Wrocław University of Technology
ECMI – Mathematics for Industry andCommerce
Krzysztof Burnecki
Computer Simulations for Random
Phenomena
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Table of contents
I Generating random variables1 Inverse transform method2 Rejection method3 Convolution method4 Composition approach5 Specific methods for particular distributions
a Box and Muller method for generating normal distributionsb Generating hyperbolic distributionsc Chambers, Mallows and Stuck method for generating stable
distributions
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Table of contents cont.
II Generating counting (point) processes1 Homogeneous Poisson process2 Non-homogeneous Poisson process3 Mixed Poisson process4 Cox (doubly stochastic Poisson) process5 Renewal process
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Table of contents cont.
III Risk process1 Risk process for different counting processes2 Simulation of risk processes
IV Modelling of the risk process1 Fitting the claim amount distribution2 Fitting the intensity of the counting process3 Visualization of the risk process
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Table of contents cont.
V Calculating ruin probability1 Ruin probability in finite time
a Exact formulasb Computer approximationsc Pollaczek–Khinchin formula
2 Ruin probability in infinite timea Exact formulasb Computer approximations
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Table of contents cont.
VI Pricing of catastrophe bonds1 Pricing model2 Fitting the model3 Dynamics of the prices via Monte Carlo simulations
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Table of contents cont.
VII Simulation of self-similar processes1 Brownian motion2 Fractional Brownian motion3 FARIMA with Gaussian innovations4 α-stable motion5 Fractional α-stable motion6 FARIMA with α-stable innovations
VIII Self-similar processes and long-range dependence1 Estimating self-similarity, tail, and memory parameters2 BMW2 computer test
IX Modelling the solar flare data with FARIMA processes
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Bibliography
L. Devroye (1984), Non-Uniform Random Variate Generation,
Springer-Verlag, New York, 1986.
http://cg.scs.carleton.ca/luc/rnbookindex.html
K. Burnecki, W. Hardle, R. Weron (2004), Simulation of risk processes,
in: Encyclopedia of Actuarial Science, Wiley, Chichester, 1564-1570.
K. Burnecki, J. Klafter, M. Magdziarz, A. Weron (2008), From solar
flare time series to fractional dynamics, Phys. A vol. 387, 1077-1097.
K. Burnecki, A. Misiorek, R. Weron (2005), Loss distributions, in:
Statistical Tools for Finance and Insurance, Springer, Berlin, 289-317.
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Bibliography cont.
K. Burnecki, P. Mista, A. Weron (2005), Ruin probabilities in finite
and infinite time, in: Statistical Tools for Finance and Insurance,
Springer, Berlin, 341–379.
K. Burnecki, R. Weron (2005), Modeling of the risk process, in:
Statistical Tools for Finance and Insurance, Springer, Berlin, 319-339.
A. Chernobai, K. Burnecki, S. Rachev, S. Trck, R. Weron (2006),
Modelling catastrophe claims with left-truncated severity distributions,
Computational Statistics vol. 21(3-4), 537-555.
A. Janicki, A.Weron (1994), Simulation and Chaotic Behavior of
Stable Stochastic Processes, Marcel Dekker, New York.
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Bibliography cont.
S.A. Klugman, H.H. Panjer, G.E. Willmot (1998), Loss Models: From
Data to Decisions, Wiley, New York.
S.M. Ross (1997), Simulation, Academic Press, San Diego, 1997.
A. A. Stanislavsky, K. Burnecki, M. Magdziarz, A. Weron, K. Weron
(2009), FARIMA modeling of solar flare activity from empirical time
series of soft X-ray solar emission, Astroph. J. vol. 693, 1877-1882.
A. Weron, K. Burnecki, Sz. Mercik, K. Weron (2005), Complete
description of all self-similar models driven by Levy stable noise, Phys.
Rev. E. vol. 71, 016113.
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Table of contents
I Generating random variables1 Inverse transform method2 Rejection method3 Convolution method4 Composition approach5 Specific methods for particular distributions
a Box and Muller method for generating normal distributionsb Generating hyperbolic distributionsc Chambers, Mallows and Stuck method for generating stable
distributions
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Generating random variables
Problem: Generate a sample of a random variable X with a given density f
or a distribution function F . (The sample is called a random variate)
The simulation can be done by:
inverse transform method,
convolution method,
composition method,
acceptance-rejection method.
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Inverse transform method. Continuous case
Assumption: We can generate U, i.e., uniform (0, 1) random variable.
Consider a continuous r.v. having distribution function F .
Theorem: For any continuous distribution F the r.v. X defined by
X = F−1(U) has distribution F , where F−1(u) = infx : F (x) = u.
Thus, the algorithm is:
Step 1: Generate a uniform random variable U.
Step 2: Set X = F−1(U).
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Inverse transform method. Example
Exponential r.v.: F (x) = 1− exp(−βx). If 1− exp(−βx) = u, then
x = − 1β log(1− u).
X = − 1β log(1− U).
Algorithm:
Step 1: Generate a uniform random variable U.
Step 2: Set X = − 1β log(U).
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Inverse transform method. Discrete case
Consider a discrete r.v. with probability function:
P(Xi = xi ) = pi .
Consider the following algorithm.
Step 1: Generate a uniform random variable U.
Step 2: Transform U into X as follows,
X = xj , if
j−1∑i=1
pi ≤ U <
j∑i=1
pi .
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Rejection method. Continuous case
Suppose we can simulate a r.v. with density function g(x). We would like
to simulate a r.v. with density function f (y). Let c be a constant such that
f (y)/g(y) ≤ c , for all y . Then the rejection algorithm is:
Step 1: Generate Y having density g and generate U.
Step 2: If U ≤ f (y)cg(y) , then X = Y ; else go to step 1.
Theorem. The r.v. X generated by the rejection method has density
function f . Moreover, the number of iterations needed to obtain X is a
geometric r.v. with mean c .
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Rejection method. Discrete case
Suppose we can simulate a r.v. with probability function P(Yi = yi ) = qi .
We would like to simulate a r.v. with probability function P(Xi = xi ) = pi .
Let c be a constant such that pi/qi ≤ c , for all i . Then the rejection
algorithm is:
Step 1: Generate Y having probability function qi and generate U.
Step 2: If U ≤ pYcqY
, then X = Y ; else go to step 1.
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Convolution method
Suppose X is a sum of independent random variables Z1,Z2, . . .Zm, i.e.
X = Z1 + Z2 + . . .Zm, where Zi ∼ Fi and are all independent.
Algorithm:
Step 1: Generate m random numbers U1,U2, . . .Um.
Step 2: Inverse transform method: Zi = F−1(Ui ).
Step 3: Set X =∑m
i=1 Zi .
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Convolution method. Example
Generate a sample from Erlang(β; m) distribution.
Algorithm:
Step 1: Generate m random numbers U1,U2, . . .Um.
Step 2: Inverse transform method: Zi = − 1β log(Ui ).
Step 3: Set X =∑m
i=1 Zi .
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Composition method
Suppose that either the distribution FX or the probability density fX can be
represented either of the following two forms:
(1) FX (x) = p1FY1 (x) + p2FY2 (x) + . . . pmFYm(x)
(2) fX (x) = p1fY1 (x) + p2fY2 (x) + . . . pmfYm(x)
where p1, . . . , pm are non-negative and sum to one (so that they form a
probability mass function).
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Composition method cont.
Then, assuming that the Yi ’s are relatively easily to generate, we can
generate X as follows:
Step 1: Generate a discrete random variable I on 1, . . . ,m, where
P(I = j) = pj for 1 ≤ j ≤ m.
Step 2: Generate YI from FYI(or fYI
).
Step 3: Return X = YI .
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Simulation of normal random variables. Box-Mulleralgorithm
Note that if Z ∼ N(0, 1), then
X = µ+ σX ∼ N(µ, σ2).
The Box-Muller algorithm for generating two i.i.d. N(0, 1) random
variables.
Step 1: Generate uniform numbers U1 and U2.
Step 2: Return X =√−2 log(U1) cos(2πU2) and
X =√−2 log(U1) sin(2πU2).
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Simulation of hyperbolic random variables
The hyperbolic distribution is defined as a normal variance-mean
mixture where the mixing distribution is the generalized inverse
Gaussian (GIG) law with parameter λ = 1.
More precisely, a random variable Z has the hyperbolic distribution if:
(Z |Y ) ∼ N (µ+ βY ,Y ) ,
where Y is a generalized inverse Gaussian GIG(λ = 1, χ, ψ) random
variable and N(m, s2) denotes the Gaussian distribution with mean m
and variance s2.
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Simulation of hyperbolic random variables cont.
The GIG law is a positive domain distribution with the pdf given by:
fGIG(x) =(ψ/χ)λ/2
2Kλ(√χψ)
xλ−1e−12 (χx−1+ψx), x > 0,
where the three parameters take values in one of the ranges: (i)
χ > 0, ψ ≥ 0 if λ < 0, (ii) χ > 0, ψ > 0 if λ = 0 or (iii) χ ≥ 0, ψ = 0
if λ > 0.
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Simulation of hyperbolic random variables cont.
The resulting algorithm reads as follows.
Step 1: Simulate a random variable Y ∼ GIG(λ, χ, ψ).
Step 2: Simulate a standard normal random variable N.
Step 3: Return X = µ+ βY +√
Y N.
The algorithm is fast and efficient if we have a handy way of simulating
GIG variates.
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Simulation of stable random variables
Definition
A random variable X is stable if there are parameters α ∈ (0; 2],σ ∈ (0;∞), β ∈ [−1; 1], µ ∈ R such that its characteristic function has thefollowing form:
E exp(itX ) =
exp(−σα|t|α(1− iβsgn(t) tan πα
2 ) + itµ) if α 6= 1,exp(−σ|t|(1− iβ 2
π sgn(t) ln |t|) + itµ) if α = 1.
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Simulation of stable random variables cont.
Stable distribution is characterized by four parameters:
0 < α ≤ 2-index of stability,
σ > 0-scale parameter,
−1 ≥ β ≤ 1-skewness parameter,
µ ∈ R-location parameter.
