Sampling per mode simulation for switching diffusions

S

AMPLING PER MODE SIMULATION FOR SWITCHING

DIFFUSIONS

Pascal Lezaud1 Jaroslav Krystul2 Franc¸ois Le Gland3 1_{DSNA/DTI/R&D and Institut de Math ´ematiques de Toulouse}

2

Twente University, Enschede

3

INRIA Rennes, Bretagne–Atlantique

8th International Conference on Rare Event Simulation, RESIM’10, Cambridge, June 21–22, 2010

1

I

NTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

_EYNMAN

-K

_{AC FORMULATION}

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

M

ODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

A

SYMPTOTIC

B

EHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

PLAN

1

I

NTRODUCTION Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

MOTIVATIONS

Many complex dynamical multi-agent systems make use of

continuous-time strong Markov processes with an hybrid state space:

I one state component evolves inRd,

I the other state component evolves in a discrete set,

I and each component may influence the evolution of the other component.

Our motivation is to estimate the probability that the continuous component hits a critical set.

We use a splitting technique adapted to the context of switching diffusions: the sampling per mode algorithm introduced by Krystul in [Krystul, 2006]

PLAN

1

I

NTRODUCTION

Motivation

Switching jump diffusion

Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0 LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0 LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0 LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0

LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0 LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

SWITCHING JUMP DIFFUSION

Strong Markov processZ = {(Xt, θt);t ≥ 0}with value inRd× Mwith a

finite setM = {1, · · · , M},

the continuous component is described as ad-dimensional SDE

dXt =b (Xt, θt)dt + σ (Xt, θt)dBt,

and the discrete mode as a pure jump process

P (θt+∆t=j|θt =i, Xt=x ) = λij(x )∆t + o(∆t), i 6= j, with jump intensities depending on the continuous component.

Zt starts att = 0inD0× Mwith known initial probabilityη0 LetA ⊂ Rd _{be a closed critical region in whichX}

t could enter but with a very small probability.

IfTAdenotes the hitting time ofA, we would like to estimateP(TA≤ T ) withT a deterministic or a stopping time.

PLAN

1

I

NTRODUCTION

Motivation

Switching jump diffusion

Splitting technique

Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

SPLITTING TECHNIQUE

Identify intermediate sets that are (sequentially) visited much more often than the rare target set:

LetA = Dn⊂ · · · ⊂ D1⊂ Rd, D0∩ D1= ∅

R

n

M

D0 D2 D1 D1 D2 A A

WithB = A × MandBk =Dk× M, we define fork = 1, · · · , n

Tk =inf{t ≥ 0 : Zt ∈ Bk} = inf{t ≥ 0 : Xt∈ Dk}, which satisfy0 = T0≤ T1≤ · · · ≤ Tn=TB. Then P(TA≤ T ) = P(TB≤ T ) = n Y k =1 P(Tk ≤ T |Tk −1≤ T ), where conditional probabilities are not very small.

SPLITTING TECHNIQUE

Identify intermediate sets that are (sequentially) visited much more often than the rare target set:

LetA = Dn⊂ · · · ⊂ D1⊂ Rd, D0∩ D1= ∅

R

n

M

D0 D2 D1 D1 D2 A A

WithB = A × MandBk =Dk× M, we define fork = 1, · · · , n

Tk =inf{t ≥ 0 : Zt ∈ Bk} = inf{t ≥ 0 : Xt∈ Dk}, which satisfy0 = T0≤ T1≤ · · · ≤ Tn=TB. Then P(TA≤ T ) = P(TB≤ T ) = n Y k =1 P(Tk ≤ T |Tk −1≤ T ), where conditional probabilities are not very small.

SPLITTING TECHNIQUE

Identify intermediate sets that are (sequentially) visited much more often than the rare target set:

LetA = Dn⊂ · · · ⊂ D1⊂ Rd, D0∩ D1= ∅

R

n

M

D0 D2 D1 D1 D2 A A

WithB = A × MandBk =Dk× M, we define fork = 1, · · · , n

Tk =inf{t ≥ 0 : Zt ∈ Bk} = inf{t ≥ 0 : Xt∈ Dk}, which satisfy0 = T0≤ T1≤ · · · ≤ Tn=TB. Then P(TA≤ T ) = P(TB≤ T ) = n Y k =1 P(Tk ≤ T |Tk −1≤ T ), where conditional probabilities are not very small.

SPLITTING TECHNIQUE

Identify intermediate sets that are (sequentially) visited much more often than the rare target set:

LetA = Dn⊂ · · · ⊂ D1⊂ Rd, D0∩ D1= ∅

R

n

M

D0 D2 D1 D1 D2 A A

WithB = A × MandBk =Dk× M, we define fork = 1, · · · , n

Tk =inf{t ≥ 0 : Zt ∈ Bk} = inf{t ≥ 0 : Xt∈ Dk}, which satisfy0 = T0≤ T1≤ · · · ≤ Tn=TB. Then P(TA≤ T ) = P(TB≤ T ) = n Y k =1 P(Tk ≤ T |Tk −1≤ T ), where conditional probabilities are not very small.

