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## Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations (1966)

Venue: | Ann. Appl. Probab |

Citations: | 9 - 2 self |

### Citations

1398 |
Numerical solution of stochastic differential equations
- Kloeden, Platen
- 1992
(Show Context)
Citation Context ... a smooth function with at most polynomially growing derivatives. Note that this question is not treated in the standard literature in computational stochastics (see, for instance, Kloeden and Platen =-=[26]-=- and Milstein [30]) which concentrates on SDEs with globally Lipschitz continuous coefficients rather than the SDE (1). The computation of statistical quantities of the form (2) for SDEs with non-glob... |

616 |
Carlo methods in financial engineering
- Glasserman
- 2004
(Show Context)
Citation Context ...on-globally Lipschitz continuous coefficients is a major issue in financial engineering, in particular, in option pricing. For details the reader is refereed to the monographs Lewis [28], Glassermann =-=[10]-=-, Higham [16] and Szpruch [37]. In order to simulate the quantity (2) on a computer, one has to discretize both the solution process X : [0, T ]×Ω→ Rd of the SDE (1) as well as the underlying probabil... |

195 |
Option Valuation under Stochastic Volatility
- Lewis
- 2000
(Show Context)
Citation Context ...2) for SDEs with non-globally Lipschitz continuous coefficients is a major issue in financial engineering, in particular, in option pricing. For details the reader is refereed to the monographs Lewis =-=[28]-=-, Glassermann [10], Higham [16] and Szpruch [37]. In order to simulate the quantity (2) on a computer, one has to discretize both the solution process X : [0, T ]×Ω→ Rd of the SDE (1) as well as the u... |

191 | Numerical Integration of Stochastic Differential Equations
- Milstein
- 1995
(Show Context)
Citation Context ... with at most polynomially growing derivatives. Note that this question is not treated in the standard literature in computational stochastics (see, for instance, Kloeden and Platen [26] and Milstein =-=[30]-=-) which concentrates on SDEs with globally Lipschitz continuous coefficients rather than the SDE (1). The computation of statistical quantities of the form (2) for SDEs with non-globally Lipschitz con... |

187 | Multilevel Monte Carlo path simulation
- Giles
- 2008
(Show Context)
Citation Context ...nt ΩN is too unlikely to occur in any of N2 Monte Carlo simulations in (4). Considerably more efficient than the Monte Carlo Euler method is the so-called multilevel Monte Carlo Euler method in Giles =-=[8]-=- (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao [9], Heinrich [13, 14], Heinrich and Sindambiwe [15] and Kebaier [23] for related results).... |

160 |
Probability Theory, an Analytic View
- Stroock
- 1993
(Show Context)
Citation Context ... ] − E [ f(Ȳ 2 (l−1),0,1) ] − f(Ȳ 2l,l,k) + f(Ȳ 2(l−1),l,k) )∥∥∥∥∥∥ Lp(Ω;R) for all N ∈ {21, 22, 23, . . .} and all p ∈ [1,∞) and the Burkholder-Davis-Gundy inequality in Theorem 6.3.10 in Stroock =-=[36]-=- shows the existence of real numbers Kp ∈ [0,∞), p ∈ [1,∞), such that∥∥∥∥∥∥E [ f(X) ] − 1 N N∑ k=1 f(Ȳ 1,0,k)− ld(N)∑ l=1 2l N N2l∑ k=1 f(Ȳ 2 l,l,k)− f(Ȳ 2(l−1),l,k) ∥∥∥∥∥∥ Lp(Ω;R) ≤ E [∣∣f(X)... |

142 | Multilevel Monte Carlo Algorithms for Lévy-driven SDEs with Gaussian Correction
- Dereich
- 2009
(Show Context)
Citation Context .... Considerably more efficient than the Monte Carlo Euler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich =-=[4]-=-, Giles [7], Giles, Higham and Mao [9], Heinrich [13, 14], Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectat... |