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Simulation of stable random variables cont.
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45alpha=2.0alpha=1.5alpha=1.1alpha=0.9alpha=0.7
Figure: The probability density function of X ∼ Sα(1, 0.1, 0).
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Simulation of stable random variables cont.
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45sigma=0.5sigma=1.0sigma=2.0
Figure: The probability density function of X ∼ S1.8(σ, 0, 0).
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Simulation of stable random variables cont.
−6 −4 −2 0 2 4 60
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2beta=−1.0beta=0.0beta=1.0
Figure: The probability density function of X ∼ S1.8(1, β, 0).
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Simulation of stable random variables cont.
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35mu=−2.0mu=0.0mu=2.0
Figure: The probability density function of X ∼ S1.8(1, 0, µ) for µ = −2, 0, 2.
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Simulation of stable random variables cont.
Due to the lack of explicit formulas of cumulative distribution function, it is
hard to apply the simplest and fastest algorithm of simulation, which is the
inversion of c.d.f.
In 1976 Chambers, Mallows and Stuck presented a method of simulating
X ∼ Sα(1, β, 0):
Let U be uniform random variable on (−π2 ; π2 ) independent of W, which is
exponentially distributed with mean equals 1.
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Simulation of stable random variables cont.
Proposition
a) for α = 1 X ∼ S1(1, β, 0) with
X =2
π
((π
2+ βU) tan U − β ln
( π2 W cos Uπ2 + βU
)),
b) for α 6= 1 X ∼ Sα(1, β, 0) with
X = Sα,βsin(α(U + Bα,β))
(cos U)1/α
(cos(U − α(U + Bα,β))
W
) 1−αα
,
where
Bα,β =arctanβ tan πα
2
α,
Sα,β =(
1 + β2 tan2 πα
2
) 12α
.
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Simulation of stable random variables cont.
Simulation of random W is obtained by inversion of cumulative
distribution, eg. for given uniform random U, we have W = − log U. The
next step of simulation of a stable random variable is to get Sα(σ, β, µ):
Proposition
Let X ∼ Sα(1, β, 0), then
Y =
σX + µ if α 6= 1,σX + 2
πσ lnσ + µ if α = 1
is Sα(σ, β, µ).
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Simulation of stable random variables cont.
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35histogramnumerical p.d.f.
Figure: The probability density function and histogram of S1.8(1, 1, 0) withsample length of 105.
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Tails of stable random variables
The tail exponent estimation method gives us the information about the
index of stability. The tails of stable are like power decay function. One can
assume about the c.d.f F (x) of stable random that, it satisfies the following
relation:
1− F (x) ∼ Cx−α,
and then by taking logarithm on both sides of this relation, we obtain:
log(1− F (x)) ∼ −α log(Cx)
That leads to an interpretation that we can estimate the index of stability
as the negative value of the slope of logarithmic values of right tail as a
function of logarithmic values of given data. This kind of estimation
depends of the size of data.
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Tails of stable random variables cont.
−6 −4 −2 0 2 4 6−8
−6
−4
−2
0
Right tail of 1.3−stable (485 values)
Estimated alpha = 1.126763
−15 −10 −5 0 5 10 15 20−15
−10
−5
0
Right tail of 0.7−stable (50063 values)
Estimated alpha = 0.685774
−14 −12 −10 −8 −6 −4 −2 0 2 4 6−15
−10
−5
0
Right tail of 1.9−stable (499535 values)
Estimated alpha = 1.848122
Figure: Log-log plot of right tails of samples of length 106, 105, 103 with differentα’s.
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Table of contents
II Generating counting (point) processes1 Homogeneous Poisson process2 Non-homogeneous Poisson process3 Mixed Poisson process4 Cox (doubly stochastic Poisson) process5 Renewal process
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Simulation of counting processes
Nt is simulated either via:
the arrival times Ti (or jump times), i.e. moments when the ith
event occurs, or
the inter-arrival times (or waiting times) Wi = Ti − Ti−1, i.e. the time
periods between successive events.
The prominent scenarios for Nt, are given by:
the homogeneous Poisson process (HPP),
the non-homogeneous Poisson process (NHPP),
the mixed Poisson process,
the Cox process,
the renewal process.
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Homogeneous Poisson process
A continuous-time stochastic process Nt : t ≥ 0 is a (homogeneous)
Poisson process with intensity (or rate) λ > 0 if:
1 Nt is a counting process, and
2 the times between events are independent and identically distributed
with an exponential(λ) distribution, i.e. exponential with mean 1/λ.
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Simulation of the HPP
Successive arrival times T1,T2, . . . ,Tn of the (homogeneous) Poisson
process can be generated by the following algorithm ([HPP1 algorithm]):
Step 1: set T0 = 0
Step 2: for i = 1, 2, . . . , n do
Step 2a: generate an exponential random variable E
with parameter λ
Step 2b: set Ti = Ti−1 + E
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Simulation of the HPP cont.
Given that N(t) = n, the n occurrence times T1,T2, . . . ,Tn have the same
distributions at the order statistics corresponding to n i.i.d. random
variables uniformly distributed on the interval (0, t].
Hence the arrival times T1,T2, . . . ,Tn of the HPP on the interval (0, t] can
be generated as follows ([HPP2 algorithm]):
Step 1: Generate a Poisson random variable N with parameter λt.
Let N = n.
Step 2: Generate n random variables Ui distributed uniformly on
(0, 1), i.e. Ui ∼ U(0, 1), i = 1, 2, . . . , n.
Step 3: (T1,T2, . . . ,Tn) = t · sortU1,U2, . . . ,Un.
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Non-homogeneous Poisson process
The non-homogeneous Poisson process (NHPP) can be thought of as a
Poisson process with a variable intensity defined by the deterministic
intensity (rate) function λ(t).
A NHPP can model situations, where event occurrence epochs are likely to
depend on the time of the year or of the week.
The increments of a NHPP do not have to be stationary.
When λ(t) = λ, the NHPP reduces to the HPP with intensity λ.
The simulation of the non-homogeneous Poisson process is slightly more
complicated than the homogeneous one.
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Simulation of the NHPP – integration method
The increment of a NHPP with rate function λ(t) is distributed as a
Poisson random variable with intensity λ =∫ t
sλ(u)du.
Hence, the distribution function Fs of the waiting time Ws satisfies:
Fs(t) = P(Ws ≤ t) = 1− P(Ws > t) = 1− P(Ns+t − Ns = 0) =
= 1− exp
(−∫ s+t
s
λ(u)du
)= 1− exp
(−∫ t
0
λ(s + v)dv
).
If we can find a formula for the inverse F−1s then for each s we can easily
generate Ws using the inverse transform method.
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Simulation of the NHPP – integration method cont.
The resulting algorithm can be summarized as follows ([NHPP1 algorithm]):
Step 1: set T0 = 0
Step 2: for i = 1, 2, . . . , n do
Step 2a: generate a random variable U distributed
uniformly on (0, 1), i.e. U ∼ U(0, 1)
Step 2b: set Ti = Ti−1 + F−1s (U)
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Simulation of the NHPP – thinning
Suppose that there exists a constant λ such that λ(t) ≤ λ for all t.
Let T ∗1 ,T∗2 , . . . be the arrival times of a HPP with intensity λ.
Accept the ith arrival time with probability λ(T ∗i )/λ, independently of all
other arrivals, as part of the thinned process (hence the name of the
method).
The sequence T1,T2, . . . of the accepted arrival times forms a sequence of
the arrival times of a NHPP with rate function λ(t).
The algorithm amounts to rejecting (hence the alternative name – rejection
method) or accepting a particular arrival as part of the thinned process.
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Simulation of the NHPP – thinning cont.
The resulting algorithm reads as follows ([NHPP2 algorithm]):
Step 1: set T0 = 0 and T ∗ = 0
Step 2: for i = 1, 2, . . . , n do
Step 2a: generate an exponential random variable E
with parameter λ
Step 2b: set T ∗ = T ∗ + E
Step 2c: generate a random variable U ∼ U(0, 1)
Step 2d: if U > λ(T ∗)/λ then return to step 2a (→reject the arrival time) else set Ti = T ∗ (→accept the arrival time)
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Simulation of the NHPP cont.
Given that N(t) = n, the n occurrence times T1,T2, . . . ,Tn have the same
distributions at the order statistics corresponding to n independent random
variables distributed on the interval (0, t], each with the common density
function f (v) = λ(v)/∫ t
0λ(u)du, v ∈ (0, t].
Hence the arrival times T1,T2, . . . ,Tn of the NHPP on the interval (0, t]
can be generated as follows ([NHPP3 algorithm]):
Step 1: Generate a Poisson random variable N with intensity∫ t
0λ(u)du. Let N = n.
Step 2: Generate n random variables Vi , i = 1, 2, . . . n given by the
densityf (v) = λ(v)/∫ t
0λ(u)du.
Step 3: (T1,T2, . . . ,Tn) = sortV1,V2, . . . ,Vn.
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Simulation of the NHPP cont.
a = 1, b = 0.01 (blue line), b = 0.1 (red), b = 1 (green)
Linear intensity (a+b*t)
0 5 10
t
010
2030
4050
N(t
)
Seasonal intensity (a+b*sin(2*pi*t))
0 5 10
t
010
2030
N(t
)
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Mixed Poisson process
In many situations the portfolio of an insurance company is diversified in
the sense that the risks associated with different groups of policy holders
are significantly different.
For example, in motor insurance we might want to make a difference
between male and female drivers or between drivers of different age.
We would then assume that the claims come from a heterogeneous group
of clients, each one of them generating claims according to a Poisson
distribution with the intensity varying from one group to another.
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Mixed Poisson process cont.
If N is a HPP with intensity 1 and Λ is a positive random variable
independent of N, then the process N = N Λ = (N(Λt))t is called a mixed
Poisson process. The random variable Λ is called a structure variable.
A mixed Poisson process has stationary increments, however, the
independent increments condition is violated.
