A Central Limit Theorem for Fleming-Viot Particle Systems Application to the Adaptive Multilevel Splitting Algorithm

Transcription

1 A Central Limit Theorem for Fleming-Viot Particle Systems Application to the Algorithm F. Cérou 1,2 B. Delyon 2 A. Guyader 3 M. Rousset 1,2 1 Inria Rennes Bretagne Atlantique 2 IRMAR, Université de Rennes 1 3 Université Pierre et Marie Curie, Paris London 218

2 Rare event simulation Dynamical rare event problem s Y s denotes a Markov continuous (diffusion) process in F ; A and B denote two sets in F. Problem: Simulate a reactive trajectory, i.e. that starts from y close to A and reaches B before going back to A. Problem: Rare event simulation p 1.

3 Rare event simulation Use of a Reaction Coordinate Reaction coordinate: one-dimensional continuous function ξ : F R Notation: {ξ > t} :={y F, ξ(y) > z} S t := inf(s ξ(y s ) = t) S A := inf(s Y s A) For simplicity assume A {ξ < 1}, B {ξ > 1}, and ξ(y ) =. so that the considered rare event becomes {S 1 > S A }, p = P(S 1 > S A )

4 Rare event simulation Reaction Coordinate Define the score of each trajectory by score = τ = sup ξ(y s ) s S A Multilevel/Importance Splitting: clone the trajectories with highest score, kill the other ones [Kahn and Harris (1951)].

5 AMS algorithm Definition (AMS algorithm) At step j of the algorithm, we define a particle system (Y 1,j,, Y N,j ) [C(R +, F )] N defined by the following set of rules. Initialization j = : consider N i.i.d. particles with law (Y s ) s SA. At step j 1, kill the particle N j with minimal score τ j Splitting: the killed particle is given the state of particle M j, uniformly picked among the (N 1) remaining particles. The particle is then resampled starting from the entrance time in level τ j : S M j,j 1 τ j. Set j 1 j. And so on until all particles reach {ξ > 1}.

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14 AMS yields Unbiased Estimator of Unormalized Quantity Denote by J t the first iteration of branchings/steps so that all particles reach {ξ > t}. Denote empirical measure of particles paths at iteration J t N η N t := 1 N j=1 δ Y n,jt s Unbiased Estimates: For any pathwise observable (test function) ϕ, we have γ t (ϕ) := E[ϕ(Y )1 St<S A ]= E[(1 1/N) Jt η N t (ϕ)]. Weak Law of Large Number? Central Limit Theorem?

15 Mapping to a Fleming-Viot particle system The level-indexed process Key Idea: Assuming Y is strong Markov, define a new Markov process called the level-indexed process in E := F { }, the state space enhanced by a cemetary, by setting (NB: ξ(y ) = ): { X t := Y St if S t < S A X t := else Can be extended to a time-homogenous process by setting X h := Y Sξ(x )+h if S ξ(x )+h < S A Can be extended to the pathwise case by setting E := C(R +, F ) { } X t := Y s St if S t < S A

16 Mapping to a Fleming-Viot particle system The level-indexed process is a killed Process Lemma The level-indexed process X = (X t ) t is a Markov process in F { }, where the cemetary / F is absorbing (a trap). Moreover, the killing time τ satisfies τ = inf{t, X t = } = sup ξ(y s ) = score. s S A Moreover the conditional distribution on state space F of interest becomes η t = L(X t τ > t) = L(Y S t < S A ), and the probability of the rare event p t = P(S t < S A ) = P(τ > t).

17 Mapping to a Fleming-Viot particle system Fleming-Viot Particle System Definition (Fleming-Viot particle system) The Fleming-Viot particle system (X 1 t,, X N t ) t [,T ] is the Markov process with state space F N defined by the rules Initialization: consider N i.i.d. particles X 1,..., X N i.i.d. η, Evolution and killing: each particle evolves independently according to the law of the underlying Markov process X until one of them hits, Splitting: the killed particle is taken from, and is given instantaneously the state of one of the (N 1) other particles (randomly uniformly chosen). and so on until final time T.

