Post on 12-Feb-2022
IntroductionEfficient Simulation of Sums
References
Efficient rare-event simulation for sums ofdependent random variables
Leonardo Rojas-Nandayapajoint work with José Blanchet
February 13, 2012MCQMC
UNSW, Sydney, Australia
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Contents
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Outline
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Rare Events
Rare EventsIndexed family of events {Au : u ∈ R} with
P(Au)→ 0, u →∞.
Probabilities difficult to estimate.
Algorithm (Estimator)
Indexed set of simulatable random variables {Zu : u ∈ R} with
E [Zu] = P(Au)
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Rare Events
Rare EventsIndexed family of events {Au : u ∈ R} with
P(Au)→ 0, u →∞.
Probabilities difficult to estimate.
Algorithm (Estimator)
Indexed set of simulatable random variables {Zu : u ∈ R} with
E [Zu] = P(Au)
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Efficiency
Efficient Algorithms
Logarithmic Efficiency:
limu→u0
Var [Zu]
P2−ε(Au)= 0, ∀ε > 0.
Bounded relative error:
limu→∞
Var [Zu]
P2(Au)<∞.
Zero Relative Error:
limu→∞
Var [zu]
P2(Au)= 0.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Efficiency
Efficient Algorithms
Logarithmic Efficiency:
limu→u0
Var [Zu]
P2−ε(Au)= 0, ∀ε > 0.
Bounded relative error:
limu→∞
Var [Zu]
P2(Au)<∞.
Zero Relative Error:
limu→∞
Var [zu]
P2(Au)= 0.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Efficiency
Efficient Algorithms
Logarithmic Efficiency:
limu→u0
Var [Zu]
P2−ε(Au)= 0, ∀ε > 0.
Bounded relative error:
limu→∞
Var [Zu]
P2(Au)<∞.
Zero Relative Error:
limu→∞
Var [zu]
P2(Au)= 0.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Why?
Approximated confidence interval for an MC estimator zu is
zu ± Φ(1− α/2)
√Var Zu
R
To keep the interval proportional to P(Au) we require
R ≈ Var zu
P2(Au).
How?Variance reduction techniques.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Why?
Approximated confidence interval for an MC estimator zu is
zu ± Φ(1− α/2)
√Var Zu
R
To keep the interval proportional to P(Au) we require
R ≈ Var zu
P2(Au).
How?Variance reduction techniques.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Outline
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Tail probabilities of sums
Tail probability of a sumLet X1, . . . ,Xn. Tail probability of the sum
P(X1 + · · ·+ Xn > u), u →∞.
Common Fact: i.i.d. case.An importance sampling algorithm with exponential change ofmeasure
Fθ(dx) := e−θx−κ(θ)F (dx)
produces an efficient algorithm if θ is such that E θ[X ] = u/n.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Tail probabilities of sums
Tail probability of a sumLet X1, . . . ,Xn. Tail probability of the sum
P(X1 + · · ·+ Xn > u), u →∞.
Common Fact: i.i.d. case.An importance sampling algorithm with exponential change ofmeasure
Fθ(dx) := e−θx−κ(θ)F (dx)
produces an efficient algorithm if θ is such that E θ[X ] = u/n.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Right tail probabilities of sums
FactSevere difficulties occur in the construction of efficientalgorithms in presence of heavy tails (Asmussen et al., 2000).
Heavy TailsThe Laplace transform (mgf) is not defined for a heavy-tailedrandom variable X .
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Right tail probabilities of sums
FactSevere difficulties occur in the construction of efficientalgorithms in presence of heavy tails (Asmussen et al., 2000).
Heavy TailsThe Laplace transform (mgf) is not defined for a heavy-tailedrandom variable X .
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Literature Review
Heavy-tailed independent random variablesAsmussen and Binswanger (1997). First efficient algorithmfor regularly varying distribution.Asmussen and Kroese (2006). A refined version which isproved to be efficient in the Lognormal and Weibull case.Boots and Shahabuddin (2001) y Juneja and Shahabuddin(2002). Importance sampling (hazard rate).Dupuis et al. (2006). Importance sampling algorithm forregularly varying distributions and based on mixtures.Blanchet and Li (2011). Adaptive method forsubexponential random variables.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Literature Review
Our contribution: Non-independent case (this talk)Blanchet et al. (2008). Strategies for Log-elliptical randomvariables.Asmussen et al. (2009). Considers correlated lognormalrandom variables.Blanchet and Rojas-Nandayapa (2012) Improved algorithmfor tail probabilities of log-elliptics. An additional moregeneral algorithm for sums.
