Monte Carlo methods for risk analysis
Stochastic simulation and numerical experiments
Overview
Some problems in risk analysis cannot be expressed in an analytical form. Others are difficult to define in a deterministic manner. Monte Carlo methods (also known as stochastic simulation techniques) consist of running “numerical experiments” to observe what happens “on average” over a large number of runs of a stochastic model. They involve repeated random sampling from input probability distributions, execution of the model with these stochastic inputs, then aggregation of the large number of executions to obtain an estimate of the quantity of interest. In some cases Monte Carlo methods are a simple and convenient alternative to analytical methods, allowing you to express your problem in a direct way and let the computer do the hard work. For complex problems which cannot be resolved analytically, they may be the only way of estimating the output of interest. These methods rely on the speed of modern computers and their ability to generate pseudorandom numbers from various relevant probability distributions.
Monte Carlo methods are widely used in risk analysis, for instance for:

propagating uncertainty through a numerical model to obtain confidence intervals on your model outputs

estimating quantile measures for performance measures

simulating evacuation from a building during the design phase

predicting failure, cost overruns and schedule overruns in project management
This submodule is a part of the risk analysis module.
Course material
Monte Carlo methods in risk analysis 

Estimating pi using a Monte Carlo simulation 

Project risk analysis with Monte Carlo simulation 

Monte Carlo simulation of failure probability in mechanical design 

Estimating failure probability of a space vessel fuel tank 

Estimating failure probability of a space vessel fuel tank (Rust kernel) 

Monte Carlo simulation of landslide risk 

Numerical integration using Monte Carlo methods 

Monte Carlo sampling methods 

Limits of stochastic simulation methods: the Saint Petersburg problem 

Various poker odds: working with discrete probability distributions 
In these course materials, applications are presented using the NumPy, SciPy and SymPy libraries for the Python programming language. One of the notebooks above shows how to implement Monte Carlo sampling in a notebookstyle interactive environment with the highperformance Rust programming language, which will typically be many thousands of times faster than implementations in Python. We also include links to relevant plugins for Microsoft Excel, if you’re into that kind of thing.
One of the Python notebooks above illustrates alternative methods to standard (naïve) random sampling:

Latin Hypercube Sampling (LHS) or stratified sampling without replacement involves splitting up each input variable into a number of equiprobable intervals and sampling separately from each interval, using the inverse cumulative distribution function for that variable. In typically leads to faster convergence than Monte Carlo procedures that implement standard random sampling.

Lowdiscrepancy sequences or quasirandom sequences such as the Halton and Sobol’ sequences, which fill the input space more quickly than pseudorandom sequences and typically lead to faster convergence than standard random sampling. These are based on deterministic mathematical sequences of points that don’t “cluster” together.
The notebook on stochastic simulation of the Saint Petersburg problem linked to above illustrates a situation where very unlikely events have an extremely high impact on the mean outcome. In such situations, Monte Carlo simulation is not a good approach to estimate the output of interest (expected value of playing the game, in this case).
Other resources
We recommend the following sources of further information on this topic:

MIT OpenCourseWare notes from the Numerical computation for mechanical engineers course

Article Principles of Good Practice for Monte Carlo Techniques, Risk Analysis, 1994

Book The Monte Carlo Simulation Method for System Reliability and Risk Analysis, Enrico Zio

An alternative geometrical method to estimate the value of pi using stochastic sampling is Buffon’s Needle, invented in 1777.
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