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. Among them, Hodges and Neuberger [72] proposed in 1989 a method based on utility maximization. The price of the contingent claim is then obtained as the smallest (resp. largest) amount leading the agent indifferent between selling (resp. buying) the claim and doing nothing. The price obtained is the indifference seller's (resp. buyer's) price. Since then, many authors have used this approach, the exponential utility function being most often used (see for instance, El Karoui and Rouge [51], Becherer [11], Delbaen et al. [39] , Musiela and Zariphopoulou [93] or Mania and Schweizer [89]...). In this chapter, we also adopt this exponential utility point of view to start with in order to nd the optimal hedge and price of a contingent claim based on a non-tradable risk. But soon, we notice that the right framework to work with is not that of the exponential utility itself but that of the certainty equivalent which is a convex functional satisfying some nice properties among which that of cash translation invariance. Hence, the results obtained in this particular framework can be immediately extended to functionals satisfying the same properties, in other words to convex risk measures as introduced by Föllmer and Schied [53] and [54]

Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.

We develop a methodology to optimally design a financial issue to hedge non-tradable risk on financial markets.The modeling involves a minimization of the risk borne by issuer given the constraint imposed by a buyer who enters the transaction if and only if her risk level remains below a given threshold. Both agents have also the opportunity to invest all their residual wealth on financial markets but they do not have the same access to financial investments. The problem may be reduced to a unique inf-convolution problem involving some transformation of the initial risk measures.

. Let (S t ) t2I be an IR d --valued adapted stochastic process on(\Omega ; F ; (F t ) t2I ; P ). A basic problem, occuring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on F equivalent to P such that (S t ) t2I is a martingale with respect to Q. It is known since the fundamental papers of Harrison--Kreps (79), Harrison--Pliska(81) and Kreps(81) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics. We introduce the intermediate concept of "no free lunch with bounded risk". This is a somewhat more precise version of the notion of "no free lunch": It requires that there should be an absolute bound of the maximal loss occuring in the trading strategies considered in the definition of "no free lunch". We shall give an argument why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process (S t ) t2I of...

Abstract. We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption/terminal wealth problems, in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of L1 to its topological bidual (L∞) ∗ , a space of finitely-additive measures. As an application, we treat the case of a constrained Itô-process market-model. 1.

We extend the analysis of the intertemporal utility maximization problem for Hindy-Huang-Kreps utilities reported in Bank and Riedel (1998) to the stochastic case. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including the cases of both complete and incomplete markets. For the complete market setting, Kuhn-Tucker-like necessary and sufficient conditions for optimality are given. Using this characterization, we show that optimal consumption plans are obtained by reflecting the associated level of satisfaction on a stochastic lower bound. When uncertainty is generated by a Lévy process and agents exhibit constant relative risk aversion, closed-form solutions are derived. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or singular to Lebesgue measure. Keywords: Hindy-Huang-Kreps preferences, non-time additive utility optimization, intertemporal utility, intertemporal substitution

Abstract. We prove that convex intersection bodies are isomorphically equivalent to unit balls of subspaces of Lq for each q ∈ (0, 1). This is done by extending to negative values of p the factorization theorem of Maurey and Nikishin which states that for any 0 < p < q < 1 every Banach subspace of Lp is isomorphic to a subspace of Lq. 1.

Abstract. The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Doléans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure. 1.

doctor rerum naturalium (dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch–Naturwissenschaftlichen Fakultät II der Humboldt–Universität zu Berlin von

Summary. We construct the Doob–Meyer decomposition of a submartingale as a pointwise superior limit of decompositions of discrete submartingales suitably built upon discretizations of the original process. This gives, in particular, a direct proof of predictability of the increasing process in the Doob–Meyer decomposition. 1 The Doob–Meyer Theorem The Doob–Meyer decomposition theorem opened the way towards the theory of stochastic integration with respect to square integrable martingales and—consequently—semimartingales, as described in the seminal paper [7]. According to Kallenberg [4], this theorem is “the cornerstone of the modern probability theory”. It is therefore not surprising that many proofs of it are known. To the author’s knowledge, all the proofs heavily depend on a result due to Doléans-Dade [3], which identifies predictable increasing processes with “natural ” increasing processes, as defined by Meyer [6]. In the present paper we develop ideas of another classical paper by K. Murali Rao [8] and construct a sequence of decompositions for which the superior limit is pointwise (in (t, ω)) equal to the desired one, and thus we obtain predictability in the easiest possible way. Let (Ω,F, {Ft} t∈[0,T],P) be a stochastic basis, satisfying the “usual ” conditions, i.e. the filtration {Ft} is right-continuous and F0 contains all P-null sets of FT. Let (D) denote the class of measurable processes {Xt} t∈[0,T] such that the family {Xτ} is uniformly integrable, where τ runs over all stopping times with respect to {Ft} t∈[0,T]. One of the variants of the Doob–Meyer theorem can be formulated as follows.