16 results on '"Emmanuel Lépinette"'
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2. Coherent Risk Measure on L 0 : NA Condition, Pricing and Dual Representation
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Emmanuel Lépinette, Duc Thinh Vu, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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Mathematical finance ,Risk measure ,010102 general mathematics ,Financial market ,Fundamental theorem of asset pricing ,Dual representation ,Characterization (mathematics) ,01 natural sciences ,Dual (category theory) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Economics ,Arbitrage ,0101 mathematics ,General Economics, Econometrics and Finance ,Mathematical economics ,Finance ,ComputingMilieux_MISCELLANEOUS - Abstract
The NA condition is one of the pillars supporting the classical theory of financial mathematics. We revisit this condition for financial market models where a dynamic risk-measure defined on [Formula: see text] is fixed to characterize the family of acceptable wealths that play the role of nonnegative financial positions. We provide in this setting a new version of the fundamental theorem of asset pricing and we deduce a dual characterization of the super-hedging prices (called risk-hedging prices) of a European option. Moreover, we show that the set of all risk-hedging prices is closed under NA. At last, we provide a dual representation of the risk-measure on [Formula: see text] under some conditions.
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- 2021
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3. Risk arbitrage and hedging to acceptability under transaction costs
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Emmanuel Lépinette and Ilya Molchanov
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Statistics and Probability ,Computer science ,01 natural sciences ,FOS: Economics and business ,Dynamic risk measure ,010104 statistics & probability ,510 Mathematics ,Computer Science::Computational Engineering, Finance, and Science ,0502 economics and business ,FOS: Mathematics ,Econometrics ,0101 mathematics ,050208 finance ,91G20, 60D05, 60G42 ,Mathematical finance ,Risk measure ,Probability (math.PR) ,05 social sciences ,Mathematical Finance (q-fin.MF) ,310 Statistics ,Quantitative Finance - Mathematical Finance ,Fixed asset ,Portfolio ,Risk arbitrage ,Arbitrage ,Statistics, Probability and Uncertainty ,Acceptance set ,Mathematics - Probability ,Finance - Abstract
The classical discrete time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing for convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate positions are converted into a single fixed asset, our framework turns into the no good deals setting. However, in general, the possibilities of assessing the risk with respect to any asset or a basket of the assets lead to a decrease of superhedging prices and the no arbitrage conditions become stronger. The mathematical technique relies on results for unbounded and possibly non-closed random sets in Euclidean space., 31 page. Accepted for publication in Finance and Stochastics
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- 2021
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4. A complement to the Grigoriev theorem for the Kabanov model
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Emmanuel Lépinette, Jun Zhao, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Polymer Science and Engineering (USTB), University of Science and Technology Beijing [Beijing] (USTB), and Lépinette, Emmanuel
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Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Liquidation value ,Transaction costs ,Financial market ,Consis-tent price systems ,01 natural sciences ,and phrases: Proportional transaction costs ,Bid and ask prices ,Set (abstract data type) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Conic section ,Arbitrage ,Financial market models ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bid price ,Absence of arbitrage opportunities ,Mathematical economics ,Complement (set theory) ,Mathematics - Abstract
International audience; We provide an equivalent characterisation of absence of arbitrage opportunity NA for the Bid and Ask financial market model analog to the Dalang--Morton--Willinger theorem formulated for discrete-time financial market models without friction. This result completes the Grigoriev theorem for conic models in the two dimensional case by showing that the set of all terminal liquidation values is closed under NA..
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- 2020
5. Diffusion Equations: Convergence of the Functional Scheme Derived from the Binomial Tree with Local Volatility for Non Smooth Payoff Functions
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Emmanuel Lépinette, Julien Baptiste, Université Paris sciences et lettres (PSL), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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Discretization ,Uniform convergence ,01 natural sciences ,010104 statistics & probability ,0502 economics and business ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Applied mathematics ,Financial market models ,0101 mathematics ,and phrases: Binomial tree model ,Mathematics ,European option pricing ,050208 finance ,Liquidation value ,Transaction costs ,Diffusion partial differential equations ,Applied Mathematics ,05 social sciences ,Finite difference method ,Finite difference ,European options ,Function (mathematics) ,Finite difference scheme ,Rate of convergence ,Local volatility ,finite element scheme ,2000 MSC: 60G44, G11-G13 ,Binomial options pricing model ,Finance - Abstract
International audience; The function solution to the functional scheme derived from the Binomial tree financial model with local volatility converges to thesolution of a diffusion equation of type ht(t, x)+ x2σ2(t,x) hxx(t, x) = 0 as the number of discrete dates n → ∞. Contrarily to classical numerical methods, in particular finite difference methods, the principle is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.