The most common choice for the distribution of the structure variable Λ is
the gamma distribution. In such a case the mixed Poisson proces is called a
negative binomial process or Polya process.
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Mixed Poisson process cont.
In the mixed Poisson process the distribution of Nt is given by a mixture
of Poisson processes.
Conditioning on the extrinsic random variable Λ, the process Nt behaves
like a HPP.
Hence, the process can be generated in the following way:
first a realization of a non-negative random variable Λ is generated,
conditioned upon its realization, Nt as a HPP with that realization
as its intensity is constructed.
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Simulation of the mixed Poisson process
Making the algorithm more formal we can write ([MPP1 algorithm]):
Step 1: generate a realization λ of the random intensity Λ
Step 2: set T0 = 0
Step 3: for i = 1, 2, . . . , n do
Step 3a: generate an exponential random variable E
with intensity λ
Step 3b: set Ti = Ti−1 + E
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Simulation of MPP cont.
Given that N(t) = n, the n occurrence times T1,T2, . . . ,Tn have the same
distributions at the order statistics corresponding to n i.i.d. random
variables uniformly distributed on the interval (0, t].
Hence the arrival times T1,T2, . . . ,Tn of the MPP on the interval (0, t]
can be generated as follows ([MPP2 algorithm]):
Step 1: Generate a mixed Poisson random variable N with parameter
Λt. Let N = n.
Step 2: Generate n random variables Ui distributed uniformly on
(0, 1), i.e. Ui ∼ U(0, 1), i = 1, 2, . . . , n.
Step 3: (T1,T2, . . . ,Tn) = t · sortU1,U2, . . . ,Un.
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Cox process
The Cox process, or doubly stochastic Poisson process, provides flexibility
by letting the intensity not only depend on time but also by allowing it to
be a stochastic process.
Cox processes seem to form a natural class for modeling risk and size
fluctuations.
IF N is a HPP with intensity 1 and Λ(t) is a stochastic process with
Λ = 0, non-decreasing sample paths and independent of N, then the
process N = N Λ = (N(Λ)) is called a Cox process or doubly stochastic
Poisson process.
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Cox process cont.
The intensity process Λ(t) is used to generate another process Nt by
acting as its intensity.
That is, Nt is a Poisson process conditional on Λ(t) which itself is a
stochastic process.
If Λ(t) is deterministic, then Nt is a NHPP. This property suggests a
simulation method.
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Simulation of the Cox process
Step 1: generate a realization λ(t) of the intensity process Λ(t)for a sufficiently large time period
Step 2: set λ = max λ(t)Step 3: set T0 = 0 and T ∗ = 0
Step 4: for i = 1, 2, . . . , n do
Step 4a: generate an exponential random variable E
with intensity λ
Step 4b: set T ∗ = T ∗ + E
Step 4c: generate a random variable U ∼ U(0, 1)
Step 4d: if U > λ(T ∗)/λ then return to step 4a (→reject the arrival time) else set Ti = T ∗ (→accept the arrival time)
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Renewal process
If the waiting times Wi are i.i.d. and nonnegative then the resulting
sequence is a renewal process.
Note, that the HPP is a renewal process with exponentially distributed
inter-arrival times. Hence, we can generate the arrival times of a renewal
process by:
Step 1: set T0 = 0
Step 2: for i = 1, 2, . . . , n do
Step 2a: generate a random variable X with an
assumed distribution function F
Step 2b: set Ti = Ti−1 + X
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Table of contents
III Risk process1 Risk process for different counting processes2 Simulation of risk processes
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Risk process
If (Ω,F ,P) is a probability space carrying:
1 a point process Ntt≥0, i.e. an integer valued stochastic process with
N0 = 0 a.s., Nt <∞ for each t <∞ and nondecreasing realizations,
and
2 an independent sequence Xk∞k=1 of positive i.i.d. random variables,
then the risk process Rtt≥0 is given by:
Rt = u + c(t)−Nt∑i=1
Xi .
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Risk process cont.
Rt = u + c(t)−Nt∑i=1
Xi ,
where:
Ntt≥0 is the claim arrival point process,
Xk∞k=1 is an independent claim sequence of positive i.i.d. random
variables with common mean µ,
u is a nonnegative constant representing the initial capital of the
company,
c(t) is the premium function.
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Homogeneous Poisson process
Since ENt = λt, it is natural to define the premium function as:
c(t) = ct = (1 + θ)µλt,
where µ = EXk and θ > 0 is the relative safety loading which ”guarantees”
survival of the insurance company.
With such a choice of the premium function we obtain the classical form of
the risk process:
Rt = u + ct −Nt∑i=1
Xi .
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Non-homogeneous Poisson process
Since ENt =∫ t
0λ(s)ds, it is natural to define the premium function in the
non-homogeneous case as:
c(t) = (1 + θ)µ
∫ t
0
λ(s)ds,
where µ = EXk and θ > 0 is the relative safety loading.
Then the risk process takes the form:
Rt = u + (1 + θ)µ
∫ t
0
λ(s)ds −Nt∑i=1
Xi .
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Mixed Poisson process
Since for each t the claim numbers Nt up to time t are Poisson with
intensity Λt, in the mixed case it is reasonable to consider the premium
function of the form:
c(t) = (1 + θ)µΛt,
where µ = EXk and θ > 0 is the relative safety loading.
Then the risk process takes the form:
Rt = u + (1 + θ)µΛt −Nt∑i=1
Xi .
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Cox process
The premium function is a generalization of the former premium functions:
c(t) = (1 + θ)µ
∫ t
0
Λ(s)ds,
where µ = EXk and θ > 0 is the relative safety loading.
Then the risk process takes the form:
Rt = u + (1 + θ)µ
∫ t
0
Λ(s)ds −Nt∑i=1
Xi .
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Renewal process
For renewal claim arrival processes a constant premium rate allows for a
constant safety loading.
Let Nt be a renewal process and assume that EW1 = 1/λ <∞.
Then the premium function is defined in a natural way as:
c(t) = (1 + θ)µλt,
like in the homogeneous Poisson process case.
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Simulation of the risk process
The simulation of the risk process Rt or the aggregated claim process
∑Nt
i=1 Xi reduces to modeling:
the claim arrival point process Nt,
the claim size sequence Xk,
Both processes are assumed to be independent, hence can be simulated
independently of each other.
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Simulation of the claim arrival point process
Nt is simulated either via:
the arrival times Ti, i.e. moments when the ith claim occurs, or
the inter-arrival times (or waiting times) Wi = Ti − Ti−1, i.e. the time
periods between successive claims.
The prominent scenarios for Nt, are given by:
the homogeneous Poisson process (HPP),
the non-homogeneous Poisson process (NHPP),
the mixed Poisson process,
the Cox process (or doubly stochastic Poisson process),
the renewal process.
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Table of contents
IV Modelling of the risk process1 Fitting the claim amount distribution2 Fitting the intensity of the counting process3 Visualization of the risk process
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Modelling of the risk process
Rt = u + c(t)−Nt∑i=1
Xi ,
the company sells insurance policies and receives a premium according
to c(t),
liabilities are represented by the aggregated claim process ∑Nt
i=1 Xi,
the claim severities are described by the random sequence Xk.
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Fitting claim size distribution
Visual techniques:
Mean excess function
Limited expected value function
Probability gates
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Two datasets
Property Claim Services (PCS) dataset, which covers losses resulting
from catastrophic events in the USA. The data includes 1990-1999
market’s loss amounts in USD adjusted for inflation using the
Consumer Price Index. Only natural perils which caused damages
exceeding 5 million dollars were taken into account.
The second dataset concerns major inflation-adjusted Danish fire
losses in profits (in Danish Krone, DKK) that occurred between 1980
and 1990 and were recorded by Copenhagen Re.
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PCS data
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Years
05
1015
Adj
uste
d PC
S ca
tast
roph
e cl
aim
s (U
SD b
illio
n)
Figure: Graph of the PCS catastrophe loss data, 1990-1999. Two largest losses inthis period were caused by Hurricane Andrew (24 August 1992) and theNorthridge Earthquake (17 January 1994).
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Mean excess function
For a claim amount random variable X , the mean excess function or mean
residual life function is the expected payment per claim on a policy with a
fixed amount deductible of x , where claims with amounts less than or equal
to x are completely ignored:
e(x) = E(X − x |X > x) =
∫∞x1− F (u) du
1− F (x).
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Mean excess function cont.
In practice, the mean excess function e is estimated by en based on a
representative sample x1, . . . , xn:
en(x) =
∑xi>x xi
#i : xi > x− x .
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Mean excess function cont.
0 5 10
x
12
34
5
e(x)
0 5 10
x
0.5
11.
52
2.5
3
e(x)
Figure: Top panel: Shapes of the mean excess function e(x) for the log-normal(green dashed line), gamma with α < 1 (red dotted line), gamma with α > 1(black solid line) and a mixture of two exponential distributions (blue long-dashedline). Bottom panel: Shapes of the mean excess function e(x) for the Pareto(green dashed line), Burr (blue long-dashed line), Weibull with τ < 1 (black solidline) and Weibull with τ > 1 (red dotted line) distributions.
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Mean excess function cont.
0 1 2 3 4 5
x (USD billion)
02
46
8
e_n(
x) (
USD
bill
ion)
0 5 10 15
x (years)*E-2
01
23
Sam
ple
mea
n ex
cess
fun
ctio
n*E
-2Figure: The empirical mean excess function en(x) for the PCS catastrophe lossamounts in billion USD (top panel) and waiting times in years (bottom panel).Comparison with the previous figure suggests that log-normal, Pareto, and Burrdistributions should provide a good fit for loss amounts, while log-normal, Burr,and exponential laws for the waiting times.
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Limited expected value function
The limited expected value function L of a claim size variable X , or of the
corresponding cdf F (x), is defined by
L(x) = Emin(X , x) =
∫ x
0
ydF (y) + x 1− F (x) , x > 0.
