Introduction to Monte Carlo (MC) Method: Random Variables in Stochastic Models

2012 ◽  
pp. 1-5
Author(s):  
Bogusław Bieda
Keyword(s):  
Monte Carlo ◽  
Stochastic Models ◽  
Random Variables ◽  
Mc Method

QMC integration errors and quasi-asymptotics

2020 ◽  
Vol 26 (3) ◽  
pp. 171-176
Author(s):  
Ilya M. Sobol ◽  
Boris V. Shukhman
Keyword(s):  
Monte Carlo ◽  
Error Estimate ◽  
Numerical Experiments ◽  
Probable Error ◽  
Quasi Monte Carlo ◽  
Integration Error ◽  
Asymptotic Error ◽  
Dimensional Cube ◽  
Integration Errors ◽  
Mc Method

AbstractA crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as {O(1/\sqrt{N})}. The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier {1/N}. However, the multiplier {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor {1/N}. However, our numerical experiments show that using quasi-random points of Sobol sequences with {N=2^{m}} with natural m makes the integration error approximately proportional to {1/N}. In our numerical experiments, {d\leq 15}, and we used {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, {d\leq 2^{14}} and {N\leq 2^{63}}.

scholarly journals Improving the Asmussen–Kroese-Type Simulation Estimators

2012 ◽  
Vol 49 (4) ◽  
pp. 1188-1193 ◽  
Author(s):  
Samim Ghamami ◽  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Nonnegative Integer ◽  
Rare Event ◽  
Random Variables ◽  
Random Variable ◽  
Heavy Tailed ◽  
Independent Identically Distributed

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

scholarly journals Monte Carlo Simulation for Economic Analysis of Hydropower Pumped Storage Project in Nepal

2013 ◽  
Vol 12 ◽  
pp. 39-44 ◽  
Author(s):  
Kaspar Vereide ◽  
Leif Lia ◽  
Laras Ødegård
Keyword(s):  
Monte Carlo ◽  
Economic Analysis ◽  
Net Present Value ◽  
Simulation Method ◽  
Construction Costs ◽  
Energy And Environment ◽  
Pumped Storage ◽  
High Degree ◽  
Mc Method

Investments in hydropower pumped storage projects (PSP) are subjected to a high degree of uncertainty. In addition to normal uncertainties in hydropower schemes, the profit of a pumped storage scheme is dependent on the margin between power prices for buying and selling, which is difficult to predict without a power purchase agreement (PPA). A PSP without a PPA and without known construction costs requires quantification of the uncertainties in order to make qualified decisions before investing in such projects. This article demonstrates the advantages of using Monte Carlo (MC) simulations as a tool in the economic analysis of PSPs. The method has been tested on a case study, namely the Tamakoshi-3 Hydropower Project (HPP) in Nepal. The MC method is used to calculate the probability distribution of the net present value of installing reversible units in the Tamakoshi-3 HPP. The calculations show that PSPs may be profitable in Nepal, given a beneficial development of the power market. The MC method is considered to be a useful tool for economic analysis of PSPs. In this case study of installing reversible units in the Tamakoshi-3 HPP, there are many uncertainties, which the MC simulation method is able to quantify. Hydro Nepal; Journal of Water, Energy and Environment Vol. 12, 2013, January Page: 39-44DOI: http://dx.doi.org/10.3126/hn.v12i0.9031 Uploaded Date : 10/29/2013

scholarly journals Monte Carlo sampling processes and incentive compatible allocations in large economies

2020 ◽  
Author(s):  
Peter J. Hammond ◽  
Lei Qiao ◽  
Yeneng Sun
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Incentive Compatibility ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Econ Theory ◽  
Monte Carlo Sampling ◽  
Link Type ◽  
Stochastic Framework ◽  
Almost All

Abstract Monte Carlo simulation is used in Hammond and Sun (Econ Theory 36:303–325, 2008. 10.1007/s00199-007-0279-7) to characterize a standard stochastic framework involving a continuum of random variables that are conditionally independent given macro shocks. This paper presents some general properties of such Monte Carlo sampling processes, including their one-way Fubini extension and regular conditional independence. In addition to the almost sure convergence of Monte Carlo simulation considered in Hammond and Sun (2008), here we also consider norm convergence when the random variables are square integrable. This leads to a necessary and sufficient condition for the classical law of large numbers to hold in a general Hilbert space. Applying this analysis to large economies with asymmetric information shows that the conflict between incentive compatibility and Pareto efficiency is resolved asymptotically for almost all sampling economies, following some similar results in McLean and Postlewaite (Econometrica 70:2421–2453, 2002) and Sun and Yannelis (J Econ Theory 134:175–194, 2007. 10.1016/j.jet.2006.03.001).

Stochastic Models of Queue Storage

1988 ◽  
Vol 2 (1) ◽  
pp. 75-93 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
I. Mitrani ◽  
L. A. Shepp ◽  
C. Knessl
Keyword(s):  
Stochastic Models ◽  
Random Variables ◽  
Single Units ◽  
Random Subset ◽  
Processing Times ◽  
Poisson Stream

We study a model of queue storage in which items (requests for single units of storage) arrive in a Poisson stream and are accommodated by the first available location in a linear scan of storage. The processing times of items are independent, exponentially distributed random variables. The set of occupied locations (identified by their indices) at time t forms a random subset Si, of [1,2,.…]. The extent of the fragmentation in Si, i.e., the alternating holes and occupied regions of storage, is measured by Wt, = max St, – |St|.

Analysis of RC Member’s Reliability Subjected to Eccentric Compression Based on Monte Carlo Simulation

2014 ◽  
Vol 487 ◽  
pp. 465-469
Author(s):  
Wen Feng Duan ◽  
Chang Liu
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Reinforced Concrete ◽  
Concrete Strength ◽  
Random Variables ◽  
Structural Member ◽  
Eccentric Compression ◽  
Compression Members ◽  
Geometric Size ◽  
Compression Member

Reinforced concrete eccentric compression member is one of the most common structural member. Eccentric compression members are divided into large eccentric compression members and small eccentric compression members. Uncertainty of calculation, geometric size and concrete strength were considered as random variables, the reliability of eccentric compression members were discussed by monte carlo simulation.

Approximations to densities in geometric probability

1980 ◽  
Vol 17 (01) ◽  
pp. 145-153 ◽  
Author(s):  
H. Solomon ◽  
M. A. Stephens
Keyword(s):  
Monte Carlo ◽  
Random Variables ◽  
Monte Carlo Sampling ◽  
Geometric Probability ◽  
Simple Approximation ◽  
Chi Square ◽  
Percentage Points ◽  
Random Polygons

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling. The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.

Sign in / Sign up

Export Citation Format

Share Document