Point Processes, and Populations Dynamics [2mm] in ...

Point Processes, and Populations Dynamics

in longevity, credit, HFT modelling

Nicole El Karoui, Alexandre Boumezoued (PhD)

UPMC(Paris VI), LPMAProbability and Random Models Laboratory, UMR-CNRS 7599

Santa Barbara, Septembre 2014

1/52 Nicole El Karoui, In honor of J.P.Fouque Santa Barbara 2014

Plan

1 Some Dates with Jean Pierre

2 Motivation to model global population

3 First look at longevity

4 Role of age

5 Macroscopic approximation


Do you know AMART?

You, probably no, but

Jean Pierre, Yes !, and me also, in principle...as his supervisor


A definition, to help you


Few years later, The tenure


From Amarts to Finance, a long way

• In France, Opening of the First financial Market in 1988

• The first Master program in Mathematical Finance is created

jointly PVI -Ecole Polytechnique in 1990.


Plan




4 Role of age



Why population dynamics ?

Marked Point Process: Renew of interest

Useful tool, under different denominations, for other domains:

I Credit risk Modelling

I Hawkes processes in HFT

I Particular methods in Simulation

I Imaging

I Brain study

I Data Mining

Population dynamics Old problems, Malthus 1798, Verhulst 1838

I First motivation, longevity risk in insurance and finance (

pension fund)

I General Model for population dynamics in Ecology

I By introducing complexity and heterogeite in the population

I The age parameter plays strategic role

I Useful for

Aim: By mixing these different points of view, and starting of

micro evolution at the individual level and demographic patterns in

birth and death to explain macro evolution of the general structure.


A lot of different denominations

I Agent Based Models (ABM) in Computational Economy

I Microsimulation models of National Agencies (Destinee in

France)

I Demography computational

I Age Period Cohort Analysis

I Interacting particle methods for simulation

Our reference: Individual-Based models in ecology:

I Population structured by traits (i.e. individual characteristics)

(Fournier-Meleard 2004, Champagnat-Ferrire-Mlard 2006)

I Extension to age-structured populations (Tran 2006,

Ferrire-Tran 2009)


Plan




4 Role of age



First look at Longevity

Developed countries are experiencing a reduction in mortality over

time

I Aging populations: new phenomenon, without past historical

reference

I Societies are facing new different challenges:

new generational equilibrium

role and place of aging population in the society

viability of shared collective systems, in particular (state or

private) pension systems


Aggregate mortality indicators

I Lifetime of an individual: τ

I Life expectancy at birth: E[τ ]

I Death rate d(a) such that P(τ > a) = e−∫ a

0 d(s)s.

I Possible quantity of interest: annual death probability

q(a) = P(τ < a + 1 | τ ≥ a)

I In practice, reduction of mortality over time:

⇒ q(a, t)=probability for an individual aged a at the

beginning of year t to die in the following year.


Mortality tables in Insurance

I q(a, t)= probability for an individual aged a at the beginning

of year t to die in the following year.

I Lexis diagram ”age - period - cohort” :

can be taken into consideration, it proves di�cult to account for censured and truncated data when

assessing national mortality. One may think for instance of migration. There are two di↵erent factors

that can be overlooked when considering migratory impacts. First of all, people leaving the synthetic

cohort are often censored, and thus still taken into account when estimating the period life tables.

Secondly, incoming immigrants can also a↵ect and significantly change the local national mortality

data of the destination country.

Far more relevant details regarding deaths can be gained from within an insured portfolio (e.g. the

cause of death). Although national data by cause of death does exist, it generally lacks consistency,

and is often useless, or of no specific interest in deriving mortality by cause (for example: in the

United States, 11% of deaths are caused by more than four diseases).

Coh

ort

Year

Age

66

65

64

63

200819431942

1

Figure 1.1: Lexis diagram: Age-year-cohort diagram.