The value of the function L at point x is equal to the expectation of the
cdf F (x) truncated at this point. In other words, it represents the expected
amount per claim retained by the insured on a policy with a fixed amount
deductible of x .
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Limited expected value function cont.
The empirical estimate is defined as follows:
Ln(x) =1
n
∑xj<x
xj +∑xj≥x
x
.
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Limited expected value function cont.
The limited expected value function (LEVF) has the following important
properties:
(i) the graph of L is concave, continuous and increasing;
(ii) L(x)→ E (X ), as x →∞;
(iii) F (x) = 1− L′(x), where L′(x) is the derivative of the
function L at point x ; if F is discontinuous at x , then the
equality holds true for the right-hand derivative L′(x+).
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Limited expected value function cont.
0 5 10 15
x (USD billion)
100
150
200
250
300
Ana
lytic
al a
nd e
mpi
rica
l LE
VFs
(U
SD m
illio
n)
Figure: The empirical (black solid line) and analytical limited expected valuefunctions (LEVFs) for the log-normal (green dashed line) and Pareto (blue dottedline) distributions for the PCS loss catastrophe data.
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Probability plot
First, the observations x1, ..., xn are ordered from the smallest to the
largest: x(1) ≤ ... ≤ x(n).
Next, they are plotted against their observed cumulative frequency, i.e.
the points correspond to the pairs (x(i),F−1([i − 0.5]/n)), for
i = 1, ..., n.
If the hypothesized distribution F adequately describes the data, the
plotted points fall approximately along a straight line.
If the plotted points deviate significantly from a straight line, especially
at the ends, then the hypothesized distribution is not appropriate.
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Probability plot cont.
0 2 4 6 8 10 12 14 16 18 0.25
0.95
0.98
0.99
0.995
0.997
0.998
Data (USD billion)
Pro
babi
lity
0 0.2 0.4 0.6 0.8
0.25
0.75
0.9
0.95
0.96
Data (USD billion)
Pro
babi
lity Hurricane Andrew
Northridge Earthquake
Figure: Pareto probability plot of the PCS loss data. Apart from the two veryextreme observations (Hurricane Andrew and Northridge Earthquake) the points(pluses) more or less constitute a straight line, validating the choice of the Paretodistribution. The inset is a magnification of the bottom left part of the originalplot.
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Probability plot cont.
16 17 18 19 20 21 22 230.001
0.003
0.01 0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98 0.99
0.997
0.999
Data
Pro
babi
lity
Northridge Earthquake
Hurricane Andrew
Figure: Log-normal probability plot of the PCS loss data. The x-axis correspondsto logarithms of the losses. The deviations from the straight line at both endsquestion the adequacy of the log-normal law.
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Empirical analysis
Danish fire losses recorded by Copenhagen Re. Losses in profits
connected with fires
Loss sizes
Lognormal, Pareto, Burr, gamma, Weibull and mixture of twoexponentials distributions
Claim counting process
Homogeneous and nonhomogeneous process
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Danish fire losses
1980 1985 1990
Time
020
4060
Los
ses
(DK
K m
illio
n)
0 4 8 12 16 20
Losses (DKK million)
-2-1
0
Log
(1-F
(x))
Figure: Left panel : Illustration of the major Danish fire losses adjusted forinflation. Right panel : Logarithm of the right tails of the empirical claim sizedistribution function (thick blue solid line) together with lognormal (red dottedline) and Burr (thin black solid line) fits.
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Danish fire losses. Loss sizes
d.f.: Lognormal Pareto Burr Weibull
Para- µ = 12.704 α = 2.4189 α = 0.8935 α = 0.6963meters: σ = 1.4271 λ = 1.0261e6 λ = 1.1219e7 λ = 8.9740e-5
τ = 1.2976
χ2 56.109 73.879 48.493 129.24KS 0.0373 0.0397 0.0413 0.0783CM 0.1687 0.2878 0.1438 1.5245AD 1.0533 2.7712 0.8221 10.638
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Danish fire losses cont. Claim counting process
0 1 2 3 4 5 6 7 8 9 10 11
Time (years)
010
020
030
040
050
060
070
0
Mea
n-va
lue
func
tion
0 5 10 15 20 25 30
Time lag (qtr)
00.
51
AC
F
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Danish fire losses cont. Claim counting process
The data reveals no seasonality
A clear increasing trend can be observed in the number of quarterly
losses
We tested different exponential and polynomial functional forms
A simple linear intensity function λ(s) = a + bs yielded the best fit
Applying a least squares procedure we arrived at the values: a = 13.97
and b = 7.57.
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Danish fire losses cont. The fitted risk process
We consider a hypothetical scenario where the insurance company
insures losses resulting from fire damage
The company’s initial capital is assumed to be u = 100 million kr
The relative safety loading used is θ = 0.5
We chose two models of the risk process: a non-homogeneous Poisson
process with lognormal claim sizes and a non-homogeneous Poisson
process with Burr claim sizes.
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Danish fire losses cont. The fitted risk process
0 1 2 3 4 5 6 7 8 9 10 11
Time (years)
010
020
030
040
050
0
Cap
ital (
DK
K m
illio
n)
0 1 2 3 4 5 6 7 8 9 10 11
Time (years)
020
040
060
080
010
0012
00
Cap
ital (
DK
K m
illio
n)
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Quantile lines
The function xp(t) is called a sample p-quantile line if for each t ∈ [t0,T ],
xp(t) is the sample p-quantile.
Recall that the sample p-quantile satisfies Fn(xp−) ≤ p ≤ Fn(xp), where Fn
is the sample distribution function.
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Visualization of the risk process
Tools:
Trajectories
Ruin probability plot
Density evolution plot
Quantile lines
Probability gates
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Trajectories of the risk process
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
Figure: Discontinuous visualization of the trajectories of a risk process. Theinitial capital u = 10 million DKK, the relative safety loading θ = 0.05, the claimsize distribution is log-normal with parameters µ = 12.6795 and σ = 1.4241, andthe driving counting process is a HPP with monthly intensity λ = 4.81.
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Trajectories of the risk process cont.
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
Figure: Alternative (continuous) visualization of the trajectories of a risk process.The bankruptcy time is denoted by a star. The parameters of the risk process arethe same as in the previous figure.
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Ruin probability plots
01
23
45
0
2
4
6
8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure: Ruin probability plot with respect to the time horizon T (left axis, inmonths) and the initial capital u (right axis, in million DKK). The relative safetyloading θ = 0.15; other parameters of the risk process are the same as in theprevious figure.
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Density evolution
Density evolution plots (and their 2-dimensional projections) are a
visually attractive method of representing the time evolution of a
process.
At each time point t = t0, t1, ..., tn, a density estimate of the
distribution of process values at this time point is evaluated.
Then the densities are plotted on a grid of t values.
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Density evolution cont.
Figure: 3-dimensional visualization of the density evolution of a risk process withrespect to the risk process value Rt (left axis) and time t (right axis). Theparameters of the risk process are the same as in the previous figure
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Density evolution cont.
Figure: 2-dimensional projection of the density evolution.
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Quantile lines
The function xp(t) is called a sample p-quantile line if for each t ∈ [t0,T ],
xp(t) is the sample p-quantile, i.e. if it satisfies Fn(xp−) ≤ p ≤ Fn(xp),
where Fn is the empirical distribution function (edf).
Recall, that for a sample of observations x1, . . . , xn the edf is defined as:
Fn(x) =1
n#i : xi ≤ x,
i.e. it is a piecewise constant function with jumps of size 1/n at points xi .
Quantile lines are a very helpful tool in the analysis of stochastic processes.
For example, they can provide a simple justification of the stationarity (or
the lack of it) of a process.
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Quantile lines cont.
0 1 2 3 4 5 6 7 8 9 104
6
8
10
12
14
16
18
20
Figure: A Poisson-driven risk process (discontinuous thin lines) and its Brownianmotion approximation (continuous thin lines). The quantile lines allow for aneasy and fast comparison of the processes. The thick solid lines represent thesample 0.1, ..., 0.9-quantile lines based on 10000 trajectories of the risk process,whereas thick dashed lines correspond to their approximation counterparts. Theparameters of the risk process are the same as in the previous figure.
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Probability gates
“Probability gates” are a graphical tool. They can be of invaluable
assistance in real-time analysis of the risk process and its models.
A “probability gate” gives the so-called cylindrical probability
PXt0 ∈ (a, b] that the simulated process Xt passes through a
specified interval (a, b] at a specified point in time t0.
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Probability gates cont.
Figure: “Probability gates” are an interactive graphical tool used for determiningthe probability that the process passes through a specified interval. The0.1, ..., 0.9-quantile lines (thick red lines) are based on 1000 simulated trajectories(thin blue lines) of the risk process originating at u = 100 billion USD. Theparameters of the α-stable Levy motion approximation of the risk process werechosen to comply with PCS data. From: SDE-Solver.
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Table of contents
V Calculating ruin probability1 Ruin probability in finite time
a Exact formulasb Computer approximationsc Pollaczek–Khinchin formula
2 Ruin probability in infinite timea Exact formulasb Computer approximations
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Basic aspects of actuarial risk theory
classical risk process
ruin probability in finite and infinite time horizon
light- and heavy-tailed distributions
adjustment coefficient
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Classical risk process
Definition
Let (Ω,F ,P) be a probability space carrying Poisson process Ntt≥0 withintensity λ, and sequence Xk∞k=1 of positive, i.i.d. random variables, withmean µ and variance σ2. Furthermore, we assume that Xk and Nt areindependent. The classical risk process Rtt≥0 is given by
Rt = u + ct −Nt∑i=1
Xi , c > 0, u ≥ 0.
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Ruin probability
To introduce the term ruin probability, first define the time to ruin as
τ(u) = inft ≥ 0 : Rt < 0.
Definition
The ruin probability in finite time T is given by
ψ(u,T ) = P(τ(u) ≤ T )
and ruin probability in infinite time is defined as
ψ(u) = P(τ(u) <∞).
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Adjustment coefficient
Definition
Let γ = supz MX (z) <∞ and let R be a positive solution of the equation
1 + (1 + θ)µR = MX (R), R < γ.