Related papersKlöppel et al. (2010) y Chan and Kroese (2009). Similarstrategies.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Literature Review
Our contribution: Non-independent case (this talk)Blanchet et al. (2008). Strategies for Log-elliptical randomvariables.Asmussen et al. (2009). Considers correlated lognormalrandom variables.Blanchet and Rojas-Nandayapa (2012) Improved algorithmfor tail probabilities of log-elliptics. An additional moregeneral algorithm for sums.
Related papersKlöppel et al. (2010) y Chan and Kroese (2009). Similarstrategies.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Rare Event SimulationSums of Random Variables
Another interesting problem
Current work with S. Asmussen and J.L. Jensen.Efficient estimation of
P(X1 + · · ·+ Xn < nx), x → 0.
Exponential Twisting.Main difficulty: approximate the Laplace transform.Bounded relative error.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Outline
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Exploratory analysis
Correlated Lognormals
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Scaled variance algorithm
Asmussen et al. (2009)Use the importance sampling distribution
LN(µ, δ(x)Σ)
where δ(x) is the scaling function.1 Under very mild conditions of δ(x): logarithmically efficient.2 Cross-entropy selection: excellent numerical results
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Scaled variance algorithm
Asmussen et al. (2009)Use the importance sampling distribution
LN(µ, δ(x)Σ)
where δ(x) is the scaling function.1 Under very mild conditions of δ(x): logarithmically efficient.2 Cross-entropy selection: excellent numerical results
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Outline
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Elliptical Distributions
DefinitionX is elliptical, denoted E(µ,Σ), if
X d= µ + R C Θ.
Location: µ ∈ Rn. Dispersion: Σ = CtC with C ∈ Rn×k .Spherical: Θ uniform random vector on the unit spheroid.Radial: R positive random variable.R and Θ independent of each other.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Elliptical Distributions
DefinitionX is elliptical, denoted E(µ,Σ), if
X d= µ + R C Θ.
Location: µ ∈ Rn. Dispersion: Σ = CtC with C ∈ Rn×k .Spherical: Θ uniform random vector on the unit spheroid.Radial: R positive random variable.R and Θ independent of each other.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Elliptical Distributions
ExampleLogelliptical distributions
Multivariate Normal.Normal Mixtures.Symmetric Generalized Hyperbolic Distributions: HyperbolicDistributions, Multivariate Normal Inverse Gaussian (NIG), GeneralizedLaplace, Bessel or Symmetric Variance-Gamma, Multivariate t .
The Symmetric Generalized Hyperbolic Distributions offerbetter adjustments than the multivariate normal distributions infinancial applications (McNeil et al., 2005).
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Elliptical Distributions
ExampleLogelliptical distributions
Multivariate Normal.Normal Mixtures.Symmetric Generalized Hyperbolic Distributions: HyperbolicDistributions, Multivariate Normal Inverse Gaussian (NIG), GeneralizedLaplace, Bessel or Symmetric Variance-Gamma, Multivariate t .
The Symmetric Generalized Hyperbolic Distributions offerbetter adjustments than the multivariate normal distributions infinancial applications (McNeil et al., 2005).
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Heavy Tails and Log-elliptical distributions
Log-elliptical Distributions
The exponential transformation (component-wise) of anelliptical random vector is known as logelliptical. Commonly themarginals are dependent heavy-tailed random variables.
ExampleSum of Logellipticals
G (r , θ) =d∑
i=1
exp (µi + r 〈Ai , θ〉) .
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Conditional Monte Carlo
EstimatorAn unbiased estimator of P(G(R,Θ) > u) is
P(G(R,Θ) > u|Θ).
Algorithm
Simulate Θ.Determine BΘ := {r > 0 : G(r ,Θ) > u}.Return P(R ∈ BΘ).
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Conditional Monte Carlo
EstimatorAn unbiased estimator of P(G(R,Θ) > u) is
P(G(R,Θ) > u|Θ).
Algorithm
Simulate Θ.Determine BΘ := {r > 0 : G(r ,Θ) > u}.Return P(R ∈ BΘ).
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Conditional Monte Carlo
Case 1: Sums of logelliptical random variables
Logarithmic efficient if
limr→∞
r fR (r)
P (R > r)1−ε = 0 ∀ε > 0.
fR is the density of F
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Conditional Monte Carlo
Efficient for more general functions
ConditionsG is continuous in the two variables and differentiable in r .
