stochastic simulation - uva · 2017-09-06 · stochastic simulation) is often used in nancial...
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Stochastic Simulation
Jan-Pieter Dorsman1 & Michel Mandjes1,2,3
1Korteweg-de Vries Institute for Mathematics, University of Amsterdam2CWI, Amsterdam
3Eurandom, Eindhoven
University of Amsterdam,Fall, 2017
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This course
◦ There will be 12 classes: on Fridays, 3pm – 5pm.
◦ The book that will be used is Stochastic Simulation, byS. Asmussen and P. Glynn (Springer, 2007). In addition, wehave lecture notes on discrete-event simulation, based on amanuscript by H. Tijms.
◦ Homework (40%): seehttps://staff.fnwi.uva.nl/j.l.dorsman/stochSim1718.html
◦ Exam (60%): written or oral, depending on number ofstudents.
![Page 3: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/3.jpg)
This course
◦ There will be 12 classes: on Fridays, 3pm – 5pm.
◦ The book that will be used is Stochastic Simulation, byS. Asmussen and P. Glynn (Springer, 2007). In addition, wehave lecture notes on discrete-event simulation, based on amanuscript by H. Tijms.
◦ Homework (40%): seehttps://staff.fnwi.uva.nl/j.l.dorsman/stochSim1718.html
◦ Exam (60%): written or oral, depending on number ofstudents.
![Page 4: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/4.jpg)
This course
◦ There will be 12 classes: on Fridays, 3pm – 5pm.
◦ The book that will be used is Stochastic Simulation, byS. Asmussen and P. Glynn (Springer, 2007). In addition, wehave lecture notes on discrete-event simulation, based on amanuscript by H. Tijms.
◦ Homework (40%): seehttps://staff.fnwi.uva.nl/j.l.dorsman/stochSim1718.html
◦ Exam (60%): written or oral, depending on number ofstudents.
![Page 5: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/5.jpg)
This course
◦ There will be 12 classes: on Fridays, 3pm – 5pm.
◦ The book that will be used is Stochastic Simulation, byS. Asmussen and P. Glynn (Springer, 2007). In addition, wehave lecture notes on discrete-event simulation, based on amanuscript by H. Tijms.
◦ Homework (40%): seehttps://staff.fnwi.uva.nl/j.l.dorsman/stochSim1718.html
◦ Exam (60%): written or oral, depending on number ofstudents.
![Page 6: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/6.jpg)
Chapter IWHAT THIS COURSE IS ABOUT
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What is this course about?
◦ Complex stochastic dynamical systems
of the discrete eventtype.
◦ Focus on systems in which no explicit results for performancemeasure are available.
◦ System is simulated. After repeating this experiment (toobtain the output of multiple runs), statistical claims can bemade about value of performance measure.
◦ I start with two motivating examples.
![Page 8: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/8.jpg)
What is this course about?
◦ Complex stochastic dynamical systems of the discrete eventtype.
◦ Focus on systems in which no explicit results for performancemeasure are available.
◦ System is simulated. After repeating this experiment (toobtain the output of multiple runs), statistical claims can bemade about value of performance measure.
◦ I start with two motivating examples.
![Page 9: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/9.jpg)
What is this course about?
◦ Complex stochastic dynamical systems of the discrete eventtype.
◦ Focus on systems in which no explicit results for performancemeasure are available.
◦ System is simulated. After repeating this experiment (toobtain the output of multiple runs), statistical claims can bemade about value of performance measure.
◦ I start with two motivating examples.
![Page 10: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/10.jpg)
What is this course about?
◦ Complex stochastic dynamical systems of the discrete eventtype.
◦ Focus on systems in which no explicit results for performancemeasure are available.
◦ System is simulated. After repeating this experiment (toobtain the output of multiple runs), statistical claims can bemade about value of performance measure.
◦ I start with two motivating examples.
![Page 11: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/11.jpg)
What is this course about?
◦ Complex stochastic dynamical systems of the discrete eventtype.
◦ Focus on systems in which no explicit results for performancemeasure are available.
◦ System is simulated. After repeating this experiment (toobtain the output of multiple runs), statistical claims can bemade about value of performance measure.
◦ I start with two motivating examples.
![Page 12: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/12.jpg)
What is this course about?
Two motivating examples:
◦ (Networks of) queueing systems.
◦ (Multivariate) ruin models.
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(Networks of) queueing systems
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(Networks of) queueing systems
Complication: link failures
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(Networks of) queueing systems
Complication: link failures
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(Networks of) queueing systems
Complication: link failures
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(Networks of) queueing systems
Complication: link failures
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(Networks of) queueing systems
◦ particles (‘customers’ in queueing lingo) move through anetwork;
◦ arrival rates and service rates are affected by an externalprocess (‘background process’) — for instance to model linkfailures, or other ‘irregularities’;
◦ queues are ‘coupled’ because they fact to commonbackground process.
◦ resulting processes applicable to communication networks,road traffic networks, chemistry, economics.
◦ in full generality, network way too complex to allow explicitanalysis — simulation comes in handy!
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(Multivariate) ruin models
Classical ruin model: reserve of insurance company given by
Xt := X0 + ct −Nt∑n=0
Bn,
with c premium rate, Nt a Poisson process (rate λ), Bn i.i.d. claimsizes.
Lots is known about this model. Goal:
P (∃t 6 T : Xt < 0) .
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(Multivariate) ruin models
Classical ruin model: reserve of insurance company given by
Xt := X0 + ct −Nt∑n=0
Bn,
with c premium rate, Nt a Poisson process (rate λ), Bn i.i.d. claimsizes.
Lots is known about this model. Goal:
P (∃t 6 T : Xt < 0) .
![Page 21: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/21.jpg)
(Multivariate) ruin models
Less is known if there are multiple insurance companies withcorrelated claims:
Xt := X0 + cX t −N
(X )t∑
n=0
B(X )n ,
Yt := Y0 + cY t −N
(Y )t∑
n=0
B(Y )n .
Here:
◦ correlated claim size sequences B(X )n and B
(Y )n (can be
achieved by e.g., letting the claims depend on a commonbackground process),
◦ N(X )t : Poisson process with rate λ(X ), and N
(Y )t Poisson
process with rate λ(Y ).
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(Multivariate) ruin models
Less is known if there are multiple insurance companies withcorrelated claims:
Xt := X0 + cX t −N
(X )t∑
n=0
B(X )n ,
Yt := Y0 + cY t −N
(Y )t∑
n=0
B(Y )n .
Here:
◦ correlated claim size sequences B(X )n and B
(Y )n (can be
achieved by e.g., letting the claims depend on a commonbackground process),
◦ N(X )t : Poisson process with rate λ(X ), and N
(Y )t Poisson
process with rate λ(Y ).
![Page 23: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/23.jpg)
(Multivariate) ruin models
Goal is to compute (both go bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∩ {Yt < 0}) .
Or (one of them goes bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∪ {Yt < 0}) .
Not known how to analyze this. Again: simulation comes in handy!
![Page 24: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/24.jpg)
(Multivariate) ruin models
Goal is to compute (both go bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∩ {Yt < 0}) .
Or (one of them goes bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∪ {Yt < 0}) .
Not known how to analyze this. Again: simulation comes in handy!
![Page 25: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/25.jpg)
(Multivariate) ruin models
Goal is to compute (both go bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∩ {Yt < 0}) .
Or (one of them goes bankrupt before time T ):
P (∃s, t 6 T : {Xs < 0} ∪ {Yt < 0}) .
Not known how to analyze this. Again: simulation comes in handy!
![Page 26: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/26.jpg)
When is simulation a viable approach?
◦ In situations in which neither explicit results are known, noralternative numerical approaches are viable. ‘Monte Carlo’ (≈stochastic simulation) is often used in financial industry and inengineering.
◦ In research, to validate conjectures of exact or asymptoticresults, or to assess the accuracy of approximations.
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Issues to be dealt with
[§I.4 of A & G]
To set up a simulation in a concrete context, various (conceptual,practical) issues have to be dealt with.
I elaborate on the issues mentioned by A & G, and I’ll add a few.
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Issue 1: proper generation of random objects
How do we generate the needed input random variables?Optimally: we generate an exact sample of the random object(variable, process) under study.
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Issue 2: number of runs needed
One replicates a simulation experiment N times, and based on theoutput the performance metric under consideration is estimated.How large should N be to obtain an estimate with a givenprecision?
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Issue 3: estimation of stationary performance measures
This is about estimation of stationary performance measures.Typically one starts the process at an initial state, and after a whilethe process tends to equilibrium.But how do we know when we ‘are’ in equilibrium? Or are thereother (clever) ways to estimate stationary performance measures?
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Issue 4: exploitation of problem structure
In many situations one could follow the naıve approach ofsimulating the stochastic process at hand at a fine time grid, butoften it suffices to consider a certain embedded process.
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Issue 5: rare-event probabilities
In many applications extremely small probabilities are relevant. Forinstance: probability of ruin of an insurance firm; probability of anexcessively high number of customers in a queue. Direct simulationis time consuming (as one does not ‘see’ rare event underconsideration. Specific rare-event-oriented techniques are needed.
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Issue 6: parameter sensitivity
A standard simulation experiment provides an estimate of aperformance measure for a given set of parameters. Often one isinterested in the sensitivity of the performance with respect tochanges of those parameters. Can simulation methods be designedthat are capable of this?
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Issue 7: simulation-based optimization
A standard simulation experiment provides an estimate of aperformance measure for a given set of parameters. Can simulationbe used (and if yes, how) to optimize a given objective function?
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Issue 8: continuous-time systems
This course is primarily on the simulation of dynamic stochasticsystems that allow a discrete-event setup. What can be done ifprocesses in continuous-time are to be simulated?
