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Optimal error estimates of Galerkin finite element methods for stochastic partical differential equations with multiplicative noise. arXiv:1103.4504v1
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Antithetic multilevel Monte Carlo estimation for multidimensional SDEs without Lévy area simulation. Arxiv preprint arXiv:1202.6283
, 2012
"... Abstract In this paper we develop antithetic multilevel Monte Carlo (MLMC) estimators for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O( ∆ t) with MLMC we can reduce the computational comp ..."
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Cited by 10 (4 self)
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Abstract In this paper we develop antithetic multilevel Monte Carlo (MLMC) estimators for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O( ∆ t) with MLMC we can reduce the computational complexity to estimate expected values of Lipschitz functionals of SDE solutions with a rootmeansquare error of ε from O(ε−3) to O(ε−2). However, in general, to obtain a rate of strong convergence higher than O( ∆ t1/2) requires simulation, or approximation, of Lévy areas. Recently, Giles and Szpruch [5] constructed an antithetic multilevel estimator that avoids the simulation of Lévy areas and still achieves an MLMC correction variance which is O( ∆ t2) for smooth payoffs and almost O( ∆ t3/2) for piecewise smooth payoffs, even though there is only O( ∆ t1/2) strong convergence. This results in an O(ε−2) complexity for estimating the value of financial European and Asian put and call options. In this paper, we extend these results to more complex payoffs based on the path minimum. To achieve this, an approximation of the Lévy areas is needed, resulting in O( ∆ t3/4) strong convergence. By modifying the antithetic MLMC estimator we are able to obtain O(ε−2 log(ε)2) complexity for estimating financial barrier and lookback options. 1
Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
 Ann. Appl. Probab
, 1966
"... The Euler–Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally onesided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from t ..."
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Cited by 9 (2 self)
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The Euler–Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally onesided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler’s method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does – in contrast to classical Monte Carlo methods – not converge in general. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally onesided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the
G.: Multilevel monte carlo simulation for lévy processes based on the Wiener–Hopf factorisation
 Stochastic Processes and their Applications
, 2014
"... In Kuznetsov et al. [28] a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the WienerHopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we ..."
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Cited by 7 (4 self)
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In Kuznetsov et al. [28] a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the WienerHopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in [28] and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Lévy process. We derive rates of convergence for both methods and show that they are uniform with respect to the “jump activity ” (e.g. characterised by the BlumenthalGetoor index). We also present a modified version of the algorithm in Kuznetsov et al. [28] which combined with the multilevel methodology obtains the optimal rate of convergence for general Lévy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.
Numerical analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
, 2013
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Optimization of mesh hierarchies in Multilevel Monte Carlo samplers. ArXiv eprints:1403.2480
, 2014
"... Abstract We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. Moreover, we discuss extensions to nonuniform discretizations based on a prio ..."
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Cited by 5 (1 self)
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Abstract We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. Moreover, we discuss extensions to nonuniform discretizations based on a priori refinements and the effect of imposing constraints on the largest and/or smallest mesh sizes. We optimize geometric and nongeometric hierarchies and compare them to each other, concluding that the geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity. We discuss how enforcing domain constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These domain constraints include an upper and lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm [13]
Vibrato Monte Carlo sensitivities
"... Abstract We show how the benefits of the pathwise sensitivity approach to computing Monte Carlo Greeks can be extended to discontinuous payoff functions through a combination of the pathwise approach and the Likelihood Ratio Method. With a variance reduction modification, this results in an estimato ..."
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Abstract We show how the benefits of the pathwise sensitivity approach to computing Monte Carlo Greeks can be extended to discontinuous payoff functions through a combination of the pathwise approach and the Likelihood Ratio Method. With a variance reduction modification, this results in an estimator which for timestep h has a variance which is O(h −1/2) for discontinuous payoffs and O(1) for continuous payoffs. Numerical results confirm the variance is much lower than the O(h −1) variance of the Likelihood Ratio Method, and the approach is also compatible with the use of adjoints to obtain multiple first order sensitivities at a fixed cost. 1
FROM ROUGH PATH ESTIMATES TO MULTILEVEL MONTE CARLO
, 1305
"... Abstract. Discrete approximations to solutions of stochastic differential equations are wellknown to converge with “strong ” rate 1/2. Such rates have played a keyrole in Giles ’ multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort n ..."
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Cited by 5 (4 self)
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Abstract. Discrete approximations to solutions of stochastic differential equations are wellknown to converge with “strong ” rate 1/2. Such rates have played a keyrole in Giles ’ multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H> 1/4 as special case). We consider implementable schemes for large classes of stochastic differential equations (SDEs) (1) dYt = V0 (Yt) dt + d∑ i=1 Vi (Yt) dX i t (ω) driven by multidimensional Gaussian signals, say X = Xt (ω) ∈ Rd. The interpretation of these equations is in Lyons ’ rough path sense [LQ02, LCL07, FV10b]. This requires smoothness/boundedness conditions on the vector fields V0 and V ≡ (V1,..., Vd); for the sake of this introduction, the reader may assume bounded vector fields with bounded derivatives of all order (but we will be more specific later). This also requires a “natural ” lift of X (·, ω) to a (random) rough path
CONVERGENCE, NONNEGATIVITY AND STABILITY OF A NEW MILSTEIN SCHEME WITH APPLICATIONS TO FINANCE
"... (Communicated by Rachel Kuske) Abstract. We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multilevel Monte Carlo simulations for meanreverting financial models ..."
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(Communicated by Rachel Kuske) Abstract. We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multilevel Monte Carlo simulations for meanreverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves nonnegativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found. 1. Introduction. The
Multilevel Monte Carlo for Basket options
 In Proceedings of the 2009 Winter Simulation Conference
, 2009
"... The multilevel Monte Carlo method has been previously introduced for the efficient pricing of options based on a single underlying quantity. In this paper we show that the method is easily extended to basket options based on a weighted average of several underlying quantities. Numerical results for ..."
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The multilevel Monte Carlo method has been previously introduced for the efficient pricing of options based on a single underlying quantity. In this paper we show that the method is easily extended to basket options based on a weighted average of several underlying quantities. Numerical results for Asian, lookback, barrier and digital basket options demonstrate that the computational cost to achieve a rootmeansquare error of is O(−2). This is achieved through a careful construction of the multilevel estimator which computes the difference in expected payoff when using different numbers of timesteps. 1