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- 2018
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6. Arbitrage theory for non convex financial market models
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Emmanuel Lépinette and Tuan Quoc Tran
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Statistics and Probability ,Convex analysis ,Transaction cost ,050208 finance ,Applied Mathematics ,05 social sciences ,Financial market ,Regular polygon ,01 natural sciences ,Liquidation value ,010104 statistics & probability ,Modeling and Simulation ,0502 economics and business ,Arbitrage ,0101 mathematics ,Fixed cost ,Mathematical economics ,Probability measure ,Mathematics - Abstract
When dealing with non linear trading costs, e.g. fixed costs, the usual tools from convex analysis are inadequate to characterize an absence of arbitrage opportunity as the mathematical model is no more convex. An unified approach is to describe a financial market model by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a large class of financial markets with transaction costs including non linear trading costs. The natural question is to which extent the results of the classical arbitrage theory are still valid when the model is not convex, in particular what does the existence of an equivalent separating probability measure mean ? Our contribution is a first attempt to characterise the absence of arbitrage opportunity in non convex financial market models.
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- 2017
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7. Consumption-investment problem with transaction costs for Lévy-driven price processes
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Dimitri De Vallière, Yuri Kabanov, and Emmanuel Lépinette
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Statistics and Probability ,Mathematical optimization ,Mathematical finance ,010102 general mathematics ,Financial market ,Hamilton–Jacobi–Bellman equation ,Optimal control ,01 natural sciences ,Dynamic programming ,010104 statistics & probability ,Uniqueness theorem for Poisson's equation ,Bellman equation ,Economics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Viscosity solution ,Mathematical economics ,Finance - Abstract
We consider an optimal control problem for a linear stochastic integro-differential equation with conic constraints on the phase variable and with the control of singular–regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs, where the prices of the assets are given by a geometric Levy process, and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of an HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the dynamic programming principle.
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- 2016
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8. Pricing without martingale measure
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Laurence Carassus, Julien Baptiste, Emmanuel Lépinette, Université Paris sciences et lettres (PSL), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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Super-hedging prices ,Computer Science::Computer Science and Game Theory ,Mathematics::Optimization and Control ,Duality (optimization) ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,Mathematics::Probability ,0502 economics and business ,Arbitrage pricing theory ,Call option ,Financial market models ,0101 mathematics ,Convex conjugate ,050205 econometrics ,Mathematics ,05 social sciences ,Stochastic game ,Essential supremum and essential infimum ,Mathematical Finance (q-fin.MF) ,Martingale (betting system) ,Conditional support ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Quantitative Finance - Mathematical Finance ,Arbitrage ,No-arbitrage condition ,Mathematical economics ,Essential supremum - Abstract
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenchel conjugate and bi-conjugate. The super-hedging problem leads endogenously to a weak condition of NA called Absence of Immediate Profit (AIP). We propose several characterizations of AIP and study the relation with the classical notions of no-arbitrage. We also give some promising numerical illustrations., 33 pages 6 figures
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- 2018
9. General financial market model defined by a liquidation value process
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Emmanuel Lépinette and Tuan Quoc Tran
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Statistics and Probability ,Solvency ,050208 finance ,05 social sciences ,Financial market ,Essential supremum and essential infimum ,01 natural sciences ,Liquidation value ,010104 statistics & probability ,Modeling and Simulation ,0502 economics and business ,Portfolio ,Mutual fund separation theorem ,Arbitrage ,0101 mathematics ,Fixed cost ,Mathematical economics ,Mathematics - Abstract
Financial market models defined by a liquidation value process generalize the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. The solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex, e.g. in presence of proportional transaction costs and fixed costs. Therefore, the classical duality principle based on the Hahn–Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. Using an alternative method based on the concepts of essential supremum and maximum, we provide a characterization of European and American contingent claim prices under the absence of arbitrage opportunity of the second kind.