91 |
Stochastic differential equations and their applications
- Mao
- 1997
(Show Context)
Citation Context ...uous sample paths solving the stochastic differential equation (SDE) dXt = µ(Xt) dt+ σ(Xt) dWt, X0 = ξ, (1) for t ∈ [0, T ] (see, e.g., Alyushina [1], Theorem 1 in Krylov [27] or Theorem 2.4.1 in Mao =-=[29]-=-). The drift coefficient µ is the infinitesimal mean of the process X and the diffusion coefficient σ is the infinitesimal standard deviation of the process X. Our goal in this introductory section is... |

82 | Strong convergence of Euler-type methods for nonlinear stochastic differential equations
- Higham, Mao, et al.
(Show Context)
Citation Context ...tional effort which is required to determine the zero of a nonlinear equation in each time step of the implicit Euler method (122). More results on implicit numerical methods for SDEs can be found in =-=[19, 17, 41, 40, 37, 39, 38]-=-, for instance. Acknowledgement This work has been partially supported by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coeffici... |

63 | Improved multilevel monte carlo convergence using the milstein scheme. In: Monte Carlo and quasi-Monte Carlo methods 2006
- Giles
- 2008
(Show Context)
Citation Context ...bly more efficient than the Monte Carlo Euler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles =-=[7]-=-, Giles, Higham and Mao [9], Heinrich [13, 14], Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectations are ap... |

62 |
Existence of strong solutions for Itô’s stochastic equations via approximations
- Gyöngy, Krylov
- 1996
(Show Context)
Citation Context ...or all p ∈ (0,∞). Theorem 2.1 immediately follows from Lemma 2.2 and Lemma 2.3 below. More results on Euler’s method for SDEs with possibly superlinearly growing nonlinearities can, e.g., be found in =-=[12, 11, 31, 32]-=- and in the references therein. Lemma 2.2 (Tails of Y N1 , N ∈ N). Assume that the above setting is fulfilled and let P [ σ(ξ) 6= 0] > 0. Then there exists a real number β ∈ (1,∞) such that P[|Y N1 | ... |

46 | Mathematical methods of statistics. Princeton, NJ: Princeton University Press Darwin C., (1859), On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life - Cramer - 1946 |

46 |
Efficient Monte Carlo Simulation of Security Prices
- Duffie, Glynn
- 1995
(Show Context)
Citation Context ...ndent copies of the Euler approximations (3) (see Section 3 for the precise definition). The Monte Carlo Euler approximation of (2) with N ∈ N time steps and N2 Monte Carlo runs (see Duffie and Glynn =-=[6]-=- for more details on this choice) is then the random real number 1 N2 N2∑ k=1 f ( Y N,kN ) . (4) Convergence of the Monte Carlo Euler approximations (4) is well-known in case of globally Lipschitz... |

43 | Multilevel monte carlo methods
- Heinrich
- 2001
(Show Context)
Citation Context ...uler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao [9], Heinrich =-=[13, 14]-=-, Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectations are approximated by the multilevel Monte Carlo metho... |

41 | Monte Carlo complexity of global solution of integral equations
- Heinrich
- 1998
(Show Context)
Citation Context ...uler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao [9], Heinrich =-=[13, 14]-=-, Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectations are approximated by the multilevel Monte Carlo metho... |

35 | Statistical Romberg extrapolation: a new variance reduction method and applications to options pricing - Kebaier - 2005 |

31 |
Stochastic numerics for mathematical physics. Scientific Computation
- Milstein, Tretyakov
- 2004
(Show Context)
Citation Context ...or all p ∈ (0,∞). Theorem 2.1 immediately follows from Lemma 2.2 and Lemma 2.3 below. More results on Euler’s method for SDEs with possibly superlinearly growing nonlinearities can, e.g., be found in =-=[12, 11, 31, 32]-=- and in the references therein. Lemma 2.2 (Tails of Y N1 , N ∈ N). Assume that the above setting is fulfilled and let P [ σ(ξ) 6= 0] > 0. Then there exists a real number β ∈ (1,∞) such that P[|Y N1 | ... |

30 |
Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme
- Talay
- 2001
(Show Context)
Citation Context ...tional effort which is required to determine the zero of a nonlinear equation in each time step of the implicit Euler method (122). More results on implicit numerical methods for SDEs can be found in =-=[19, 17, 41, 40, 37, 39, 38]-=-, for instance. Acknowledgement This work has been partially supported by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coeffici... |