If there exists a non-zero solution to the above equation, we call such R anadjustment coefficient (or Lundberg exponent).
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Adjustment coefficient cont.
0
x
0.95
11.
051.
11.
151.
2
y
R
Figure: Illustration of the existence of the adjustment coefficient. The solid blueline represents the curve y = 1 + (1 + θ)µz and the dotted red one y = MX (z).
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Light- and heavy-tailed distributions
We distinguish here between light- and heavy-tailed distributions.
Definition
A distribution FX (x) is said to be light-tailed, if there exist constantsa > 0, b > 0 such that FX (x) = 1− FX (x) ≤ ae−bx or, equivalently, ifthere exists z > 0, such that MX (z) <∞, where MX (z) is the momentgenerating function. Distribution FX (x) is said to be heavy-tailed, if for alla > 0, b > 0 FX (x) > ae−bx , or, equivalently, if ∀z > 0 MX (z) =∞.
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Light- and heavy-tailed distributions cont.
Table: Typical claim size distributions. In all cases x ≥ 0.
Light-tailed distributions
Name Parameters pdfExponential β > 0 fX (x) = β exp(−βx)
Gamma α > 0, β > 0 fX (x) = βα
Γ(α) xα−1 exp(−βx)
Weibull β > 0, τ ≥ 1 fX (x) = βτxτ−1 exp(−βxτ )
Mixed exp’s βi > 0,n∑
i=1
ai = 1 fX (x) =n∑
i=1
aiβi exp(−βix)
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Light- and heavy-tailed distributions cont.
Table: Typical claim size distributions. In all cases x ≥ 0.
Heavy-tailed distributions
Name Parameters pdfWeibull β > 0, 0 < τ < 1 fX (x) = βτxτ−1 exp(−βxτ )
Log-normal µ ∈ R, σ > 0 fX (x) = 1√2πσx
exp− (ln x−µ)2
2σ2
Pareto α > 0, λ > 0 fX (x) = α
λ+x
(λλ+x
)αBurr α > 0, λ > 0, τ > 0 fX (x) = ατλαxτ−1
(λ+xτ )α+1
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Ruin probability in finite time horizonExact ruin probabilities in finite time
Exponential loss amounts (β = 1, c=1)
ψ(u,T ) = λ exp −(1− λ)u − 1
π
∫ π
0
f1(x)f2(x)
f3(x)dx ,
where
f1(x) = λ exp
2√λT cos x − (1 + λ)T + u
(√λ cos x − 1
),
f2(x) = cos(
u√λ sin x
)−cos
(u√λ sin x + 2x
), and f3(x) = 1+λ−2
√λ cos x .
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Approximations of the ruin probability in finite time
Monte Carlo method
Segerdahl normal approximation
Diffusion approximation
Corrected diffusion approximation
Finite time De Vylder approximation
The idea of the De Vylder approximation – replace the claim surplus
process with the one exponential claims fitting first three moments:
β =3µ(2)
µ(3), λ =
9λµ(2)3
2µ(3)2 , and θ =2µµ(3)
3µ(2)2 θ.
Next, employ the exact, exponential case formula.
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Numerical comparison of the finite time approximations
5 approximations – mixture of 2 exponentials case, θ = 30%.
0 3 6 9 12 15 18 21 24 27 30
u (USD billion)
00.
10.
20.
30.
40.
50.
60.
7
psi(u
,T)
0 3 6 9 12 15 18 21 24 27 30
u (USD billion)
-0.8
-0.6
-0.4
-0.2
00.
20.
40.
6
(psi(
u,T)
-psi_
(MC)
(u,T
))/ps
i_(M
C)(u
,T)
Figure: Monte Carlo (left panel), the relative error (right panel). Segerdahl (short-dashed blueline), diffusion (dotted red line), corrected diffusion (solid black line) and finite time De Vylder(long-dashed green line). T fixed and u varying.
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Numerical comparison of the finite time approximations
0 2 4 6 8 10 12 14 16 18 20
T (years)
00.
010.
020.
030.
040.
050.
06
psi(
u,T
)
0 2 4 6 8 10 12 14 16 18 20
T (years)
-1-0
.8-0
.6-0
.4-0
.20
0.2
0.4
0.6
0.8
1
(psi
(u,T
)-ps
i_(M
C)(
u,T
))/p
si_(
MC
)(u,
T)
Figure: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relativeerror of the approximations (right panel). The Segerdahl (short-dashed blue line), diffusion (dottedred line), corrected diffusion (solid black line) and finite time De Vylder (long-dashed green line)approximations. The mixture of two exponentials case with u fixed and T varying.
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Infinite horizonExact ruin probabilities
No initial capital. (u = 0)
Exponential claims. (explicit, analytical)
Gamma claims. (numerical integration from 0 to ∞)
Mixture of n exponentials claims. (analytical result for n = 2)
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A survey of approximations
Cramer–Lundberg approximation
Exponential approximation
Lundberg approximation
Beekman–Bowers approximation
Renyi approximation
De Vylder approximation
Heavy traffic approximation
Light traffic approximation
Heavy-light traffic approximation
Heavy-tailed claims approximation
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4-moment gamma De Vylder approximation
Parameters determining the new process with gamma claims
λ =λ(µ(3))2(µ(2))3
(µ(2)µ(4) − 2(µ(3))2)(2µ(2)µ(4) − 3(µ(3))2), θ =
θµ(2(µ(3))2 − µ(2)µ(4))
(µ(2))2µ(3),
µ =3(µ(3))2 − 2µ(2)µ(4)
µ(2)µ(3), µ
(2) =(µ(2)µ(4) − 2(µ(3))2)(2µ(2)µ(4) − 3(µ(3))2)
(µ(2)µ(3))2.
4MG approximation
ψ4MG (u) =θ(1− R
α )e−βRα u
1 + (1 + θ)R − (1 + θ)(1− Rα )
+αθ sin(απ)
π· I ,
where
I =
∫ ∞0
x αe−(x+1)βu dx[x α(1 + α(1 + θ)(x + 1)
)− cos(απ)
]2+ sin2(απ)
,
and α = µ2
µ(2)−µ2 , β = µµ(2)−µ2 .
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Computer approximation via Pollaczek–Khinchinformula
ψ(u) = P(M > u) =θ
1 + θ
∞∑n=0
(1
1 + θ
)n
B∗n0 (u),
B0 – tail of the distribution corresponding to the density b0(x) = FX (x)µ .
Since ψ(u) = EZ , where Z = 1(M > u), it may be generated as follows.
SIMULATION ALGORITHM
1 Generate a random variable K from the geometric distribution withp = 1
1+θ,
2 Generate random variables X1,X2, · · · ,XK from the density b0(x),3 Calculate M = X1 + X2 + · · · + XK ,4 If M > u, let Z = 1, otherwise let Z = 0,
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Pollaczek–Khinchin formula
Proposition
The density b0(x) has a closed form only for four of the considereddistributions:
exponential =⇒ b0(x) exponential,
mixture of exponentials =⇒ b0(x) mixture of exponentials with
the weights
(a1β1∑n
i=1(aiβi
), · · · ,
anβn∑n
i=1(aiβi
)
),
Pareto =⇒ b0(x) Pareto with (α− 1, ν),
Burr =⇒ b0(x) transformed beta.
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Numerical comparison of the methods
The relative error of 12 methods w.r.t. exact values.
0 5 10 15 20 25 30 35 40 45 50
u (USD billion)
-0.3
-0.2
-0.1
00.
10.
20.
3
(psi
(u)-
psi_
exa
ct(
u))/
psi_
exa
ct(
u)
0 5 10 15 20 25 30 35 40 45 50
u (USD billion)
-1-0
.8-0
.6-0
.4-0
.20
0.2
0.4
0.6
0.8
1
(psi
(u)-
psi_
exa
ct(
u))/
psi_
exa
ct(
u)
Figure: More effective methods (left): the Cramer–Lundberg (solid blue line), exponential(short-dashed brown line), Beekman–Bowers (dotted red line), De Vylder (medium-dashed black line)and 4-moment gamma De Vylder (long-dashed green line). Less effective (right): Lundberg(short-dashed red line), Renyi (dotted blue line), heavy traffic (solid magenta line), light traffic(long-dashed green line) and heavy-light traffic (medium-dashed brown line). The mixture of twoexponentials case.
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Numerical comparison of the methods
The relative error of 12 methods w.r.t. Pollaczek–Khinchin approximation
as a reference method.
0 1 2 3 4 5 6 7 8 9 10
u (USD billion)
-1-0
.8-0
.6-0
.4-0
.20
0.2
0.4
0.6
(psi
(u)-
psi_
exa
ct(
u))/
psi_
exa
ct(
u)
0 1 2 3 4 5 6 7 8 9 10
u (USD billion)
-1-0
.8-0
.6-0
.4-0
.20
0.2
0.4
0.6
(psi
(u)-
psi_
exa
ct(
u))/
psi_
exa
ct(
u)
Figure: More effective methods (left panel): the exponential (dotted blue line), Beekman–Bowers(short-dashed brown line), heavy-light traffic (solid red line), De Vylder (medium-dashed black line)and 4-moment gamma De Vylder (long-dashed green line). Less effective methods (right panel):Lundberg (short-dashed red line), heavy traffic (solid magenta line), light traffic (long-dashed greenline), Renyi (medium-dashed brown line) and subexponential (dotted blue line). The log-normal case.
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Table of contents
VI Pricing of catastrophe bonds1 Pricing model2 Fitting the model3 Dynamics of the prices via Monte Carlo simulations
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Catastrophe (CAT) bonds
CAT bonds
Compound doubly stochastic Poisson pricing model
CAT bond prices
Calibration of the pricing model. The PCS catastrophe data
Unconditional and conditional approaches
Impact of the presence of left-truncation of the loss data on the CAT
bond prices
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CAT bonds
The sponsor establishes a special purpose vehicle (SPV) as an issuer of
bonds and as a source of reinsurance protection.
