Motivations The population process Population with two timescales Reduction of complexity in an heterogeneous population with two timescales Colloque Jeunes Probabilistes et Statisticiens, avril 2016 Sarah Kaakai, joint work Nicole El Karoui LPMA, Universit´ e Paris 6 18 avril 2016 1/20
Motivations The population process Population with two timescales Plan 1 Motivations 2 The population process 3 Population with two timescales 2/20
Motivations The population process Population with two timescales Motivations Initial motivation: model and better understand human longevity. ◮ Longevity data are often aggregated data such as national data. ◮ Issue: strong heterogeneity in populations. Ex: Expected lifetime at old ages for individuals with high school diploma is 20% higher than for individuals with no diploma (INSEE). ◮ Social structure of populations is not stable and can change at fast pace.( Evolution of French diploma repartition for 30-45 years old) 3/20
Motivations The population process Population with two timescales Motivations More ”micro” description of the population needed to understand aggregated variables. ◮ Aggregation methods are important in several fields (economics, biology, operations research..) ◮ Goal : model complex link between the micro and macro. ◮ Separation of timescales between different types of phenomenons: permits to make some averaging approximations and thus to better understand the aggregation process. Our framework: Study of a stochastic model of population when ”social changes” occur at a fast pace in comparison to the demographic time scale of a life time. 4/20
Motivations The population process Population with two timescales Plan 1 Motivations 2 The population process Setup Representation of the population process as a sum of counting processes Markov framework 3 Population with two timescales 5/20
Motivations The population process Population with two timescales Population process ◮ We consider a population divided in p homogeneous subgroups: individuals in the same subgroup are indistinguishable and have the same ”demographic” behavior. ◮ Population process: describes the evolution of the number of ag process Z = ( Z 1 , .., Z p ) ∈ N p , individuals in each subgroup. c` adl` adapted to a given filtration ( F t ) t ≥ 0 where Z i t is the number of individual in subgroup i at time t . Z = < Z , 1 > = � Z i . ◮ Total size of the the population: ¯ 6/20
Motivations The population process Population with two timescales Assumption and remark ◮ Two kinds of events occur: Demographic events: birth or death in one of the subgroups. ∆ ¯ Z t = ± 1, ∆ Z i t = ± 1. Mixing events: An individual changes of characteristics and moves from a subgroup i to j . ∆ ¯ Z t = 0, ∆ Z j t = 1,∆ Z i t = − 1. ◮ Assumption: No events happen simultaneously. ◮ ( Z i t ) only increase/decrease of one when an event happens in the subpopulation i . But Z is not a multivariate birth and death process since components have simultaneous jumps when mixing events happen. 7/20
Motivations The population process Population with two timescales Notations for the set of all possible events ◮ Canonical basis in R p : ( e i ) 1 ≤ i ≤ p . e ∞ = 0 . ◮ Mixing events: natural notation ( i , j ) for mixing event from i to j . Set of all mixing events: I mix = { ( i , j ) ∈ � 1 , p � 2 ; i � = j } . ◮ Demographic events: we define the ”metaphysical” infinite population {∞} of individuals not born yet and who died. Notation ( ∞ , i ) (( i , ∞ )) for the event birth (death) in subgroup i . Set of demographic event: I dem = ( � 1 , p � × {∞} ) ∪ ( {∞} × � 1 , p � ). ◮ Set of all events: I = I dem ∪ I mix . ◮ Jump associated to event of type ( i , j ): φ ( i , j ) = e j − e i . { ( i , j ) happens at time t } = { ∆ Z t = φ ( i , j ) } . 8/20
Motivations The population process Population with two timescales Generating counting processes Definition For all ( i , j ) ∈ I , we denote by N = ( N ij ) ( i , j ) ∈I the a multivariate counting process with: N ij � t = t ≥ 0 , (1) 1 { ∆ Z s = φ ( i , j ) } s ≤ t Counting processes easily characterized by their intensity. Gives martingale property and makes computations very simple. Markov framework Assumption (Intensity of the counting processes) For all ( i , j ) ∈ I the counting process N ij has the {F t } -intensity q ij ( Z s ) : � t M ij t = N ij 0 q ij ( Z s ) ds is a {F t } -local martingale. t − We assume that there are no explosions of the processes. N is a strong Markov process. 9/20