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- 2015
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10. Risk Arbitrage and Hedging to Acceptability
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Emmanuel Lépinette, Ilya Molchanov, Institute of Mathematical Statistics and Actuarial Science [Bern] (IMSV), University of Bern, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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Mathematical optimization ,021103 operations research ,Computer science ,Risk measure ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,Infimum and supremum ,Dynamic risk measure ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Discrete time and continuous time ,Portfolio ,Arbitrage ,Risk arbitrage ,0101 mathematics ,Acceptance set - Abstract
The classical discrete time model of transaction costs relies on the assumption that the increments of the feasible portfolio process belong to the solvency set at each step. We extend this setting by assuming that any such increment belongs to the sum of an element of the solvency set and the family of acceptable positions, e.g. with respect to a dynamic risk measure.We formulate several no risk arbitrage conditions and explore connections between them. If the acceptance sets consist of non-negative random vectors, that is the underlying dynamic risk measure is the conditional essential infimum, we extend many classical no arbitrage conditions in markets with transaction costs and provide their natural geometric interpretation. The mathematical technique relies on results for unbounded and possibly non-closed random sets in the Euclidean space.
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- 2016
11. Robust No Arbitrage of the Second Kind with a Continuum of Assets and Proportional Transaction Costs
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Emmanuel Lépinette
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Transaction cost ,Numerical Analysis ,050208 finance ,Continuum (topology) ,Applied Mathematics ,05 social sciences ,Financial market ,Fundamental theorem of asset pricing ,01 natural sciences ,Microeconomics ,010104 statistics & probability ,Covered interest arbitrage ,0502 economics and business ,Econometrics ,Economics ,Bond market ,Portfolio ,Risk arbitrage ,Arbitrage ,0101 mathematics ,Mathematical economics ,Finance ,Index arbitrage - Abstract
We study the criteria of robust absence of arbitrage opportunity (RNA2) of the second kind as initially introduced by Rasony M. in the case of a continuous-time and infinite dimensional financial market model with proportional transaction costs allowing for bond market modeling. Robust no arbitrage criteria seems to be unavoidable to assure closedness of the set of attainable claims. This allows us to relate the (RNA2) condition to the richness of the solvency cones (Kt)t∈[0,T] of all solvent portfolio positions, or equivalently, to the existence of (many) consistent price systems starting from an arbitrary selector of the interior of the positive dual of K under efficient friction.
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- 2015
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12. Approximate hedging for nonlinear transaction costs on the volume of traded assets
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Emmanuel Lépinette, Romuald Elie, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Dauphine-PSL-Centre National de la Recherche Scientifique (CNRS)
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Statistics and Probability ,Malliavin calculus ,Order book ,01 natural sciences ,010104 statistics & probability ,0502 economics and business ,Economics ,0101 mathematics ,[MATH]Mathematics [math] ,050208 finance ,Transaction costs ,Mathematical finance ,05 social sciences ,Financial market ,Leland–Lott strategy ,Local volatility ,Replicating portfolio ,Replicating strategy ,Delta hedging ,Statistics, Probability and Uncertainty ,Greeks ,Mathematical economics ,Finance ,91G20, 60G44, 60H07 - Abstract
This paper is dedicated to the replication of a convex contingent claim h(S 1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h n of the payoff function, we first introduce the frictionless replicating portfolio of $h^{n}(S^{n}_{1})$ , where S n is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in $\mathbb{L}^{2}$ to the claim h(S 1) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.