18 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

19 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

20 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

21 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

22 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

23 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

24 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

25 Mapping to a Fleming-Viot particle system Lemma The AMS algorithm mapped with the level-indexed process and stopped at iteration J t (the first iteration when all particles reach {ξ > t}) is actually a Fleming-Viot particle system. Proof Picture.

26 Mapping to a Fleming-Viot particle system Some Important Estimators Normalized distribution proba of rare event un-normalized distribution η t = L(X t X t ) p t = P(X t ) γ t := p t η t Empirical Empirical Empirical ηt N = 1 N N i=1 δ Xt i t = ( 1 1 ) NNt N, γt N = pt N ηt N. N T := J T /N is the average number of branchings per particle system.

27 Main Results Regularity Assumption For x F { }, t T, consider the sub-markovian semi-group Q t (ϕ)(x) := E[ϕ(X t )1 τ >t X = x]. Assumption (Non-synchronous jumps) For any initial condition X t = x F and any ϕ C b (F ): (i) the jump times of the càdlàg version of the martingale process t L t := Q T t (ϕ)(x t ) have an atomless distribution: P(L t L t X = x) = t. (ii) The killing time has also an atomless distribution.

28 Main Results Non-explosion Assumption The non-synchronous jumps Assumption is morally equivalent to: martingale jumps and branchings in the Fleming-Viot system are never simultaneous. In addition we ask: Assumption (Non-explosion) The Fleming-Viot system is non-explosive in the sense that the number of branching at any finite time is almost surely finite P(N T < + ) = 1.

29 Main Results Example with Hard Obstacle (The originality of our result!) t X t process in E, and let F be an open domain with boundary F = F \ F. Let τ be the hitting time of E \ F. Proposition (i) The hitting time τ on boundary has an atomless distribution. (ii) The process starting from boundary F exits in open E \ F immediately almost surely. Then Assumption no synchronous jumps is fullfilled. Proposition (Grigorescu and Kang, 212) Assume that F is open and bounded in R d with smooth boundary F, and that X is a diffusion with smooth and uniformly elliptic coefficients. Then Assumption non explosion holds true, as well as Assumption no synchronuous jumps.

30 Main Results Central Limit Theorems Theorem Under Assumptions non-synchronous jumps and non-explosion, for any ϕ C b (F ) one has the convergence in distribution ( ) N γt N (ϕ) γ D T (ϕ) N (, N σ2 T (ϕ)), where by definition σ 2 T (ϕ) =p2 T Var η T (ϕ) p 2 T ln(p T ) η T (ϕ) 2 2 T Var ηt (Q T t (ϕ))p t dp t.

31 Main Results Central Limit Theorems By Slutsky Lemma: Corollary Under Assumptions non-synchronous jumps and non-explosion, for any ϕ C b (F ), one has the convergence in distribution ( ) N ηt N (ϕ) η D T (ϕ) N (, N σ2 T (ϕ η T (ϕ))/pt 2 ). Besides, where ( ) N pt N p D T N (, N σ2 ), σ 2 = σ 2 T (1 F ) = p 2 T ln(p T ) 2 T Var ηt (Q T t (1 F ))p t dp t.

32 Main Results Remarks on Asymptotic Variances The variance is the limit of the CLT of a discrete time Fleming-Viot particle system (CLT in Del Moral s book). Bounds on the probability estimator p 2 T log(p T ) σ 2 2p T (1 p T ) + p 2 T log(p T ) Lower bound: (soft) killing time independent of the state. Upper bound: killing time mainly dependent on the initial condition. Dominant term 2p T (1 p T ) is twice the naive Monte Carlo variance.

33 Proof Stochastic calculus with jumps Recall that one can integrate with respect ot semi-martingales X = montonous processes + martingales as follows: Y t dx t Y t (X t+dt X t ) We then have the chain rule d(x t Y t ) = Y t dx t + X t dy t + d[x, Y ] t where t [X, X ] t is an increasing process, bilinear with respect to vector space structure on X called the quadratic variation. Broadly speaking [X, X ] t = P lim t i+1 t i (X ti+1 X ti ) 2 If X is monotonous, [X, X ] t is the sum of the squares of the jumps. i

