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convex regularization of local volatility models from option prices: Convergence analysis and rates

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第4期75卷 2398-2415页

作者： De Cezaro, A. Scherzer, O. Zubelli, J. P. IMPA BR-22460320 Rio De Janeiro RJ Brazil Univ Vienna Computat Sci Ctr A-1090 Vienna Austria Austrian Acad Sci Radon Inst Computat & Appl Math A-4040 Linz Austria

We study a convex regularization of the local volatility surface identification problem for the Black-Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid-ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero. We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Local volatility surface identification convex regularization Convergence rates Source condition interpretation convex risk measures

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convex regularization-based hardware/software co-design for real-time enhancement of remote sensing imagery

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JOURNAL OF REAL-TIME IMAGE PROCESSING 2009年第3期4卷 261-272页

作者： Castillo Atoche, Alejandro Shkvarko, Y. Torres Roman, D. Perez Meana, H. IPN CINVESTAV Unidad Guadalajara Zapopan 45015 Jalisco Mexico Univ Autonoma Yucatan Merida Yucatan Mexico IPN ESIME Unidad Culhuacan Mexico City 07738 DF Mexico

In this paper, we address a new approach for high-resolution reconstruction and enhancement of remote sensing (RS) imagery in near-real computational time based on the aggregated hardware/software (HW/SW) co-design paradigm. The software design is aimed at the algorithmic-level decrease of the computational load of the large-scale RS image enhancement tasks via incorporating into the fixed-point iterative reconstruction/enhancement procedures the convex convergence enforcement regularization by constructing the proper projectors onto convex sets (POCS) in the solution domain. The established POCS-regularized iterative techniques are performed separately along the range and azimuth directions over the RS scene frame making an optimal use of the sparseness properties of the employed sensor system modulation format. The hardware design is oriented on employing the Xilinx Field Programmable Gate Array XC4VSX35-10ff668 and performing the image enhancement/reconstruction tasks in a computationally efficient parallel fashion that meets the near-real time imaging system requirements. Finally, we report some simulation results and discuss the implementation performance issues related to enhancement of the real-world RS imagery indicative of the significantly increased performance efficiency gained with the developed approach.

关键词： Remote sensing imagery convex regularization Synthetic aperture radar Hardware/software co-design FPGA

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convex regularization FOR HIGH-DIMENSIONAL MULTIRESPONSE TENSOR REGRESSION

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ANNALS OF STATISTICS 2019年第3期47卷 1554-1584页

作者： Raskutti, Garvesh Yuan, Ming Chen, Han Univ Wisconsin Dept Stat 330 N Orchard St Madison WI 53715 USA Columbia Univ Stat Dept 1255 Amsterdam Ave New York NY 10027 USA

In this paper, we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable regularizers assuming that the underlying tensor lies in an unknown low-dimensional subspace. Within our framework, we derive general risk bounds of the resulting estimate under fairly general dependence structure among covariates. Our framework leads to upper bounds in terms of two very simple quantities, the Gaussian width of a convex set in tensor space and the intrinsic dimension of the low-dimensional tensor subspace. To the best of our knowledge, this is the first general framework that applies to multiple response problems. These general bounds provide useful upper bounds on rates of convergence for a number of fundamental statistical models of interest including multiresponse regression, vector autoregressive models, low-rank tensor models and pairwise interaction models. Moreover, in many of these settings we prove that the resulting estimates are minimax optimal. We also provide a numerical study that both validates our theoretical guarantees and demonstrates the breadth of our framework.

关键词： Tensor regression convex regularization Gaussian width intrinsic dimension low-rank sparsity

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Effective Condition Number Bounds for convex regularization

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第4期66卷 2501-2516页

作者： Amelunxen, Dennis Lotz, Martin Walvin, Jake City Univ Hong Kong Dept Math Hong Kong Peoples R China Univ Warwick Math Inst Zeeman Bldg Coventry CV4 7AL W Midlands England Univ Manchester Sch Math Alan Turing Bldg Manchester M13 9PL Lancs England

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the l(1)-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.

关键词： convex regularization compressed sensing integral geometry convex optimization dimension reduction

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ON REPRESENTER THEOREMS AND convex regularization

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SIAM JOURNAL ON OPTIMIZATION 2019年第2期29卷 1260-1281页

作者： Boyer, Claire Chambolle, Antonin De Castro, Yohann Duval, Vincent De Gournay, Frederic Weiss, Pierre Sorbonne Univ LPSM Paris France ENS Paris DMA Paris France Ecole Polytech CNRS CMAP Palaiseau France Ecole Ponts ParisTech CERMICS Marne La Vallee France INRIA Rocquencourt France Univ Paris 09 Univ PSL CNRS CEREMADE Paris France Univ Toulouse INSA IMT Toulouse France Univ Toulouse INSA ITAV Toulouse France Univ Toulouse CNRS ITAV Toulouse France Univ Toulouse CNRS IMT Toulouse France

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions that are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. An extension to a broader class of quasi-convex regularizers is also discussed. As a side result, we characterize the minimizers of the total gradient variation, which was previously an unresolved problem.

