A sequential particle algorithm that keeps the particle system alive

(1)

HAL Id: inria-00001090

https://hal.inria.fr/inria-00001090

Submitted on 3 Feb 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

system alive

François Le Gland, Nadia Oudjane

To cite this version:

François Le Gland, Nadia Oudjane. A sequential particle algorithm that keeps the particle system alive. [Research Report] PI 1783, 2006, pp.35. �inria-00001090�

(2)

I R I ^S

A

I^N STIT^U

T^D E R

E^C HERC^H

E EN IN^F ORM^A

TIQUEE^T SY^ST

ÈME^S ALÉA^T

O^IRES

P U B L I C A T I O N I N T E R N E

N^o

I R I S A

1783

A SEQUENTIAL PARTICLE ALGORITHM THAT KEEPS THE PARTICLE SYSTEM ALIVE

FRANÇOIS LE GLAND, NADIA OUDJANE

(3)

(4)

http://www.irisa.fr

A Sequential Particle Algorithm that Keeps the Particle System Alive

Fran¸cois Le Gland* , Nadia Oudjane** ***

Syst`emes num´eriques Projet ASPI

Publication interne n˚1783 — F´evrier 2006 — 35 pages

Abstract: A sequential particle algorithm proposed by Oudjane (2000) is studied here, which uses an adaptive random number of particles at each generation and guarantees that the particle system never dies out. This algorithm is especially useful for approximating a nonlinear (normalized) Feynman–Kac flow, in the special case where the selection functions can take the zero value, e.g. in the simulation of a rare event using an importance splitting approach. Among other results, a central limit theorem is proved by induction, based on the result of R´enyi (1957) for sums of a random number of independent random variables. An alternate proof is also given, based on an original central limit theorem for triangular arrays of martingale increments spread across generations with different random sizes.

Key-words: Feynman–Kac flow, nonnegative selection function, binary selection function, interacting particle system, extinction time, random number of particles, central limit theorem, sequential analysis, stopping time, Anscombe theorem, triangular array of martingale increments.

(R´esum´e : tsvp)

This work was partially supported by the European Commission, under the IST project Distributed Control and Stochastic Analysis of Hybrid Systems Supporting Safety Critical Real-Time Systems Design(HYBRIDGE), project number IST–2001–32460, and by CNRS, under theMathSTIC projectChaˆınes de Markov Cach´ees et Filtrage Particulaire, and under theAS–STICproject M´ethodes Particulaires(AS 67).

*IRISA / INRIA, Campus de Beaulieu, 35042 RENNES C´edex, France. —legland@irisa.fr

**EDF, Division R&D, 92141 CLAMART C´edex, France. —nadia.oudjane@edf.fr

***Département de Mathématiques, Université Paris XIII, 93430 VILLETANEUSE, France. —n oudjane@math.univ-paris13.fr

Centre National de la Recherche Scientifique Institut National de Recherche en Informatique

(5)

garantissant la survie du syst`eme de particules

Résumé : Nous étudions un algorithme particulaire séquentiel proposé par Oudjane (2000), qui utilise un nombre aléatoire de particules à chaque génération et qui garantit que le système de particules ne s’éteint jamais. Cet algorithme est spécialement utile pour approcher un flot non–linéaire (normalisé) de Feynman–Kac, dans le cas particulier où les fonctions de sélection peuvent prendre la valeur zéro, e.g. dans la simulation d’un

évènement rare parimportance splitting. Nous prouvons par récurrence un théorème central limite, reposant sur le résultat de Rényi (1957) pour la somme d’un nombre aléatoire de variables aléatoires indépendantes. Nous donnons aussi une autre preuve, reposant sur un théorème central limite original pour un tableau triangulaire d’accroissements de martingales répartis sur des générations de tailles aléatoires différentes.

Mots clés : flot de Feynman–Kac, fonction de sélection positive ou nulle, fonction de sélection binaire, système de particules en interaction, temps d’extinction, nombre aléatoire de particules, théorème central limite, analyse séquentielle, temps d’arrêt, théorème d’Anscombe, tableau triangulaire d’accroissements de martingales.