34 Proof Stochastic calculus with jumps If t M t is a (local) martingale, then t M 2 t [M, M] t is again a local martingale. In the presence of jumps there ar plenty of quadratic variation -like increasing processes t i(m) t such that t M 2 t i(m) t is a local martingale. For instance there is a unique i(m) t =< M, M > t which is predictable. Example: let t M t { 1, 1} be Poisson-like Markov + Martingale random walk process jumping up or down with proba 1/2 at indep. expo. times. Then [M, M] t = jumps Var(jump) = 1 < M, M > t = dt

35 Proof CLT for martingales with jumps Theorem (Martinagle CLT (Ethier-Kurtz)) On a filtered probability space, let t mt N denote a sequence of càdlàg local martingales indexed by N 1. Assume moreover that (i) m N D µ, where µ is a given probability on R. N + (ii) Vanishing jumps: One has lim N + E[sup t [,T ] m N t mt N 2 ] =. (iii) For each N, there exists an increasing càdlàg process t it N such that ( ) t mt N m N 2 it N is a local martingale. (iv) Vanishing jump: The process t ] it N satisfies lim N + E [sup t [,T ] it N i Nt =.

36 Proof CLT for martingales with jumps Theorem (Martinagle CLT (Ethier-Kurtz)) For the increasing càdlàg process t it N (with vanishing jumps) such that ( ) t mt N m N 2 it N is a local martingale: (v)!! Main Assumption!!: There is a cont. and incr. det. function t i t s. t., t [, T ], it N P i t. N + Then (m N t ) t [,T ] converges in law (under the Skorokhod topology) to (M t ) t [,T ], where M µ and (M t M ) t [,T ] is a Gaussian martingale, independent of M, with independent increments and variance function i t (time changed Brownian motion).

37 Proof CLT for martingales with jumps In short, we need to construct martingales of order 1/ N from the particle system and ensure the convergence of a quadratic variation of those martingales of order 1/N.

38 Proof Overview of the proof Key object: the càdlàg martingale t γ N t (Q) := γ N t ( ) Q T t (ϕ). Initial condition treated separately (easy). We will handle the distribution of γt N (Q) in the limit N by using a Central Limit Theorem for continuous time martingales. Not straightforward: the convergence of the quadratic variation N[γ N (Q), γ N (Q)] t is difficult (lots of IPPs!!).

39 Proof Martingale decomposition [Villemonais 214] The key martingale decomposition is the following: γ N t (Q) = γ N (Q) + 1 N t p N u (dm u + dm u ). With M t := 1 N N n=1 Mn t and M t := 1 N N n=1 Mn t. M n t is the martingale contribution except for branching times of particle n. M n t is the martingale contribution at branchings only of particle n. No ambiguity, natural way to do this.

40 Proof Orthogonality The 2N martingales {M n t, M m t } 1 n,m N are mutually orthogonal. More specifically (i) [M, M] t is a local martingale, (ii) [M, M] t = 1 N N [M n, M n ] t, n=1 (iii) Moreover, if we note the intermediate quadratic variation (M, M) t = 1 N N [M n, M n ] t, n=1 then the process [M, M] t (M, M) t is also a local martingale.

41 Proof Ingredient (i): A key formula Lemma The quadratic variation of martingales associated with the particles dynamics outside branchings can be related to through the key formula γ N t (Q 2 ) = γ N t ([Q T t (ϕ)] 2 ) p N t d(m, M) t = dγ N t (Q 2 ) + Martingale

42 Proof Ingredient (ii): L 2 apriori estimates Proposition (Villemonais 214, CDGR 217) For any ϕ D, we have [( ) ] E γt N (ϕ) γ T (ϕ) 2 6 ϕ 2 N. Proof. γ N T (ϕ) γ T (ϕ) = 1 N T p N t dm t + 1 N T + γ N (Q T ϕ) γ (Q T ϕ), p N t dm t (i) Initial condition is OK by independence.

43 Proof L 2 estimates (ii) M-terms. Using Ito s isometry and d[m, M] t 4 ϕ 2 dn t, we obtain [ ( T ) 2 ] [ T ] E pt N ( ) dm t = E p N 2d[M, t M]t 4 ϕ 2 1 ( ) 1 1 2(j 1) N N 4 ϕ 2. (iii) M-terms. In the same way, applying Ito s isometry and the key formula pt N d(m, M) t = dγt N (Q 2 ) + Martingale, we get [ ( T ) 2 ] [ T ] E pt N ( ) dm t = E p N 2d[M, t M]t [ T E j=1 ] [ ] pt N d(m, M) t = E γt N (Q2 ) ϕ 2.