29 |
Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients
- Hutzenthaler, Jentzen, et al.
(Show Context)
Citation Context ... [26] and Section 12 in Milstein [30]). The case of superlinearly growing and hence non-globally Lipschitz continuous coefficients of the SDE is more subtle. Indeed, Theorem 2.1 in the recent article =-=[22]-=- shows in the presence of noise that Euler’s method diverges to infinity both in the strong and numerically weak sense if the coefficients of the SDE grow superlinearly (see Theorem 2.1 below for a ge... |

28 | A note on Euler’s approximations - Gyöngy - 1998 |

26 |
Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients
- Hutzenthaler, Jentzen, et al.
(Show Context)
Citation Context ... n+ 1− tN T ) Y N,l,kn (111) for all t ∈ [nTN , (n+1)TN ], n ∈ {0, 1, . . . , N − 1}, N ∈ N, l ∈ N0 and all k ∈ N. The following corollary is a direct consequence of Hutzenthaler, Jentzen and Kloeden =-=[21]-=- and Müller-Gronbach [33] (see also Ritter [35]). It asserts that the piecewise linear approximations Ȳ N , N ∈ N, converge in the strong sense to the exact solution. The convergence order is 12 exc... |

24 | Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff
- Giles, Higham, et al.
(Show Context)
Citation Context ... Monte Carlo Euler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao =-=[9]-=-, Heinrich [13, 14], Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectations are approximated by the multileve... |

24 |
The pathwise convergence of approximation schemes for stochastic differential equations
- Kloeden, Neuenkirch
- 2007
(Show Context)
Citation Context ...1∥∥c L2pc(Ω;C([0,T ],Rd)) ) (1 + ld(N)) 3 2 √ N for all N ∈ {21, 22, 23, . . .} and all p ∈ [1,∞). This shows (119). Inequality (120) then immediately follows from Lemma 2.1 in Kloeden und Neuenkirch =-=[25]-=-. This completes the proof of Proposition 6.2. It is well-known that the multilevel Monte Carlo method combined with the (fully) implicit Euler method converges too. The following simulation indicates... |

23 | Infinitedimensional quadrature and approximation of distributions, Found
- Creutzig, Dereich, et al.
(Show Context)
Citation Context ...ations in (4). Considerably more efficient than the Monte Carlo Euler method is the so-called multilevel Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter =-=[3]-=-, Dereich [4], Giles [7], Giles, Higham and Mao [9], Heinrich [13, 14], Heinrich and Sindambiwe [15] and Kebaier [23] for related results). In this method, time is discretized through the Euler method... |

23 |
Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In Stochastic analysis and related topics V
- Hu
- 1996
(Show Context)
Citation Context ...tional effort which is required to determine the zero of a nonlinear equation in each time step of the implicit Euler method (122). More results on implicit numerical methods for SDEs can be found in =-=[19, 17, 41, 40, 37, 39, 38]-=-, for instance. Acknowledgement This work has been partially supported by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coeffici... |

23 | The Euler scheme with irregular coefficients. The Annals of Probability
- Yan
(Show Context)
Citation Context ...gham and Mao [9] and Dörsek and Teichmann [5]. Moreover, numerical approximation results for SDEs with non-globally Lipschitz continuous and at most linearly growing coefficients can be found in Yan =-=[42]-=-, for instance. Proof of Proposition 6.2. The triangle inequality gives∥∥∥∥∥∥E [ f(X) ] − 1 N N∑ k=1 f(Ȳ 1,0,k)− ld(N)∑ l=1 2l N N2l∑ k=1 f(Ȳ 2 l,l,k)− f(Ȳ 2(l−1),l,k) ∥∥∥∥∥∥ Lp(Ω;R) ≤ ∣∣∣E[f(... |

21 | E.: Monte Carlo complexity of parametric integration
- Heinrich, Sindambiwe
- 1999
(Show Context)
Citation Context ...level Monte Carlo Euler method in Giles [8] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao [9], Heinrich [13, 14], Heinrich and Sindambiwe =-=[15]-=- and Kebaier [23] for related results). In this method, time is discretized through the Euler method and expectations are approximated by the multilevel Monte Carlo method. More formally, let Y N,l,kn... |