The issuer sells bonds to investors. The proceeds from the sale are
invested in a collateral account.
The sponsor pays a premium to the issuer; this and the investment of
bond proceeds are a source of interest paid to investors.
If the specified catastrophic risk is not triggered, investors are paid
generous interest rate; but if the event occurs, investors sacrifice their
principal and interest.
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CAT bond triggers
There are three major types of CAT triggers: indemnity, index and
parametric.
An indemnity trigger involves the actual losses of the bond-issuing
insurer.
An industry index trigger involves, in the US for example, an index
created from property claim service (PCS) loss estimates.
A parametric trigger is based on, for example, the Richter scale
readings of the magnitude of an earthquake at specified data stations.
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Compound doubly stochastic Poisson pricing model
Doubly stochastic Poisson process Ns (s∈ [0,T ]) describing the flow
of natural events. The intensity of this process is assumed to be a
predictable bounded process λ(s).
Losses Xii∈N which are i.i.d. with F (x) = PXi<x. Moreover, X
and N are independent.
Aggregate loss process Lt =∑Nt
i=1 Xi .
Define a new process Mt = I (Lt ≥ D).
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Compound doubly stochastic Poisson pricing modelcont.
Progressive continuous a.e. process of discounting rates r . This
process describes the value at time s of USD 1 paid at time t > s by
exp (−R(s, t))=exp(−
t∫s
r(ξ) dξ).
Maturity time T and threshold level D.
Threshold time τ = inft : Lt ≥ D.
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Compound doubly stochastic Poisson pricing modelcont.
In the case of a zero-coupon bond: payment of a certain Z (random)
amount at maturity time T contingent on threshold time τ > T .
In the case of a bond paying only coupons: coupon payments Ct which
stop immediately at τ .
In the case of a coupon bond: payment of the principal value (PV) at
maturity time T contingent on threshold time τ > T and coupon
payments Ct which stop immediately at τ .
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The zero-coupon CAT bond price
Proposition
The non-arbitrage price of the zero-coupon CAT bond associated with thethreshold D, catastrophic flow Ns , the distribution function of the incurredlosses F , paying Z at maturity is given by
V 1t = E [Z exp −R(t,T ) (1−MT )|F t ] .
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The coupon CAT bond price for the bond paying onlycoupons
Proposition
The no-arbitrage price of the CAT bond associated with a threshold D,catastrophic flow Ns , a distribution function of incurred losses F , withcoupon payments Cs which terminate at time τ is given by
V 2t = E
[∫ T
t
exp −R(t, s)Cs(1−Ms)ds|F t
].
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The coupon-bearing CAT bond price
Proposition
The no-arbitrage price of the CAT bond associated with a threshold D,catastrophic flow Ns , a distribution function of incurred losses F , payingPV at maturity, and coupon payments Cs which cease at the thresholdtime τ is given by
V 3t = E
[PV exp −R(t,T ) (1−MT )
+
∫ T
t
exp −R(t, s)Cs(1−Ms)ds|F t
].
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Calibration of the pricing model
Loss distributions
exponential, lognormal, generalized Pareto, Burr, gamma, Weibull,log-αstable
Counting processes
homogeneous and non-homogeneous Poisson
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The data
We analyse losses resulting from natural catastrophic events in the
United States. Estimates of such losses are provided by ISO’s
(Insurance Services Office Inc.) Property Claim Services (PCS).
The term “natural catastrophe” denotes a natural disaster that affects
many insurers and when claims are expected to reach a certain dollar
threshold. Initially the threshold was set to $5 million.
In 1997 ISO increased its dollar threshold to $25 million.
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Calibration of the pricing model cont.
Figure: Graph of the PCS catastrophe loss data, 1990-1999.
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Calibration of the pricing model cont.
Notation:
Fγ is the distribution function of X
fγ is the density function of X
γ is a parameter vector or a scalar
λ(t) is the intensity function of Nt
H is a pre-specified threshold
the superscripts ”o” and ”c” refer to ”observed” (the incomplete data
set), and ”complete” or ”conditional” (the complete data set),
respectively
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Calibration of the pricing model cont.
1 The first approach involves using the observed frequency estimate
λo(t) and fitting the unconditional distribution to the truncated data.
2 An alternative approach would be to find the estimates λc(t) and
γcMLE for the unknown function λc(t) and parameter γc .
γcMLE = arg maxγ
log
(n∏
k=1
fγ(xk)
1− Fγ(H)
)λc(t) = λo(t)/(1− Fγc (H))
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Goodness-of-fit tests
Adjusted Kolmogorov-Smirnov (D), Kuiper (V ), Anderson-Darling (A2)
and Cramer-von Mises (W 2) statistics.
D = max(D+,D−),
V = D+ + D−,
A2 = n
∫ ∞−∞
(Fn(x)− F (x))2
F (x)(1− F (x))dF (x),
W 2 = n
∫ ∞−∞
(Fn(x)− F (x))2dF (x),
where D+ =√
n supxFn(x)− F (x), D− =√
n supxF (x)− Fn(x),Fn(x) is the adjusted empirical d.f. and F (x) is the fitted d.f.
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Conditional vs unconditional approach
γ, F (H) Unconditional Conditional
Weibull β 2.8091·10−6 0.0187τ 0.6663 0.2656
F (H) 21.23% 82.12%
Burr α 0.1816 0.1748
β 3.0419·1035 1.4720·1035
τ 4.6867 4.6732F (H) 2.58% 3.87%
Table: Estimated parameters and F (H) of the fitted distribution to the PCS data.
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Fitting intensity function
1990 1991 1992 1993 1994 1995 1996 1997
2
4
6
8
10
12
14
16
18
20
Time [years]
Num
ber
of e
vent
s
0 0.1 0.2 0.3 0.4 0.50
10
20
30
40
50
60
70
80
FrequencyP
erio
dogr
am
Figure: Left panel : The quarterly number of losses for the PCS data. Rightpanel : Periodogram of the PCS quarterly number of losses. A distinct peak isvisible at frequency ω = 0.25 implying a period of 1/ω = 4 quarters, i.e. one year.
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Fitting intensity function cont.
The observed intensity function of the form:
λo(t) = a + b · 2π · sin2π(t − c).The least square method used to quarterly number of losses. Results:
a b c MSE MAE
30.8750 1.6840 0.3396 18.9100 3.8385
In the homogeneous Poisson process case: MSE = 115.5730
and MAE = 10.1308.
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Calibrated model
Assumptions for the illustration purposes:
Discount rate r = ln(1.025) corresponding to LIBOR = 2.5%.
T ∈ [90, 720] days.
D ∈ USD [2.3, 27.5] billion (quarterly — 3*annual average loss).
Principal value = 1 USD.
In the zero-coupon case we assume that the bond is priced at 3.5%
over LIBOR.
For the bond paying only coupons and coupon bond we assume
Cs ≡ 0.06.
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Dynamics of the zero-coupon CAT bond price
Figure: The difference between zero-coupon CAT bond prices in the unconditionaland conditional cases with respect to the threshold level and time to expiry.
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Dynamics of the CAT bond price for the bond payingonly coupons
Figure: The difference between CAT bond prices, for the bonds paying onlycoupons, in the unconditional and conditional cases with respect to the thresholdlevel and time to expiry.
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Dynamics of the coupon CAT bond price
Figure: The difference between coupon-bearing CAT bond prices in theunconditional and conditional cases with respect to the threshold level and timeto expiry.
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Table of contents
VII Simulation of self-similar processes1 Brownian motion2 Fractional Brownian motion3 FARIMA with Gaussian innovations4 α-stable motion5 Fractional α-stable motion6 FARIMA with α-stable innovations
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Self-similar processes
A stochastic process X = (X (t))t>0 is called self-similar (ss) if for
some H > 0,
X (at)d= aHX (t)
for every a > 0, whered= denotes equality of finite dimensional
distributions of the processes. (H is the self-similarity index or
exponent of the self-similar process X.)
If we interpret t as ’time’ and X (t) as ’space’ then the above equation
tells us that every change of time scale a > 0 corresponds to a change
of space scale aH .
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Self-similar processes with stationary increments
X = (X (t))t>0 is said to have stationary increments (si) if for any
b > 0,
(X (t + b)− X (b))d= (X (t)− X (0)).
Any H-sssi X = (X (t))t∈R induces a stationary sequence
Y = (Y (t))t∈Z, where Yj = X (j + 1)− X (j); j = . . . ,−1, 0, 1, . . . The
sequence Y corresponding to the H-self-similar process X is called
noise.
The self-similarity is very closely related to stationarity: a logarithmic
time transform translates shift invariance of the stationary process into
scale invariance of the self-similar process – Lamperti theorem.
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Illustration of self-similarity
Trajectories of the fractional stable motion for H = 0.8 and α = 1.6 (thin
lines).
The thick lines stand for quantile lines, the bottom one for p = 0.1 and the
top one for 1− p = 0.9. The lines determine the subdomain of R2 to which
the trajectories of the approximated process should belong with
probabilities 0.8 at any fixed moment of time.
Due to self-similarity the quantile lines f (t) have the form f (t) = const · tH .
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Illustration of self-similarity cont.
0 10 20 30 40 50 60 70 80 90 100−150
−100
−50
0
50
100
150
t
Z 1.6
0.8 (t
)
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Illustration of stationarity
Trajectories of the stationary process obtained from the fractional stable
motion by the Lamperti transformation (thin lines).
The thick lines stand for quantile lines, the bottom one for p = 0.1 and the
top one for 1− p = 0.9. The lines determine the subdomain of R2 to which
the trajectories of the approximated process should belong with
probabilities 0.8 at any fixed moment of time.
Due to stationarity the quantile lines f (t) have the form f (t) = const.
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Illustration of stationarity cont.
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4−4
−3
−2
−1
0
1
2
3
4
t
Y(t)
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Self-similar and stable processes
Most prominent examples of self-similar processes belong to the class
of Levy stable processes, namely Brownian motion (H = 1/2),
fractional Brownian motion (0 < H < 1), Levy stable motion
(0 < α ≤ 2) and fractional Levy stable motion (0 < H < 1 and
0 < α ≤ 2).
