Concept Architecture
Concept
Theoretically, Stochastic Simulation is a mathematical modelling technique that represents systems influenced by random variation by repeatedly sampling from probability distributions. Unlike deterministic models, stochastic simulations recognise uncertainty in model inputs and generate a distribution of possible outcomes rather than a single estimate. The method is founded on probability theory, random variable generation, and statistical inference.
Mathematically, stochastic simulation repeatedly draws random values from specified probability distributions and evaluates the model for each iteration. Thousands or millions of iterations are typically performed using Monte Carlo methods, allowing estimation of expected values, confidence intervals, probabilities, and outcome distributions. Convergence is achieved as the number of simulations increases according to the Law of Large Numbers.
In practice, stochastic simulation is extensively applied in health economics to evaluate uncertainty in cost-effectiveness models, disease progression models, pharmacoeconomic evaluations, transmission models, operational research, and resource planning. It forms the mathematical foundation of probabilistic sensitivity analysis (PSA) and supports evidence-based healthcare decision-making under uncertainty.
Purpose
Stochastic simulation quantifies uncertainty by repeatedly sampling random variables from probability distributions to estimate the distribution of possible outcomes, expected values, risks, and probabilities for complex healthcare systems and economic models.
Mathematical Formulae
Primary Formula
X? ~ F(?)
Where:
- X? = simulated random variable
- F = probability distribution
- ? = distribution parameters
Estimated expected outcome:
E(Y) � (1/N) �Y?
Where:
- N = number of simulations
- Y? = outcome from simulation i
Supporting Formulae
Sample Mean
? = (1/N) �Y?
Sample Variance
s� = (1/(N?1)) �(Y? ? ?)�
Monte Carlo Standard Error
MCSE = s / �N
Confidence Interval
? � 1.96 ? (s / �N)
Related Mathematical Methods
- Monte Carlo Simulation
- Probability Theory
- Random Number Generation
- Markov Models
- Bayesian Analysis
- Probabilistic Sensitivity Analysis
- Bootstrapping
- Discrete Event Simulation
- Agent-Based Simulation
- Statistical Sampling
Example
A probabilistic cost-effectiveness model estimates the incremental cost-effectiveness ratio of a new treatment.
Model parameters are assigned probability distributions:
- Cost ~ Gamma Distribution
- Utility ~ Beta Distribution
- Relative Risk ~ Lognormal Distribution
The model performs 10,000 simulations.
Mean incremental cost:
�2,480
Mean incremental QALYs:
0.18
Probability of being cost-effective at �30,000/QALY:
87.4%
The simulation captures uncertainty that would not be represented by a deterministic analysis.
Excel Implementation
Random Uniform Variable
=RAND()
Random Normal Variable
=NORM.INV(RAND(),Mean,SD)
Random Beta Distribution
=BETA.INV(RAND(),Alpha,Beta)
Average Simulation Result
=AVERAGE(B2:B10001)
Simulation Standard Deviation
=STDEV.S(B2:B10001)
95% Confidence Interval Width
=1.96*STDEV.S(B2:B10001)/SQRT(COUNT(B2:B10001))
VBA (Optional)
Stochastic simulation can be automated in VBA by:
- Running thousands of Monte Carlo iterations
- Sampling from statistical distributions
- Performing probabilistic sensitivity analysis
- Calculating confidence intervals
- Producing cost-effectiveness acceptability curves
- Monitoring convergence of simulation results
Sources
Rubinstein RY, Kroese DP. Simulation and the Monte Carlo Method.
Fishman GS. Monte Carlo: Concepts, Algorithms, and Applications.
Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
Robert CP, Casella G. Monte Carlo Statistical Methods.
Law AM. Simulation Modeling and Analysis.
Related Concepts (2)
Library
Publications
3
Markov Chain Monte Carlo in Practice — W. R. Gilks, S. Richardson & D. J. Spiegelhalter (editors), 1st Edition ed., 1996 (Chapman & Hall)
A foundational collection on Markov chain Monte Carlo methodology and practice, linking stochastic simulation with Bayesian inference and Gibbs sampling.
BookView source →Markov Chains — J. R. Norris, 1st Edition ed., 1997 (Cambridge University Press)
A rigorous introduction to discrete- and continuous-time Markov chains, transition structures, recurrence and long-run behaviour.
BookView source →Markov Processes for Stochastic Modeling — Masaaki Kijima, 1st Edition ed., 1997 (Chapman & Hall)
An applied mathematical treatment of discrete- and continuous-time Markov processes for modeling the transient behaviour of stochastic systems.
BookView source →
Frequently Asked Questions (6)
What is stochastic simulation?
A simulation method that explicitly incorporates random variation to represent uncertainty in the behaviour of a system or process.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
How does stochastic simulation represent uncertainty?
Stochastic simulation represents uncertainty by explicitly incorporating random variation into the model of a system or process. Rather than treating quantities as fixed, it lets them vary according to chance so the simulation reflects real variability. Building random variation into the model to capture uncertainty is what stochastic simulation does By letting quantities vary by chance from run to run, the method produces a spread of outcomes that reflects real uncertainty, rather than the single answer a deterministic model would give.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
What does stochastic simulation incorporate?
Stochastic simulation incorporates random variation, deliberately including chance in the model to represent uncertainty in the behaviour of a system or process. This randomness is explicit rather than an afterthought. Incorporating random variation to reflect uncertainty is the defining feature of stochastic simulation By letting quantities vary by chance from run to run, the method produces a spread of outcomes that reflects real uncertainty, rather than the single answer a deterministic model would give.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
What does stochastic simulation model?
Stochastic simulation models the behaviour of a system or process under uncertainty, using random variation to represent the ways that behaviour can vary. It produces outcomes that differ from run to run, reflecting the chance built into the model. Modelling uncertain system or process behaviour is the purpose of stochastic simulation By letting quantities vary by chance from run to run, the method produces a spread of outcomes that reflects real uncertainty, rather than the single answer a deterministic model would give.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
Why does stochastic simulation include random variation explicitly?
Stochastic simulation includes random variation explicitly because the uncertainty in a system or process is part of what it aims to represent, and treating quantities as fixed would hide that variability. Making the randomness explicit lets the simulation show a range of possible outcomes. This deliberate inclusion of random variation is central to stochastic simulation By letting quantities vary by chance from run to run, the method produces a spread of outcomes that reflects real uncertainty, rather than the single answer a deterministic model would give.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
How does stochastic simulation relate to Monte Carlo simulation?
Stochastic simulation explicitly incorporates random variation to represent uncertainty in a system or process, while Monte Carlo simulation repeatedly samples random values to estimate the behaviour or outcomes of uncertain systems. Monte Carlo simulation is a common way to carry out stochastic simulation. The two are connected through the use of randomness to represent and estimate uncertainty By letting quantities vary by chance from run to run, the method produces a spread of outcomes that reflects real uncertainty, rather than the single answer a deterministic model would give.
Source: Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004. doi:10.1007/978-0-387-21617-1.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Jun 2026
Content version: 1.0.0
Canonical Identity
- Term code
- CS-SM-SM-001
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