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Issue 9: programming environment
Various packages are available, some really intended to dosimulations with, others have a more general purpose.Which packages are good for which purposes?
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Issue 10: data structure
For a given simulation experiment, what is the best data structureto store the system’s key quantities?
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Chapter IIGENERATING RANDOM OBJECTS
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Generating random objects
◦ Generating uniformly distributed random variables
◦ Generating generally distributed random variable
◦ Generating random vectors
◦ Generating elementary stochastic processes
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Generating random objects
◦ Generating uniformly distributed random variables
◦ Generating generally distributed random variable
◦ Generating random vectors
◦ Generating elementary stochastic processes
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Generating random objects
◦ Generating uniformly distributed random variables
◦ Generating generally distributed random variable
◦ Generating random vectors
◦ Generating elementary stochastic processes
![Page 42: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/42.jpg)
Generating random objects
◦ Generating uniformly distributed random variables
◦ Generating generally distributed random variable
◦ Generating random vectors
◦ Generating elementary stochastic processes
![Page 43: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/43.jpg)
Generating random objects
◦ Generating uniformly distributed random variables
◦ Generating generally distributed random variable
◦ Generating random vectors
◦ Generating elementary stochastic processes
![Page 44: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/44.jpg)
Generating uniform numbers
[§II.1 of A & G]
We want to develop machine to generate i.i.d. numbers U1,U2, . . .that are uniformly distributed on [0, 1].
As we will see: building block for nearly any simulation experiment.
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Generating uniform numbers
[§II.1 of A & G]
We want to develop machine to generate i.i.d. numbers U1,U2, . . .that are uniformly distributed on [0, 1].
As we will see: building block for nearly any simulation experiment.
![Page 46: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/46.jpg)
Generating uniform numbers
Physical devices:
Classical approach: time between emission of particles byradioactive source is essentially exponential. Say the mean of thisexponential time T is λ−1, then e−λT has a Uniform distributionon the interval [0, 1].
[Check!]
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Generating uniform numbers
Physical devices:
Classical approach: time between emission of particles byradioactive source is essentially exponential. Say the mean of thisexponential time T is λ−1, then e−λT has a Uniform distributionon the interval [0, 1].
[Check!]
![Page 48: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/48.jpg)
Generating uniform numbers
Physical devices:
Classical approach: time between emission of particles byradioactive source is essentially exponential. Say the mean of thisexponential time T is λ−1, then e−λT has a Uniform distributionon the interval [0, 1].
[Check!]
![Page 49: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/49.jpg)
Generating uniform numbers
Physical devices:
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Generating uniform numbers
Alternative: deterministic recursive algorithms (also known as:pseudo-random number generators).
Typical form: with A,C ,M ∈ N,
un :=snM, where sn+1 := (Asn + C ) mod M.
It is the art to make the period of the recursion as long as possible.Early random number generators work with M = 231 − 1, A = 75,and C = 0.
By now, way more sophisticated algorithms have beenimplemented. In all standard software packages (Matlab,Mathematica, R) implementations are used that yield nearly i.i.d.uniform numbers.
![Page 51: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/51.jpg)
Generating uniform numbers
Alternative: deterministic recursive algorithms (also known as:pseudo-random number generators).
Typical form: with A,C ,M ∈ N,
un :=snM, where sn+1 := (Asn + C ) mod M.
It is the art to make the period of the recursion as long as possible.Early random number generators work with M = 231 − 1, A = 75,and C = 0.
By now, way more sophisticated algorithms have beenimplemented. In all standard software packages (Matlab,Mathematica, R) implementations are used that yield nearly i.i.d.uniform numbers.
![Page 52: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/52.jpg)
Generating uniform numbers
Alternative: deterministic recursive algorithms (also known as:pseudo-random number generators).
Typical form: with A,C ,M ∈ N,
un :=snM, where sn+1 := (Asn + C ) mod M.
It is the art to make the period of the recursion as long as possible.Early random number generators work with M = 231 − 1, A = 75,and C = 0.
By now, way more sophisticated algorithms have beenimplemented. In all standard software packages (Matlab,Mathematica, R) implementations are used that yield nearly i.i.d.uniform numbers.
![Page 53: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/53.jpg)
Generating uniform numbers
Alternative: deterministic recursive algorithms (also known as:pseudo-random number generators).
Typical form: with A,C ,M ∈ N,
un :=snM, where sn+1 := (Asn + C ) mod M.
It is the art to make the period of the recursion as long as possible.Early random number generators work with M = 231 − 1, A = 75,and C = 0.
By now, way more sophisticated algorithms have beenimplemented. In all standard software packages (Matlab,Mathematica, R) implementations are used that yield nearly i.i.d.uniform numbers.
![Page 54: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/54.jpg)
Generating uniform numbers
All kinds of tests have been proposed to test a random numbergenerator’s performance.
Clearly not sufficient to show that the samples stem from a uniformdistribution on [0, 1],
as also the independence is important.
Test of uniformity can be done by bunch of standard tests: χ2,Kolmogorov-Smirnov, etc. Test on independence by run test.
![Page 55: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/55.jpg)
Generating uniform numbers
All kinds of tests have been proposed to test a random numbergenerator’s performance.
Clearly not sufficient to show that the samples stem from a uniformdistribution on [0, 1], as also the independence is important.
Test of uniformity can be done by bunch of standard tests: χ2,Kolmogorov-Smirnov, etc. Test on independence by run test.
![Page 56: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/56.jpg)
Generating generally distributed numbers
[§II.2 of A & G]
What is the game?
Suppose I provide you with a machine that canspit out i.i.d. uniform numbers (on [0, 1], that is). Can you give mea sample from a general one-dimensional distribution?
![Page 57: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/57.jpg)
Generating generally distributed numbers
[§II.2 of A & G]
What is the game? Suppose I provide you with a machine that canspit out i.i.d. uniform numbers (on [0, 1], that is).
Can you give mea sample from a general one-dimensional distribution?
![Page 58: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/58.jpg)
Generating generally distributed numbers
[§II.2 of A & G]
What is the game? Suppose I provide you with a machine that canspit out i.i.d. uniform numbers (on [0, 1], that is). Can you give mea sample from a general one-dimensional distribution?
![Page 59: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/59.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial Normal, LognormalGeometric ExponentialNeg. binomial Gamma (and Erlang)Poisson Weibull, Pareto
![Page 60: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/60.jpg)
Generating generally distributed numbers
Two simple examples:
◦ Let X be binomial with parameters n and p. Let Xi := 1 ifUi < p and 0 else. Then
∑ni=1 Xi has desired distribution.
◦ Let X be geometric with parameter p. Sample Ui untilUi < p; let us say this happens at the N-th attempt. Then Nhas desired distribution.
The negative binomial distributed distribution can be sampledsimilarly. [How?]
![Page 61: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/61.jpg)
Generating generally distributed numbers
Two simple examples:
◦ Let X be binomial with parameters n and p. Let Xi := 1 ifUi < p and 0 else. Then
∑ni=1 Xi has desired distribution.
◦ Let X be geometric with parameter p. Sample Ui untilUi < p; let us say this happens at the N-th attempt. Then Nhas desired distribution.
The negative binomial distributed distribution can be sampledsimilarly. [How?]
![Page 62: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/62.jpg)
Generating generally distributed numbers
Two simple examples:
◦ Let X be binomial with parameters n and p. Let Xi := 1 ifUi < p and 0 else. Then
∑ni=1 Xi has desired distribution.
◦ Let X be geometric with parameter p. Sample Ui untilUi < p; let us say this happens at the N-th attempt. Then Nhas desired distribution.
The negative binomial distributed distribution can be sampledsimilarly. [How?]
![Page 63: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/63.jpg)
Generating generally distributed numbers
Two simple examples:
◦ Let X be binomial with parameters n and p. Let Xi := 1 ifUi < p and 0 else. Then
∑ni=1 Xi has desired distribution.
◦ Let X be geometric with parameter p. Sample Ui untilUi < p; let us say this happens at the N-th attempt. Then Nhas desired distribution.
The negative binomial distributed distribution can be sampledsimilarly. [How?]
![Page 64: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/64.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, LognormalGeometric X ExponentialNeg. binomial X Gamma (and Erlang)Poisson Weibull, Pareto
![Page 65: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/65.jpg)
Generating generally distributed numbers
Easiest case: random variables with finite support (say X ).
Say, itlives on {i1, . . . , iN} (ordered), with respective probabilitiesp1, . . . , pN that sum to 1.
Define pi :=∑i
j=1 pi (with p0 := 0); observe that pN = 1.Let U be uniform on [0, 1]. Then
X := ij if U ∈ [pj−1, pj).
Proof:P(X = ij) = pj − pj−1 = pj ,
as desired. Procedure easily extended to case of countable support.
![Page 66: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/66.jpg)
Generating generally distributed numbers
Easiest case: random variables with finite support (say X ). Say, itlives on {i1, . . . , iN} (ordered), with respective probabilitiesp1, . . . , pN that sum to 1.
Define pi :=∑i
j=1 pi (with p0 := 0); observe that pN = 1.Let U be uniform on [0, 1]. Then
X := ij if U ∈ [pj−1, pj).
Proof:P(X = ij) = pj − pj−1 = pj ,
as desired. Procedure easily extended to case of countable support.
![Page 67: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/67.jpg)
Generating generally distributed numbers
Easiest case: random variables with finite support (say X ). Say, itlives on {i1, . . . , iN} (ordered), with respective probabilitiesp1, . . . , pN that sum to 1.
Define pi :=∑i
j=1 pi (with p0 := 0); observe that pN = 1.Let U be uniform on [0, 1]. Then
X := ij if U ∈ [pj−1, pj).
Proof:P(X = ij) = pj − pj−1 = pj ,
as desired.
Procedure easily extended to case of countable support.