(6)

1 Introduction

The problem considered here is the particle approximation of the linear (unnormalized) flow and of the associated nonlinear (normalized) flow, defined by

hγn, φi=E[φ(Xn) Yn k=0

gk(Xk) ] and hµn, φi= hγn, φi hγn,1i ,

for any bounded measurable functionφ, where{Xk, k= 0,1,· · · , n}is a Markov chain with initial probability distribution η0 and transition kernels{Qk, k = 1,· · ·, n}, and where {gk, k= 0,1,· · · , n}are given bounded measurable nonnegative functions, known as selection or fitness functions. This problem has been widely studied [3] and the focus here is on the special case where the selection functions can possibly take the zero value. Clearly

γk =gk(γk−1Qk) =gk(µk−1Qk)hγk−1,1i and γ0=g0η0 , (1) or equivalently γk = γk−1Rk where the nonnegative (unnormalized) kernel Rk is defined by Rk(x, dx⁰) = Qk(x, dx⁰)gk(x⁰), hence

hγk,1i=hµk−1Qk, gki hγk−1,1i and hγ0,1i=hη0, g0i, (2) and the minimal assumption made throughout this paper is that hγn,1i>0, or equivalently thathη0, g0i>0 andhµk−1Qk, gki>0 for anyk= 1,· · · , n, otherwise the problem is not well defined. There are many practical situations where the selection functions can possibly take the zero value

• simulation of a rare event using animportance splitting approach [3, Section 12.2], [1, 6],

• simulation of a Markov chain conditionned or constrained to visit a given sequence of subspaces of the state space (this includes tracking a mobile in the presence of obstacles : when the mobile is hidden behind an obstacle, occlusion occurs and no observation is available at all, however this information can still be used, with a selection function equal to the indicator function of the region hidden by the obstacle),

• simulation of a r.v. in the tail of a given probability distribution,

• nonlinear filtering with bounded observation noise,

• implementation of a robustification approach in nonlinear filtering, using truncation of the likelihood function [10, 15],

• algorithms of approximate nonlinear filtering, where hidden state and observation are simulated jointly, and where the simulated observation is validated against the actual observation [4, 5, 17, 18], e.g. when there is no explicit expression available for the likelihood function, or when a likelihood function does not even exist (nonadditive observation noise, noise–free observations, etc.).

This work has been announced in [12], and it is organized as follows. In Section 2, the (usual) nonsequential particle algorithm is presented, and the potential difficulty that arises if the selection functions can possibly take the zero value, i.e. the possible extinction of the particle system, is addressed. RefinedL¹ error estimates are stated in Theorem 2.2, for the purpose of comparison with the sequential particle algorithm, and the central limit theorem proved in [3, Section 9.4] is recalled. In Section 3, the sequential particle algorithm already proposed in [14, 11] is introduced, which uses an adaptive random number of particles at each generation and automatically keeps the particle system alive, i.e. which ensures its non–extinction. The main contributions of this work are L¹ error estimates stated in Theorem 3.4 and a central limit theorem stated in Theorem 3.7. An interesting feature of the sequential particle algorithm is that a fixed performance can be guaranteed in advance, at the expense of a random computational effort : this could be seen as an adaptive rule to automatically choose the number of particles. To get a fair comparison of the nonsequential and sequential particle algorithms, the time–averaged random number of simulated particles, which is an indicator of how much computational effort has been used, is used as a normalizing factor. The different behaviour of the two particle algorithms is illustrated on the simple example of binary selection functions (taking only the value 0 or 1). In Section 4, some basic properties are proved for sums of a random number of i.i.d. random variables, especially when this random number is a stopping time, and a conditional version of the central limit theorem, known in sequential analysis as the Anscombe theorem and proved in [16], is stated in Theorem 4.7. A central limit theorem for

(7)

triangular arrays of martingale increments spread across generations with different random sizes, is stated in Theorem 4.10.

The remaining part of this work is devoted to proofs of the main results. L¹ error estimates for the nonsequential and sequential particle algorithms, stated in Theorems 2.2 and 3.4, are proved in Sections 5 and 6, respectively. The central limit theorem for the sequential particle algorithm, stated in Theorem 3.7, is proved in Section 7 by induction, based on the central limit theorem stated in Theorem 4.7 for sums of a random number of i.i.d. random variables, and an alternate proof is given in Section 8, based on the central limit theorem stated in Theorem 4.10 for triangular arrays of martingale increments spread across generations with different random sizes.

Finally, Theorems 4.7 and 4.10 are proved in Appendices A and B, respectively, and some elementary results are proved in Appendix C.

