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Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
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Euler scheme for SDE’s with nonLipschitz diffusion coefficient: Strong convergence
 ESAIM Probab. Statist
, 2008
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HIGH ORDER DISCRETIZATION SCHEMES FOR THE CIR PROCESS: APPLICATION TO AFFINE TERM STRUCTURE AND HESTON MODELS
"... Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya a ..."
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Cited by 32 (3 self)
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Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.
Balanced Milstein methods for ordinary SDEs, Monte Carlo Methods & Applications 12
, 2006
"... Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linearimplicit schemes which generate mean s ..."
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Cited by 16 (3 self)
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Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linearimplicit schemes which generate mean square converging numerical approximations with qualitative improvements and global rate 1.0 of mean square convergence, compared to commonly known numerical methods for SDEs. 1
Gamma Expansion of the Heston Stochastic Volatility Model
"... We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We ..."
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Cited by 14 (1 self)
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We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the PitmanYor decomposition of Bessel bridges. We combine this representation with the BroadieKaya exact simulation method and use it to circumvent the most timeconsuming step in that method.
Multilevel Monte Carlo Quadrature of Discontinuous Payoffs in the Generalized Heston Model using Malliavin Integration by Parts
, 2013
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 14 (0 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.
An eulertype method for the strong approximation of the cox–ingersoll–ross process
 Proceedings of the Royal Society A Engineering Science
"... Abstract. We analyze the strong approximation of the CoxIngersollRoss (CIR) process in the regime where the process does not hit zero by a positivity preserving driftimplicit Eulertype method. As an error criterion we use the pth mean of the maximum distance between the CIR process and its appr ..."
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Cited by 12 (2 self)
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Abstract. We analyze the strong approximation of the CoxIngersollRoss (CIR) process in the regime where the process does not hit zero by a positivity preserving driftimplicit Eulertype method. As an error criterion we use the pth mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations (SDEs) with Lipschitz coefficients – despite the fact that the CIR process has a nonLipschitz diffusion coefficient.
TIME DEPENDENT HESTONMODEL
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 11 (0 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Diffusion Monte Carlo method: Numerical analysis in a simple case
 ESAIM: M2AN
, 2007
"... The Diffusion Monte Carlo method is devoted to the computation of electronic groundstate energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a Stochastic Diffe ..."
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Cited by 8 (0 self)
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The Diffusion Monte Carlo method is devoted to the computation of electronic groundstate energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple onedimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to + ∞ while the discretization step of the SDE tends to 0. We confirm our theoretical rates of convergence by numerical experiments.
Exact simulation of prices and greeks: application to cir
, 2008
"... Acknowledgment: The authors would like to thank gratefully Calyon and Inria. This work was done during an o cial collaboration between their teams. We generalize the exact simulation algorithm of one dimensional solution of SDE proposed by Beskos et al. [6]. We apply Malliavin Calculus to simulate e ..."
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Cited by 6 (1 self)
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Acknowledgment: The authors would like to thank gratefully Calyon and Inria. This work was done during an o cial collaboration between their teams. We generalize the exact simulation algorithm of one dimensional solution of SDE proposed by Beskos et al. [6]. We apply Malliavin Calculus to simulate exactly the greeks, that is the derivatives with respect to the initial condition. We obtain estimations of these derivatives without time discretization or space discretization. We detail the method for the CIR process and give numerical results.