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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
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.
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.
Numerical analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
, 2013
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Fractional smoothness and applications in Finance
, 2010
"... This overview article concerns the notion of fractional smoothness of random variables of the form g(XT), where X = (Xt)t∈[0,T] is a certain diffusion process. Wereviewtheconnectiontothereal interpolation theory,giveexamples and applications of this concept. The applications in stochastic finance ma ..."
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Cited by 5 (3 self)
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This overview article concerns the notion of fractional smoothness of random variables of the form g(XT), where X = (Xt)t∈[0,T] is a certain diffusion process. Wereviewtheconnectiontothereal interpolation theory,giveexamples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.
Multilevel Monte Carlo methods
"... An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas ..."
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Cited by 2 (1 self)
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An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas
THE EULERMARUYAMA APPROXIMATION FOR THE ABSORPTION TIME OF THE CEV DIFFUSION
"... (Communicated by Peter E. Kloeden) Abstract. The standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires nondegeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffus ..."
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Cited by 1 (0 self)
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(Communicated by Peter E. Kloeden) Abstract. The standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires nondegeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the EulerMaruyama scheme.
Finance Stoch (2009) 13: 403–413 DOI 10.1007/s0078000900921 Analysing multilevel Monte Carlo for options with nonglobally Lipschitz payoff
"... method for approximating the expected value of a function of a stochastic differential equation solution. A key application is to compute the expected payoff of a financial option. This new method improves on the computational complexity of standard Monte Carlo. Giles analysed globally Lipschitz pa ..."
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method for approximating the expected value of a function of a stochastic differential equation solution. A key application is to compute the expected payoff of a financial option. This new method improves on the computational complexity of standard Monte Carlo. Giles analysed globally Lipschitz payoffs, but also found good performance in practice for nonglobally Lipschitz cases. In this work, we show that the multilevel Monte Carlo method can be rigorously justified for nonglobally Lipschitz payoffs. In particular, we consider digital, lookback and barrier options. This requires nonstandard strong convergence analysis of the Euler–Maruyama method.