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30
Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
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On the discretization schemes for the CIR (and Bessel squared) processes
 Monte Carlo Methods Appl
, 2005
"... In this paper, we focus on the simulation of the CIR processes and present several discretization schemes of both the implicit and explicit types. We study their strong and weak convergence. We also examine numerically their behaviour and compare them to the schemes already proposed by Deelstra and ..."
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Cited by 50 (2 self)
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In this paper, we focus on the simulation of the CIR processes and present several discretization schemes of both the implicit and explicit types. We study their strong and weak convergence. We also examine numerically their behaviour and compare them to the schemes already proposed by Deelstra and Delbaen [5] and Diop [6]. Finally, we gather all the results obtained and recommend, in the standard case, the use of one of our explicit schemes. 1
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.
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.
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.
On Filtering in Markovian Term Structure Models (An approximation approach)
, 2001
"... We study a nonlinear filtering problem to estimate, on the basis of noisy observations of forward rates, the market price of interest rate risk as well as the parameters in a particular term structure model within the HeathJarrowMorton family. An approximation approach is described for the actual ..."
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Cited by 6 (4 self)
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We study a nonlinear filtering problem to estimate, on the basis of noisy observations of forward rates, the market price of interest rate risk as well as the parameters in a particular term structure model within the HeathJarrowMorton family. An approximation approach is described for the actual computation of the filter. Key words : Filter approximations, HeathJarrowMorton model, market price of interest rate risk, Markovian representations, measure transformation, nonlinear filtering, term structure of interest rates.
Approximation of the distribution of a stationary Markov process with application to option pricing
, 2009
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High order discretization schemes for stochastic volatility models
, 2009
"... In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, ..."
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Cited by 5 (3 self)
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In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, a scheme, based on the NinomiyaVictoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an OrsteinUhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a,b].