2 Nonsequential particle algorithm

The evolution of the normalized (nonlinear) flow{µk, k= 0,1,· · · , n} is described by the following diagram µk−1−−−−−−−−−→ηk=µk−1Qk−−−−−−−−−→µk=gk·ηk ,

with initial conditionµ0=g0·η0, where the notation·denotes the projective product. It follows from (2) and from the definition that

E[

Yn k=0

gk(Xk) ] =hγn,1i= Yn k=0

hηk, gki,

i.e. the expectation of a product is replaced by the product of expectations. Notice that the ratio

ρk = sup

x∈E

gk(x) hηk, gki

is an indicator of how difficult a given problem is : indeed, a large value of ρk means that regions where the selection function gk is large have a small probability underηk. The idea behind the particle approach is to look for an approximation

µk ≈µ^N_k = XN i=1

w_kⁱδξ_kⁱ ,

in the form of the weighted empirical probability distribution associated with the particle system (ξ_kⁱ, wⁱ_k, i= 1,· · ·, N), where N denotes the number of particles. The weights and positions of the particles are chosen is such a way that the evolution of the approximate sequence {µ^N_k , k= 0,1,· · ·, n}is described by the following diagram

µ^N_k−1−−−−−−−−−→η^N_k =S^N(µ^N_k−1Qk)−−−−−−−−−→µ^N_k =gk·η_k^N ,

with initial condition defined by µ^N₀ = g0·η^N₀ and η₀^N = S^N(η0), where the notation S^N(µ) denotes the empirical probability distribution associated with an N–sample with common probability distribution µ. In practice, particles

• are selected according to their respective weights (wⁱ_k−1, i= 1,· · · , N) (selection step),

• move according to the Markov kernelQk (mutation step),

• are weighted by evaluating the fitness functiongk (weighting step).

Starting from (1) and introducing the particle approximation

γ_k^N =gkS^N(µ^N_k−1Qk)hγ_k−1^N ,1i=gkη^N_k hγ_k−1^N ,1i, and

γ₀^N =g0S^N(η0) =g0η^N₀ ,

Irisa

(8)

for the unnormalized (linear) flow, it is easily seen that

hγ_k^N,1i=hη^N_k , gki hγ_k−1^N ,1i and hγ₀^N,1i=hη₀^N, g0i, (3) hence

γ_k^N

hγ_k^N,1i =gk·η^N_k =µ^N_k and γ₀^N

hγ₀^N,1i =g0·η^N₀ =µ^N₀ .

However, if the function gk can possibly take the zero value, and even if hηk, gki > 0, it can happen that hη^N_k , gki= 0, i.e. it can happen that the evaluation of the functiongk returns the zero value for all the particles generated at the end of the mutation step : in such a situation, the particle systems dies out and the algorithm cannot continue. A reinitialization procedure has been proposed and studied in [5], in which the particle system is generated afresh from an arbitrary restarting probability distributionν. Alternatively, one could be interested by the behavior of the algorithm until the extinction time of the particle system, defined by

τ^N = inf{k≥0 : hη_k^N, gki= 0} .

Under the assumption thathγn,1i>0, the probabilityP[τ^N ≤n] that the algorithm cannot continue up to the time instant ngoes to zero with exponential rate [3, Theorem 7.4.1].

Example 2.1. [Binary selection] In the special case of binary selection functions (taking only the value 0 or 1), i.e. indicator functionsgk= 1Ak of Borel subsets for anyk= 0,1,· · · , n, it holds

p0=hη0, g0i=P[X0∈A0], and

pk =hηk, gki=P[Xk∈Ak|X0∈A0,· · · , Xk−1∈Ak−1], for anyk= 1,· · ·, n, and it follows from (2) that

Pn =P[X0∈A0,· · ·, Xn∈An] =hγn,1i= Yn k=0

hηk, gki.

On thegood set {τ^N > n}, the nonsequential particle algorithm results in the following approximations pk≈p^N_k =hη^N_k , gki=|I_k^N|

N where I_k^N ={i= 1,· · · , N : ξ_kⁱ ∈Ak} ,

denotes the set of successful particles within anN–sample with common probability distributionη0(fork= 0) andµ^N_k−1Qk (fork= 1,· · ·, n), and it follows from (3) that

Pn ≈P_n^N =hγ_n^N,1i= Yn k=0

hη^N_k , gki= Yn k=0

|I_k^N| N .