![Page 68: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/68.jpg)
Generating generally distributed numbers
Easiest case: random variables with finite support (say X ). Say, itlives on {i1, . . . , iN} (ordered), with respective probabilitiesp1, . . . , pN that sum to 1.
Define pi :=∑i
j=1 pi (with p0 := 0); observe that pN = 1.Let U be uniform on [0, 1]. Then
X := ij if U ∈ [pj−1, pj).
Proof:P(X = ij) = pj − pj−1 = pj ,
as desired. Procedure easily extended to case of countable support.
![Page 69: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/69.jpg)
Generating generally distributed numbers
This algorithm is a simple form of inversion. Relies on theleft-continuous version of the inverse distribution function: withF (·) the distribution function of the random variable X ,
F←(u) := min{x : F (x) > u}.
F←(u) known as quantile function.
![Page 70: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/70.jpg)
Generating generally distributed numbers
This algorithm is a simple form of inversion. Relies on theleft-continuous version of the inverse distribution function: withF (·) the distribution function of the random variable X ,
F←(u) := min{x : F (x) > u}.
F←(u) known as quantile function.
![Page 71: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/71.jpg)
Generating generally distributed numbers
This algorithm is a simple form of inversion. Relies on theleft-continuous version of the inverse distribution function: withF (·) the distribution function of the random variable X ,
F←(u) := min{x : F (x) > u}.
F←(u) known as quantile function.
![Page 72: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/72.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .
(b) F←(U) has CDF F .(c) if F is continuous, then F (X ) ∼ U.
Proof: Part (a) follows from definitions. Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 73: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/73.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .(b) F←(U) has CDF F .
(c) if F is continuous, then F (X ) ∼ U.
Proof: Part (a) follows from definitions. Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 74: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/74.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .(b) F←(U) has CDF F .(c) if F is continuous, then F (X ) ∼ U.
Proof: Part (a) follows from definitions. Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 75: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/75.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .(b) F←(U) has CDF F .(c) if F is continuous, then F (X ) ∼ U.
Proof:
Part (a) follows from definitions. Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 76: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/76.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .(b) F←(U) has CDF F .(c) if F is continuous, then F (X ) ∼ U.
Proof: Part (a) follows from definitions.
Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 77: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/77.jpg)
Generating generally distributed numbers
Proposition Let U be uniform on [0, 1].(a) u 6 F (x) ⇔ F←(u) 6 x .(b) F←(U) has CDF F .(c) if F is continuous, then F (X ) ∼ U.
Proof: Part (a) follows from definitions. Part (b) follows from (a):
P(F←(U) 6 x) = P(U 6 F (x)) = F (x).
Part (c) follows from (a) and continuity of F :
P(F (X ) > u) = P(X > F←(u)) = P(X > F←(u)) = 1−F (F←(u)),
which equals (as desired) 1− u. This is because F (F←(u)) > ufrom (a); also, by considering sequence yn such that yn ↑ F←(u),such that F (yn) < u, and by continuity F (F←(u)) 6 u.
![Page 78: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/78.jpg)
Generating generally distributed numbers
Example: how to generate exponentially distributed numbers (withmean λ−1) from U?
F (x) = 1− e−λx ⇔ F←(u) = − log(1− u)
λ.
Proposition yields: − log(1− U)/λ has an exp(λ) distribution.
We may as well use − logU/λ. [Check!]
Erlang(k, λ) now follows directly as well. [How?]
![Page 79: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/79.jpg)
Generating generally distributed numbers
Example: how to generate exponentially distributed numbers (withmean λ−1) from U?
F (x) = 1− e−λx ⇔ F←(u) = − log(1− u)
λ.
Proposition yields: − log(1− U)/λ has an exp(λ) distribution.
We may as well use − logU/λ. [Check!]
Erlang(k, λ) now follows directly as well. [How?]
![Page 80: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/80.jpg)
Generating generally distributed numbers
Example: how to generate exponentially distributed numbers (withmean λ−1) from U?
F (x) = 1− e−λx ⇔ F←(u) = − log(1− u)
λ.
Proposition yields: − log(1− U)/λ has an exp(λ) distribution.
We may as well use − logU/λ. [Check!]
Erlang(k, λ) now follows directly as well. [How?]
![Page 81: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/81.jpg)
Generating generally distributed numbers
Example: how to generate exponentially distributed numbers (withmean λ−1) from U?
F (x) = 1− e−λx ⇔ F←(u) = − log(1− u)
λ.
Proposition yields: − log(1− U)/λ has an exp(λ) distribution.
We may as well use − logU/λ. [Check!]
Erlang(k, λ) now follows directly as well. [How?]
![Page 82: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/82.jpg)
Generating generally distributed numbers
Example: how to generate exponentially distributed numbers (withmean λ−1) from U?
F (x) = 1− e−λx ⇔ F←(u) = − log(1− u)
λ.
Proposition yields: − log(1− U)/λ has an exp(λ) distribution.
We may as well use − logU/λ. [Check!]
Erlang(k, λ) now follows directly as well. [How?]
![Page 83: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/83.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, LognormalGeometric X Exponential XNeg. binomial X Gamma (and Erlang X)Poisson Weibull, Pareto
![Page 84: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/84.jpg)
Generating generally distributed numbers
Now we also have a fast way to sample a geometric randomvariable:
X :=
⌈logU
log p
⌉.
Justification:
P(⌈
logU
log p
⌉> k
)= P
(logU
log p> k − 1
),
which equals
P(logU 6 (k − 1) log p) = P(U 6 pk−1) = pk−1.
Advantage: requires just one random number. Disadvantage: logis slow operator.
![Page 85: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/85.jpg)
Generating generally distributed numbers
Now we also have a fast way to sample a geometric randomvariable:
X :=
⌈logU
log p
⌉.
Justification:
P(⌈
logU
log p
⌉> k
)= P
(logU
log p> k − 1
)
,
which equals
P(logU 6 (k − 1) log p) = P(U 6 pk−1) = pk−1.
Advantage: requires just one random number. Disadvantage: logis slow operator.
![Page 86: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/86.jpg)
Generating generally distributed numbers
Now we also have a fast way to sample a geometric randomvariable:
X :=
⌈logU
log p
⌉.
Justification:
P(⌈
logU
log p
⌉> k
)= P
(logU
log p> k − 1
),
which equals
P(logU 6 (k − 1) log p) = P(U 6 pk−1) = pk−1.
Advantage: requires just one random number. Disadvantage: logis slow operator.
![Page 87: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/87.jpg)
Generating generally distributed numbers
Now we also have a fast way to sample a geometric randomvariable:
X :=
⌈logU
log p
⌉.
Justification:
P(⌈
logU
log p
⌉> k
)= P
(logU
log p> k − 1
),
which equals
P(logU 6 (k − 1) log p) = P(U 6 pk−1) = pk−1.
Advantage: requires just one random number. Disadvantage: logis slow operator.
![Page 88: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/88.jpg)
Generating generally distributed numbers
Now that we can sample exponential random variable, we can alsosample from Poisson distribution.
Let S0 := 0 and Xi i.i.d. exponentials.
Sn :=n∑
i=1
Xi .
Define:N := max{n ∈ N : Sn 6 1}.
Observe: Sn is the number of arrivals of a Poisson process at time1, which has a Poisson distribution with mean λ · 1 = λ.
To avoid the log operation, we can also use
N := max
{n ∈ N :
n∏i=1
Ui > e−λ
}.
![Page 89: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/89.jpg)
Generating generally distributed numbers
Now that we can sample exponential random variable, we can alsosample from Poisson distribution.Let S0 := 0 and Xi i.i.d. exponentials.
Sn :=n∑
i=1
Xi .
Define:N := max{n ∈ N : Sn 6 1}.
Observe: Sn is the number of arrivals of a Poisson process at time1, which has a Poisson distribution with mean λ · 1 = λ.
To avoid the log operation, we can also use
N := max
{n ∈ N :
n∏i=1
Ui > e−λ
}.
![Page 90: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/90.jpg)
Generating generally distributed numbers
Now that we can sample exponential random variable, we can alsosample from Poisson distribution.Let S0 := 0 and Xi i.i.d. exponentials.
Sn :=n∑
i=1
Xi .
Define:N := max{n ∈ N : Sn 6 1}.
Observe: Sn is the number of arrivals of a Poisson process at time1, which has a Poisson distribution with mean λ · 1 = λ.
To avoid the log operation, we can also use
N := max
{n ∈ N :
n∏i=1
Ui > e−λ
}.
![Page 91: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/91.jpg)
Generating generally distributed numbers
Now that we can sample exponential random variable, we can alsosample from Poisson distribution.Let S0 := 0 and Xi i.i.d. exponentials.
Sn :=n∑
i=1
Xi .
Define:N := max{n ∈ N : Sn 6 1}.
Observe: Sn is the number of arrivals of a Poisson process at time1, which has a Poisson distribution with mean λ · 1 = λ.
To avoid the log operation, we can also use
N := max
{n ∈ N :
n∏i=1
Ui > e−λ
}.
![Page 92: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/92.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, LognormalGeometric X Exponential XNeg. binomial X Gamma (and Erlang X)Poisson X Weibull, Pareto
![Page 93: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/93.jpg)
Generating generally distributed numbers
Weibull distribution: for some α > 0 and λ > 0,
F (x) = P(X 6 x) = 1− e−λxα.
Pareto distribution: for some α > 0,
F (x) = P(X 6 x) = 1− (x + 1)−α.
Both can be done with inverse-CDF technique.
![Page 94: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/94.jpg)
Generating generally distributed numbers
Weibull distribution: for some α > 0 and λ > 0,
F (x) = P(X 6 x) = 1− e−λxα.
Pareto distribution: for some α > 0,
F (x) = P(X 6 x) = 1− (x + 1)−α.
