scholarly journals Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

2013 ◽  
Vol 23 (5) ◽  
pp. 1913-1966 ◽  
Author(s):  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Peter E. Kloeden
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Euler Method ◽  
Multilevel Monte Carlo ◽  
Nonlinear Stochastic Differential Equations

Multilevel Monte Carlo by using the Halton sequence

2020 ◽  
Vol 26 (3) ◽  
pp. 193-203
Author(s):  
Shady Ahmed Nagy ◽  
Mohamed A. El-Beltagy ◽  
Mohamed Wafa
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Computational Cost ◽  
Random Numbers ◽  
Multilevel Monte Carlo ◽  
Halton Sequence ◽  
Random Generator ◽  
Mc Simulation ◽  
Multilevel Monte Carlo Method

AbstractMonte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

scholarly journals Multilevel Monte Carlo for Stochastic Differential Equations with Small Noise

2016 ◽  
Vol 54 (2) ◽  
pp. 505-529 ◽  
Author(s):  
David F. Anderson ◽  
Desmond J. Higham ◽  
Yu Sun
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Multilevel Monte Carlo ◽  
Small Noise

scholarly journals Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140679 ◽  
Author(s):  
Eike H. Müller ◽  
Rob Scheichl ◽  
Tony Shardlow
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Operator Splitting ◽  
Random Variables ◽  
Weak Approximation ◽  
Multilevel Monte Carlo ◽  
Discrete Random Variables ◽  
Modified Equations

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

scholarly journals Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 481
Author(s):  
Dong An ◽  
Noah Linden ◽  
Jin-Peng Liu ◽  
Ashley Montanaro ◽  
Changpeng Shao ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Monte Carlo Methods ◽  
Quantum Algorithms ◽  
General Setting ◽  
Quantum Algorithm ◽  
Mathematical Finance ◽  
Classical Solutions ◽  
Multilevel Monte Carlo

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.

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