scholarly journals Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

2018 ◽  
Vol 143 ◽  
pp. 99-113 ◽  
Author(s):  
Annika Lang ◽  
Andreas Petersson
Keyword(s):  
Monte Carlo ◽  
Multilevel Monte Carlo ◽  
Weak Error

scholarly journals Weak Error for Nested Multilevel Monte Carlo

2020 ◽  
Vol 22 (3) ◽  
pp. 1325-1348
Author(s):  
Daphné Giorgi ◽  
Vincent Lemaire ◽  
Gilles Pagès
Keyword(s):  
Monte Carlo ◽  
Multilevel Monte Carlo ◽  
Weak Error

scholarly journals Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris–Lecar model

2020 ◽  
Vol 52 (1) ◽  
pp. 138-172
Author(s):  
Vincent Lemaire ◽  
MichÉle Thieullen ◽  
Nicolas Thomas
Keyword(s):  
Monte Carlo ◽  
Markov Processes ◽  
Building Blocks ◽  
Not Given ◽  
Multilevel Monte Carlo ◽  
The Third ◽  
Piecewise Deterministic Markov Processes ◽  
Deterministic Processes ◽  
Piecewise Deterministic Processes ◽  
Weak Error

AbstractIn the first part of this paper we study approximations of trajectories of piecewise deterministic processes (PDPs) when the flow is not given explicitly by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for piecewise deterministic Markov processes (PDMPs). These estimates are the building blocks of the multilevel Monte Carlo (MLMC) method, which we study in the second part. The coupling required by the MLMC is based on the thinning procedure. In the third part we apply these results to a two-dimensional Morris–Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms classical Monte Carlo.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

2021 ◽  
Vol 433 ◽  
pp. 110164
Author(s):  
S. Ben Bader ◽  
P. Benedusi ◽  
A. Quaglino ◽  
P. Zulian ◽  
R. Krause
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Cardiac Electrophysiology ◽  
Space Time ◽  
Multilevel Monte Carlo

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 Smoothing via Transport Methods

2018 ◽  
Vol 40 (4) ◽  
pp. A2315-A2335 ◽  
Author(s):  
Jeremie Houssineau ◽  
Ajay Jasra ◽  
Sumeetpal S. Singh
Keyword(s):  
Monte Carlo ◽  
Multilevel Monte Carlo

scholarly journals A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

2017 ◽  
Vol 14 (03) ◽  
pp. 415-454 ◽  
Author(s):  
Ujjwal Koley ◽  
Nils Henrik Risebro ◽  
Christoph Schwab ◽  
Franziska Weber
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Convergence Rate ◽  
Finite Volume ◽  
Monte Carlo Algorithm ◽  
Diffusion Equations ◽  
Entropy Solutions ◽  
Finite Volume Scheme ◽  
Multilevel Monte Carlo ◽  
Convection Diffusion

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

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