Both can be done with inverse-CDF technique.
![Page 95: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/95.jpg)
Generating generally distributed numbers
Weibull distribution: for some α > 0 and λ > 0,
F (x) = P(X 6 x) = 1− e−λxα.
Pareto distribution: for some α > 0,
F (x) = P(X 6 x) = 1− (x + 1)−α.
Both can be done with inverse-CDF technique.
![Page 96: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/96.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, LognormalGeometric X Exponential XNeg. binomial X Gamma (and Erlang X)Poisson X Weibull, Pareto X
![Page 97: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/97.jpg)
Generating generally distributed numbers
Acceptance-rejection: to be used when (i) no explicit expression forCDF is available, or
(ii) the CDF does not allow explicit inversion.
Setup:
◦ X has PDF f (x), and Y has PDF g(x) such that, for somefinite constant C and all x ,
f (x) 6 Cg(x).
◦ We know how to sample Y (but we don’t know how tosample X ).
![Page 98: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/98.jpg)
Generating generally distributed numbers
Acceptance-rejection: to be used when (i) no explicit expression forCDF is available, or (ii) the CDF does not allow explicit inversion.
Setup:
◦ X has PDF f (x), and Y has PDF g(x) such that, for somefinite constant C and all x ,
f (x) 6 Cg(x).
◦ We know how to sample Y (but we don’t know how tosample X ).
![Page 99: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/99.jpg)
Generating generally distributed numbers
Acceptance-rejection: to be used when (i) no explicit expression forCDF is available, or (ii) the CDF does not allow explicit inversion.
Setup:
◦ X has PDF f (x), and Y has PDF g(x) such that, for somefinite constant C and all x ,
f (x) 6 Cg(x).
◦ We know how to sample Y (but we don’t know how tosample X ).
![Page 100: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/100.jpg)
Generating generally distributed numbers
Acceptance-rejection: to be used when (i) no explicit expression forCDF is available, or (ii) the CDF does not allow explicit inversion.
Setup:
◦ X has PDF f (x), and Y has PDF g(x) such that, for somefinite constant C and all x ,
f (x) 6 Cg(x).
◦ We know how to sample Y (but we don’t know how tosample X ).
![Page 101: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/101.jpg)
Generating generally distributed numbers
Algorithm:
◦ Sample Y , and in addition U that is uniform on [0, 1].
◦ If
U 6f (Y )
Cg(Y ),
then X := Y , and otherwise you try again.
Define the event of acceptance by
A :=
{U 6
f (Y )
Cg(Y )
}.
![Page 102: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/102.jpg)
Generating generally distributed numbers
Algorithm:
◦ Sample Y , and in addition U that is uniform on [0, 1].
◦ If
U 6f (Y )
Cg(Y ),
then X := Y , and otherwise you try again.
Define the event of acceptance by
A :=
{U 6
f (Y )
Cg(Y )
}.
![Page 103: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/103.jpg)
Generating generally distributed numbers
Algorithm:
◦ Sample Y , and in addition U that is uniform on [0, 1].
◦ If
U 6f (Y )
Cg(Y ),
then X := Y , and otherwise you try again.
Define the event of acceptance by
A :=
{U 6
f (Y )
Cg(Y )
}.
![Page 104: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/104.jpg)
Generating generally distributed numbers
Algorithm:
◦ Sample Y , and in addition U that is uniform on [0, 1].
◦ If
U 6f (Y )
Cg(Y ),
then X := Y , and otherwise you try again.
Define the event of acceptance by
A :=
{U 6
f (Y )
Cg(Y )
}.
![Page 105: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/105.jpg)
Generating generally distributed numbers
Proof:
P(X 6 x) = P(X 6 x |A) =P(X 6 x , A)
P(A)
Denominator:
P(A) = P(U 6
f (Y )
Cg(Y )
)= E
(f (Y )
Cg(Y )
)=
∫ ∞−∞
f (y)
Cg(y)g(y)dy =
∫ ∞−∞
f (y)
Cdy =
1
C.
Here we use that
P(U 6 X ) =
∫P(U 6 x)P(X ∈ dx) =
∫xP(X ∈ dx) = EX .
![Page 106: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/106.jpg)
Generating generally distributed numbers
Proof:
P(X 6 x) = P(X 6 x |A) =P(X 6 x , A)
P(A)
Denominator:
P(A) = P(U 6
f (Y )
Cg(Y )
)= E
(f (Y )
Cg(Y )
)=
∫ ∞−∞
f (y)
Cg(y)g(y)dy =
∫ ∞−∞
f (y)
Cdy =
1
C.
Here we use that
P(U 6 X ) =
∫P(U 6 x)P(X ∈ dx) =
∫xP(X ∈ dx) = EX .
![Page 107: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/107.jpg)
Generating generally distributed numbers
Proof, continued: Numerator can be dealt with similarly.
P(X 6 x , A) = P(U 6
f (Y )
Cg(Y )6 x
)= E
(f (Y )
Cg(Y )1
{f (Y )
Cg(Y )6 x
})=
∫ x
−∞
f (y)
Cg(y)g(y)dy =
∫ x
−∞
f (y)
Cdy =
F (x)
C.
Ratio: F (x), as desired.
![Page 108: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/108.jpg)
Generating generally distributed numbers
Proof, continued: Numerator can be dealt with similarly.
P(X 6 x , A) = P(U 6
f (Y )
Cg(Y )6 x
)= E
(f (Y )
Cg(Y )1
{f (Y )
Cg(Y )6 x
})=
∫ x
−∞
f (y)
Cg(y)g(y)dy =
∫ x
−∞
f (y)
Cdy =
F (x)
C.
Ratio: F (x), as desired.
![Page 109: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/109.jpg)
Generating generally distributed numbers
Can be used to sample from Normal distribution. Without loss ofgenerality [Why?] we show how to sample from Normaldistribution conditioned on being positive.
Has density
f (x) =
√2
πe−x
2/2.
We now find a C such that, with g(x) = e−x (exp. distr. withmean 1!),
f (x) 6 Cg(x).
Observe that we may pick C :=√
2e/π, because
f (x)
g(x)=
√2
πe−x
2/2+x =
√2
πe−(x−1)
2/2e1/2 6
√2e
π.
![Page 110: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/110.jpg)
Generating generally distributed numbers
Can be used to sample from Normal distribution. Without loss ofgenerality [Why?] we show how to sample from Normaldistribution conditioned on being positive.
Has density
f (x) =
√2
πe−x
2/2.
We now find a C such that, with g(x) = e−x (exp. distr. withmean 1!),
f (x) 6 Cg(x).
Observe that we may pick C :=√
2e/π, because
f (x)
g(x)=
√2
πe−x
2/2+x =
√2
πe−(x−1)
2/2e1/2 6
√2e
π.
![Page 111: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/111.jpg)
Generating generally distributed numbers
Can be used to sample from Normal distribution. Without loss ofgenerality [Why?] we show how to sample from Normaldistribution conditioned on being positive.
Has density
f (x) =
√2
πe−x
2/2.
We now find a C such that, with g(x) = e−x (exp. distr. withmean 1!),
f (x) 6 Cg(x).
Observe that we may pick C :=√
2e/π, because
f (x)
g(x)=
√2
πe−x
2/2+x =
√2
πe−(x−1)
2/2e1/2 6
√2e
π.
![Page 112: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/112.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, Lognormal XGeometric X Exponential XNeg. binomial X Gamma (and Erlang X)Poisson X Weibull, Pareto X
![Page 113: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/113.jpg)
Generating generally distributed numbers
On the efficiency of acceptance-rejection: probability of acceptanceis 1/C . More precisely: the number of attempts before a samplecan be accepted is geometrically distributed with successprobability 1/C .
As a consequence you want to pick C as small as possible.
![Page 114: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/114.jpg)
Generating generally distributed numbers
On the efficiency of acceptance-rejection: probability of acceptanceis 1/C . More precisely: the number of attempts before a samplecan be accepted is geometrically distributed with successprobability 1/C .
As a consequence you want to pick C as small as possible.
![Page 115: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/115.jpg)
Generating generally distributed numbers
Gamma distribution can be done with acceptance rejection;Example 2.8 in A & G.‘Dominating density’ (i.e., the density of Y ) is ‘empirically found’.
![Page 116: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/116.jpg)
Generating generally distributed numbers
Discrete random variables Continuous random variables
Binomial X Normal, Lognormal XGeometric X Exponential XNeg. binomial X Gamma X (and Erlang X)Poisson X Weibull, Pareto X
![Page 117: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/117.jpg)
Generating generally distributed numbers
A cool specific algorithm for the Normal distribution (Box &Muller):With U1 and U2 independent uniforms on [0, 1],
Y1 :=√−2 logU1 sin(2πU2), Y2 :=
√−2 logU1 cos(2πU2)
yields two independent standard Normal random variables.
![Page 118: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/118.jpg)
Generating generally distributed numbers
Proof: first observe
U1 ≡ U1(Y1,Y2) = e−(Y21 +Y 2
2 )/2,
U2 ≡ U2(Y1,Y2) =1
2πarctan
(Y2
Y1
).
Hence, with J(y1, y2) the determinant of the Jacobian,
fY1,Y2(y1, y2) = J(y1, y2) · fU1,U2(u1(y1, y2), u2(y1, y2))
= J(y1, y2).
![Page 119: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/119.jpg)
Generating generally distributed numbers
Proof: first observe
U1 ≡ U1(Y1,Y2) = e−(Y21 +Y 2
2 )/2,
U2 ≡ U2(Y1,Y2) =1
2πarctan
(Y2
Y1
).
Hence, with J(y1, y2) the determinant of the Jacobian,
fY1,Y2(y1, y2) = J(y1, y2) · fU1,U2(u1(y1, y2), u2(y1, y2))
= J(y1, y2).