Every self-similar process with stationary and independent increments
is Levy stable.
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Fractional stable motion
The process ZHα = (ZH
α (t))t∈R defined by the integral representation
ZHα (t) =
∫ 0
−∞
(|t − u|H− 1
α − |u|H− 1α
)dZα(u)
+
∫ t
0
|t − u|H− 1α dZα(u),
where Zα is a Levy stable motion, 0 < H < 1 and 0 < α ≤ 2 is called a
fractional Levy stable motion.
When α = 2 it becomes a fractional Brownian motion.
When H = 1/α it becomes a Levy stable motion.
When both H = 1/α and α = 2 it becomes a Brownian motion.
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FARIMA processes
Let d–fractional, d < 1− 1/α. X (n) : n ∈ Z–FARIMA(p, d , q) if
Φp(B)X (n) = Θq(B)(1− B)−dZα(n), n ∈ Z.
where
B–backward operator, i.e. BX (n) = X (n − 1)
Φp(z) = 1− φ1z − φ2z2 − . . .− φpzp–AR polynomial
Θq(z) = 1 + θ1z + θ2z2 + . . .+ θqzq–MA polynomial
Zα(n) : n ∈ Z–SαS noise with index of stability α ∈ (0, 2]
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FARIMA processes
(1− B)−d–fractional integrating operator with series expansion
(1− B)−d =∞∑j=0
bd(j)B j
with coefficients
bd(j) =Γ(j + d)
Γ(d)Γ(j + 1), j = 0, 1, . . .
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Series expansion
Proposition (Kokoszka and Taqqu)
Φp, Θq do not have common roots and Φp has no roots inz : |z | ≤ 1.α(d − 1) < −1 ⇐⇒ d < 1− 1
α .
FARIMA(p, d , q) X (n) : n ∈ Z has the form
X (n) = Cd(B)Zα(n) =∞∑j=0
c(j)Zα(n − j),
where the coefficients c(j)’s are defined by
Cd(z) :=Θq(z)
Φp(z)(1− z)−d =
∞∑j=0
c(j)z j .
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Simulation of fractional Brownian motion
The fractional Gaussian process (fGp) method introduced by Davies
and Harte uses the fast Fourier transform algorithm to transform
i.i.d. standard normal random variables into the correlated series.
The method operates on the order of N log2 N calculations and
enables to simulate a fractional Gaussian noise Y = Yjj∈Z whose the
autocovariance function is given by
γ(τ) ≡ γτ =VarY1
2
(|τ + 1|2H − 2|τ |2H + |τ − 1|2H
), τ = 0,±1,±2, . . .
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Simulation algorithm
The fGp algorithm can be divided into four steps.
1 Let N be a power of 2 and let M = 2N. For j = 0, 1, . . . ,M/2, we
compute the exact spectral power expected for this autocovariance
function, Sj , from the discrete Fourier transform of the following
sequence of γ : γ0, γ1, . . . , γM/2−1, γM/2.
Sj ≡M/2∑τ=0
γτe−i2πj(τ/M) +M−1∑
τ=M/2+1
γM−τe−i2πj(τ/M).
2 We check whether Sj ≥ 0 for all j . This should be true for the
fractional Brownian motion. Negativity indicates that the sequence is
corrupt.
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Simulation algorithm cont.
3 Let Wk , where k ∈ 0, 1, . . . ,M − 1, be a set of i.i.d. Gaussian
random variables with zero mean and unit variance. Now, we calculate
the randomized spectral amplitudes Vk :
V0 =√
S0W0,
Vk =
√1
2Sk(W2k−1 + iW2k) for 1 ≤ k <
M
2,
VM/2 =√
SM/2WM−1,
Vk = V ∗M−k forM
2< k ≤ M − 1,
where ∗ denotes that Vk and VM−k are complex conjugates.
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Simulation algorithm cont.
4 We compute the simulated time series Yn using first N elements of the
discrete Fourier transform of V :
Yn =1√M
M−1∑k=0
Vke−i2πk(n/M),
where n = 0, 1, . . . ,N − 1.
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FARIMA. Simulation algorithm
Approximation of FARIMA path n = 0, 1, . . . ,N − 1
∞∑j=0
c(j)Zα(n − j) = X (n) ≈ XM(n) :=M−1∑j=0
c(j)Zα(n − j)
R.H.S. has form like finite discrete convolution
So apply convolution theorem for DFT
DM(a)(k)DM(b)(k) = DM(a ∗ b)(k), k ∈ Z,
where
(a ∗ b)(n) :=M−1∑j=0
a(n − j)b(j), n ∈ Z
and a, b–M–periodic
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FARIMA. Simulation algorithm cont.
Define (M + N)–periodic sequences:
Zα(j) : j = 0, 1, . . . ,M + N − 1–generate
Zα(j + k(M + N)) := Zα(j),
j = 0, 1, . . . ,M + N − 1, k ∈ Z
c(j + k(M + N)) = c(j)–compute
c(j) =
c(j), for j = 0, 1, . . . ,M − 1,0, for j = M,M + 1, . . . ,M + N − 1.
Then
X (n) ≈M+N−1∑
j=0
c(j)Zα(n − j), n = 0, 1, . . . ,N − 1
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Table of contents
VIII Self-similar processes and long-range dependence1 Estimating self-similarity, tail, and memory parameters2 BMW2 computer test
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Self-similar processes and long-range dependence
Any H-self-similar process with stationary increments X (t)t∈R induces a
stationary sequence Yjj∈Z, where
Yj = X (j + 1)− X (j); j = . . . ,−1, 0, 1, . . ..
The sequence Yj corresponding to the fractional Brownian motion is called
fractional Gaussian noise (FGN). It is called a standard fractional Gaussian
noise if VarYj = 1 for every j ∈ Z.
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Self-similar processes and long-range dependence cont.
If H = 1/2, then its autocovariance function r(k) = R(0, k) = 0 for k 6= 0
and hence it is the sequence of independent identically distributed (i.i.d.)
Gaussian random variables.
The situation is quite different when H 6= 1/2, namely the Yj ’s are
dependent and the time series has the autocovariance function of the form
r(k) ∼ VarY1 H(2H − 1)k2H−2, as k →∞.
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Long-range dependence
The autocovariance function r(k) tends to 0 as k →∞ for all 0 < H < 1,
but when 1/2 < H < 1 it tends to zero so slowly that the sum∑∞k=−∞ r(k) diverges.
We say that in this case the increment process exhibits long-memory or
”long-range dependence.
Formula implies that the spectral density h(λ) of the stationary process
FGN has a pole at zero. A phenomenon often referred to as ”1/f noise”.
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Long-range dependence cont.
If 0 < H < 1/2, then∑∞
k=−∞ r(k) = 0 and the spectral density tends to
zero as |λ| → 0.
We say in that case that the sequence displays a short-memory.
Furthermore, as the coefficient H(2H − 1) is negative, the r(j)’s are
negative for all large j , a behaviour referred to as ”negative dependence”.
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Self-similarity and long range dependence
Both self-similar and long-range dependent processes are two most
important kinds of random processes that can be used to model scale
invariance observed in diverse fields covering natural phenomena
(biology, physics) and human activity (telecommunications network
traffic, finance).
The increments of any finite variance H-sssi process have long-range
dependence as long as 1/2 < H < 1.
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Time series with heavy-tails and long range dependence
Well known examples of models that display strong dependence in
time (long-range dependence) are fractional Gaussian noise (fGn) and
Gaussian fractional autoregressive moving average (FARIMA) (0, d , 0).
Most prominent examples of models that exhibit both long-range
dependence and large fluctuations (heavy-tailed distributions) are
fractional Levy stable noise (fLsn) and Levy stable FARIMA (0, d , 0).
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Fractional stable noise
The increment process corresponding to the fractional Levy stable
process is called a fractional Levy stable noise. By analogy with the
case α = 2, we say that it has the long-range dependence when
H > 1/α and the negative dependence when H < 1/α.
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Fractional stable noise. Illustration
A sample path of the fractional Levy stable noise for H = 0.6 and
α = 1/1.8.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Fractional stable noise. Illustration cont.
A sample path of the fractional Levy stable noise for H = 0.9 and
α = 1/1.8.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Self-similarity components
There are two components of self-similarity property: memory of the
process and the distribution of the process increments.
The self-similarity index for stable processes reads
H = d + 1/α,
where the parameter d measures the memory of the investigated
process and the parameter α (0 < α ≤ 2) is the stability index of the
process increments distribution.
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Self-similarity components cont.
Examples
Brownian motion has no memory and its increments are Gaussian; so,
d = 0 and α = 2. The self-similarity index reads
H = 1/2.
Fractional Brownian motion has memory and its increments are
Gaussian; so, d 6= 0 and α = 2. The self-similarity index reads
H = d + 1/2 6= 1/2.
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Self-similarity components cont.
Examples
Levy motion has no memory and its increments are α-stable; so, d = 0
and 0 < α < 2. The self-similarity index reads
H = 1/α > 1/2.
Fractional Levy stable motion has memory and its increments are
α-stable; so, d 6= 0 and 0 < α < 2. The self-similarity index reads
H = d + 1/α 6= 1/α.
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Testing self-similarity
There are many methods providing tests of long-range dependence
using various estimators.
It is important to know whether an estimator is estimating H or d .
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Testing self-similarity cont.
Estimators:
The R/S and DFA exponents give information on memory; thus on the
d component of the Hurst index, namely they yield d + 1/2.
The absolute value (AV) method exponent estimates the Hurst index
H, namely it yields H − 1.
Finite impulse response transformation (FIRT) method exponent yields
the Hurst index H.
We can combine these two facts to obtain both the memory component d
and distribution component α.
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Testing self-similarity cont.
In an alternative approach we use the concept of surrogate data and apply
the absolute value (AV) method (HAV = H − 1 = d + 1/α− 1).