PLAN

1

I

NTRODUCTION

Motivation

Switching jump diffusion Splitting technique

Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

SOME ISSUES

Splitting technique applies since a switching process is a strong Markov process, but

this approach fails to produce a reasonable estimate, since each resampling step tends

I to sample more particles from mode with higher probability,

I to discard the particles in the ”light” modes.

Increasing the number of particles should improve the estimate but only at the cost of increased simulation time,

Idea: keep constant the number of particles in each visited mode at each resampling step,

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN

-K

AC FORMULATION Multilevel Feynman-Kac distributions

Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

To capture the behaviour ofZ between each thresholds, we consider the random excursionsZk ofZ betweenTk −1andTk ∧ T

Zk = ((Xt, θt),Tk −1 ≤ t ≤ Tk∧ T ) ,

and we introduce the selection functions,

gk(Zk) =1{Z_{Tk ∧T}∈Bk}, g

j

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

To capture the behaviour ofZ between each thresholds, we consider the random excursionsZk ofZ betweenTk −1andTk ∧ T

Zk = ((Xt, θt),Tk −1 ≤ t ≤ Tk∧ T ) , and we introduce the selection functions,

gk(Zk) =1{Z_{Tk ∧T}∈Bk}, g

j

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

To capture the behaviour ofZ between each thresholds, we consider the random excursionsZk ofZ betweenTk −1andTk ∧ T

Zk = ((Xt, θt),Tk −1 ≤ t ≤ Tk∧ T ) , and we introduce the selection functions,

gk(Zk) =1{Z_{Tk ∧T}∈Bk}, g

j

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

To capture the behaviour ofZ between each thresholds, we consider the random excursionsZk ofZ betweenTk −1andTk ∧ T

Zk = ((Xt, θt),Tk −1 ≤ t ≤ Tk∧ T ) , and we introduce the selection functions,

gk(Zk) =1{Z_{Tk ∧T}∈Bk}, g

j

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

Clearly,

1{Tk ≤ T } = gk(Zk), and 1{Tk ≤ T , θTk =j} = g

j k(Zk).

We can interpret the rare event probability in terms of the Feynman-Kac measures defined by γk(f ) = E [f (Zk)gk −1(Zk −1)] = Ef ((Xt, θt), Tk −1≤ t ≤ Tk∧ T )1{Tk −1≤T }  b γk(f ) = E [f (Zk)gk(Zk)] = E h f ((Xt, θt), Tk −1 ≤ t ≤ Tk)1{T k ≤ T } i ,

and the corresponding normalized measures defined by

η_k(f ) = γk(f ) γk(1) = E [f ((X t, θt),Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T ] b ηk(f ) = b γ_k(f ) b γk(1) = E [f ((Xt , θt), Tk −1≤ t ≤ Tk)|Tk ≤ T ] .

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

Clearly,

1{Tk ≤ T } = gk(Zk), and 1{Tk ≤ T , θTk =j} = g

j k(Zk).

We can interpret the rare event probability in terms of the Feynman-Kac measures defined by γk(f ) = E [f (Zk)gk −1(Zk −1)] = Ef ((Xt, θt), Tk −1≤ t ≤ Tk ∧ T )1{Tk −1≤T }  b γk(f ) = E [f (Zk)gk(Zk)] = E h f ((Xt, θt), Tk −1 ≤ t ≤ Tk)1{T k ≤ T } i ,

and the corresponding normalized measures defined by

η_k(f ) = γk(f ) γk(1) = E [f ((X t, θt),Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T ] b ηk(f ) = b γ_k(f ) b γk(1) = E [f ((Xt , θt), Tk −1≤ t ≤ Tk)|Tk ≤ T ] .

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

Clearly,

1{Tk ≤ T } = gk(Zk), and 1{Tk ≤ T , θTk =j} = g

j k(Zk).

We can interpret the rare event probability in terms of the Feynman-Kac measures defined by γk(f ) = E [f (Zk)gk −1(Zk −1)] = Ef ((Xt, θt), Tk −1≤ t ≤ Tk ∧ T )1{Tk −1≤T }  b γk(f ) = E [f (Zk)gk(Zk)] = E h f ((Xt, θt), Tk −1 ≤ t ≤ Tk)1{T k ≤ T } i ,

and the corresponding normalized measures defined by

η_k(f ) = γk(f ) γk(1) = E [f ((Xt , θ_t),Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T ] b ηk(f ) = b γk(f ) b γk(1) = E [f ((Xt , θt), Tk −1≤ t ≤ Tk)|Tk ≤ T ] .

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In particular, forf ≡ 1

γk(1) = P(Tk −1≤ T ), bγk(1) = P(Tk ≤ T ),

and forf = gk orf = g_kj, it holds

ηk(gk) = P[Tk ≤ T |Tk −1 ≤ T ], ηk(g_kj) = P[Tk ≤ T , θTk =j|Tk −1 ≤ T ].

We have the key formulas

γ_k(f ) = ηk(f ) k −1 Y p=0 ηp(gp) and bγk(f ) =ηbk(f ) k Y p=0 ηp(gp). Then, we recover P(Tn≤ T ) = n Y p=0 ηp(gp),

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In particular, forf ≡ 1

γk(1) = P(Tk −1≤ T ), bγk(1) = P(Tk ≤ T ),

and forf = gk orf = g_kj, it holds

ηk(gk) = P[Tk ≤ T |Tk −1 ≤ T ], ηk(g_kj) = P[Tk ≤ T , θTk =j|Tk −1 ≤ T ].