![Page 120: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/120.jpg)
Generating generally distributed numbers
du1dy1
= −y1u1,du2dy1
=1
2π· 1
1 + y22 /y21
· − y2y21
=1
2π· −y2y21 + y22
du1dy2
= −y2u1,du2dy2
=1
2π· 1
1 + y22 /y21
· 1
y1=
1
2π· y1y21 + y22
.
Determinant is (2π)−1u1 = (2π)−1 e−(y21+y2
2 )/2.Hence, as desired,
fY1,Y2(y1, y2) =1
2πe−(y
21+y2
2 )/2.
![Page 121: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/121.jpg)
Generating generally distributed numbers
du1dy1
= −y1u1,du2dy1
=1
2π· 1
1 + y22 /y21
· − y2y21
=1
2π· −y2y21 + y22
du1dy2
= −y2u1,du2dy2
=1
2π· 1
1 + y22 /y21
· 1
y1=
1
2π· y1y21 + y22
.
Determinant is (2π)−1u1 = (2π)−1 e−(y21+y2
2 )/2.
Hence, as desired,
fY1,Y2(y1, y2) =1
2πe−(y
21+y2
2 )/2.
![Page 122: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/122.jpg)
Generating generally distributed numbers
du1dy1
= −y1u1,du2dy1
=1
2π· 1
1 + y22 /y21
· − y2y21
=1
2π· −y2y21 + y22
du1dy2
= −y2u1,du2dy2
=1
2π· 1
1 + y22 /y21
· 1
y1=
1
2π· y1y21 + y22
.
Determinant is (2π)−1u1 = (2π)−1 e−(y21+y2
2 )/2.Hence, as desired,
fY1,Y2(y1, y2) =1
2πe−(y
21+y2
2 )/2.
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Generating generally distributed numbers
Apart from inversion-CDF (uses CDF) and acceptance-rejection(uses PDF) there are quite a few alternative techniques.
In some cases, CDF or PDF is not available, but the transform
F [s] :=
∫esxP(X ∈ dx)
is. Then algorithms of following type may work.
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Other techniques
Inversion formula, with ψ(s) := F [is]:
f (x) =1
2π
∫ ∞−∞
ψ(s)e−isxds.
Algorithm:
◦ Fix n. Evaluate f (xi ) for x1, . . . , xn using inversion formula.
◦ Construct approximate distribution with probability mass
pi :=f (xi )∑nj=1 f (xj)
in xi .
◦ Sample from this approximate distribution function.
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Other techniques
Inversion formula, with ψ(s) := F [is]:
f (x) =1
2π
∫ ∞−∞
ψ(s)e−isxds.
Algorithm:
◦ Fix n. Evaluate f (xi ) for x1, . . . , xn using inversion formula.
◦ Construct approximate distribution with probability mass
pi :=f (xi )∑nj=1 f (xj)
in xi .
◦ Sample from this approximate distribution function.
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Other techniques
Inversion formula, with ψ(s) := F [is]:
f (x) =1
2π
∫ ∞−∞
ψ(s)e−isxds.
Algorithm:
◦ Fix n. Evaluate f (xi ) for x1, . . . , xn using inversion formula.
◦ Construct approximate distribution with probability mass
pi :=f (xi )∑nj=1 f (xj)
in xi .
◦ Sample from this approximate distribution function.
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Other techniques
Inversion formula, with ψ(s) := F [is]:
f (x) =1
2π
∫ ∞−∞
ψ(s)e−isxds.
Algorithm:
◦ Fix n. Evaluate f (xi ) for x1, . . . , xn using inversion formula.
◦ Construct approximate distribution with probability mass
pi :=f (xi )∑nj=1 f (xj)
in xi .
◦ Sample from this approximate distribution function.
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Generating random vectors
[§II.3 of A & G]
The most relevant example is: how to draw a sample from amultivariate Normal distribution? Make us of Choleskydecomposition.
Multivariate Normal distribution characterized through meanvector µ (of length p) and covariance matrix Σ (of dimensionp × p). Without loss of generality: µ = 0 [Why?]
Σ is positive definite, hence can be written as CCT, for a lowertriangular matrix C .
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Generating random vectors
[§II.3 of A & G]
The most relevant example is: how to draw a sample from amultivariate Normal distribution? Make us of Choleskydecomposition.
Multivariate Normal distribution characterized through meanvector µ (of length p) and covariance matrix Σ (of dimensionp × p). Without loss of generality: µ = 0 [Why?]
Σ is positive definite, hence can be written as CCT, for a lowertriangular matrix C .
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Generating random vectors
[§II.3 of A & G]
The most relevant example is: how to draw a sample from amultivariate Normal distribution? Make us of Choleskydecomposition.
Multivariate Normal distribution characterized through meanvector µ (of length p) and covariance matrix Σ (of dimensionp × p). Without loss of generality: µ = 0 [Why?]
Σ is positive definite, hence can be written as CCT, for a lowertriangular matrix C .
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Generating random vectors
Now the Xi (for i = 1, . . . , p) can be sampled as follows.Let Y a p-dimensional Normal vector with independent standardNormal components (i.e., with covariance matrix I ). LetX := CY .
Then X has the right covariance matrix CCT = Σ (use standardrules for multivariate Normal distributions).
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Generating random vectors
Now the Xi (for i = 1, . . . , p) can be sampled as follows.Let Y a p-dimensional Normal vector with independent standardNormal components (i.e., with covariance matrix I ). LetX := CY .
Then X has the right covariance matrix CCT = Σ (use standardrules for multivariate Normal distributions).
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Generating elementary stochastic processes
[§II.4 of A & G]
I. Discrete-time Markov chains. Characterized by transition matrixP = (pij)i ,j∈E , with E the state space.
For ease we take E = {1, 2, . . .}. Trivially simulated, relyingprocedure that uses pij :=
∑jk=1 pik .
II. Continuous-time Markov chains. Characterized by transitionrate matrix Q = (qij)i ,j∈E , with E the state space. Sample timesspent in each of states from exponential distribution, and then useprocedure above to sample next state.
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Generating elementary stochastic processes
[§II.4 of A & G]
I. Discrete-time Markov chains. Characterized by transition matrixP = (pij)i ,j∈E , with E the state space.
For ease we take E = {1, 2, . . .}. Trivially simulated, relyingprocedure that uses pij :=
∑jk=1 pik .
II. Continuous-time Markov chains. Characterized by transitionrate matrix Q = (qij)i ,j∈E , with E the state space. Sample timesspent in each of states from exponential distribution, and then useprocedure above to sample next state.
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Generating elementary stochastic processes
III. Poisson process (with rate λ). Interarrival times are exponential(with mean λ−1).
IV. Inhomogeneous Poisson process (with rate λ(t) at time t).Assume λ(t) 6 λ across all values of t. Un: i.i.d. uniform numberson [0, 1].
Algorithm: (where n corresponds to a homogeneous Poissonprocess with rate β, and n? to the inhomogeneous Poisson process)
◦ Step 1: n := 0, n? := 0, σ := 0.
◦ Step 2: n := n + 1, Tn ∼ exp(λ) (i.e., Tn := − logUn/λ),σ := σ + Tn.
◦ Step 3: If U ′n 6 λ(σ)/λ, then n? := n? + 1.
◦ Step 4: Go to Step 2.
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Generating elementary stochastic processes
III. Poisson process (with rate λ). Interarrival times are exponential(with mean λ−1).
IV. Inhomogeneous Poisson process (with rate λ(t) at time t).Assume λ(t) 6 λ across all values of t. Un: i.i.d. uniform numberson [0, 1].
Algorithm: (where n corresponds to a homogeneous Poissonprocess with rate β, and n? to the inhomogeneous Poisson process)
◦ Step 1: n := 0, n? := 0, σ := 0.
◦ Step 2: n := n + 1, Tn ∼ exp(λ) (i.e., Tn := − logUn/λ),σ := σ + Tn.
◦ Step 3: If U ′n 6 λ(σ)/λ, then n? := n? + 1.
◦ Step 4: Go to Step 2.
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IntermezzoDISCRETE-EVENT SIMULATION
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Chapter IIIOUTPUT ANALYSIS
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Simulation as a computational tool
[§III.1 of A & G]
Idea: we want to estimate the performance measure z := EZ .Perform independent samples Z1, . . . ,ZR of Z .Estimate:
zR :=1
R
R∑r=1
Zr .
Question: what is the performance of this estimator?
Two issues play a role: bias (is expectation equal to parameter ofinterest?) and accuracy (what is variance of estimator?).When unbiased, we would like to have a minimal variance. Oftenthe objective is to approximate its distribution (e.g. asymptoticNormality).
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Simulation as a computational tool
[§III.1 of A & G]
Idea: we want to estimate the performance measure z := EZ .Perform independent samples Z1, . . . ,ZR of Z .Estimate:
zR :=1
R
R∑r=1
Zr .
Question: what is the performance of this estimator?
Two issues play a role: bias (is expectation equal to parameter ofinterest?) and accuracy (what is variance of estimator?).When unbiased, we would like to have a minimal variance. Oftenthe objective is to approximate its distribution (e.g. asymptoticNormality).
![Page 141: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/141.jpg)
Simulation as a computational tool
[§III.1 of A & G]
Idea: we want to estimate the performance measure z := EZ .Perform independent samples Z1, . . . ,ZR of Z .Estimate:
zR :=1
R
R∑r=1
Zr .
Question: what is the performance of this estimator?
Two issues play a role: bias (is expectation equal to parameter ofinterest?) and accuracy (what is variance of estimator?).When unbiased, we would like to have a minimal variance. Oftenthe objective is to approximate its distribution (e.g. asymptoticNormality).
