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Concept

Convexité (finance)

Résumé

La convexité (en anglais : bond convexity) est un indicateur du risque de taux lié à un instrument à taux fixe, comme une obligation, qui complète la sensibilité ou la duration Définition En utilisant le théorème de Taylor, on peut approcher la variation du prix d'une obligation en fonction de son taux actuariel. P(r) \simeq P(r_0) + P'(r_0)(r-r_0) + \frac{P^{(2)}(r_0)}{2!}(r - r_0)^2 Avec :

r,! le taux actuariel,
P(r),! le prix de l'instrument en fonction du taux actuariel,
P'(r) = \frac {dP(r)}{dr} la dérivée du prix de l'instrument par rapport au taux actuariel.
P^{(2)}(r) = \frac {d^2P(r)}{d^2r} la dérivée seconde du prix de l'instrument.

dP(r) \simeq P'(r)dr+ \frac{P^{(2)}(r)}{2!}dr^2 dP(r)/P(r) \simeq \frac{P'(r)}{P(r)} dr+ \frac{P^{(2)}(r)}{2P(r)}dr^2 Soit en utilisant la définition de la sensibilité S. dP(r)/P(r) \simeq -Sdr+ \frac{P^{(2)}(r)}{2P(r)

Source officielle

https://fr.wikipedia.org/wiki/Convexit%C3%A9_(finance)

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Séances de cours associées

Chargement

In this thesis we study calculus of variations for differential forms. In the first part we develop the framework of direct methods of calculus of variations in the context of minimization problems for functionals of one or several differential forms of the type,

\int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).

We introduce the appropriate convexity notions in each case, called \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega),

\emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})

and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{ext-int. one convexity} for functionals of the type

\int_{\Omega} f(d\omega, \delta\omega).

We study their interrelationships and the connections of these convexity notions with the classical notion of polyconvexity, quasiconvexity and rank one convexity in classical vectorial calculus of variations. We also study weak lower semicontinuity and weak continuity of these functionals in appropriate spaces, address coercivity issues and obtain existence theorems for minimization problems for functionals of one differential forms.\smallskip In the second part we study different boundary value problems for linear, semilinear and quasilinear Maxwell type operator for differential forms. We study existence and derive interior regularity and

L^{2}

boundary regularity estimates for the linear Maxwell operator

\delta (A(x)d\omega) = f

with different boundary conditions and the related Hodge Laplacian type system

\delta (A(x)d\omega) + d\delta\omega = f,

with appropriate boundary data. We also deduce, as a corollary, some existence and regularity results for div-curl type first order systems. We also deduce existence results for semilinear boundary value problems \begin{align*} \left\lbrace \begin{gathered} \delta ( A (x) ( d\omega ) ) + f( \omega ) = \lambda\omega \text{ in } \Omega, \ \nu \wedge \omega = 0 \text{ on } \partial\Omega. \end{gathered} \right. \end{align*} Lastly, we briefly discuss existence results for quasilinear Maxwell operator \begin{align*} \delta ( A ( x, d \omega ) ) = f , \end{align*} with different boundary data.

EPFL2016

Chargement

We study the relation between various notions of exterior convexity introduced in [S. Bandyopadhyay, B. Dacorogna and S. Sil, J. Eur. Math. Soc. 17 (2015) 1009-1039.] with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in [S. Bandyopadhyay, B. Dacorogna and S. Sil, J. Eur. Math. Soc. 17 (2015) 1009-1039.].

Edp Sciences S A2016

Chargement

Convergence without Convexity: Sampling, Optimization, and Games

Ya-Ping Hsieh

Many important problems in contemporary machine learning involve solving highly non- convex problems in sampling, optimization, or games. The absence of convexity poses significant challenges to convergence analysis of most training algorithms, and in some cases (such as min-max games) it is not even known whether common training algorithms converge or not. In this thesis, we aim to partially bridge the gap by

Proposing a new sampling framework to transform non-convex problems in to convex ones.
Characterizing the convergent sets of a wide family of popular algorithms for min-max optimization.
Devising provably convergent algorithms for finding mixed Nash Equilibria of infinite-dimensional bi-affine games.

Our theory has several important implications. First, we resolve a decade-old open problem in Bayesian learning via our non-convex sampling framework. Second, our algorithms for bi-affine games apply to the formidably difficult training of generative adversarial networks and robust reinforcement learning, and on both examples we demonstrate promising empirical performance. Finally, our results on min-max optimization lead to a series of negative results for state-of-the-art algorithms, suggesting that one requires fundamentally new tools to advance the theory.

EPFL2020

Chargement

\int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).

We introduce the appropriate convexity notions in each case, called \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega),

\emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})

and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{ext-int. one convexity} for functionals of the type

\int_{\Omega} f(d\omega, \delta\omega).

L^{2}

boundary regularity estimates for the linear Maxwell operator

\delta (A(x)d\omega) = f

with different boundary conditions and the related Hodge Laplacian type system

\delta (A(x)d\omega) + d\delta\omega = f,

EPFL2016