We have the key formulas

γ_k(f ) = ηk(f ) k −1 Y p=0 ηp(gp) and bγk(f ) =ηbk(f ) k Y p=0 ηp(gp). Then, we recover P(Tn≤ T ) = n Y p=0 ηp(gp),

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In particular, forf ≡ 1

γk(1) = P(Tk −1≤ T ), bγk(1) = P(Tk ≤ T ),

and forf = gk orf = g_kj, it holds

ηk(gk) = P[Tk ≤ T |Tk −1 ≤ T ], ηk(g_kj) = P[Tk ≤ T , θTk =j|Tk −1 ≤ T ].

We have the key formulas

γ_k(f ) = ηk(f ) k −1 Y p=0 ηp(gp) and bγk(f ) =ηbk(f ) k Y p=0 ηp(gp). Then, we recover P(Tn≤ T ) = n Y p=0 ηp(gp),

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In particular, forf ≡ 1

γk(1) = P(Tk −1≤ T ), bγk(1) = P(Tk ≤ T ),

and forf = gk orf = g_kj, it holds

ηk(gk) = P[Tk ≤ T |Tk −1 ≤ T ], ηk(g_kj) = P[Tk ≤ T , θTk =j|Tk −1 ≤ T ].

We have the key formulas

γ_k(f ) = ηk(f ) k −1 Y p=0 ηp(gp) and bγk(f ) =ηbk(f ) k Y p=0 ηp(gp). Then, we recover P(Tn≤ T ) = n Y p=0 ηp(gp),

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In order to keep trace of the discrete mode, we construct for anyj ∈ M γ_kj(_{f ) = E}hf (Zk)gj_{k −1}(Zk −1) i = Ehf (Zt,Tk −1≤ t ≤ Tk ∧ T )1{Tk −1≤T ,θ_Tk−1=j} i , b γ_kj(f ) = Ehf (Zk)gjk(Zk) i = Ehf (Zt,Tk −1 ≤ t ≤ Tk)1{T k ≤ T , θTk =j} i .

and the normalized measures

η_kj(f ) = γ j k(f ) γ_kj(1) = Ef (Zt, Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T , θTk −1=j , b η_kj(f ) = bγ j k(f ) b γ_kj(1) = E [f (Zt, Tk −1≤ t ≤ Tk)|Tk ≤ T , θTk =j] .

We have the decompositions

b ηk = X j∈M ωj_kη_bkj, ηk +1= X j∈M ω_kjη_{k +1}j , where ω_kj =η_bk(g_kj) = P(θTk =j|Tk ≤ T ).

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In order to keep trace of the discrete mode, we construct for anyj ∈ M γ_kj(_{f ) = E}hf (Zk)gj_{k −1}(Zk −1) i = Ehf (Zt,Tk −1≤ t ≤ Tk ∧ T )1{Tk −1≤T ,θ_Tk−1=j} i , b γ_kj(f ) = Ehf (Zk)gjk(Zk) i = Ehf (Zt,Tk −1 ≤ t ≤ Tk)1{T k ≤ T , θTk =j} i .

and the normalized measures

η_kj(f ) = γ j k(f ) γj_k(1) = Ef (Zt, Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T , θTk −1=j , b η_kj(f ) = bγ j k(f ) b γj_k(1) = E [f (Zt, Tk −1≤ t ≤ Tk)|Tk ≤ T , θTk =j] .

We have the decompositions

b ηk = X j∈M ωj_kη_bkj, ηk +1= X j∈M ω_kjη_{k +1}j , where ω_kj =η_bk(g_kj) = P(θTk =j|Tk ≤ T ).

MULTILEVEL

FEYNMAN-KAC DISTRIBUTIONS

In order to keep trace of the discrete mode, we construct for anyj ∈ M γ_kj(_{f ) = E}hf (Zk)gj_{k −1}(Zk −1) i = Ehf (Zt,Tk −1≤ t ≤ Tk ∧ T )1{Tk −1≤T ,θ_Tk−1=j} i , b γ_kj(f ) = Ehf (Zk)gjk(Zk) i = Ehf (Zt,Tk −1 ≤ t ≤ Tk)1{T k ≤ T , θTk =j} i .

and the normalized measures

η_kj(f ) = γ j k(f ) γj_k(1) = Ef (Zt, Tk −1≤ t ≤ Tk ∧ T )|Tk −1 ≤ T , θTk −1=j , b η_kj(f ) = bγ j k(f ) b γj_k(1) = E [f (Zt, Tk −1≤ t ≤ Tk)|Tk ≤ T , θTk =j] .

We have the decompositions

b ηk = X j∈M ωj_kbη_kj, ηk +1= X j∈M ω_kjη_{k +1}j , where ω_kj =η_bk(g_kj) = P(θTk =j|Tk ≤ T ).