![Page 142: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/142.jpg)
Simulation as a computational tool
Standard approach: if σ2 := VarZ <∞, then the central limittheorem provided us with: as R →∞,
√R(zR − z
)→ N (0, σ2).
This suggests for finite R the approximation, with V ∼ N (0, 1),
zRd≈ z +
σV√R.
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Simulation as a computational tool
Standard approach: if σ2 := VarZ <∞, then the central limittheorem provided us with: as R →∞,
√R(zR − z
)→ N (0, σ2).
This suggests for finite R the approximation, with V ∼ N (0, 1),
zRd≈ z +
σV√R.
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Simulation as a computational tool
Based on this convergence in distribution, one could use confidenceintervals if the type(
zR − qασ√R, zR + qα
σ√R
),
with qα reflecting the α/2-quantile of N (0, 1), in the sense that
1− Φ(qα) = P(N (0, 1) > qα) =1− α
2
(for example confidence level α equalling 0.95 leads to qα = 1.96).
Are we done now? NO! We don’t know σ2.
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Simulation as a computational tool
Based on this convergence in distribution, one could use confidenceintervals if the type(
zR − qασ√R, zR + qα
σ√R
),
with qα reflecting the α/2-quantile of N (0, 1), in the sense that
1− Φ(qα) = P(N (0, 1) > qα) =1− α
2
(for example confidence level α equalling 0.95 leads to qα = 1.96).
Are we done now?
NO! We don’t know σ2.
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Simulation as a computational tool
Based on this convergence in distribution, one could use confidenceintervals if the type(
zR − qασ√R, zR + qα
σ√R
),
with qα reflecting the α/2-quantile of N (0, 1), in the sense that
1− Φ(qα) = P(N (0, 1) > qα) =1− α
2
(for example confidence level α equalling 0.95 leads to qα = 1.96).
Are we done now? NO! We don’t know σ2.
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Simulation as a computational tool
Idea: estimate σ2.
Traditional estimator:
s2 :=1
R − 1
R∑r=1
(Zr − zR)2 =1
R − 1
(R∑
r=1
Z 2r − Rz2R
).
This is an unbiased estimator. [Check!]
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Simulation as a computational tool
Idea: estimate σ2.
Traditional estimator:
s2 :=1
R − 1
R∑r=1
(Zr − zR)2 =1
R − 1
(R∑
r=1
Z 2r − Rz2R
).
This is an unbiased estimator. [Check!]
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Computing smooth functions of expectations
[§III.3 of A & G]
Example: estimation of standard deviation of random variable W .
Then: z = (z1, z2)T, with zi = EZ (i), where
Z (1) = W 2, Z (2) = W .
Standard deviation σ = f (z), with
f (z) =√z1 − z22 .
Here our aim to estimate a smooth function of the expectationsEW 2 and EW .
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Computing smooth functions of expectations
[§III.3 of A & G]
Example: estimation of standard deviation of random variable W .
Then: z = (z1, z2)T, with zi = EZ (i), where
Z (1) = W 2, Z (2) = W .
Standard deviation σ = f (z), with
f (z) =√z1 − z22 .
Here our aim to estimate a smooth function of the expectationsEW 2 and EW .
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Computing smooth functions of expectations
[§III.3 of A & G]
Example: estimation of standard deviation of random variable W .
Then: z = (z1, z2)T, with zi = EZ (i), where
Z (1) = W 2, Z (2) = W .
Standard deviation σ = f (z), with
f (z) =√z1 − z22 .
Here our aim to estimate a smooth function of the expectationsEW 2 and EW .
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Computing smooth functions of expectations
Another example: Let An and Bn be independent sequences ofi.i.d. random variables.Perpetuity is:
Y =∞∑n=0
Bn
n∏m=1
Am
(think of Bn as amounts put on an account in slot n, and An theinterest in slot n).
Then
EY =EB
1− EA.
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Computing smooth functions of expectations
Another example: Let An and Bn be independent sequences ofi.i.d. random variables.Perpetuity is:
Y =∞∑n=0
Bn
n∏m=1
Am
(think of Bn as amounts put on an account in slot n, and An theinterest in slot n). Then
EY =EB
1− EA.
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Computing smooth functions of expectations
We wish to estimate
% := EY =EB
1− EA.
Then: z = (z1, z2)T, with zi = EZ (i), where
Z (1) = B, Z (2) = A.
Goal is to estimate % = f (z), with
f (z) =z1
1− z2.
Again, our aim to estimate a smooth function of expectations, inthis case EB and EA.
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Computing smooth functions of expectations
We wish to estimate
% := EY =EB
1− EA.
Then: z = (z1, z2)T, with zi = EZ (i), where
Z (1) = B, Z (2) = A.
Goal is to estimate % = f (z), with
f (z) =z1
1− z2.
Again, our aim to estimate a smooth function of expectations, inthis case EB and EA.
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Computing smooth functions of expectations
How to to estimate a smooth function of expectations?
Naıve idea: estimate f (z) by f (zR), with
zR :=1
R
R∑r=1
Z r .
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Computing smooth functions of expectations
First example: recall Zr (1) = W 2r and Zr (2) = Wr , and
f (z) =√
z1 − z22 .
So estimator is
σ =
√√√√ 1
R
R∑r=1
Zr (1)−
(1
R
R∑r=1
Zr (2)
)2
.
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Computing smooth functions of expectations
Second example: recall Zr (1) = Br and Zr (2) = Ar , and
f (z) =z1
1− z2.
So estimator is
% =
(1
R
R∑r=1
Zr (1)
)/(1− 1
R
R∑r=1
Zr (2)
).
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Computing smooth functions of expectations
Question: is this procedure any good?
Evident: consistent as R →∞ — at least, in case f (z) iscontinuous at z .
Can we again quantify the rate of convergence? Is thereapproximate Normality? This can be assessed by applying theso-called delta-method.
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Computing smooth functions of expectations
Question: is this procedure any good?
Evident: consistent as R →∞
— at least, in case f (z) iscontinuous at z .
Can we again quantify the rate of convergence? Is thereapproximate Normality? This can be assessed by applying theso-called delta-method.
![Page 161: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/161.jpg)
Computing smooth functions of expectations
Question: is this procedure any good?
Evident: consistent as R →∞ — at least, in case f (z) iscontinuous at z .
Can we again quantify the rate of convergence? Is thereapproximate Normality? This can be assessed by applying theso-called delta-method.
![Page 162: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/162.jpg)
Computing smooth functions of expectations
Question: is this procedure any good?
Evident: consistent as R →∞ — at least, in case f (z) iscontinuous at z .
Can we again quantify the rate of convergence? Is thereapproximate Normality?
This can be assessed by applying theso-called delta-method.
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Computing smooth functions of expectations
Question: is this procedure any good?
Evident: consistent as R →∞ — at least, in case f (z) iscontinuous at z .
Can we again quantify the rate of convergence? Is thereapproximate Normality? This can be assessed by applying theso-called delta-method.
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Computing smooth functions of expectations
Using the usual Taylor arguments, with ∇f (z) the row vector ofpartial derivatives,
f (zR)− f (z) = ∇f (z) · (zR − z) + o(||zR − z ||)
=1
R
R∑r=1
Vr + o(||zR − z ||),
where Vr denotes ∇f (z) · (Z r − z). Is there again a central limittheorem?
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Computing smooth functions of expectations
Yes! √R(f (zR)− f (z)
)→ N (0, σ2),
with σ2 := VarV1.
How to evaluate σ2? Define Σij := Cov(Zi ,Zj). Then
σ2 = ∇f (z) · Σ · ∇f (z)T.
(Later we’ll see an application of this technique: regenerativemethod to compute steady-state quantities.)
For d = 1 (where d is dimension of vector Z r ) , we just getσ2 = (f ′(z))2VarZ ; for f (z) = z this gives us back our earlierprocedure. But there are two differences as well.
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Computing smooth functions of expectations
Yes! √R(f (zR)− f (z)
)→ N (0, σ2),
with σ2 := VarV1.
How to evaluate σ2? Define Σij := Cov(Zi ,Zj). Then
σ2 = ∇f (z) · Σ · ∇f (z)T.
(Later we’ll see an application of this technique: regenerativemethod to compute steady-state quantities.)
For d = 1 (where d is dimension of vector Z r ) , we just getσ2 = (f ′(z))2VarZ ; for f (z) = z this gives us back our earlierprocedure. But there are two differences as well.
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Computing smooth functions of expectations
Yes! √R(f (zR)− f (z)
)→ N (0, σ2),
with σ2 := VarV1.
How to evaluate σ2? Define Σij := Cov(Zi ,Zj). Then
σ2 = ∇f (z) · Σ · ∇f (z)T.
(Later we’ll see an application of this technique: regenerativemethod to compute steady-state quantities.)
For d = 1 (where d is dimension of vector Z r ) , we just getσ2 = (f ′(z))2VarZ ; for f (z) = z this gives us back our earlierprocedure. But there are two differences as well.
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Computing smooth functions of expectations
Yes! √R(f (zR)− f (z)
)→ N (0, σ2),
with σ2 := VarV1.
How to evaluate σ2? Define Σij := Cov(Zi ,Zj). Then
σ2 = ∇f (z) · Σ · ∇f (z)T.
(Later we’ll see an application of this technique: regenerativemethod to compute steady-state quantities.)
For d = 1 (where d is dimension of vector Z r ) , we just getσ2 = (f ′(z))2VarZ ; for f (z) = z this gives us back our earlierprocedure. But there are two differences as well.
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Computing smooth functions of expectations
Difference 1: estimator f (zR) is generally biased.
Effect can bequantified (include one more term in expansion): with H(z) theHessian in z ,
f (zR)− f (z) = ∇f (z) · (zR − z)
+1
2(zR − z)TH(z)(zR − z) + o(||zR − z ||2).