There exists δ0 > 0, s∗ ∈ Sd , r0 > 0 and v > 0 such that for all0 < δ ≤ δ0 and all r > r0 it holds
supθ∈S
G (r , θ)1−vδ ≤ infθ∈D(δ,s∗)
G (r , θ)
supθ∈D(δ0,s∗)
G (r , θ) = supθ∈Sd
G (r , θ) ,
where D(δ, s∗) = {θ ∈ Sd : ‖θ − s∗‖ < δ}.0 < δ1 < 1 chosen such that for all r > r0 and θ ∈ D(δ, s∗) it holds
δ1 ≤d log G (r , θ)
dr≤ 1δ1.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Conditional Monte Carlo
Interesting cases where we have proved efficiency
Sums, maxima, norms and portfolios of options with SymmetricGeneralized Hyperbolic Distributions.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Outline
1 IntroductionRare Event SimulationEfficient Simulation for Sums of Random Variables
2 Efficient simulation of sums of dependent random variablesScaled variance for sums of correlated lognormalsConditional Monte Carlo for functions of logellipticalsSums with general dependent heavy tails
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Exploratory analysis
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Importance Sampling Distribution
Auxiliary IS distribution
Set b = log(x/n).Take the distribution Gb of any efficient IS algorithm for
E[ d∑
i=1
I (Yi > b)
].
Use Gb as an IS distribution for estimatingP(eY1 + · · ·+ eYn > x).
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Efficiency
EfficiencyThe last algorithm is efficient if
limb→∞
logP (Yi > b − c)
logP (Yi > b)= 1.
We say Yi are Logarithmically Long Tailed.
ObservationThis condition includes the most practical heavy-taileddistributions.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Efficiency
EfficiencyThe last algorithm is efficient if
limb→∞
logP (Yi > b − c)
logP (Yi > b)= 1.
We say Yi are Logarithmically Long Tailed.
ObservationThis condition includes the most practical heavy-taileddistributions.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
Scaled variance algorithmConditional MC for LogellipticalDependent heavy tails
Thanks!
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables
IntroductionEfficient Simulation of Sums
References
BibliographyAsmussen, S. and K. Binswanger (1997). Simulation of ruin probabilities for subexponential claims. ASTIN
Bulletin 27, 297–318.Asmussen, S., K. Binswanger, and B. Højgaard (2000). Rare events simulation for heavy-tailed distributions.
Bernoulli 6, 303–322.Asmussen, S., J. Blanchet, S. Juneja, and L. Rojas-Nandayapa (2009). Efficient simulation of tail probabilities of
sums of correlated lognormals. Annals of Operations Research. To appear.Asmussen, S. and D. P. Kroese (2006). Improved algorithms for rare event simulation with heavy tails. Advances in
Applied Probability 38, 545–558.Blanchet, J., S. Juneja, and L. Rojas-Nandayapa (2008). Efficient tail estimation for sums of correlated lognormals.
In Proceedings of the 2008 Winter Simulation Conference, Miami, FL., USA, pp. 607–614. IEEE.Blanchet, J. and C. Li (2011). Efficient rare-event simulation for heavy-tailed compound sums. ACM TOMACS 21.
Forthcoming.Blanchet, J. and L. Rojas-Nandayapa (2012). Efficient simulation of tail probabilities of sums of dependent random
variables. Journal of Applied Probability 48A, 147–164.Boots, N. K. and P. Shahabuddin (2001). Simulating ruin probabilities in insurance risk processes with
subexponential claims. In Proceedings of the 2001 Winter Simulation Conference, Arlington, VA., USA, pp.468–476. IEEE.
Chan, J. C. C. and D. Kroese (2009). Rare-event probability estimation with conditional monte carlo. Annals ofOperations Research. To appear.
Dupuis, P., K. Leder, and H. Wang (2006). Importance sampling for sums of random variables with regularly varyingtails. ACM TOMACS 17, 1–21.
Juneja, S. and P. Shahabuddin (2002). Simulating heavy-tailed processes using delayed hazard rate twisting. ACMTOMACS 12, 94–118.
Klöppel, S., R. Reda, and W. Scharchermayer (2010). A rotationally invariant technique for rare event simulation.Risk Magazine 22, 90–94.
McNeil, A., R. Frey, and P. Embrechts (2005). Quantitative Risk Management: Concepts, Techniques and Tools.New Jersey: Princeton University Press.
Leonardo Rojas Nandayapa Efficient rare-event simulation for sums of dependent random variables