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN

-K

AC FORMULATION

Multilevel Feynman-Kac distributions

Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

DYNAMICAL EVOLUTION

Using the Markov property ofZ(with Markov kernelMk), we obtain

γ_k(f ) = γk −1(gk −1Mk f )and γkj = γk −1(gk −1j Mk f ).

and the nonlinear measure-valued transformations

b ηk(f ) = ηk(fgk) ηk(gk) := Ψk(ηk)(f ), ηb j k(f ) = η_k(fg_kj) ηk(g_kj) := Ψj_k(ηk)(f ).

so, the following two separate selection/mutation transitions

ηk

selection

−−−−−→η_bk := Ψk(ηk) mutation

DYNAMICAL EVOLUTION

Using the Markov property ofZ(with Markov kernelMk), we obtain

γ_k(f ) = γk −1(gk −1Mk f )and γkj = γk −1(gk −1j Mk f ).

and the nonlinear measure-valued transformations

b ηk(f ) = ηk(fgk) ηk(gk) := Ψk(ηk)(f ), ηb j k(f ) = η_k(fg_kj) ηk(g_kj) := Ψj_k(ηk)(f ).

so, the following two separate selection/mutation transitions

ηk

selection

−−−−−→η_bk := Ψk(ηk) mutation

DYNAMICAL EVOLUTION

Using the Markov property ofZ(with Markov kernelMk), we obtain

γ_k(f ) = γk −1(gk −1Mk f )and γkj = γk −1(gk −1j Mk f ).

and the nonlinear measure-valued transformations

b ηk(f ) = ηk(fgk) ηk(gk) := Ψk(ηk)(f ), ηb j k(f ) = η_k(fg_kj) ηk(g_kj) := Ψj_k(ηk)(f ).

so, the following two separate selection/mutation transitions

ηk −−−−−→selection ηbk := Ψk(ηk)

mutation

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

M

ODE ALGORITHM Particle Methods

Sampling per Mode algorithm

4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

PARTICLE METHODS

Particle methods are a kind of stochastic linearisation technique for solving nonlinear equation in measure space.

Using two sequences ofNparticlesξ = (ξ1, · · · , ξN)andξ = (bb ξ1, · · · , bξN),

we approximate the two step transitions

ηk selection −−−−−→η_bk := Ψk(ηk) mutation −−−−−→ ηk +1=ηbkMk +1, by η_kN:= 1 N N X i=1 δ_ξi k selection −−−−−→η_bkN:= 1 N N X i=1 δ b ξi k mutation −−−−−→ η_{k +1}= 1 N N X i=1 δ_ξi k +1.

PARTICLE METHODS

Particle methods are a kind of stochastic linearisation technique for solving nonlinear equation in measure space.

Using two sequences ofNparticlesξ = (ξ1_{, · · · , ξ}N_)and

b

ξ = (bξ1_{, · · · , bξ}N_), we approximate the two step transitions

ηk selection −−−−−→η_bk := Ψk(ηk) mutation −−−−−→ ηk +1=bηkMk +1, by η_kN:= 1 N N X i=1 δ_ξi k selection −−−−−→η_bkN:= 1 N N X i=1 δ b ξi k mutation −−−−−→ η_{k +1}= 1 N N X i=1 δ_ξi k +1.

PARTICLE METHODS

Particle methods are a kind of stochastic linearisation technique for solving nonlinear equation in measure space.

Using two sequences ofNparticlesξ = (ξ1_{, · · · , ξ}N_)and

b

ξ = (bξ1_{, · · · , bξ}N_), we approximate the two step transitions

ηk selection −−−−−→η_bk := Ψk(ηk) mutation −−−−−→ ηk +1=bηkMk +1, by η_kN:= 1 N N X i=1 δ_ξi k selection −−−−−→η_bkN:= 1 N N X i=1 δ b ξi k mutation −−−−−→ η_{k +1}= 1 N N X i=1 δ_ξi k +1.

PARTICLE METHODS

Particle methods are a kind of stochastic linearisation technique for solving nonlinear equation in measure space.

Using two sequences ofNparticlesξ = (ξ1_{, · · · , ξ}N_)and

b

ξ = (bξ1_{, · · · , bξ}N_), we approximate the two step transitions

ηk selection −−−−−→η_bk := Ψk(ηk) mutation −−−−−→ ηk +1=bηkMk +1, by η_kN:= 1 N N X i=1 δ_ξi k selection −−−−−→η_bkN:= 1 N N X i=1 δ b ξi k mutation −−−−−→ η_{k +1}= 1 N N X i=1 δ_ξi k +1.

PARTICLE METHODS

Particle methods are a kind of stochastic linearisation technique for solving nonlinear equation in measure space.

Using two sequences ofNparticlesξ = (ξ1_{, · · · , ξ}N_)and

b

ξ = (bξ1_{, · · · , bξ}N_), we approximate the two step transitions

ηk selection −−−−−→η_bk := Ψk(ηk) mutation −−−−−→ ηk +1=bηkMk +1, by η_kN:= 1 N N X i=1 δ_ξi k selection −−−−−→η_bkN:= 1 N N X i=1 δ b ξi k mutation −−−−−→ η_{k +1}= 1 N N X i=1 δ_ξi k +1.

