On the Robustness of the Snell envelope

Transcription

1 On the Robustness of the Snell envelope Pierre Del Moral, Peng Hu, adia Oudjane, Bruno Rémillard To cite this version: Pierre Del Moral, Peng Hu, adia Oudjane, Bruno Rémillard On the Robustness of the Snell envelope [Research Report] RR-7303, 2010, pp35 <inria v1> HAL Id: inria Submitted on 28 May 2010 v1, last revised 15 Jan 2011 v4 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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

2 ISTITUT ATIOAL DE RECHERCHE E IFORMATIQUE ET E AUTOMATIQUE On the Robustness of the Snell envelope Pierre Del Moral, Peng Hu, adia Oudjane, Bruno Rémillard 7303 May 2010 Stochastic Methods and Models apport de recherche ISS ISR IRIA/RR FR+EG

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4 On the Robustness of the Snell envelope Pierre Del Moral, Peng Hu, adia Oudjane, Bruno Rémillard Theme : Stochastic Methods and Models Applied Mathematics, Computation and Simulation Équipe-Projet ALEA Rapport de recherche n 7303 May pages Abstract: We analyze the robustness properties of the Snell envelope bacward evolution equation for discrete time models We provide a general robustness lemma, and we apply this result to a series of approximation methods, including cut-off type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the Stochastic Mesh method of Broadie-Glasserman In each situation, we provide non asymptotic convergence estimates, including L p -mean error bounds and exponential concentration inequalities In particular, this analysis allows us to recover existing convergence results for the quantization tree method and to improve significantly the rates of convergence obtained for the Stochastic Mesh estimator of Broadie-Glasserman In the final part of the article, we propose a genealogical tree based algorithm based on a mean field approximation of the reference Marov process in terms of a neutral type genetic model In contrast to Broadie-Glasserman Monte Carlo models, the computational cost of this new stochastic particle approximation is linear in the number of sampled points Key-words: Snell envelope, optimal stopping, American option pricing, genealogical trees, interacting particle model Centre IRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération Talence cedex, France, PierreDel- Moral@inriafr Centre IRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération Talence cedex, France, PengHu@inriafr EDF R & D Clamart adia OUDJAE <nadiaoudjane@edffr> HEC Montréal brunoremillard@hecca Centre de recherche IRIA Bordeaux Sud Ouest Domaine Universitaire - 351, cours de la Libération Talence Cedex Téléphone :

5 Sur la robustesse de l enveloppe de Snell Résumé : On analyse les propriétés de la robustesse de l enveloppe de Snell pour les modèles de temps discret On fournit un lemme de robustesse générale, et on applique ce résultat à une série de méthodes d approximation, y compris les approximations de type Cut-off, Euler discrétization schémas, modèles d interpolation, modèles de quantification et la méthode stochastique de Broadie- Glasserman Dans chaque situation, on fournit des estimations de convergence non asymptotique, y compris les limites d erreur de norme L p et les inégalités de concentration exponentielle En particulier, cette analyse permet de récupérer les résultats de convergence existant pour la méthode de quantification et d améliorer considérablement le taux de convergence obtenu pour l estimateur de maillage stochastique du Broadie-Glasserman Dans la dernière partie de l article, on propose un algorithme d arbre généalogique basé sur une approximation mean-field de la référence de processus de Marov en termes d un modèle génétique de type neutre Contrairement aux modèles Broadie- Glasserman Monte-Carlo, le coût de calcul de cette nouvelle approximation stochastique particulaire est linéaire par rapport le nombre de points échantillonnés Mots-clés : enveloppe de Snell, arrêt optimal, évaluation de l option américain, arbre génétique, modèle particulaire d interaction