1.2. Heterogeneity, inter-age dependence and basis risk

It is expected that any given population will display some degree of heterogeneous mortality.

Heterogeneity often arises due to a number of observable factors, which include age, gender, occupation

and physiological factors. As far as longevity risk is concerned, policyholders that are of higher socio-

economic status (assessed by occupation, income or education) tend to experience lower levels of

mortality. However, significant di↵erences also exist within the same socio-economic levels according

to gender. Females tend to outlive males and have lower mortality rates at all ages. In addition, some

heterogeneity arises due to features of the living environment, such as: climate, pollution, nutritional

standards, population density and sanitation (see Section 2.2 for a more detailed discussion).

When considering insured portfolios, insurers tend to impose selective criteria that limits contractual

access to those considered to possess no explosive risk (by level of health and medical history). This

4


Age Pyramid in France

Initial population for N=10 000 in 2008

Age pyramid in 2008 Age pyramid in 2008

100 50 0 50 100

07

1524

3342

5160

6978

8796

107

119

Number of males Number of females

Age

Source: The French National Institute for Statistics and Economic Studies (INSEE


Human Mortality Database

Human Mortality Database. University of California, Berkeley

(USA), and Max Planck Institute for Demographic Research

(Germany).

Available at www.mortality.org or www.humanmortality.de


Mortality structure (France)

0

20

40

6080 1900

1950

2000

0.00

0.05

0.10

0.15

0.20

probabilités de décès (FR)


National mortality by gender (France)

0

20

40

6080 1900

1950

2000

0.00

0.05

0.10

0.15

0.20

probabilités de décès (femmes, FR)

0

20

40

6080 1900

1950

2000

0.00

0.05

0.10

0.15

0.20

probabilités de décès (hommes, FR)

Femmes Hommes


National mortality: log q(a,t)

I Looking at log q(a, t) age a in [0,100]

I for different years t (1950,1965,1980,1995,2005)

0 20 40 60 80 100

-10

-8-6

-4-2

0

age a

log(

q(. ,

t,))

19501965198019952005

Figure: Logarithm of annual death probabilities (national population)18/52 Nicole El Karoui, In honor of J.P.Fouque Santa Barbara 2014

Age pyramid 2009-2019

Age pyramid 2009-2019

Aims of microscopic models

I Provide population evolution at the scale of the individual

I Allows to understand patterns of aggregate indicators

Two examples in this talk

1 Impact of aging

2 How individual birth patterns in heterogenous populations can

create artificial mortality changes (”cohort effect”) ?


Individual-based model

Human population extension: Bensusan 2010 (Phd thesis),

Boumezoued & El Karoui (working paper)

I Possibility for an individual to evolve during life (marriage,

divorce, professional evolution,...)

I The population is subject to a stochastic environment

(Yt)t≥0, evolving through time

This environment refers to random changes increasing or

decreasing mortality

Natural catastrophes, pandemics, wars, emerging diseases,...

Medical environment


Individual-based model

Demographic rates: an individual of traits xt ∈ X ⊂ Rd and age

at ∈ [0, a] at time t, (born at time 0)

I Dies at rate d (xt , at , t,Y )

P(Tdeath ≥ t | Y ) = exp(−∫ t

0d(xs , as , s,Y ) ds

)I Gives birth at rate b (xt , at , t,Y )

and the new individual has traits x ′ ∼ Kb(xt , at , x.′)

I Evolves during life at rate e (xt , at , t,Y )

from traits xt to x ′ ∼ K e(xt , at , dx ′)

I Demographic rates depend on characteristics, age, time and

on the stochastic environment Y

I Conditionally on the environment Y , the events for a given

individual are jumps of a counting process


Non homogenous process simulation

Simulation by Thinning of a counting process with rate λt ,

assuming that λt ≤ λ

I Let (Tn) the jump times of a Poisson process Nt with rate λ:

Tn+1 − Tn ∼ Exp(λ)

I Recursively, at time Tn pick a Bernouilli independent r.v. Un

s.t. P(Un = 1) = λTn/λ.