关键词： inverse problems convex regularization representer theorem vector space total variation

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Finite-dimensional approximation of convex regularization via hexagonal pixel grids

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APPLICABLE ANALYSIS 2015年第3期94卷 612-636页

作者： Kirisits, Clemens Poeschl, Christiane Resmerita, Elena Scherzer, Otmar Univ Vienna Computat Sci Ctr A-1090 Vienna Austria Alpen Adria Univ Klagenfurt Inst Math A-9020 Klagenfurt Austria Radon Inst Computat & Appl Math A-4040 Linz Austria

This work extends the existing convergence analysis for discrete approximations of minimizers of convex regularization functionals. In particular, some solution concepts are generalized, namely the standard minimum norm solutions for squared-norm regularizers and the -minimizing solutions for general convex regularizers, respectively. A central part of the manuscript addresses finite-dimensional approximations of solutions of ill-posed operator equations with basis functions defined on hexagonal grids, which require the novel solution concept.

关键词： convex regularization ill-posed problem total variation hexagonal pixel 65J20 47J06 (65M60)

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RLS Algorithm With convex regularization

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IEEE SIGNAL PROCESSING LETTERS 2011年第8期18卷 470-473页

作者： Eksioglu, Ender M. Tanc, A. Korhan Istanbul Tech Univ Dept Elect & Commun Engn TR-80626 Istanbul Turkey

In this letter, the RLS adaptive algorithm is considered in the system identification setting. The RLS algorithm is regularized using a general convex function of the system impulse response estimate. The normal equations corresponding to the convex regularized cost function are derived, and a recursive algorithm for the update of the tap estimates is established. We also introduce a closed-form expression for selecting the regularization parameter. With this selection of the regularization parameter, we show that the convex regularized RLS algorithm performs as well as, and possibly better than, the regular RLS when there is a constraint on the value of the convex function evaluated at the true weight vector. Simulations demonstrate the superiority of the convex regularized RLS with automatic parameter selection over regular RLS for the sparse system identification setting.

关键词： Adaptive filter convex regularization l1 norm l0 norm RLS sparsity

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convex regularization OF LOCAL VOLATILITY ESTIMATION

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INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE 2017年第1期20卷 1750006-1750006页

作者： Albani, Vinicius De Cezaro, Adriano Zubelli, Jorge P. Univ Fed Santa Catarina Dept Math BR-88040900 Florianopolis SC Brazil Fed Univ Rio Grande Inst Math Stat & Phys Av Italia Km 8 BR-96201900 Rio Grande RS Brazil Inst Pure & Appl Math Estr Dona Castorina 110 BR-22460320 Rio De Janeiro Brazil

We apply convex regularization techniques to the problem of calibrating Dupire's local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quanti-zation of the quoted prices and lack of liquidity of option prices for strikes far away from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors, thus, allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov's discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

关键词： convex regularization local volatility surfaces regularization convergence rates numerical methods for volatility calibration

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A theory of optimal convex regularization for low-dimensional recovery

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INFORMATION AND INFERENCE-A JOURNAL OF THE IMA 2024年第2期13卷 iaae013-iaae013页

作者： Traonmilin, Yann Gribonval, Remi Vaiter, Samuel Univ Bordeaux CNRS UMR 5251 Bordeaux INPIMB F-33400 Talence France Univ Lyon CNRS ENS Lyon UCBLInriaLIP F-69342 Lyon France Univ Cote Azur CNRS LJAD F-06108 Nice France

We consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model, we ask ourselves what is the 'best' convex regularizer to perform its recovery. To answer this question, we define an optimal regularizer as a function that maximizes a compliance measure with respect to the model. We introduce and study several notions of compliance. We give analytical expressions for compliance measures based on the best-known recovery guarantees with the restricted isometry property. These expressions permit to show the optimality of the $\ell <^>{1}$ -norm for sparse recovery and of the nuclear norm for low-rank matrix recovery for these compliance measures. We also investigate the construction of an optimal convex regularizer using the examples of sparsity in levels and of sparse plus low-rank models.

关键词： inverse problems convex regularization low-dimensional modeling sparse recovery low-rank matrix recovery

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CONVERGENCE OF HEURISTIC PARAMETER CHOICE RULES FOR convex TIKHONOV regularization

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2020年第3期58卷 1773-1800页

作者： Kindermann, Stefan Raik, Kemal Johannes Kepler Univ Linz Ind Math Inst Altenbergerstr 69 A-4040 Linz Austria

We investigate the convergence theory of several known as well as new heuristic parameter choice rules for convex Tikhonov regularization. The success of such methods is dependent on whether certain restrictions on the noise are satisfied. In the linear theory, such conditions are well understood and hold for typically irregular noise. In this paper, we extend the convergence analysis of heuristic rules using noise restrictions to the convex setting and prove convergence of the aforementioned methods therewith. The convergence theory is exemplified for the case of an ill-posed problem with a diagonal forward operator in l(q) spaces. Numerical examples also provide further insight.

关键词： ill-posed problems convex regularization heuristic parameter choice rules

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