6 On the Robustness of the Snell envelope 3 1 Introduction The calculation of optimal stopping time in random processes based on a given optimality criteria is one of the major problems in stochastic control and optimal stopping theory, and particularly in financial mathematics with American option pricing and hedging In discrete time setting, these problems are related to Bermuda options and are defined in terms of given real valued stochastic process Z 0 n, adapted to some increasing filtration F = F 0 n that represents the available information at any time 0 n For any {0,,n}, we let T be the set of all stopping times τ taing values in {,,n} The Snell envelope of Z 0 n, is the stochastic process U 0 n defined for any 0 < n by the following bacward equation U = Z EU +1 F, with the terminal condition U n = Z n The main property of this stochastic process is that U = sup EZ τ F = EZ τ F with τ = min { l n : U l = Z l } T τ T 11 At this level of generality, in the absence of any additional information on the sigma-fields F n, or on the terminal random variable Z n, no numerical computation of the Snell envelop are available To get one step further, we assume that F n n 0 is the natural filtration associated with some Marov chain X n n 0 taing values in some sequence of measurable state spaces E n, E n n 0 We let η 0 = LawX 0 be the initial distribution on E 0, and we denote by M n x n 1, dx n the elementary Marov transition of the chain from E n 1 into E n We also assume that Z n = f n X n, for some collection of non negative measurable functions f n on E n In this situation 1, the computation of the Snell envelope amounts to solve the following bacward functional equation u = f M +1 u +1, 12 for any 0 < n, with the terminal value u n = f n In the above displayed formula, M +1 u +1 stands for the measurable function on E defined for any x E by the conditional expectation formula M +1 u +1 x = M +1 x, dx +1 u +1 x +1 = Eu +1 X +1 X = x E +1 One can chec that a necessary and sufficient condition for the existence of the Snell envelope u 0 n is that M,l f l x < for any 1 l n, and any state x E To chec this claim, we simply notice that f u f + M +1 u +1 = f u M,l f l 13 l n Even if it loos innocent, the numerical solving of the recursion 12 often requires extensive calculations The central problem is to compute the conditional expectations M +1 u +1 on the whole state space E, at every time step 1 Consult the last paragraph of this section for a statement of the notation used in this article

7 On the Robustness of the Snell envelope 4 0 < n For Marov chain models taing values in some finite state spaces with a reasonably large cardinality, the above expectations can be easily computed by a simple bacward inspection of the whole realization tree that lists all possible outcomes and every transition of the chain In more general situations, we need to resort to some approximation strategy Most of the numerical approximation schemes amount to replacing the pair of functions and Marov transitions f, M 0 n by some approximation model f, M 0 n on some possibly reduced measurable subsets Ê E We let û be the Snell envelope on Ê of the functions f associated with the sequence of integral operators M from Ê 1 into Ê û = f M +1 û Using the elementary inequality a a b b a b + a b which is valid for any a, a, b, b R, one readily obtains, for any 0 < n u û f f + M +1 u +1 M +1 û +1 M +1 u +1 M +1 û +1 M +1 M +1 u +1 + M +1 u +1 û +1 Iterating the argument, one finally gets the following robustness lemma Lemma 11 For any 0 < n, on the state space Ê, we have that u û n l= M,l f l f n 1 l + l= We quote a direct consequence of the above lemma sup x E E u x û x := u û be M,l M l+1 M l+1 u l+1 n f l f l bel + l= n l=+1 M l M l u l bel 1 Lemma 11 provides a simple and natural way to analyze the robustness properties of the Snell equation 12 with respect to the pair parameters f, M It also provides a simple framewor to analyze in unison most of the numerical approximation models currently used in practice based on the approximation of the dynamic programming formula 12 by 14, including cut-off techniques, Euler type discrete time approximations, quantization tree models, interpolation type approximations, and Monte Carlo importance sampling approximations otice that this framewor could also apply to approximations based on regression methods such as proposed in [?] and [?], however it does not directly apply to approximations based on estimation of the optimal stopping time τ using the characterization 11 to deduce the Snell envelope U, such as the Longstaff-Schwartz algorithm see [?] We emphasize that this non asymptotic robustness analysis also allows to combine in a natural way several approximation model For instance, under appropriate tightness conditions, cut-off techniques can be used to reduce the numerical analysis of 12 to compact state spaces Ên and bounded functions f n In the same line of ideas, in designing any type of Monte Carlo approximation models, we can suppose that the transitions of the chain X n are nown based on a preliminary analysis of Euler type approximation models