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

M

ODE ALGORITHM

Particle Methods

Sampling per Mode algorithm 4

ASYMPTOTIC

BEHAVIOUR

Asymptotic Behaviour Law of Large Numbers Central Limit Theorem

SAMPLING PER

MODE ALGORITHM

The main idea ofSampling per mode algorithmconsists in maintaining a fixed number of particles in each mode, at each resampling step.

So, instead of starting the algorithm withNparticles randomly distributed, we draw in each modej, a fixed numberNj _{particles and at each}

resampling step, the same number of particles is sampled for each visited mode.

Obviously, the total number of particles can change at each time some mode is not visited, or empty mode is visited afresh.

LetNb_k andN_k denote the total numbers of particlesξb_k andξ_k, andω_kj,N

the weights associated with the modes, we have the evolution scheme

(Nk, (ω_{k −1}j,N )j∈Jk −1, ξk) → ( bNk, (ω

j

k)j∈Jk, bξk) → (Nk +1, (ω

j,N

k )j∈Jk, ξk +1)

SAMPLING PER

MODE ALGORITHM

The main idea ofSampling per mode algorithmconsists in maintaining a fixed number of particles in each mode, at each resampling step.

So, instead of starting the algorithm withNparticles randomly distributed, we draw in each modej, a fixed numberNj _{particles and at each}

resampling step, the same number of particles is sampled for each visited mode.

Obviously, the total number of particles can change at each time some mode is not visited, or empty mode is visited afresh.

LetNb_k andN_k denote the total numbers of particlesξb_k andξ_k, andω_kj,N

the weights associated with the modes, we have the evolution scheme

(Nk, (ω_{k −1}j,N )j∈Jk −1, ξk) → ( bNk, (ω

j

k)j∈Jk, bξk) → (Nk +1, (ω

j,N

k )j∈Jk, ξk +1)

SAMPLING PER

MODE ALGORITHM

The main idea ofSampling per mode algorithmconsists in maintaining a fixed number of particles in each mode, at each resampling step.

So, instead of starting the algorithm withNparticles randomly distributed, we draw in each modej, a fixed numberNj _{particles and at each}

resampling step, the same number of particles is sampled for each visited mode.

Obviously, the total number of particles can change at each time some mode is not visited, or empty mode is visited afresh.

LetNb_k andN_k denote the total numbers of particlesξb_k andξ_k, andω_kj,N

the weights associated with the modes, we have the evolution scheme

(Nk, (ω_{k −1}j,N )j∈Jk −1, ξk) → ( bNk, (ω

j

k)j∈Jk, bξk) → (Nk +1, (ω

j,N

k )j∈Jk, ξk +1)

SAMPLING PER

MODE ALGORITHM

The main idea ofSampling per mode algorithmconsists in maintaining a fixed number of particles in each mode, at each resampling step.

So, instead of starting the algorithm withNparticles randomly distributed, we draw in each modej, a fixed numberNj _{particles and at each}

resampling step, the same number of particles is sampled for each visited mode.

Obviously, the total number of particles can change at each time some mode is not visited, or empty mode is visited afresh.

LetNb_k andN_k denote the total numbers of particlesξb_k andξ_k, andωj,N_k

the weights associated with the modes, we have the evolution scheme

(Nk, (ω_{k −1}j,N )j∈Jk −1, ξk) → ( bNk, (ω

j

k)j∈Jk, bξk) → (Nk +1, (ω

j,N

k )j∈Jk, ξk +1)

INITIALIZATION

In each modej, we sampleNj _particlesξκ 0 = bξ κ 0 = (0, (X κ 0,j)) ∼ η j 0.

R

n

M

D0 D2 D1 D1 D2 A A Letωj_{0= P(θ0}=j), thenηN 0 andηb N 0 are given by ηN 0 = P j∈Mω j 0η j,N 0 , ηb N 0 = P j∈Mω j 0ηb j,N 0 , with ηj,N₀ = _N1j P κ∈Jj₀δξκ0, ηb j,N 0 = N1j P κ∈Jj₀δξbκ₀. HereJ₀j is the set of the indices of the particles in the modej.

INITIALIZATION

In each modej, we sampleNj _particlesξκ 0 = bξ κ 0 = (0, (X κ 0,j)) ∼ η j 0.

R

n

M

D0 D2 D1 D1 D2 A A Letωj_{0= P(θ0}=j), thenηN 0 andηb N 0 are given by ηN 0 = P j∈Mω j 0η j,N 0 , ηb N 0 = P j∈Mω j 0ηb j,N 0 , with ηj,N₀ = _N1j P κ∈J₀j δξκ0, ηb j,N 0 = N1j P κ∈Jj₀δξbκ₀.

INITIALIZATION

In each modej, we sampleNj _particlesξκ 0 = bξ κ 0 = (0, (X κ 0,j)) ∼ η j 0.

R

n

M

D0 D2 D1 D1 D2 A A Letωj_{0= P(θ0}=j), thenηN 0 andηb N 0 are given by ηN 0 = P j∈Mω j 0η j,N 0 , ηb N 0 = P j∈Mω j 0ηb j,N 0 , with ηj,N₀ = _N1j P κ∈J₀j δξκ0, ηb j,N 0 = N1j P κ∈Jj₀δξbκ₀. HereJ₀j is the set of the indices of the particles in the modej.