Now take means, to obtain the bias, as R →∞,
E(f (zR)− f (z)
)=
1
2R
∑i ,j
ΣijHij(z) + o(R−1),
with as before Σij := Cov(Zi ,Zj).
Remedy: adapted estimator (with Σij the obvious estimator of Σij)
f (zR)− 1
2R
∑i ,j
ΣijHij(zR).
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Computing smooth functions of expectations
Difference 1: estimator f (zR) is generally biased. Effect can bequantified (include one more term in expansion): with H(z) theHessian in z ,
f (zR)− f (z) = ∇f (z) · (zR − z)
+1
2(zR − z)TH(z)(zR − z) + o(||zR − z ||2).
Now take means, to obtain the bias, as R →∞,
E(f (zR)− f (z)
)=
1
2R
∑i ,j
ΣijHij(z) + o(R−1),
with as before Σij := Cov(Zi ,Zj).
Remedy: adapted estimator (with Σij the obvious estimator of Σij)
f (zR)− 1
2R
∑i ,j
ΣijHij(zR).
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Computing smooth functions of expectations
Difference 1: estimator f (zR) is generally biased. Effect can bequantified (include one more term in expansion): with H(z) theHessian in z ,
f (zR)− f (z) = ∇f (z) · (zR − z)
+1
2(zR − z)TH(z)(zR − z) + o(||zR − z ||2).
Now take means, to obtain the bias, as R →∞,
E(f (zR)− f (z)
)=
1
2R
∑i ,j
ΣijHij(z) + o(R−1),
with as before Σij := Cov(Zi ,Zj).
Remedy: adapted estimator (with Σij the obvious estimator of Σij)
f (zR)− 1
2R
∑i ,j
ΣijHij(zR).
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Computing smooth functions of expectations
Difference 2: σ2 harder to estimate.
Evident candidate:
σ2 :=1
R − 1
R∑r=1
(∇f (zR) (Z r − zR)
)2.
This requires computation of the gradient of f (if this is explicitlyavailable, then there is obviously no problem).
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Computing smooth functions of expectations
Difference 2: σ2 harder to estimate. Evident candidate:
σ2 :=1
R − 1
R∑r=1
(∇f (zR) (Z r − zR)
)2.
This requires computation of the gradient of f (if this is explicitlyavailable, then there is obviously no problem).
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Computing roots of equations defined by expectations
[§III.4 of A & G]
Let f : Rd+1 7→ R be known, z := EZ ∈ Rd . Then we want tofind the (a?) θ such that
f (z , θ) = 0;
call the root θ?.
If an explicit ζ : Rd 7→ R is known such that θ? = ζ(z), then we’rein the framework of the previous section. So focus on case ζ is notknown.
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Computing roots of equations defined by expectations
[§III.4 of A & G]
Let f : Rd+1 7→ R be known, z := EZ ∈ Rd . Then we want tofind the (a?) θ such that
f (z , θ) = 0;
call the root θ?.
If an explicit ζ : Rd 7→ R is known such that θ? = ζ(z), then we’rein the framework of the previous section. So focus on case ζ is notknown.
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Computing roots of equations defined by expectations
Procedure: (i) estimate z by zR , and (ii) find θR by solving
f (zR , θR) = 0.
How to get confidence intervals?
Trivial:
0 = f (zR , θR)− f (z , θ?)
= f (zR , θR)− f (z , θR) + f (z , θR)− f (z , θ?).
Apply delta-method again: in usual notation,
∇fz(z , θ?)(zR − z
)+ fθ(z , θ?)(θR − θ?) + O(||zR − z ||2) = 0.
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Computing roots of equations defined by expectations
Procedure: (i) estimate z by zR , and (ii) find θR by solving
f (zR , θR) = 0.
How to get confidence intervals?
Trivial:
0 = f (zR , θR)− f (z , θ?)
= f (zR , θR)− f (z , θR) + f (z , θR)− f (z , θ?).
Apply delta-method again: in usual notation,
∇fz(z , θ?)(zR − z
)+ fθ(z , θ?)(θR − θ?) + O(||zR − z ||2) = 0.
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Computing roots of equations defined by expectations
From
∇fz(z , θ?)(zR − z
)+ fθ(z , θ?)(θR − θ?) + O(||zR − z ||2) = 0
we obtain √R(θR − θ?
)→ N (0, σ2),
with (recalling ∇fz(z , θ?) is d-dimensional row vector)
σ2 =Var
(∇fz(z , θ?) · Z
)(fθ(z , θ?)
)2 .
As before σ2 can be estimated by
σ2 =
1
R − 1
R∑r=1
(∇fz(zR , θR) · (Z r − z)
)2(fθ(zR , θR)
)2 .
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Chapter IVSTEADY-STATE SIMULATION
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Complications arising when estimating steady-state quantities
[§IV.1 of A & G]
We now focus on ergodic stochastic processes Y (·). This meansthat a limiting time-average limit Y exists — think a stable queue.
Our objective is to estimate
z := limt→∞
1
t
∫ t
0Y (s)ds.
How to do this?
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Complications arising when estimating steady-state quantities
[§IV.1 of A & G]
We now focus on ergodic stochastic processes Y (·). This meansthat a limiting time-average limit Y exists — think a stable queue.
Our objective is to estimate
z := limt→∞
1
t
∫ t
0Y (s)ds.
How to do this?
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Complications arising when estimating steady-state quantities
Naıve approach: let t grow large, and simulate process Y (·) forlong time:
zT :=1
T
∫ T
0Y (s)ds,
for T ‘large’.
Inherent problems:
◦ how large should T be? (Depends on speed of convergence.)
◦ how to construct confidence intervals? (Observe that thereare now no i.i.d. observations.)
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Complications arising when estimating steady-state quantities
Naıve approach: let t grow large, and simulate process Y (·) forlong time:
zT :=1
T
∫ T
0Y (s)ds,
for T ‘large’.
Inherent problems:
◦ how large should T be? (Depends on speed of convergence.)
◦ how to construct confidence intervals? (Observe that thereare now no i.i.d. observations.)
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Complications arising when estimating steady-state quantities
Naıve approach: let t grow large, and simulate process Y (·) forlong time:
zT :=1
T
∫ T
0Y (s)ds,
for T ‘large’.
Inherent problems:
◦ how large should T be? (Depends on speed of convergence.)
◦ how to construct confidence intervals? (Observe that thereare now no i.i.d. observations.)
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Complications arising when estimating steady-state quantities
Naıve approach: let t grow large, and simulate process Y (·) forlong time:
zT :=1
T
∫ T
0Y (s)ds,
for T ‘large’.
Inherent problems:
◦ how large should T be? (Depends on speed of convergence.)
◦ how to construct confidence intervals? (Observe that thereare now no i.i.d. observations.)
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Complications arising when estimating steady-state quantities
Under ‘rather general conditions’, as T grows large,
√T(zT − z)→ N (0, σ2),
for a σ2 characterized below.
This means that we can approximate, with V ∼ N (0, 1),
zTd= z +
σV√T.
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Complications arising when estimating steady-state quantities
Under ‘rather general conditions’, as T grows large,
√T(zT − z)→ N (0, σ2),
for a σ2 characterized below.
This means that we can approximate, with V ∼ N (0, 1),
zTd= z +
σV√T.
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Complications arising when estimating steady-state quantities
Essentially for the CLT to hold, we should have that
Var(∫ T
0Y (s)ds
)scales linearly in T as T grows large, with σ2 being theproportionality constant.
Let’s compute this constant
σ2 := limT→∞
1
TVar
(∫ T
0Y (s)ds
).
Definec(s) := Covπ(Y (0),Y (s)),
with subscript π denoting the system was in stationarity at time 0.In addition, [Why?]
c(s, v) := Covπ(Y (s),Y (v)) = c(|s − v |).
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Complications arising when estimating steady-state quantities
Essentially for the CLT to hold, we should have that
Var(∫ T
0Y (s)ds
)scales linearly in T as T grows large, with σ2 being theproportionality constant.Let’s compute this constant
σ2 := limT→∞
1
TVar
(∫ T
0Y (s)ds
).
Definec(s) := Covπ(Y (0),Y (s)),
with subscript π denoting the system was in stationarity at time 0.In addition, [Why?]
c(s, v) := Covπ(Y (s),Y (v)) = c(|s − v |).
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Complications arising when estimating steady-state quantities
Essentially for the CLT to hold, we should have that
Var(∫ T
0Y (s)ds
)scales linearly in T as T grows large, with σ2 being theproportionality constant.Let’s compute this constant
σ2 := limT→∞
1
TVar
(∫ T
0Y (s)ds
).
Definec(s) := Covπ(Y (0),Y (s)),
with subscript π denoting the system was in stationarity at time 0.In addition, [Why?]
c(s, v) := Covπ(Y (s),Y (v)) = c(|s − v |).
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Complications arising when estimating steady-state quantities
Using in the first step a standard rule for variance of integrals,
1
TVarπ
(∫ T
0Y (s)ds
)=
1
T
∫ T
0
∫ T
0Covπ(Y (s),Y (v))ds dv
=2
T
∫ T
0
∫ s
0Covπ(Y (s),Y (v))dv ds
=2
T
∫ T
0
∫ s
0c(s − v)dv ds
=2
T
∫ T
0
∫ s
0c(v)dv ds =
2
T
∫ T
0
∫ T
vc(v)ds dv
= 2
∫ T
0
(1− v
T
)c(v)dv → 2
∫ ∞0
c(v)dv ,
as T →∞. (Last step: dominated convergence.)
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Complications arising when estimating steady-state quantities
[Notice an inconsistency in A & G: proof is for initial distributionπ, whereas claim is stated for general initial distribution.]
Result:
σ2 = 2
∫ ∞0
c(v)dv .