In other words, the probability Pn of a successful sequence is approximated as the product of the fraction of successful particles at each generation, and each transition probability pk separately is approximated as the fraction of successful particles at the corresponding generation. Notice that the computational effort, i.e. the number N of simulated particles at each generation, isfixed in advance, whereas the number|I_k^N|of successful particles at thek–th generation israndom, and could even be zero.

The following results have been obtained for the nonsequential particle algorithm with a constant number N of particles : a nonasymptotic estimate [3, Theorem 7.4.3]

sup

φ:kφk=1E|1

{τ^N > n} hµ^N_n, φi − hµn, φi | ≤ c⁰_n

√N +P[τ^N ≤n] , and a central limit theorem (see [3, Section 9.4] for a slightly different algorithm)

√N[ 1

{τ^N > n} hµ^N_n, φi − hµn, φi] =⇒N(0, v⁰_n(φ)),

in distribution asN ↑ ∞, with an explicit expression for the asymptotic variance. In the simple case where the fitness functions are positive, i.e. cannot take the zero value, these results are well–known and can be found in [7, Proposition 2.9, Corollary 2.20], where the proof relies on a central limit theorem for triangular arrays of martingale increments, or in [9, Theorem 4], where the same central limit theorem is obtained by induction.

For the purpose of comparison with the sequential particle algorithm, the following nonasymptotic error estimates are proved in Section 5.

(9)

Theorem 2.2. With the extinction timeτ^N defined by

τ^N = inf{k≥0 : hη_k^N, gki= 0} , it holds

E|1

{τ^N > n}

hγ^N_n,1i

hγn,1i −1| ≤z^N_n +P[τ^N ≤n], (4) and

sup

φ:kφk=1E|1

{τ^N > n} hµ^N_n, φi − hµn, φi | ≤2z_n^N+P[τ^N ≤n], (5) where the sequence {z^N_k , k= 0,1,· · ·, n}satisfies the linear recursion

z_k^N ≤ρk(1 +

√ρk

√N)z_k−1^N +

√ρk

√N and z^N₀ ≤

√ρ0

√N . (6)

Remark 2.3. The forcing term in (6) is√ρk/√ N, and limsup

N↑∞

[√

N z_k^N]≤ρk limsup

N↑∞

[√

N z_k−1^N ] +√ρk , and

limsup

N↑∞

[√

N z₀^N]≤√ρ0.

Notice that with a fixed numberN of simulated particles, the performance is√ρk/√

N and depends onρk : as a result, it is not possible to guarantee in advance a fixed performance, sinceρk is not known.

For completeness, the central limit theorem obtained in [3, Section 9.4] for a slightly different algorithm is recalled below.

Theorem 2.4 (Del Moral). With the extinction time τ^N defined by τ^N = inf{k≥0 : hη_k^N, gki= 0} , it holds

√N[ 1

{τ^N > n} hγ_n^N,1i

hγn,1i −1 ] =⇒N(0, V_n⁰),

and √

N[ 1

{τ^N > n} hµ^N_n, φi − hµn, φi] =⇒N(0, v_n⁰(φ)),

in distribution as N ↑ ∞, for any bounded measurable functionφ, with the asymptotic variance

V_n⁰= Xn k=0

var(gkRk+1:n1, ηk) hηk, gkRk+1:n1i² , and

v_n⁰(φ) = Xn k=0

var(gkRk+1:n(φ− hµn, φi), ηk) hηk, gkRk+1:n1i² , respectively, where

Rk+1:nφ(x) =Rk+1· · ·Rnφ(x) =E[φ(Xn) Yn p=k+1

gp(Xp)|Xk=x] , for any k= 0,1,· · · , n, with the convention Rn+1:nφ(x) =φ(x), for anyx∈E.

Irisa

(10)

Remark 2.5. Notice that

hη0, g0R1:n(φ− hµn, φi)i=hγ0R1:n, φ− hµn, φi i=hγn, φ− hµn, φi i= 0, and

hηk, gkRk+1:n(φ− hµn, φi)i=hµk−1Rk:n, φ− hµn, φi i= hγn, φ− hµn, φi i hγk−1,1i = 0, for anyk= 1,· · ·, n, hence the following equivalent expression holds for the asymptotic variance

v_n⁰(φ) = Xn k=0

hηk,|gkRk+1:n[φ− hµn, φi]|²i hηk, gkRk+1:n1i² .