MUTATION

( bNk, ωkN, bξk) → (Nk +1, ωkN, ξk +1)

IfNb_k =0the particle system dies, otherwise independently of each other,

each particleξbκ_k evolves randomly according to the Markov transition Mk +1

R

n

M

D0 D2 D1 D1 D2 A A

Neither the total number of particles nor the weight of each particle change (Nk +1= bNk). SoηN k +1= P j∈Jkω j,N k η j,N k +1,withη j,N k +1= 1 Nj P κ∈J_kjδξk +1κ ,whereJ j k is the set of the labels of the particles in modej ∈ Jk.

MUTATION

( bNk, ωkN, bξk) → (Nk +1, ωkN, ξk +1)

IfNb_k =0the particle system dies, otherwise independently of each other,

each particleξbκ_k evolves randomly according to the Markov transition Mk +1

R

n

M

D0 D2 D1 D1 D2 A A

Neither the total number of particles nor the weight of each particle change (Nk +1= bNk). SoηN k +1= P j∈Jkω j,N k η j,N k +1,withη j,N k +1= 1 Nj P κ∈J_kjδξk +1κ ,whereJ j k is the set of the labels of the particles in modej ∈ Jk.

MUTATION

( bNk, ωkN, bξk) → (Nk +1, ωkN, ξk +1)

IfNb_k =0the particle system dies, otherwise independently of each other,

each particleξbκ_k evolves randomly according to the Markov transition Mk +1

R

n

M

D0 D2 D1 D1 D2 A A

Neither the total number of particles nor the weight of each particle change (Nk +1= bNk). SoηN k +1= P j∈Jkω j,N k η j,N k +1,withη j,N k +1= 1 Nj P κ∈Jkj δξκ k +1,whereJ j k is the set of the labels of the particles in modej ∈ Jk.

SELECTION/R

ESAMPLING

(Nk +1, ωkN, ξk +1) → ( bNk +1, ωk +1N , bξk +1)

Select only the particlesξκ_{k +1}having reached the desired setBk +1;

R

n

M

D0 D2 D1 D1 D2 A A LetIN

k +1denote the set of (indices of) good particles; ifIk +1N = ∅the algorithm is stopped. Otherwise,

for each non empty modej, resampleNj particles according to

Ψj_{k +1}(η_{k +1}N ), and set b ηN k +1= P j∈Jk +1ω j,N k +1ηb j,N k +1, withηb j,N k +1= 1 Nj P κ∈bJ_{k +1}j δξbκ_{k +1}and ωj,N_{k +1}=η_bN_{k +1}(g_{k +1}j ) =η N k +1(g j k +1) ηN k +1(gk +1) .

The total number of particles isNb_{k +1}=P_j∈J k +1N

SELECTION/R

ESAMPLING

(Nk +1, ωkN, ξk +1) → ( bNk +1, ωk +1N , bξk +1)

Select only the particlesξκ_{k +1}having reached the desired setBk +1;

R

n

M

D0 D2 D1 D1 D2 A A

LetI_{k +1}N denote the set of (indices of) good particles; ifI_{k +1}N = ∅the algorithm is stopped. Otherwise,

for each non empty modej, resampleNj particles according to

Ψj_{k +1}(η_{k +1}N ), and set b ηN k +1= P j∈Jk +1ω j,N k +1ηb j,N k +1, withηb j,N k +1= 1 Nj P κ∈bJ_{k +1}j δξbκ_{k +1}and ωj,N_{k +1}=η_bN_{k +1}(g_{k +1}j ) =η N k +1(g j k +1) ηN k +1(gk +1) .

The total number of particles isNb_{k +1}=P_j∈J k +1N

SELECTION/R

ESAMPLING

(Nk +1, ωkN, ξk +1) → ( bNk +1, ωk +1N , bξk +1)

Select only the particlesξκ_{k +1}having reached the desired setBk +1;

R

n

M

D0 D2 D1 D1 D2 A A

LetI_{k +1}N denote the set of (indices of) good particles; ifI_{k +1}N = ∅the algorithm is stopped. Otherwise,

for each non empty modej, resampleNj particles according to

Ψj_{k +1}(η_{k +1}N ), and set b ηN k +1= P j∈Jk +1ω j,N k +1ηb j,N k +1, withηb j,N k +1= 1 Nj P κ∈bJ_{k +1}j δξbκ_{k +1}and ωj,N_{k +1}=η_bN_{k +1}(g_{k +1}j ) =η N k +1(g j k +1) ηN k +1(gk +1) .

The total number of particles isNb_{k +1}=P_j∈J k +1N

SELECTION/R

ESAMPLING

(Nk +1, ωkN, ξk +1) → ( bNk +1, ωk +1N , bξk +1)

Select only the particlesξκ_{k +1}having reached the desired setBk +1;

R

n

M

D0 D2 D1 D1 D2 A A

LetI_{k +1}N denote the set of (indices of) good particles; ifI_{k +1}N = ∅the algorithm is stopped. Otherwise,

for each non empty modej, resampleNj particles according to

Ψj_{k +1}(η_{k +1}N ), and set b ηN k +1= P j∈Jk +1ω j,N k +1ηb j,N k +1, withηb j,N k +1= 1 Nj P κ∈bJj_{k +1}δξbκ_{k +1}and ωj,N_{k +1}=η_bN_{k +1}(g_{k +1}j ) =η N k +1(g j k +1) ηN k +1(gk +1) .

