ECMI { Mathematics for Industry and Commerce Krzysztof ...prac.im.pwr.wroc.pl/~burnecki/csfrf.pdf · Wrocaw University ofTechnology Table of contents cont. VII Simulation of self-similar

Wrocław University of Technology

ECMI – Mathematics for Industry andCommerce

Krzysztof Burnecki

Computer Simulations for Random

Phenomena


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


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



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



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



VI Pricing of catastrophe bonds1 Pricing model2 Fitting the model3 Dynamics of the prices via Monte Carlo simulations



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


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.


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.


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.


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


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.


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).


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 2: Set X = − 1β log(U).


Inverse transform method. Discrete case

Consider a discrete r.v. with probability function:

P(Xi = xi ) = pi .

Consider the following algorithm.


Step 2: Transform U into X as follows,

X = xj , if

j−1∑i=1

pi ≤ U <

j∑i=1

pi .


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 .


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.


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 .


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 .


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).


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 .


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).


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.


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.


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.


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.


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.



−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).



−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).



−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).



−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.



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.



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

.



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).



−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.


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.


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



−14 −12 −10 −8 −6 −4 −2 0 2 4 6−15

−10

−5

0



Figure: Log-log plot of right tails of samples of length 106, 105, 103 with differentα’s.


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


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.


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.


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


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.


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.


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.


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)


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.


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


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)


Simulation of the NHPP cont.


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.


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

)


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.


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.


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.


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


with intensity λ

Step 3b: set Ti = Ti−1 + E


Simulation of MPP cont.


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.


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.


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.


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


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)


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


Table of contents

III Risk process1 Risk process for different counting processes2 Simulation of risk processes


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 .


Risk process cont.


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.


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 .


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 .


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,



Rt = u + (1 + θ)µΛt −Nt∑i=1

Xi .


Cox process

The premium function is a generalization of the former premium functions:

c(t) = (1 + θ)µ

∫ t

0

Λ(s)ds,



Rt = u + (1 + θ)µ

∫ t

0

Λ(s)ds −Nt∑i=1

Xi .


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.


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.


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.


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


Modelling of the risk process


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.


Fitting claim size distribution

Visual techniques:

Mean excess function

Limited expected value function

Probability gates


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.


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).


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).


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 .



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.



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.


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 .


Limited expected value function cont.

The empirical estimate is defined as follows:

Ln(x) =1

n

∑xj<x

xj +∑xj≥x

x

.



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+).



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.


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.


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.


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.


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


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.


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


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


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.


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.


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)


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.


Visualization of the risk process

Tools:

Trajectories

Ruin probability plot

Density evolution plot

Quantile lines

Probability gates


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.


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.


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.


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.


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


Density evolution cont.

Figure: 2-dimensional projection of the density evolution.


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.


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.


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.


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.


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


Basic aspects of actuarial risk theory

classical risk process

ruin probability in finite and infinite time horizon

light- and heavy-tailed distributions

adjustment coefficient


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.


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) <∞).


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).


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).


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) =∞.


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)


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


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 .


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.


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.


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.


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)


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


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 .


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,


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.


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.


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.


Table of contents

VI Pricing of catastrophe bonds1 Pricing model2 Fitting the model3 Dynamics of the prices via Monte Carlo simulations


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


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.


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.


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).


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.


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 .


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 ] .


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

].


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

].


Calibration of the pricing model

Loss distributions

exponential, lognormal, generalized Pareto, Burr, gamma, Weibull,log-αstable

Counting processes

homogeneous and non-homogeneous Poisson


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.


Calibration of the pricing model cont.

Figure: Graph of the PCS catastrophe loss data, 1990-1999.



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



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))


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.


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.


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.


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.


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.


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.


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.


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.


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


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 .


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.


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 .


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

)


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.


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)


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.


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.


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]


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, . . .


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 .


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, . . .


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.


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.


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.


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


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


Table of contents

VIII Self-similar processes and long-range dependence1 Estimating self-similarity, tail, and memory parameters2 BMW2 computer test


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.


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 →∞.


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”.


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”.


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.


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).


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.


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


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


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.


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.


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.


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 .


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.


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.


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.


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


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


Table of contents

IX Modelling the solar flare data with FARIMA processes


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.


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 ]


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


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.


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)| =∞.


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.



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).



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.


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,


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.


FARIMA(0, d , 0) model

Set p = q = 0.

The model is described by

∆dX (n) = εn.


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 .


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.


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


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.


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


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.


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


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


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


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.


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.


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.


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.


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.


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.


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.


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.


Thanks

Documents

ECMI { Mathematics for Industry and Commerce Krzysztof ...prac.im.pwr.wroc.pl/~burnecki/csfrf.pdf · Wrocaw University ofTechnology Table of contents cont. VII Simulation of self-similar