Example 2.6. [Binary selection] In the special case of binary selection functions, it holds Rk+1:n1(x) =P[Xk+1∈Ak+1,· · ·, Xn ∈An|Xk =x],

for anyk = 0,1,· · · , n, with the convention Rn+1:n1(x) = 1, for anyx∈E, and it follows from Theorem 2.4

that √

N[ 1

{τ^N > n} P_n^N

Pn −1] =⇒N(0, V_n⁰), in distribution as N↑ ∞, with the asymptotic variance

V_n⁰= Xn k=0

( 1

pk −1) + Xn k=0

1 pk

var(Rk+1:n1, µk) hµk, Rk+1:n1i² . Indeed, since g²_k=gk, it holds

var(gkRk+1:n1, ηk)

hηk, gkRk+1:n1i² = hηk, gk|Rk+1:n1|²i hηk, gkRk+1:n1i² −1

= 1 pk

hµk,|Rk+1:n1|²i hµk, Rk+1:n1i² −1

= ( 1

pk −1) + 1

pk [hµk,|Rk+1:n1|²i hµk, Rk+1:n1i² −1], for anyk= 0,1,· · ·, n.

3 Sequential particle algorithm

The purpose of this work is to study a sequential particle algorithm, already proposed in [14, 11], which automatically keeps the particle system alive, i.e. which ensures its non–extinction. For any level H >0, and for anyk= 0,1,· · ·, n, define the random number of particles

N_k^H= inf{N≥1 : XN i=1

gk(ξ_kⁱ)≥H sup

x∈E

gk(x)},

where the random variables ξ₀¹,· · ·, ξⁱ₀,· · · are i.i.d. with common probability distribution η0 (fork= 0), and where, conditionally w.r.t. theσ–algebraH_k−1^H generated by the particle system until the (k−1)–th generation, the random variablesξ_k¹,· · ·, ξ_kⁱ,· · · are i.i.d. with common probability distributionµ^H_k−1Qk (fork= 1,· · · , n).

The particle approximation {µ^H_k , k = 0,1,· · · , n} is now parameterized by the levelH >0, and its evolution is described by the following diagram

µ^H_k−1−−−−−−−−−→η^H_k =S^N^k^H(µ^H_k−1Qk)−−−−−−−−−→µ^H_k =gk·η_k^H ,

(11)

with initial condition defined by µ^H₀ = g0·η^H₀ and η₀^H = S^N⁰^H(η0). Starting from (1) and introducing the particle approximation

γ_k^H =gkS^N^k^H(µ^H_k−1Qk)hγ_k−1^H ,1i=gkη^H_k hγ_k−1^H ,1i, and

γ₀^H=g0S^N⁰^H(η0) =g0η₀^H , for the unnormalized (linear) flow, it is easily seen that

hγ_k^H,1i=hη^H_k , gki hγ_k−1^H ,1i and hγ₀^H,1i=hη₀^H, g0i, (7) hence

γ_k^H

hγ_k^H,1i =gk·η^H_k =µ^H_k and γ₀^H

hγ₀^H,1i =g0·η^H₀ =µ^H₀ . Clearly,N_k^H≥H and ifhµ^H_k−1Qk, gki>0 — a sufficient condition for which is

b

gk(x) =Qkgk(x) =E[gk(Xk)|Xk−1=x]>0,

for anyxin the support ofµ^H_k−1 — then the random numberN_k^H of particles is a.s. finite, see Section 4 below.

Moreover

hη^H₀, g0i=hS^N⁰^H(η0), g0i= 1 N₀^H

N0^H

X

i=1

g0(ξ₀ⁱ)≥ H N₀^H sup

x∈Eg0(x)>0, and

hη_k^H, gki=hS^N^k^H(µ^H_k−1Qk), gki= 1 N_k^H

N_k^H

X

i=1

gk(ξ_kⁱ)≥ H N_k^H sup

x∈E

gk(x)>0,

for anyk= 1,· · ·, n, i.e. the particle system never dies out and the algorithm can always continue, by construc- tion.

Remark 3.1. It follows from Lemma 4.3 below that N₀^H

H →ρ0 in probability, and in view of Remark 4.4 (ii) below, if hµ^H_k−1Qk, gki>0 then N_k^H

H ρ^H_k →1 in probability asH↑ ∞, where

ρ^H_k = sup

x∈E

gk(x) hµ^H_k−1Qk, gki , for anyk= 1,· · ·, n.