The total number of particles isNb_{k +1}=P_j∈J k +1N

SELECTION/R

ESAMPLING

(Nk +1, ωkN, ξk +1) → ( bNk +1, ωk +1N , bξk +1)

Select only the particlesξκ_{k +1}having reached the desired setBk +1;

R

n

M

D0 D2 D1 D1 D2 A A

LetI_{k +1}N denote the set of (indices of) good particles; ifI_{k +1}N = ∅the algorithm is stopped. Otherwise,

for each non empty modej, resampleNj particles according to

Ψj_{k +1}(η_{k +1}N ), and set b ηN k +1= P j∈Jk +1ω j,N k +1ηb j,N k +1, withηb j,N k +1= 1 Nj P κ∈bJj_{k +1}δξbκ_{k +1}and ωj,N_{k +1}=η_bN_{k +1}(g_{k +1}j ) =η N k +1(g j k +1) ηN k +1(gk +1) .

The total number of particles isNb_{k +1}=P_j∈J k +1N

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

A

SYMPTOTIC

B

EHAVIOUR Asymptotic Behaviour

Law of Large Numbers Central Limit Theorem

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj :=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj :=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj :=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj:=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj:=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj:=Nj/Nare ”preserved”

ASYMPTOTIC

BEHAVIOUR

Now, we are addressing the asymptotic behaviour of our estimator as

N → ∞.

To obtain a law of large numbers, we followed the [Del Moral 2004]’s approach based on a martingale decomposition,

and for the central limit theorem, we used a CLT for triangular arrays developed in [Le Gland & Oudjane, 2006]

Before the statement of the two theorems, we need some notations:

I Ninf=infj∈MNj

I letN → ∞in such a way that eachρj:=Nj/Nare ”preserved” I this implies thatNinf→ ∞.

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

A

SYMPTOTIC

B

EHAVIOUR

Asymptotic Behaviour

Law of Large Numbers

Central Limit Theorem

MAIN

THEOREMS: LAW OF

LARGE

NUMBERS

THEOREM(LAW OFLARGENUMBERS)

For anyn ≥ 0and any bounded functionf, we have

E(γnN(f )1{N_{n>0}) = γ}n(f ), sup f :kf k∞≤1 E  [_1{N n>0}γ N n(f ) − γn(f )]2  ≤b 2_(n) Ninf .

LLNFOR NORMALIZED MEASURES

For anyn ≥ 0and any bounded functionf, we have

sup f :kf k∞≤1    E h ηN_n(_{f )1{N} n>0} − ηn(f ) i   ≤ b(n) 2 Ninf +a(n)e−Ninf/c(n)_. sup f : kf k∞≤1  E   1{N_n>0}η N n(f ) − ηn(f )    21/2 ≤ 2b 2_(n) |γn(1)|2Ninf .

MAIN

THEOREMS: LAW OF

LARGE

NUMBERS

THEOREM(LAW OFLARGENUMBERS)

For anyn ≥ 0and any bounded functionf, we have

E(γnN(f )1{N_{n>0}) = γ}n(f ), sup f :kf k∞≤1 E  [_1{N n>0}γ N n(f ) − γn(f )]2  ≤b 2_(n) Ninf .

LLNFOR NORMALIZED MEASURES

For anyn ≥ 0and any bounded functionf, we have

sup f :kf k∞≤1    E h ηN_n(_{f )1{N} n>0} − ηn(f ) i   ≤ b(n) 2 Ninf +a(n)e−Ninf/c(n)_. sup f : kf k∞≤1  E   1{N_n>0}η N n(f ) − ηn(f )    21/2 ≤ 2b 2_(n) |γn(1)|2Ninf .

MAIN

THEOREMS: LAW OF

LARGE

NUMBERS

THEOREM(LAW OFLARGENUMBERS)

For anyn ≥ 0and any bounded functionf, we have

E(γnN(f )1{N_{n>0}) = γ}n(f ), sup f :kf k∞≤1 E  [_1{N n>0}γ N n(f ) − γn(f )]2  ≤b 2_(n) Ninf .

LLNFOR NORMALIZED MEASURES

For anyn ≥ 0and any bounded functionf, we have

sup f :kf k∞≤1    E h ηN_n(_{f )1{N} n>0} − ηn(f ) i   ≤ b(n) 2 Ninf +a(n)e−Ninf/c(n)_. sup f : kf k∞≤1  E   1{N_n>0}η N n(f ) − ηn(f )    21/2 ≤ 2b 2_(n) |γn(1)|2Ninf .

MAIN

THEOREMS: LAW OF

LARGE

NUMBERS

THEOREM(LAW OFLARGENUMBERS)

For anyn ≥ 0and any bounded functionf, we have

E(γnN(f )1{N_{n>0}) = γ}n(f ), sup f :kf k∞≤1 E  [_1{N n>0}γ N n(f ) − γn(f )]2  ≤b 2_(n) Ninf .