I Then Nt := cardinal{k : Uk = 1, Tk ≤ t} is a counting

process with rate λt

I (Tn) are interpreted as inspection times for the system

I The thinning method makes easier the simulation of counting

processes with complex rates


Simulation algorithm

I Assumption of bounded demographic rates with b, d and e

⇒ use of the Thinning method (∼ Inspection times)

1 Start with N indiv. at T , generate τ ∼ Exp(N(d + b + e)

)The bigger the population, the more it is inspected

2 Select an individual (x I , aI ) uniformly and compute:

p1 = b(x I ,aI ,T+τ,Y )

b+d+e, p2 = d(x I ,aI ,T+τ,Y )

b+d+e, p3 = e(x I ,aI ,T+τ,Y )

b+d+e

Only one individual is checked (not exhaustive)

3 Determine the nature of inspection at time T + τ

I Birth: add a new individual with probability p1

I Death: remove (x I , aI ) with probability p2

I Evolution: change traits of (x I , aI ) with probability p3

I No event with probability 1− p1 − p2 − p3


Measure-valued stochastic process

I At time t, the population is described by the following point

measure on X × [0, a]:

Zt(dx , da) =∑Nt

i=1 δ(x it ,ait)

(dx , da).

A way to describe the boxes, and their evolution

I Computation of various quantities of interest is made by

testing some functions (x , a) 7→ ft(x , a):

〈Zt , ft〉 =∫X×[0,a] ft(x , a)Zt(dx , da) =

∑Nti=1 ft(x i

t , ait).

I Example: population size Nt = 〈Zt , 1〉

I Role of the composition of the initial population at time 0


Plan




4 Role of age



Role of age: a simple example from Alex.B.

Pure birth process with age structure

I each individual has the same birth rate b(a) (d=e=0)

I λt =∑

i∈Zt−b(ai ) = 〈Zt−, b〉

I recursive evolution of the box : individuals in Zt− are all

individual born up to t−

Self-exciting process (Nt) = (〈Zt , 1〉) is a counting process with

intensity

λt = N0b(t) +

∫(0,t)

b(t − s) dNs = N0b(t) +∑Ti<t

b(t − Ti ),

where the (Ti ) are the times of jump of N.

I This leads to an equation on Nt : how to construct it ?


Two examples

Birth rates Hawkes: b(a) = exp−a, Human b(a) = exp−(a− c)2/2


Representation of pure birth process:

Hawkes process

I Let Q(ds, dθ) be a Poisson point measure with intensity

measure ds dθ on R+ × R+

I need the whole population Zt(da) =∑Nt

i=1 δAi (Zt) counting

the birth but also taking track of the ages distribution.

I Representation of the birth process by Thinning:

Zt(da) =

N0∑i=1

δt(da) +

∫ t

0

∫R+

1θ≤〈Zs−,b〉δt−s(da)Q(ds, dθ)

I Note that Zt(da) is Markov but Nt = 〈Zt , 1〉 is not.


Number of events

Hawkes (left) Human (right)


Population size


Age pyramid


Cohort effect

Birth cohort for the period [t1, t2]: group of individuals born

between t1 and t2.

I Individuals of the same birth cohort share similar demographic

characteristics (”cohort effect”)

I Age, Period, Cohort analysis put a lot of problems in practice,

in different domains, medecine, sociology,...due to the lag in

data,..insurance...

I Huge literature on APC problems


Golden cohort

Golden cohort: generations born between 1925 and 1945Cairns et al. (2009) ra,t = (qa,t−1 − qa,t)/qa,t

The Golden cohort has experienced more rapid improvements than

earlier and later generations.