Motivations The population process Population with two timescales Link between the counting processes and the population process Functionals of population process can be written as sum of their jumps: � f ( Z t ) = f ( Z 0 ) + ( f ( Z s − + ∆ Z s ) − f ( Z s − )) 1 { ∆ Z s � =0 } s ≤ t 10/20
Motivations The population process Population with two timescales Link between the counting processes and the population process Functionals of population process can be written as sum of their jumps: � f ( Z t ) = f ( Z 0 ) + ( f ( Z s − + ∆ Z s ) − f ( Z s − )) 1 { ∆ Z s � =0 } s ≤ t � t � ( f ( Z s − + φ ( i , j )) − f ( Z s − )) dN ij f ( Z t ) = f ( Z 0 ) + s 0 ( i , j ) ∈I 10/20
Motivations The population process Population with two timescales Link between the counting processes and the population process Functionals of population process can be written as sum of their jumps: � f ( Z t ) = f ( Z 0 ) + ( f ( Z s − + ∆ Z s ) − f ( Z s − )) 1 { ∆ Z s � =0 } s ≤ t � t � ( f ( Z s − + φ ( i , j )) − f ( Z s − )) dN ij f ( Z t ) = f ( Z 0 ) + s 0 ( i , j ) ∈I Martingale problem � t K mix + K dem � � f ( Z t ) − f ( Z 0 ) − f ( Z s ) ds is a martingale, with 0 � K mix ( dem ) f ( z ) = q ij ( z ) ( f ( z + φ ( i , j )) − f ( z )) i , j ∈I mix ( dem ) Z is a continuous time Markov chain. 10/20
Motivations The population process Population with two timescales Plan 1 Motivations 2 The population process 3 Population with two timescales Mixing excursions Average aggregated framework 11/20
Motivations The population process Population with two timescales Instantaneous mixing with respect to demographic timescale Separation of timescale ◮ Let’s define the nth demographic event T d n : n − 1 ; ∆ ¯ T d n = inf { t > T d Z t � = 0 } ( T 0 = 0). ◮ Hypothesis: mixing intensities >> demographic intensities. T d T d T d 0 1 2 3 t Idea: ◮ Decouple mixing and demographic events by isolating ”mixing excursions” which evolve on finite spaces between two demographic events. ◮ Process reinterpreted as a Markov chain of killed trajectories, reborn after a demographic event. 12/20
Motivations The population process Population with two timescales Mixing excursions T d T d T d 0 1 2 3 t ◮ Time between two demographic events: D n = T d n +1 − T d n ◮ nth mixing excursion of the population process ; process after n th demographic event and killed at the ( n + 1) th : � if t < D n Z t + T d E n t = n (3) ∂ if t ≥ D n . Population process is equivalent to the sequence of Mixing excursions ( E n ) n ∈ N : � n ≤ t } E n Z t = 1 { T d ∀ t ≥ 0 . (4) t − T d n n ≥ 0 13/20
Motivations The population process Population with two timescales Population before the first demographic event Population killed at the first demographic event: ◮ Only mixing events happen on { t < T d 1 } � t � ( f ( Z s − + φ ( i , j )) − f ( Z s − )) dN ij f ( Z t ) = f ( Z 0 ) + s . 0 ( i , j ) ∈I mix N dem = 1 is the first jump time of ¯ ◮ T d ( i , j ) ∈I dem N ij . � ◮ Happens with intensity q dem ( Z t ) 1 { t < T d 1 } with p q dem ( · ) = q ∞ i ( · ) + q i ∞ ( · ). � i =1 Mixing excursions are ”pure mixing” processes, killed at a rate , of generator: K ∂ f ( z ) = K mix f ( z ) − q dem ( z ) f ( z ) . 14/20
Motivations The population process Population with two timescales Killed pure mixing process Pure mixing process ◮ X = ( X x ) x ∈ N p processes with semigroup P mix generated by K mix . ◮ Conservative process: for all initial population z ∈ N p , X z has value in the finite space U ¯ z of populations of size ¯ z = < z , 1 > : z = { x ∈ N p ¯ U ¯ z = ¯ x } . Subordinated process ◮ The semigroup P ∂ is subordinated to the pure mixing process,i.e: ∀ f : N p → R + , s ≥ 0. P ∂ s f ≤ P mix f s ◮ P ∂ can be realized by the expectation of a pure mixing process discounted by a multiplicative functional of the process. Here: � s 0 q dem ( X z s ) e − u ) du ] . P ∂ s f ( z ) = E X [ f ( X z (5) 15/20
Motivations The population process Population with two timescales Fast mixing process Hypothesis: mixing intensities >> demographic intensities. ◮ The population depends on a small parameter ǫ . The two timescale population process Z ǫ has generator 1 ǫ K mix + K dem . ◮ Killed ”fast” mixing representation: � s P ∂,ǫ f ( z ) = E X ǫ [ f ( X z ,ǫ 0 q dem ( X z ,ǫ ) exp( − u ) du )] s s with X ǫ some ”fast” pure mixing process of generator 1 ǫ K mix . 16/20
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