Apparently, for the claim to hold, one should have that thecovariances of Y (0) and Y (t) (and hence the autocorrelationfunction) has a finite integral. We sometimes call this: the processY (·) is short-range dependent.
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Complications arising when estimating steady-state quantities
[Notice an inconsistency in A & G: proof is for initial distributionπ, whereas claim is stated for general initial distribution.]
Result:
σ2 = 2
∫ ∞0
c(v)dv .
Apparently, for the claim to hold, one should have that thecovariances of Y (0) and Y (t) (and hence the autocorrelationfunction) has a finite integral. We sometimes call this: the processY (·) is short-range dependent.
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Complications arising when estimating steady-state quantities
[Notice an inconsistency in A & G: proof is for initial distributionπ, whereas claim is stated for general initial distribution.]
Result:
σ2 = 2
∫ ∞0
c(v)dv .
Apparently, for the claim to hold, one should have that thecovariances of Y (0) and Y (t) (and hence the autocorrelationfunction) has a finite integral. We sometimes call this: the processY (·) is short-range dependent.
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Finite-state continuous-time Markov chain
[§IV.2 of A & G]
(X (t))t>0 irreducible Markov chain on {1, . . . , d}, with transitionrate matrix Q.Define Y (t) = f (X (t)). To be computed:
σ2 = 2
∫ ∞0
Covπ(Y (0),Y (s))ds.
First rewrite expression to, with pij(t) := P(X (t) = j |X (0) = i),
2
∫ ∞0
d∑i=1
d∑j=1
(πipij(s)− πiπj)f (i)f (j)ds.
Call
Dij :=
∫ ∞0
(pij(s)− πj)ds.
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Finite-state continuous-time Markov chain
[§IV.2 of A & G]
(X (t))t>0 irreducible Markov chain on {1, . . . , d}, with transitionrate matrix Q.Define Y (t) = f (X (t)). To be computed:
σ2 = 2
∫ ∞0
Covπ(Y (0),Y (s))ds.
First rewrite expression to, with pij(t) := P(X (t) = j |X (0) = i),
2
∫ ∞0
d∑i=1
d∑j=1
(πipij(s)− πiπj)f (i)f (j)ds.
Call
Dij :=
∫ ∞0
(pij(s)− πj)ds.
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Finite-state continuous-time Markov chain
We obtain that, with ‘•’ denoting the componentwise product,
σ2 = 2 (f •π)TDf .
The matrix D is referred to as the deviation matrix, and essentiallymeasures the speed of convergence to the invariant distribution.
D can be alternatively evaluated as F − Π, with Π := 1πT
(rank-one matrix), and
F := (Π− Q)−1.
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Finite-state continuous-time Markov chain
We obtain that, with ‘•’ denoting the componentwise product,
σ2 = 2 (f •π)TDf .
The matrix D is referred to as the deviation matrix, and essentiallymeasures the speed of convergence to the invariant distribution.
D can be alternatively evaluated as F − Π, with Π := 1πT
(rank-one matrix), and
F := (Π− Q)−1.
![Page 199: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/199.jpg)
Finite-state continuous-time Markov chain
We obtain that, with ‘•’ denoting the componentwise product,
σ2 = 2 (f •π)TDf .
The matrix D is referred to as the deviation matrix, and essentiallymeasures the speed of convergence to the invariant distribution.
D can be alternatively evaluated as F − Π, with Π := 1πT
(rank-one matrix), and
F := (Π− Q)−1.
![Page 200: Stochastic Simulation - UvA · 2017-09-06 · stochastic simulation) is often used in nancial industry and in engineering. In research, to validate conjectures of exact or asymptotic](https://reader034.vdocuments.us/reader034/viewer/2022050102/5f41993022669155c165a3ae/html5/thumbnails/200.jpg)
Regenerative method
[§IV.4 of A & G]
Recall: our objective is to estimate
z := limt→∞
1
t
∫ t
0Y (s)ds.
Let T be a regenerative point. Then (regeneration ratio formula):
z =EI (T )
ET, I (T ) :=
∫ T
0Y (s)ds.
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Regenerative method
For discrete-state-space irreducible ergodic Markov process X (t),one could define a ‘return state’ i?. Suppose X (0) = i?. Then ther -th return time is defined recursively: τ0 := 0 and
τr := inf {t > 0 : X (t + τr−1) = i?} ,
and
Ir :=
∫ τr
τr−1
Y (s)ds.
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Regenerative method
Consequence for simulation: we need to estimate EI (τ) and Eτ(and find confidence intervals for the resulting estimator).
Idea: simulate R regenerative cycles, providing observationsI1, . . . , IR and τ1, . . . , τR .
Estimator: estimating numerator and denominator separately,
zR =1
R
R∑r=1
Ir
/1
R
R∑r=1
τr =R∑
r=1
Ir
/R∑
r=1
τr .
Confidence intervals?
[Many typos in A & G: τr and Yr need to be swapped. I choseslightly different, and more transparent, notation.]
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Regenerative method
Consequence for simulation: we need to estimate EI (τ) and Eτ(and find confidence intervals for the resulting estimator).
Idea: simulate R regenerative cycles, providing observationsI1, . . . , IR and τ1, . . . , τR .
Estimator: estimating numerator and denominator separately,
zR =1
R
R∑r=1
Ir
/1
R
R∑r=1
τr =R∑
r=1
Ir
/R∑
r=1
τr .
Confidence intervals?
[Many typos in A & G: τr and Yr need to be swapped. I choseslightly different, and more transparent, notation.]
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Regenerative method
Define Zr := Ir − zτr , and Zd= Zr , I
d= Ir , τ
d= τr ,
η2 :=EZ 2
(Eτ)2.
Then, as R →∞,
√R(zR − z
)→ N (0, η2).
Confidence intervals can be constructed as before, after estimatingη2 by
η2R :=
(1
R − 1
R∑r=1
(Ir − zRτr
)2)/(1
R
R∑r=1
τr
)2
.
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Regenerative method
Proof: an application of the delta-method. Let z1 := EI andz2 := Eτ . Define f (z1, z2) = z1/z2.As we have seen, variance of estimator:
σ2 = ∇f (z) · Σ · ∇f (z)T.
Can be rewritten to(1
Eτ,− EI
(Eτ)2
)(VarI Cov(I , τ)
Cov(I , τ) Varτ
)(1
Eτ,− EI
(Eτ)2
)T
,
or(Eτ
(Eτ)2,− EI
(Eτ)2
)(VarI Cov(I , τ)
Cov(I , τ) Varτ
)(Eτ
(Eτ)2,− EI
(Eτ)2
)T
.
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Regenerative method
Going through the computations, we obtain
1
(Eτ)4·((Eτ)2Var I − 2EI Eτ Cov(I , τ) + (EI )2Var τ
),
or equivalently,
1
(Eτ)2·(Var I − 2z Cov(I , τ) + z2Var τ
)=
VarZ(Eτ)2
=EZ 2
(Eτ)2.
Second claim (estimator for η2) is obvious.
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Regenerative method
Going through the computations, we obtain
1
(Eτ)4·((Eτ)2Var I − 2EI Eτ Cov(I , τ) + (EI )2Var τ
),
or equivalently,
1
(Eτ)2·(Var I − 2z Cov(I , τ) + z2Var τ
)=
VarZ(Eτ)2
=EZ 2
(Eτ)2.
Second claim (estimator for η2) is obvious.
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Regenerative method
Actually, a CLT for the related estimator
zT :=1
T
∫ T
0Y (s)ds,
for T deterministic, can be found along the same lines.
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Batch-means method
[§IV.5 of A & G]
Recall: our objective is to estimate
z := limt→∞
1
t
∫ t
0Y (s)ds.
Prerequisite is weak convergence to BM:(√nt ·
(1
nt
∫ nt
0Y (s)ds − z
))t>0
→ (σ B(t))t>0.
[First factor is not√n t, as in A & G!]
Such a functional centrallimit theorem typically holds when there is weak dependence in theY (·) process.
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Batch-means method
[§IV.5 of A & G]
Recall: our objective is to estimate
z := limt→∞
1
t
∫ t
0Y (s)ds.
Prerequisite is weak convergence to BM:(√nt ·
(1
nt
∫ nt
0Y (s)ds − z
))t>0
→ (σ B(t))t>0.
[First factor is not√n t, as in A & G!] Such a functional central
limit theorem typically holds when there is weak dependence in theY (·) process.
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Batch-means method
Define contributions of intervals of length t/R:
Yr (t) =1
t/R
∫ rt/R
(r−1)t/RY (s)ds;
there are R of these.
Estimator:
zR :=1
R
R∑r=1
Yr (t).
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Batch-means method
Define contributions of intervals of length t/R:
Yr (t) =1
t/R
∫ rt/R
(r−1)t/RY (s)ds;
there are R of these.
Estimator:
zR :=1
R
R∑r=1
Yr (t).
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Batch-means method
How to construct confidence intervals?For any R,
√R
(1
t
∫ t
0Y (s)ds − z
)/sR(t)→ TR−1
as t →∞. Here TR is a Student-t distribution with R degrees offreedom, and
SR(t) :=1
R − 1
R∑r=1
(1
t/R
∫ rt/R
(r−1)t/RY (s)ds − 1
t
∫ t
0Y (s)ds
)2
.
(And when R is large as well, this behaves as N (0, 1).)
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Batch-means method
How to construct confidence intervals?For any R,
√R
(1
t
∫ t
0Y (s)ds − z
)/sR(t)→ TR−1
as t →∞. Here TR is a Student-t distribution with R degrees offreedom, and
SR(t) :=1
R − 1
R∑r=1
(1
t/R
∫ rt/R
(r−1)t/RY (s)ds − 1
t
∫ t
0Y (s)ds
)2
.
(And when R is large as well, this behaves as N (0, 1).)
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Chapter VVARIANCE REDUCTION