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- 2015
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13. Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs
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Bruno Bouchard, Emmanuel Lépinette, Erik Taflin, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Centre de Recherche en Économie et Statistique (CREST), Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] (ENSAI)-École polytechnique (X)-École Nationale de la Statistique et de l'Administration Économique (ENSAE Paris)-Centre National de la Recherche Scientifique (CNRS), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), and Bouchard, Bruno
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[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,No-arbitrage ,01 natural sciences ,transaction costs ,FOS: Economics and business ,010104 statistics & probability ,MSC2010 91B25, 60G44 ,0502 economics and business ,Econometrics ,Economics ,FOS: Mathematics ,Trading strategy ,0101 mathematics ,91B25, 60G44 ,No free lunch with vanishing risk ,continuous time bond market ,050208 finance ,Actuarial science ,Continuum (topology) ,Bond ,05 social sciences ,Financial market ,Probability (math.PR) ,Fundamental theorem of asset pricing ,Price system ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Bond market ,Pricing of Securities (q-fin.PR) ,Quantitative Finance - Pricing of Securities ,Mathematics - Probability - Abstract
We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, R\'asonyi and L\'epinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general c\`adl\`ag case for further studies., Comment: 41 pages
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- 2014
14. The Fundamental Theorem of Asset Pricing under Transaction Costs
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Paolo Guasoni, Emmanuel Lépinette, Miklós Rásonyi, Boston School of Management, Boston University [Boston] (BU), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Computer and Automation Research Institute [Budapest] (MTA SZTAKI ), Guasoni P, Lepinette E, and Rasonyi M
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Statistics and Probability ,Computer Science::Computer Science and Game Theory ,Arbitrage ,Fundamental Theorem of Asset Pricing ,01 natural sciences ,Transaction Costs ,010104 statistics & probability ,Finite Variation ,0502 economics and business ,Economics ,0101 mathematics ,Arbitrage, Fundamental theorem of asset pricing, Transaction costs, Admissible strategies, Finite variation ,Transaction cost ,050208 finance ,Finite variation ,Mathematical finance ,05 social sciences ,TheoryofComputation_GENERAL ,Fundamental theorem of asset pricing ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Bounded function ,Statistics, Probability and Uncertainty ,Bid price ,Mathematical economics ,Admissible Strategies ,Finance - Abstract
This paper proves the fundamental theorem of asset pricing with transaction costs, when bid and ask prices follow locally bounded càdlàg (right-continuous, left-limited) processes. The robust no free lunch with vanishing risk condition (RNFLVR) for simple strategies is equivalent to the existence of a strictly consistent price system (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The RNFLVR condition implies that admissible strategies are predictable processes of finite variation. The Appendix develops an extension of the familiar Stieltjes integral for càdlàg integrands and finite-variation integrators, which is central to modelling transaction costs with discontinuous prices.
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- 2012
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15. Vector-valued risk measure processes
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Imen Ben Tahar, Emmanuel Lépinette-Denis, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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050208 finance ,Actuarial science ,Risk measure ,05 social sciences ,Characterization (mathematics) ,Space (mathematics) ,01 natural sciences ,Dual (category theory) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,0502 economics and business ,Coherent risk measure ,Economics ,0101 mathematics ,Mathematical economics ,Axiom - Abstract
Introduced by Artzner, Delbaen, Eber and Heath (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini, Meddeb and Touzi (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results and examples of vector valued risk measure processes.
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- 2012
16. Parabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculus
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Emmanuel Lépinette-Denis, Sébastien Darses, Laboratoire d'Analyse, Topologie, Probabilités (LATP), Université Paul Cézanne - Aix-Marseille 3-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
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Statistics and Probability ,Feynman-Kac Formula ,Stochastic calculus ,Smooth solutions ,Stochastic Calculus ,Girsanov's Theorem ,01 natural sciences ,Upper and lower bounds ,Hyperbolic systems ,010104 statistics & probability ,Initial value problem ,Order (group theory) ,0101 mathematics ,Mathematics ,Vanishing viscosity method ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Quasi-linear Parabolic PDEs ,Feynman–Kac formula ,Cauchy distribution ,Lipschitz continuity ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Statistics, Probability and Uncertainty ,60H30, 35K, 35L - Abstract
International audience; We consider two quasi-linear initial-value Cauchy problems on Rd: a parabolic system and an hyperbolic one. They both have a rst order non-linearity of the form (t; x; u) ru, a forcing term h(t; x; u) and an initial condition u0 2 L1(Rd) \ C1(Rd), where (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t; x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but recursive parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.
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- 2010
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