2 DATA 7

1970 1980 1990 2000

2040

6080

Year

Age

−2%

−1%

0%

1%

2%

3%

4%

Annu

al im

prov

emen

t rat

e (%

)

Figure 3: Improvement rates in mortality for England & Wales by calendar yearand age relative to mortality rates at the same age in the previous year. Red cellsimply that mortality is deteriorating; green small rates of improvement, and blueand white strong rates of improvement. The black diagonal line follows the progressof the 1930 cohort.

2.1.2 The cohort eÆect

Some of the models we employ incorporate what is commonly called the “cohorteÆect”. The rationale for its incorporation lies in an analysis of the rates at whichmortality has been improving at diÆerent ages and in diÆerent years. Rates ofimprovement are plotted in Figure 3 (see, also Willets, 2004, and Richards et al.,2006). A black and white version of this graph can be found in the Appendix, Figure38.

In line with previous authors (see, for example, Willets, 2004, Richards et al., 2006)we can note the following points. In certain sections of the plot, we can detectstrong diagonals of similar colours. Most obviously, cohorts born around 1930 havestrong rates of improvement between ages 40 and 70 relative to, say, cohorts born10 years earlier or 10 years later. The cohort born around 1950 seems to have worsemortality than the immediately preceeding cohorts.

There are other ways to illustrate the cohort eÆect and these can be found in Ap-pendix A.


Analysis of R. C. Willets, 2004

Some possible explanations:

I Impact of World War II on previous generations,

I Changes on smoking prevalence: tobacco consumption in next

generations,

I Impact of diet in early life,

I Post World War II welfare state,

I Patterns of birth rates


Cohort effect and Fertility


Cohort effect (UK)

”One possible consequence of rapidly changing birth rates is that

the average child is likely to be different in periods where birth

rates are very different. For instance, if trends in fertility vary by

socio-economic class, the class mix of a population will change.”

The Cohort Effect: Insights And Explanations, 2004, R. C. Willets


Simple toy model

I Reference death rate d(a) = A exp(Ba)

I Parameters A ∼ 0.0004 and B ∼ 0.073 estimated on French

national data for year 1925 to capture a proper order of

magnitude

I ”Upper class”: time independent death rate d1(a) = d(a) and

birth rate b1(a) = c1[20,40](a) (c=0.1)

I ”Lower class”: time independent death rate d2(a) = 2d(a)

but birth rate

b2(a, t) = 4c1[20,40](a)1[0,t1]∪[t3,∞)(t) + 2c1[20,40](a)1[t2,t3](t)

Constant death rates but reduction in overall fertility between

times t1 (=10) and t2 (=20)

I Aim: compute standard demographic indicators


Aggregate fertility

I One trajectory with 20000 individuals (randomly) splitted

between groups. Estimation of aggregate fertility

0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

year

num

ber o

f chi

ldre

n pe

r ind

ivid

ual a

ged

[20,

40]


Life expectancy by year of birth

I ”Cohort effect” for aggregate life expectancy

0 5 10 15 20 25 30

5556

5758

5960

year of birth

life

expe

ctan

cy (a

t birt

h)


0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

year

num

ber o

f chi

ldre

n pe

r ind

ivid

ual a

ged

[20,

40]

0 5 10 15 20 25 30

5556

5758

5960

year of birth

life

expe

ctan

cy (a

t birt

h)

Figure: Observed fertility (left) and estimated life expectancy by year of

birth (right)

I Death rates by specific group remain the same

I But reduction in fertility for ”lower class” during 10-20

modifies the generations composition

⇒ ”upper class” is more represented among those born

between 10 and 20

Wealth and longevity

The role of heterogenity

I by birth or emigration

I by evolution (swap-mutation) often faster than the

demographic events

I by the environment

Identification problem

I In spatial Birth Death Point process (Huber 2010), swapping

is used to accelerate the convergence, and shortening the

mixing time of the chain.


