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182
A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations
, 2009
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Deterministic Multilevel Algorithms for INFINITEDIMENSIONAL INTEGRATION ON R^N
 PREPRINT 40, DFGSPP 1324
, 2010
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Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
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Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and QuasiMonte Carlo methods 2006
, 2007
"... Summary. In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic dierential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a roo ..."
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Cited by 64 (14 self)
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Summary. In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic dierential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a rootmeansquare error of is reduced to O(2). This is achieved through a careful construction of the multilevel estimator which computes the dierence in expected payo when using dierent numbers of timesteps. 1
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 25 (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.
Analysing Multilevel Monte Carlo for Options with Nonglobally Lipschitz Payoff
, 2008
"... 607–617) introduced a multilevel Monte Carlo 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 Mont ..."
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Cited by 25 (9 self)
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607–617) introduced a multilevel Monte Carlo 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.
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 24 (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.
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 (5 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.