20 |
The optimal uniform approximation of systems of stochastic differential equations, The Annals of Applied Probability
- Müller-Gronbach
(Show Context)
Citation Context ...ecewise linear 24 interpolations of the tamed Euler approximations converge in the strong sense with the optimal convergence order according to Müller-Gronbach’s lower bound in the Lipschitz case in =-=[33]-=-. Theorem 6.2 then establishes almost sure and strong convergence of the multilevel Monte Carlo method combined with the tamed Euler method. The payoff function is allowed to depend on the whole path.... |

19 |
2004), An Introduction to Financial Option Valuation
- Higham
(Show Context)
Citation Context ...ipschitz continuous coefficients is a major issue in financial engineering, in particular, in option pricing. For details the reader is refereed to the monographs Lewis [28], Glassermann [10], Higham =-=[16]-=- and Szpruch [37]. In order to simulate the quantity (2) on a computer, one has to discretize both the solution process X : [0, T ]×Ω→ Rd of the SDE (1) as well as the underlying probability space (Ω,... |

19 |
A simple proof of the existence of a solution to the Itô equation with monotone coefficients. Theory Probab
- krylov
- 1990
(Show Context)
Citation Context ... : [0, T ]× Ω→ Rd with continuous sample paths solving the stochastic differential equation (SDE) dXt = µ(Xt) dt+ σ(Xt) dWt, X0 = ξ, (1) for t ∈ [0, T ] (see, e.g., Alyushina [1], Theorem 1 in Krylov =-=[27]-=- or Theorem 2.4.1 in Mao [29]). The drift coefficient µ is the infinitesimal mean of the process X and the diffusion coefficient σ is the infinitesimal standard deviation of the process X. Our goal in... |

16 |
Strong convergence rates for backward Euler-Maruyama method for nonlinear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics
- Mao, Szpruch
(Show Context)
Citation Context |

14 |
Probability theory. Universitext. Springer-Verlag London Ltd
- Klenke
- 2008
(Show Context)
Citation Context ...we have ∞∑ n=1 P [ A (2) 2n ] <∞. (61) 13 Proof of Lemma 4.10. Subadditivity of the probability measure P and the inequality P[σ̄−1|ξ0,1| ≥ x] ≤ 1x exp (− x22 ) for all x ∈ (0,∞) (e.g., Lemma 22.2 in =-=[24]-=-) imply P [ A (2) N ] = P [ ∃ l ∈ {0, 1, 2, . . . , ld(N)} ∃ k ∈ {1, 2, . . . , N 2l } : |ξl,k| ≥ 2 (l−1)4 T− 14N ] ≤ ld(N)∑ l=0 N 2l∑ k=1 P [ |ξl,k| ≥ 2 (l−1)4 T− 14N ] = ld(N)∑ l=0 N 2l · P [ |ξ0,1|... |

12 |
Approximation and optimization on the wiener space
- Ritter
- 1990
(Show Context)
Citation Context ...(n+1)TN ], n ∈ {0, 1, . . . , N − 1}, N ∈ N, l ∈ N0 and all k ∈ N. The following corollary is a direct consequence of Hutzenthaler, Jentzen and Kloeden [21] and Müller-Gronbach [33] (see also Ritter =-=[35]-=-). It asserts that the piecewise linear approximations Ȳ N , N ∈ N, converge in the strong sense to the exact solution. The convergence order is 12 except for a logarithmic term. Corollary 6.1 (Stron... |

9 | A semigroup point of view on splitting schemes for stochastic (partial) differential equations
- Doersek, Teichmann
(Show Context)
Citation Context ... with globally Lipschitz continuous coefficients but under less restrictive smoothness assumption on the payoff function, the reader is referred to Giles, Higham and Mao [9] and Dörsek and Teichmann =-=[5]-=-. Moreover, numerical approximation results for SDEs with non-globally Lipschitz continuous and at most linearly growing coefficients can be found in Yan [42], for instance. Proof of Proposition 6.2. ... |

