may occur also with utility functions …nite on the entire real line.

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A unified framework for utility maximization problems: An Orlicz space approach

When the price processes of the financial assets are described by possibly unbounded semimartingales, the classical concept of admissible trading strategies may lead to a trivial utility maximization problem because the set of stochastic integrals bounded from below may be reduced to the zero process. However, it could happen that the investor is willing to trade in such a risky market, where potential losses are unlimited, in order to increase his/her expected utility. We translate this attitude into mathematical terms by employing a class $\mathcal{H}^{W}$ of W-admissible trading strategies which depend on a loss random variable W. These strategies enjoy good mathematical properties and the losses they could generate in trading are compatible with the preferences of the agent.

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Utility maximization in incomplete markets for unbounded processes

We consider a stochastic financial incomplete market where the price
processes are described by a vector valued semimartingale that is possibly
non locally bounded. We face the classical problem of the utility
maximization from terminal wealth, with utility functions that are finite
valued over $(a,\infty )$, $a\in \lbrack -\infty ,\infty )$, and satisfy
weak regularity assumptions. We adopt a class of trading strategies that
allows for non locally bounded stochastic integrals. The embedding of the
utility maximization problem in Orlicz spaces permits to formulate the
problem in a unified way for both the cases: $a\in \mathbb{R}$ or $a=-\infty
$. By duality methods we prove the existence of the optimal solutions to the
primal and dual problems and show that a singular component in the pricing
functionals may occur also with utility functions finite on the entire real
line.

A Unified Framework for Utility Maximization Problems: an Orlicz space approach

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family inline image of possible physical measures. A robust notion inline image 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: inline image holds if and only if every inline image admits a martingale measure that 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.

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Sara Biagini

Papers

On the Extension of the Namioka-Klee Theorem and on the Fatou Property for Risk Measures

A unified framework for utility maximization problems: An Orlicz space approach

Utility maximization in incomplete markets for unbounded processes

A Unified Framework for Utility Maximization Problems: an Orlicz space approach

Robust Fundamental Theorem for Continuous Processes