Remark 3.2. For anyk= 0,1,· · · , nand any integeri≥1, letF_k,i^H =F_k,0^H ∨σ(ξ_k¹,· · ·, ξⁱ_k), whereF_0,0^H ={∅,Ω} (for k = 0) and F^H_k,0 =H^H_k−1 (for k= 1,· · ·, n) by convention. The random numberN_k^H is a stopping time w.r.t.F^H_k ={F^H_k,i, i≥0}, which allows to define theσ–algebraF^H_k,NH

k =H^H_k : clearly N_k^H is measurable w.r.t.

H^H_k, and therefore the random variable

σ^H_k =N₀^H+· · ·+N_k^H , is measurable w.r.t.H^H_k.

Example 3.3. [Binary selection] In the special case of binary selection functions, the sequential particle algorithm results in the following approximations

pk ≈p^H_k =hη_k^H, gki= H

N_k^H where N_k^H = inf{N ≥1 : |I_k^N|=H}, for any integer H ≥1, and where for any integer N≥1

I_k^N ={i= 1,· · ·, N : ξⁱ_k∈Ak},

Irisa

(12)

denotes the set of successful particles within anN–sample with common probability distributionη0(fork= 0) andµ^H_k−1Qk (fork= 1,· · ·, n), and it follows from (7) that

Pn≈P_n^H =hγ_n^H,1i= Yn k=0

hη_k^H, gki= Yn k=0

H N_k^H .

Notice that the approximation µ^H_k = gk ·η^H_k obtained here is exactly the empirical probability distribution associated with an H–sample that would be obtained using the rejection method, with common probability distribution g0·η0 (for k = 0) and gk·(µ^H_k−1Qk) (for k = 1,· · ·, n). Here again, the probability Pn of a successful sequence is approximated as the product of the fraction of successful particles at each generation, and each transition probability pk separately is approximated as the fraction of successful particles at the corresponding generation. In opposition to the nonsequential particle algorithm, notice that the number H of successful particles at each generation is fixed in advance, whereas the computational effort, i.e. the number N_k^H of simulated particles needed to get H successful particles exactly at thek–th generation, israndom.

The main contributions of this paper are the following results for the sequential particle algorithm with a random number of particles, defined by the levelH >0 : a nonasymptotic estimate (which was already obtained in [11, Theorem 5.4] in a different context), see Theorem 3.4 below

sup

φ:kφk=1

E| hµ^H_n −µn, φi | ≤ cn

√H ,

and a central limit theorem, see Theorem 3.7 below

√Hhµ^H_n −µn, φi=⇒N(0, vn(φ)),

in distribution as H↑ ∞, with an explicit expression for the asymptotic variance.

Theorem 3.4. Ifhµ^H_k−1Qk, gki>0 for any k= 1,· · ·, n— a sufficient condition for which is b

gk(x) =Qkgk(x) =E[gk(Xk)|Xk−1=x]>0 , for any xin the support ofµ^H_k−1 — then

E|hγ_n^H,1i

hγn,1i −1| ≤z_n^H and sup

φ:kφk=1

E| hµ^H_n −µn, φi | ≤2z_n^H , (8) where the sequence {z^H_k , k= 0,1,· · ·, n}satisfies the linear recursion

z_k^H≤ρk(1 +ωH+ω_H²)z^H_k−1+ωH(1 +ωHρk) , (9) and

z^H₀ ≤ωH(1 +ωHρ0), where ωH= 1

H

√H+ 1 is of order1/√ H.

Remark 3.5. Up to higher order terms, the forcing term in (9) is 1/√

H exactly (which is equivalent to

qρ^H_{k /}^qN_k^H, in view of Remark 3.1), and limsup

H↑∞

[√

H z^H_k ]≤ρk limsup

H↑∞

[√

H z^H_k−1] + 1 and limsup

H↑∞

[√

H z₀^H]≤1.

In opposition to the nonsequential particle algorithm, notice that it is possible here to guarantee in advance a fixed performance of 1/√

H exactly, without any knowledge ofρk, at the expense of using an adaptive random number N_k^H of simulated particles : this could be seen as an adaptive rule to automatically choose the number of particles.

Remark 3.6. It follows from Theorem 3.4 that hµ^H_k−1Qk, gki → hηk, gki in probability, hence ρ^H_k → ρk in probability, for any k= 1,· · ·, n, and it follows from Remark 3.1 that N_k^H

H →ρk in probability asH ↑ ∞, for any k= 0,1,· · ·, n.