44 Proof INgredient (iii): Time uniform a priori estimate of p N t Lemma One has sup t [,T ] pt N P p t N. Proof. Independent of the context.t p t is continuous on [, T ] by construction, it is cleat that t p N t is decreasing for all N 2. The Lemma results of last Proposition and a from a probabilistic version of Second Dini (or Pólya) theorem: if a sequence of monotone functions converges pointwise on a compact interval and if the limit function is also continuous, then the convergence is uniform on that interval.

45 Proof Increasing Process in general CLT In order to use martingale CLT, we need a càdlàg increasing process it N such that (γ N (Q) t ) 2 it N is a martingale (quadratic variation - like). After tedious computations and trials and errors we chose a quadratic variation with only the branching jumps integrated. i N t = t n=1 k=1 ( p N u ) 2 d(m, M) u + 1 N t ( ) p N 2dRu u. t NB: with the rest term is O(1/N) with ( N + (1 ) R t = 1 2 N Var (n) η τ n,k Var η N u (Q)pN u dp N u (Q) Var η N (Q) τ n,k ) 1 t τn,k.

46 Proof Integration by parts formulas Let t zt N be any càdlàg semi-martingale, c > a deterministic constant, and assume for any branching time τ j, j 1: If zτ N j c/n,then t p N s dz N s = p N t z N t z N If z N c(1 1/N) j, τ j t z N s (p N s ) 1 dp N s = If z N τ j c(1 1/N) j /N, t t z N s dp N s z N s d ln p N s + O(1/N). + O(1/N). t z N s d ln p N s = z N t ln p N t t ln p N s dz N s + O(1/N).

47 Proof Integration by parts for i N t Using the intergation by parts formula abvoe and the key formula : Lemma The increasing process it N can be integrated by parts and be rewritten as it N = pt N γt N ( Q 2 ) γ N ( Q 2 ) [ + γt (Q)] N 2 ln pt N 2 t γ N u (Q 2 )dp N u + O ( 1 N ).

48 Proof Convergence of i N t and final proof of the CLT By the non synchronous jump Assumption all the vanishing jump assumptions in the martingale CLT are verified. The only remaining remaining part to prove is the following: Proposition For any t [, T ], one has i N t = p N t γ N t i N t ( Q 2 ) γ N P N i t(ϕ). where ( Q 2 ) +[ γ N t (Q)] 2 ln p N t 2 t γ N u (Q 2 )dp N u +O ( 1 N t i t (ϕ) = p t γ t (Q 2 ) γ (Q 2 ) + [γ t (Q)] 2 ( ln p t 2 γ u Q 2 ) dp u.

49 Proof Proof of Convergence of i N t After some calculations, all but one term can be treated using the L 2 a priori convergence estimate. The only remaing problem is the following: i N t i t (ϕ) =easy converging terms with L 2 -estimate and IPP+ 2 t (p N u p u ) d γ N u (Q 2 ) It is difficult to prove its convergence to because the L 2 estimate is only pointwise. Hence handling the integrator is cumbersome!!.

50 Proof Convergence of i N t This difficult term is then treated as follows. Use the key formula t t (pu N p u ) d γu N (Q 2 ) = (pu N p u )pu N d(m, M) u +O ( 1 ) N Since (M, M) is an increasing process, it comes t ( t (pu N p u )pu N d(m, M) u sup pu N p u u The key formula back again implies: [ t ] [ ] E pu N d(m, M) u = E γt N (Q 2 ) ϕ 2. The a priori uniform estimate implies convergence of sup u p N u p u. p N u d(m, M) u ).

51 Proof Arxiv Bernard Delyon, Frédéric Cérou, Arnaud Guyader, Mathias Rousset. A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing. 217 Bernard Delyon, Frédéric Cérou, Arnaud Guyader, Mathias Rousset. Asymptotic normality of the AMS algorithm. 218