If the self-similarity results from the process memory only (e.g.
fractional Brownian motion), then the values of the applied estimator
should change to −1/2 for the surrogate data independently on the
initial values.
If the self-similarity results only from the process’ increments infinite
variance (e.g. Levy stable motion), then the estimator values should
be the same for the original and surrogate data.
The self-similarity resulting from both origins (e.g. fractional Levy
stable motion) should be observed as a partial change in the
estimators values.
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Simulated data
The calculations were performed for two cases: Levy stable motion with H
taking values 0.6, 0.7, 0.8, 0.9 and fractional Levy stable motion for
α = 1.8 and H taking values 0.6, 0.7, 0.8, 0.9.
We simulated 10000 realizations of the processes.
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Simulated data cont.
Levy stable motion
The values of the estimator (circle) should form the line H − 1 (red
dashed line).
The estimator values for the surrogate data (plus sign) obtained from
Levy motion should not change.
0.5 0.60.6 0.7 0.8 0.9 1−0.5
−0.45
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Index of self−similarity
Abs
olut
e va
lue
expo
nent
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Simulated data cont.
Fractional Levy stable motion
The values of the estimator (circle) should form the line H − 1 (red
dashed line).
The estimator values for the surrogate data (plus sign) obtained from
the fractional Levy motion should be equal to 1/1.8− 1 ∼ −0.44.
0.5 0.6 0.7 0.8 0.9 1−0.5
−0.45
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Index of self−similarity
Abs
olut
e va
lue
expo
nent
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Table of contents
IX Modelling the solar flare data with FARIMA processes
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Solar flares. Motivation
In 1989, during a massive mass ejection from the Sun’s surface, a large
part of Canada and the U.S. suffered a power loss for more than nine
hours.
Again, in 1998, in a similar chain of events, the satellite Galaxy 4 was
left nonoperational, causing a wide communication breakup.
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Solar flare data
Strong solar activity intervals (last one: 1999–2003).
X-ray flare data from GOES satellite
(http://www.ngdc.noaa.gov/stp/SOLAR/ftpsolarflares.html).
Energy values aggregated on a daily basis.
00 01 02 030
0.5
1
1.5
2
2.5x 10
−3
Date (years)
E[W
/m2 ]
Wrocław University of Technology
Processes with long-range dependence (long memory).Examples
Increments of the fractional Brownian motion YH(t) (1/2 < H < 1)
Increments of the fractional Levy stable motion YH,α(t)
(H > 1− 1/α, 1 < α ≤ 2)
FARIMA time series with light-tailed innovations (e.g. Gaussian)
(d > 0)
FARIMA time series with heavy-tailed innovations (e.g. Pareto or
stable) (d > 1− 2/α)
Fractional Ornstein-Uhlenbeck process
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Long-range dependence. Gaussian case (α = 2)
Cov(n) = E [X (n)X (0)]− E [X (n)]E [X (0)] ∼ n2d−1.
For d ≥ 0∞∑n=0
|Cov(n)| =∞.
The spectral density (Fourier transform of Cov(n))
f (ω) ∼ c |ω|−2d , as ω → 0.
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Long-range dependence. Stable non-Gaussian case(0 < α < 2)
Codifference τX ,Y of two jointly α-stable random variables X and Y
τX ,Y = ln Ee i(X−Y ) − ln Ee iX − ln Ee−iY .
For d > 1− 2/α
∞∑n=0
|τ(n)| =∞.
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FARIMA(p, d , q) model
Fractional autoregressive integrated moving average (FARIMA)
Φ(B)∆dX (n) = Θ(B)εn, n ∈ Z ,
autoregressive (AR(p)) part
Φ(B) = 1− a1B − a2B2 − · · · − apBp
moving average (MA(q)) part
Θ(B) = 1− b1B − b2B2 − · · · − bqBq
d < 1− 1/α, εj (i.i.d.) random variables.
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FARIMA(p, d , q) model
BX (n) = X (n − 1), ∆X (n) = X (n)− X (n − 1),
∆d = (1− B)d =∞∑j=0
(dj
)(−B)j =
∞∑j=0
πjBj ,
πj =Γ(j − d)
Γ(j + 1)Γ(−d).
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FARIMA(p, d , q) model
Linear solution
X (n) =∞∑j=0
cjεn−j ,
where cj are defined by the equation
Θq(z)(1− z)−d
Φp(z)=∞∑j=0
cjzj , |z | < 1.
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Prediction in the model
Linear predictor based on the finite past Xn, . . . ,X0:
Xn+h =n∑
j=0
ajXn−j ,
aj = −k−1∑t=0
cthj+k−t ,
Θq(z)(1− z)−d
Φp(z)=∞∑j=0
cjzj ,
Φp(z)(1− z)d
Θq(z)=∞∑j=0
hjzj , |z | < 1,
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FARIMA and fractional Levy stable motion.Relationship
FARIMA is asymptotically self-similar with
H = d +1
α
and
N−H[Nt]∑j=1
X (j), t ≥ 0 d=⇒ CY (t), t ≥ 0,
where Y (t), t ≥ 0 is either fractional Brownian or Levy stable motion.
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FARIMA(0, d , 0) model
Set p = q = 0.
The model is described by
∆dX (n) = εn.
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Fractional Langevin equation
Continuous-time analogue of equation describing FARIMA(0, d , 0):
dd
dtdZ (t) = lα(t), t ∈ R
∆d is replaced by fractional derivative operator of the Riemann-Liouville
type dd
dtd,
sequence of i.i.d. variable εt (innovations) is replaced by the α-stable Levy
noise lα(t) = dLα(t)dt .
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CFARIMA model (Magdziarz and Weron (2007))
Stationary solution of the fractional Langevin equation
Z (t) =1
Γ(d)
∫ t
−∞(t − s)d−1dLα(s).
We introduce the perturbation parameter ε > 0 and define the
continuous-time FARIMA process
Z (t) =1
Γ(d)
∫ t
−∞(t − s + ε)d−1dLα(s), t ∈ R.
FARIMA and CFARIMA processes have long-memory for the same range of
parameter d :
d > 1− 2/α.
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Fitting heavy-tail exponent
P(X > x) = 1− F (x) ∼ cx−α, as x →∞
Hill estimator
Max-spectrum estimator
Meerschaert-Scheffler estimator
McCulloch’s estimator
α = 1.25
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Fitting heavy-tail exponent
Figure: The evolution of the tail index α during the last solar cycles 1974-2006(top picture) obtained via the max spectrum estimator, the bottom picture showsWolf numbers in this period.
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Fitting long-range dependence parameters
Finite impulse response transformation (FIRT) −→ H.
Variance of residuals method (VR) −→ H
Rescaled range (R/S) method −→ d
Absolute value method −→ H
Wavelet transform method −→ H
Variance method −→ d
H = d +1
α
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TEST
If the process is fractional Brownian motion, FARIMA or CFARIMA
with Gaussian noise, then the values of the applied estimator should
change to 1/2 for the surrogate data independently on the initial
values.
If the process is Brownian motion or Levy stable motion, then the
estimator values should be the same for the original and surrogate
data.
If the process is fractional Levy stable motion, FARIMA or CFARIMA
with α-stable noise for α < 2, then the values of the estimator should
change to 1/α for the surrogate data.
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Long-range dependence estimators
Table: Values of the FIRT, VR and RS estimators for the original time series andthe shuffled (surrogate) solar flare data.
Data set HFIRT HVR dRS
Original time seriesSolar flares 1.1424 1.0665 0.2408
Surrogate dataSolar flares 0.8452 0.7722 0.0507
d = 0.19
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Mean-squared errors of the estimators
00.03
0.060.09
0.120.15
0.180.2
00.03
0.060.09
0.120.15
0.180.20
0.02
0.04
0.06
a2
a1
MS
E
Figure: Combined mean-squared error of the calculated FIRT, VR and RSestimators for the simulated FARIMA(2, 0.19, 0) time series with respect to theones calculated for the solar flare data for different linear and quadraticcoefficients of the AR(2) part. The parameter α = 1.25. The minimum of theerror for the values: a1 = 0.02 and a2 = 0.03
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Model for the simulations
FARIMA(2,0.19,0) with Pareto innovations with α = 1.25 with AR(2)
coefficients a1 = 0.02 and a2 = 0.03.
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3
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Heavy-tail exponent
0.5
1
1.5
2
α MC
Figure: Values of the calculated α estimators for the simulated FARIMA timeseries. Solid line represents the value of the estimator for the analyzed data.
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FIRT and VR estimators
0.8
1
1.2
1.4
1.6
1.8
2
2.2
HF
IRT
0.8
1
1.2
1.4
1.6
1.8
2
2.2
HV
R
Figure: Values of the FIRT and VR estimators for the simulated FARIMA timeseries.
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FIRT and VR estimators cont.
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
HF
IRT
S
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
HV
RS
Figure: Values of the FIRT and VR estimators for the surrogate data of thesimulated FARIMA time series.
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R/S estimator
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
d RS
Figure: Values of the R/S estimator HRS for the simulated FARIMA time series.
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R/S estimator
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
d RS
S
Figure: Values of the R/S estimator HRS for the surrogate data of the simulatedFARIMA time series.
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Prediction in the model
0 50 100 150 200 250 300 3500
1
2
3
4
5
6x 10
−4
2002 (days)
E[W
/m2 ]
Figure: Solar flare data and 1-day-ahead prediction in the FARIMA(2, 0.19, 0)model. The prediction applies the linear predictor.
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Summary of the model
FARIMA(2,0.19,0) model with
Pareto innovations with α = 1.25
AR(2) coefficients a1 = 0.02 and a2 = 0.03
reconstructs the statistical properties of the solar flare data.
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Procedure
1 To examine that a given empirical time series is generated by an
α-stable noise.
2 To check that the time series has the self-similarity property (Test).
3 To check that the time series satisfies the long-range dependence
inequality: d > 1− 2/α.
4 To build a corresponding CFARIMA model, which leads to the
fractional Langevin equation and to the fractional Fokker-Planck
equation.
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Thanks