LLNFOR NORMALIZED MEASURES

For anyn ≥ 0and any bounded functionf, we have

sup f :kf k∞≤1    E h ηN_n(_{f )1{N} n>0} − ηn(f ) i   ≤ b(n) 2 Ninf +a(n)e−Ninf/c(n)_. sup f : kf k∞≤1  E   1{N_n>0}η N n(f ) − ηn(f )    21/2 ≤ 2b 2_(n) |γn(1)|2Ninf .

PLAN

1

INTRODUCTION

Motivation

Switching jump diffusion Splitting technique Some issues

2

F

EYNMAN-KAC FORMULATION

Multilevel Feynman-Kac distributions Dynamical evolution

3

S

AMPLING PER

MODE ALGORITHM

Particle Methods

Sampling per Mode algorithm

4

A

SYMPTOTIC

B

EHAVIOUR

Asymptotic Behaviour Law of Large Numbers

Central Limit Theorem 5

CONCLUSION

MAIN

THEOREMS: CENTRAL

LIMIT

THEOREM

THEOREM(CENTRALLIMITTHEOREM)

LetN → ∞in such a way thatρj =Nj/Nare ”preserved” for allj ∈ M. Then, the random variable

√ N_1{N n+1>0}γ N n+1(1) − P(Tn<T ) 

converges in law to a Gaussian random variable with mean0and variance

Wn+1, where Wn+1 P2(Tn<T ) = n+1 X q=0 Ωq  1 Pq − 1  + n+1 X q=0 Ωq Pq " b η_q [∆n q◦ π]2
b ηq2 ∆nq◦ π
− 1 # , with Ωq= X j∈M (ω_q−1j )2ρ−1_j =1 + χ2(ωq−1, ρ), and ∆n_q(_{t, z) = P(T}n≤ T |Tq=t, ZTq =z).

MAIN

THEOREMS: CENTRAL

LIMIT

THEOREM

THEOREM(CENTRALLIMITTHEOREM)

LetN → ∞in such a way thatρj =Nj/Nare ”preserved” for allj ∈ M. Then, the random variable

√ N_1{N n+1>0}γ N n+1(1) − P(Tn<T ) 

converges in law to a Gaussian random variable with mean0and variance

Wn+1, where Wn+1 P2(Tn<T ) = n+1 X q=0 Ωq  1 Pq − 1  + n+1 X q=0 Ωq Pq " b η_q [∆n q◦ π]2
b ηq2 ∆nq◦ π
− 1 # , with Ωq= X j∈M (ω_q−1j )2ρ−1_j =1 + χ2(ωq−1, ρ), and ∆n_q(_{t, z) = P(T}n≤ T |Tq=t, ZTq =z).

MAIN

THEOREMS: CENTRAL

LIMIT

THEOREM

THEOREM(CENTRALLIMITTHEOREM)

LetN → ∞in such a way thatρj =Nj/Nare ”preserved” for allj ∈ M. Then, the random variable

√ N_1{N n+1>0}γ N n+1(1) − P(Tn<T ) 

converges in law to a Gaussian random variable with mean0and variance

Wn+1, where Wn+1 P2(Tn<T ) = n+1 X q=0 Ωq  1 Pq − 1  + n+1 X q=0 Ωq Pq " b η_q [∆n q◦ π]2
b ηq2 ∆nq◦ π
− 1 # , with Ωq= X j∈M (ω_q−1j )2ρ−1_j =1 + χ2(ωq−1, ρ), and ∆n_q(_{t, z) = P(T}n≤ T |Tq=t, ZTq =z).

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties.

However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent,

I or aggregate the modes in order to decrease the complexity (for large scale distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

CONCLUSION

The ”sampling per mode” algorithm has ”good” properties. However, to gain more time of simulation, we can:

I use importance sampling technique to make rare switches more frequent, I or aggregate the modes in order to decrease the complexity (for large scale

distributed hybrid systems).

Thus, we need to extend the previous results.

A better comprehension of the expression ofWn+1could help the choice of theNj _{regarding the cost of the algorithm.}

This algorithm is implemented in a software developed by National Aerospace Laboratory (NLR) and used to evaluate the safety

characteristics of an arbitrary (new) operational Air Traffic Management concept [Blom, 2009].

R ´

EF

ERENCES

´

Henk Blom, Bert Bakker, and Jaroslav Krystul.,

Rare event estimation for a large-scale stochastic hybrid system with air traffic application. Rare Event Simulation using Monte Carlo Methods, pp. 194–214. Wiley, 2009.

Krystul, J.,

Modelling of Stochastic Hybrid Systems with Applications to Accident Risk Assessment (2006). PhD Dissertation: University of Twente.

Del Moral, P. (2004).

Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004). Springer-Verlag.

Le Gland, F. and Oudjane, N.,

A sequential particle algorithm that keeps the particle system alive. (2006) in Stochastic Hybrid Systems : Theory and Safety Critical Applications, pp. 351-389, Springer, 2006

F. C ´erou, P. Del Moral, F. Le Gland and P. Lezaud,

Genetic genealogical models in rare event analysis, ALEA, volume 1, paper 8, 2006