8 On the Robustness of the Snell envelope 5 All the series of applications presented above are in discussed in Section 2, in terms robustness properties of the Snell envelope In each situation, we provide a stochastic model that expresses the approximation scheme in terms of some pair of functions and transitions f n, M n n 0 We also deduce from Lemma 11 non asymptotic convergence theorems, including L p -mean error bounds and related exponential inequalities for the deviations of Monte Carlo type approximation models In the present article, three type of Monte Carlo particle models are developed: The first one is the importance sampling type stochastic mesh method introduced by M Broadie and P Glasserman in their seminal paper [?] see also [?], for some recent refinements As any full Monte Carlo type technique, the main advantage of their approach is that it applies to high dimensional American options with a finite possibly large, number of exercise dates In [?], the authors provide a set of conditions under which the Monte Carlo importance scheme converges as the computational effort increases The wor of the algorithm is quadratic in the number of sampled points in the stochastic mesh In this context, in Section 32, we provide new non asymptotic estimates, including L p -mean error bounds and exponential concentration inequalities To give a flavor of these results, we assume that there exists some collection of probability measures η n such that M n x, η n, for any n 1 and any x E n 1 We further assume that the functions f n, as well as the Radon iodym derivatives, R n x, y = dmnx, dη n y, are computable pointwise and it is easy to sample independent and identically distributed random variables ξn i i 1 with common distribution η n In this situation 12 can be rewritten as follows u x = f x η +1 dy R +1 x, y u +1 y 15 E +1 for any 0 < n, and any x E We let û be the solution of the bacward equation 15 defined as above on the whole state space E, by replacing the measures η +1 by their -empirical approximations η +1 := 1 i=1 δ ξ+1 i otice that the bacward recursive calculation of the functions û on the stochastic mesh ξ i 1 i is given below û ξ i = f ξ i 1 R +1 ξ i, ξ j +1 û +1ξ j for any 0 < n, any 1 i j=1 Theorem 12 For any p 1, 0 n, and any x E we have E u x û x p 1 p 2ap { M,l x, dy η l+1 [R 1 l+1 y,u l+1 p ]} p l<n with the smallest even integer p greater than p, and the constants ap p 0 given below p 0 a2p 2p = 2p p 2 p and a2p + 1 2p+1 = 2p + 1 p+1 p + 1/2 2 p+1/2 with q p = q!/q p!, for any 1 p q 17

9 On the Robustness of the Snell envelope 6 The second type of Monte Carlo particle model discussed in this article is a slight variation of the Broadie-Glasserman model The main advantage of this new strategy comes from the fact that the sampling distribution η n can be chosen as the distribution of the random states X n of the reference Marov chain, when the Radon iodym derivatives, R n x, y = dmnx, dη n y is not nown explicitely We only assume that the Marov transitions M n x, are absolutely continuous with respect to some measures λ n on E n, with positive Radon iodym derivatives H n x, y = dmnx, dλ n y Using the fact that η n λ n, with dηn dλ n y = η n 1 H n, y we notice that the bacward recursion 12 can be rewritten as follows u x = f x η +1 dy E +1 H +1 x, y η H +1, y u +1y 18 for any 0 < n, and any x E Arguing as before, we let û be the solution of the bacward equation 18 defined as above on the whole state space E, by replacing the measures η by the occupation measure η = 1 i=1 δ ξ i associated with independent copies ξ = ξ i 1 i of the Marov chain X, from the origin = 0 up to the final time horizon = n Hence, we recover a similar approximation to 16, except that the Radon iodym derivatives, R +1 ξ i, ξj +1 is replaced by the approximation, ˆR +1 ξ i, ξj +1 = H +1 ξ i, ξj +1 i=1 H +1ξ i, ξj +1 1 The stochastic analysis of this particle model follows essentially the same line of arguments as the one of the Broadie-Glasserman model For further details on the convergence analysis of this scheme, with the extended version of Theorem 12 to this class of models, we refer the reader to the second part of Section 32 Several rather crude estimates can be derived from these L p -mean error bounds For instance, let us suppose that R n x, y r n y, for any x E n 1 and some measurable functions r n L p E n, η n, for any p 1 In this situation sup E u x û x p 1 p x E 2 ap l<n η l+1 r l+1 u l+1 p 1/p Using 13 and Hölder s inequalities, we prove that the rhs in the above display is finite as soon as f L q E, η l M l,, for any q 1 and l n When the functions r n and f n are bounded, we have sup E u x û x p 1 p ap b n with b n 2 x E l<n r l+1 u l+1 for some finite constant b n <, whose values do not depend on the parameter p In this situation, we deduce the following exponential concentration inequality sup P u x û x > b n + ǫ exp ǫ 2 /2b n 2 19 x E