Wealth and longevity: complex dependence


Plan




4 Role of age



Stochastic Equations


Martingale problem

(No evolution for simplicity of notations)

Conditionally on the complete path of Y , in conditionally

independency framewor

Mt(f ) =< Zt , ft > − < Z0, f0 > −∫ t

0

ds

∫Zs(dx , da)

[(∂fs∂a

+∂fs∂s

)(x , a)

+b(x , a, s,Y )

∫χ

fs(x ′, 0)K b(x , a, dx ′)− d(x , a, s,Y )fs(x , a)]

(1)

is a square integrable martingale with quadratic variation

< M(f ) >t=

∫ t

0

ds

∫χ×[0,a]

Zs(dx , da)×

[b(x , a, s,Y )

∫χ

f 2s (x ′, 0)K b(x , a, dx ′) + d(x , a, s,Y )f 2

s (x , a)]

(2)(3)


Macroscopic approximation

Assumptions

I Renormalization: pop. described by the measure

Znt (dx , da) = 1

n

∑Nnt

i=1 δ(x it ,ait)

(each individual has weight 1/n)

I Weak convergence of the initial population as n→ +∞:

Zn0 (dx , da)⇒ g0(x , a)γ(dx)da (initial size is of order n)

I By homogeneity, the quadratic variation of Mn(f ) is of order1n and so goes to 0

I Cv in distribution on the canonical space of cadlag measure

valued process of the process (Znt ))


Conditionnaly to Y , limit PDE

I Law of Large Numbers (Heuristic): the noise vanishes as the

size n of the initial population goes to infinity

⇒ deterministic behavior in time for large populations

Limit PDE

I Limit process as the size → +∞I Weak convergence of (Zn

t (dx , da))t≥0 to the solution(gt(x , a)γ(dx , da

)t≥0

of conditional (wrt Y) deterministic

PDE

Link between two description in a given environment:

I microscopic: stochastic behavior of each individual

I macroscopic: deterministic evolution of the whole population


Deterministic equations in demography

I Malthus (1798), Verhulst (1838): pop. structured by traits

g(x , t): density of individuals of trait x at time t

∂g

∂t(x , t) =

∫χ

g(y , t)b(y)kb(y , x)dy − d(x)g(x , t),

g(x , 0) = g0(x).

I McKendrick (1926), VonFoerster (1959): structured by age

g(a, t): density of individuals of age a at time t

∂g

∂t(a, t) +

∂g

∂a(a, t)︸︷︷︸

transport

= −d(a)g(a, t), g(0, t) =

∫ +∞

0b(a)g(a, t)da︸︷︷︸

renewal

g(a, 0) = g0(a).


Large population limit

PDE for the population density g(x , a, t): approximation (a.s) for

large populations with

I stochastic environment Y

I evolution during life(∂g

∂t+∂g

∂a

)(x , a, t) = −d

(x , a, t,Y

)g(x , a, t)

−e(x , a, t,Y

)g(x , a, t) +

∫χ

e(x′, a, t,Y )ke(x

′, a, x)g(x

′, a, t)γ(dx

′)

g(x , 0, t) =

∫X×[0,a]

b(x

′, a, t,Y

)kb(x

′, a, x)g(x

′, a, t)γ(dx

′)da

g(x , a, 0) = g0(x , a)

I Take advantage of the impact of pure environment noise


Conclusion and Perspectives

I Empirical aggregate mortality is the result of complex

demographic mechanisms

I Intrinsic mortality and fertility (+evolution) for different

characteristics leads to a dynamic heterogeneity: the

composition of the population changes over time

In progress or further research:

I Test many assumptions on demographic patterns

I Calibration and estimation of the intensities, and identifcation

isssues

I Improve numerical efficiency, based on competences of the

other actors in these aera

Conclusion: Tool for hypotheses testing for a better understanding

of demographic patterns


A benevolent presence

Sau, Gold of health and longevity wish you

Un Joyeux Anniversaire


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