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61
A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations
, 2009
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Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
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Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients
"... We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Mo ..."
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Cited by 46 (15 self)
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We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain circumstances the asymptotic cost of solving the stochastic problem is a constant (but moderately large) multiple of the cost of solving the deterministic problem. Numerical calculations demonstrating the effectiveness of the method for one and twodimensional model problems arising in groundwater flow are presented. 1
On irregular functionals of SDEs and the Euler scheme
 Fractional smoothness and applications in Finance 19
"... Abstract. We prove a sharp upper bound for the approximation error�g(X) − g ( ˆ X)  p in terms of moments of X − ˆ X, where X and ˆ X are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution of a stochastic differential ..."
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Cited by 24 (1 self)
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Abstract. We prove a sharp upper bound for the approximation error�g(X) − g ( ˆ X)  p in terms of moments of X − ˆ X, where X and ˆ X are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution of a stochastic differential equation at time T by the Euler scheme, and show that the approximation of the payoff of the binary option has asymptotically sharp strong convergence rate 1/2. This has consequences for multilevel Monte Carlo methods. 1.
Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and its Application to Multilevel Monte Carlo Methods
"... We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as lognormal rand ..."
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Cited by 24 (6 self)
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We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as lognormal random fields with exponential covariance, have only very limited spatial regularity, and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.
MULTILEVEL MONTE CARLO FOR CONTINUOUS TIME MARKOV CHAINS, WITH APPLICATIONS IN BIOCHEMICAL KINETICS
, 2012
"... We show how to extend a recently proposed multilevel Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the system to a specified accuracy. The extension is nontrivia ..."
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Cited by 23 (16 self)
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We show how to extend a recently proposed multilevel Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the system to a specified accuracy. The extension is nontrivial, exploiting a coupling of the requisite processes that is easy to simulate while providing a small variance for the estimator. Further, and in a stark departure from other implementations of multilevel Monte Carlo, we show how to produce an unbiased estimator that is significantly less computationally expensive than the usual unbiased estimator arising from exact algorithms in conjunction with crude Monte Carlo. We thereby dramatically improve, in a quantifiable manner, the basic computational complexity of current approaches that have many names and variants across the scientific literature, including the Bortz–Kalos–Lebowitz algorithm, discrete event simulation, dynamic Monte Carlo, kinetic Monte Carlo, the nfold way, the next reaction method, the residencetime algorithm, the stochastic simulation algorithm, Gillespie’s algorithm, and tauleaping. The new algorithm applies generically, but we also give an example where the coupling idea alone, even without a multilevel discretization, can be used to improve efficiency by exploiting system structure. Stochastically modeled chemical reaction networks provide a very important application for this work. Hence, we use this context for our notation, terminology, natural scalings, and computational examples.
Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
 SIAM Journal of Financial Mathematics
"... Abstract. In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretization on an unbounded domain is convergent of first order in the timestep and s ..."
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Cited by 18 (2 self)
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Abstract. In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretization on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretization is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretization of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.
Multilevel monte carlo algorithms for infinitedimensional integration
 on RN, J. Complexity (2010
"... 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 (4 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.