7 | Convergence of the stochastic Euler scheme for locally Lipschitz coefficients
- Hutzenthaler, Jentzen
(Show Context)
Citation Context ...in Kloeden and Platen [26] and Section 12 in Milstein [30]). Recently, convergence of the Monte Carlo Euler approximations (4) has also been established for the SDE (1). More formally, Theorem 2.1 in =-=[20]-=- implies lim N→∞ ∣∣∣∣∣∣E [ f(XT ) ] − 1 N2 N2∑ k=1 f ( Y N,kN )∣∣∣∣∣∣ = 0 (5) P-almost surely (see also Theorem 3.1 below). The Monte Carlo Euler method is thus strongly consistent (see, e.g., Nik... |

5 |
Step size control for the uniform approximation of systems of stochastic differential equations with additive noise
- Hofmann, Müller-Gronbach, et al.
(Show Context)
Citation Context ...in (112) is sharp according to Müller-Gronbach’s lower bound established in Theorem 3 in [33] in the case of globally Lipschitz continuous coefficients (see also Hofmann, Müller-Gronbach and Ritter =-=[18]-=-). Proof of Corollary 6.1. Let Ỹ N : [0, T ]× Ω→ Rd, N ∈ N, be stochastic processes defined by Ỹ Nt := Y N,0,1 n + µ(Y N,0,1n ) · ( t− nTN ) 1 + ‖µ(Y N,0,1n ) · TN ‖Rd + σ(Y N,0,1n ) ( W 0,1t −W 0,1... |

4 |
Euler polygonal lines for Itô equations with monotone coefficients. Teor. Veroyatnost. i Primenen
- Alyushina
- 1987
(Show Context)
Citation Context ...pted stochastic process X : [0, T ]× Ω→ Rd with continuous sample paths solving the stochastic differential equation (SDE) dXt = µ(Xt) dt+ σ(Xt) dWt, X0 = ξ, (1) for t ∈ [0, T ] (see, e.g., Alyushina =-=[1]-=-, Theorem 1 in Krylov [27] or Theorem 2.4.1 in Mao [29]). The drift coefficient µ is the infinitesimal mean of the process X and the diffusion coefficient σ is the infinitesimal standard deviation of ... |

4 |
Numerical Approximations of Nonlinear Stochastic Systems
- Szpruch
- 2010
(Show Context)
Citation Context ...us coefficients is a major issue in financial engineering, in particular, in option pricing. For details the reader is refereed to the monographs Lewis [28], Glassermann [10], Higham [16] and Szpruch =-=[37]-=-. In order to simulate the quantity (2) on a computer, one has to discretize both the solution process X : [0, T ]×Ω→ Rd of the SDE (1) as well as the underlying probability space (Ω,F ,P). The simple... |

3 |
Strong convergence and stability of numerical methods for non-linear stochastic differential equations under monotone conditions
- Szpruch, Mao
- 2010
(Show Context)
Citation Context |

2 | Strongly nonlinear Ait-Sahalia-Type interest rate model and its numerical approximation
- Szpruch, Mao, et al.
(Show Context)
Citation Context ...values in the case of SDEs with superlinearly growing nonlinearities. This is particularly unfortunate as SDEs with superlinearly growing nonlinearities are very important in applications (see, e.g., =-=[28, 40, 37]-=- for applications in financial engineering). We recommend not to use the multilevel Monte Carlo Euler method for applications with such nonlinear SDEs. Nonetheless, the multilevel Monte Carlo method c... |

1 |
Consistent estimator
- Nikulin
- 2001
(Show Context)
Citation Context ...ies lim N→∞ ∣∣∣∣∣∣E [ f(XT ) ] − 1 N2 N2∑ k=1 f ( Y N,kN )∣∣∣∣∣∣ = 0 (5) P-almost surely (see also Theorem 3.1 below). The Monte Carlo Euler method is thus strongly consistent (see, e.g., Nikulin =-=[34]-=-, Cramér [2] or Appendix A.1 in Glassermann [10]). The reason why convergence (5) of the Monte Carlo Euler method does hold although the Euler approximations diverge is as follows. The events ΩN , N ... |