scholarly journals Efficient Steady-State Simulation of High-Dimensional Stochastic Networks

2021 ◽  
Author(s):  
Jose Blanchet ◽  
Xinyun Chen ◽  
Nian Si ◽  
Peter W. Glynn
Keyword(s):  
Monte Carlo ◽  
Steady State ◽  
Linear Complexity ◽  
Stochastic Networks ◽  
High Dimensional ◽  
Regularity Conditions ◽  
Multilevel Monte Carlo ◽  
Reflected Brownian Motion ◽  
Asymptotically Optimal ◽  
Steady State Simulation

We propose and study an asymptotically optimal Monte Carlo estimator for steady-state expectations of a d-dimensional reflected Brownian motion (RBM). Our estimator is asymptotically optimal in the sense that it requires [Formula: see text] (up to logarithmic factors in d) independent and identically distributed scalar Gaussian random variables in order to output an estimate with a controlled error. Our construction is based on the analysis of a suitable multilevel Monte Carlo strategy which, we believe, can be applied widely. This is the first algorithm with linear complexity (under suitable regularity conditions) for a steady-state estimation of RBM as the dimension increases.

Conditions Under Which a Markov Chain Converges to its Steady State in Finite Time

1988 ◽  
Vol 2 (3) ◽  
pp. 377-382 ◽  
Author(s):  
Peter W. Glynn ◽  
Donald L. Iglehart
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Steady State ◽  
Discrete Time ◽  
Continuous Time ◽  
Finite Time ◽  
Transition Matrix ◽  
Continuous Time Markov Chain ◽  
Transient Problem ◽  
Steady State Simulation

Analysis of the initial transient problem of Monte Carlo steady-state simulation motivates the following question for Markov chains: when does there exist a deterministic Tsuch that P{X(T) = y|(0) = x} = ®(y), where ρ is the stationary distribution of X? We show that this can essentially never happen for a continuous-time Markov chain; in discrete time, such processes are i.i.d. provided the transition matrix is diagonalizable.

scholarly journals A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

2017 ◽  
Vol 341 ◽  
pp. 121-139 ◽  
Author(s):  
Hillary R. Fairbanks ◽  
Alireza Doostan ◽  
Christian Ketelsen ◽  
Gianluca Iaccarino
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Uncertain Systems ◽  
Low Rank ◽  
High Dimensional ◽  
Multilevel Monte Carlo ◽  
Control Variate

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
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