10 On the Robustness of the Snell envelope 7 This result is a direct consequence from the fact that, for any non negative random variable U, b < st r 1 EU r 1 r ar b P U b + ǫ exp ǫ 2 /2b 2 To chec this claim, we develop the exponential to chec that t 0 E e tu bt 2 exp + bt P U b + ǫ exp 2 sup t 0 ǫt bt2 2 In the final part of the article, Section 33, we present an alternative Monte Carlo method based on the genealogical tree evolution models associated with a neutral genetic model with mutation given by the Marov transitions M n The main advantage of this new strategy comes from the fact that the computational effort of the algorithm is now linear in the number of sampled points We recall that a neutral genetic is a Marov chain with a selection/mutation transition denoted by: ξ n = ξ i n 1 i selection ξ n = ξ i n 1 i mutation ξ n+1 = ξ i n+1 1 i The neutral genetic selection simply consists in simulating independent particles ξ n i 1 i wrt the same distribution 1 1 i δ ξn i During the mutation phase, the particles explore the state space independently the interactions between the various particles being created by the selection steps according to the Marov transitions M n+1 x, dy In other terms, we have ξ n i ξn+1 i where ξn+1 i stands for a random variable with the law M n+1 ξ n, i This type of model is frequently used in biology, and genetic algorithms literature see for instance [?], and references therein An important observation concerns the genealogical tree structure of the previously defined genetic particle model If we interpret the selection transition as a birth and death process, then arises the important notion of the ancestral line of a current individual More precisely, when a particle ξ n 1 i ξn i evolves to a new location ξn, i we can interpret ξ n 1 i as the parent of ξn i Looing bacwards in time and recalling that the particle ξ n 1 i has selected a site ξj n 1 in the configuration at time n 1, we can interpret this site ξ j n 1 as the parent of ξ n 1 i and therefore as the ancestor denoted ξn 1,n i at level n 1 of ξn i Running bacwards in time we may trace the whole ancestral line ξ i 0,n ξi 1,n ξi n 1,n ξi n,n = ξi n 110 The genealogical tree model is summarized in the following synthetic picture that corresponds to the case, n = 3, 4: ξ 1 0,4 = ξ2 0,4 = ξ3 0,4 ξ2,4 1 ξ3,4 1 ξ1,4 1 = ξ2 1,4 = ξ3 1,4 ξ3,4 2 ξ2,4 2 = ξ3 2,4 ξ3,4 3 ξ 1 4,4 = ξ1 4 ξ 2 4,4 = ξ2 4 ξ 3 4,4 = ξ3 4

11 On the Robustness of the Snell envelope 8 The main advantage of this path particle model comes from the fact that the occupation measure of the ancestral tree model converges in some sense to the distribution of the path of the reference Marov chain P n := 1 1 i δ ξ 1 0,n,ξ 1 1,n,,ξi n,n P n := LawX 0,, X n It is also well nown that the Snell envelope associated with a Marov chain evolving on some finite state space is easily computed using the tree structure of the chain evolution Therefore, replacing the reference distribution P n by its -approximation P n, we define an -approximated Marov model whose evolutions are described by the genealogical tree model defined above Let û be the Snell envelope associated with this -approximated Marov chain For finite state space models, we shall prove the following result sup sup 0 n 1 i E u û ξ i,n p 1/p cp n/ for any p 1, with some collection of finite constants c p n < whose values only depend on the parameters p and n For the convenience of the reader, we end this introduction with some notation used in the present article We denote respectively by PE, and BE, the set of all probability measures on some measurable space E, E, and the Banach space of all bounded and measurable functions f equipped with the uniform norm f Given a subset A E, we set f A := sup x A fx We also denote by Osc 1 E the set of functions f with oscillations oscf = sup x,y fx fy less than 1 We let µf = µdx fx, be the Lebesgue integral of a function f BE, with respect to a measure µ PE For any p 1, we also set L p E, η the set of functions f such that η f p <, equipped with the norm f p,η = η f p 1/p We recall that a bounded integral ernel Mx, dy from a measurable space E, E into an auxiliary measurable space E, E is an operator f Mf from BE into BE such that the functions x Mfx := Mx, dyfy E are E-measurable and bounded, for any f BE In the above displayed formulae, dy stands for an infinitesimal neighborhood of a point y in E Sometimes, for indicator functions f = 1 A, with A E, we also use the notation Mx, A := M1 A x The ernel M also generates a dual operator µ µm from ME into ME defined by µmf := µmf A Marov ernel is a positive and bounded integral operator M with M1 = 1 Given a pair of bounded integral operators M 1, M 2, we let M 1 M 2 the composition operator defined by M 1 M 2 f = M 1 M 2 f Given a sequence of bounded integral operator M n from some state space E n 1 into another E n, we set M,l := M +1 M +2 M l, for any l, with the convention M, = Id, the identity operator For time homogenous state spaces, we denote by M m = M m 1 M = MM m 1 the m-th composition of a given bounded integral operator M, with m 1 In the context of finite state spaces, these integral operations coincides with the traditional matrix operations on multidimensional state spaces

12 On the Robustness of the Snell envelope 9 We also assume that the reference Marov chain X n with initial distribution η 0 PE 0 and elementary transitions M n x n 1, dx n from E n 1 into E n is defined on some filtered probability space Ω, F, P η0, and we use the notation E Pη0 to denote the expectations with respect to P η0 In this notation, for all n 1 and for any f n BE n, we have that E Pη0 {f n X n F n 1 } = M n f n X n 1 := M n X n 1, dx n f n x n E n with the σ-field F n = σx 0,, X n generated by the sequence of random variables X p, from the origin p = 0 up to the time p = n When there are no possible confusion, we write X Lp = E X p 1/p, the L p -norm of a given real valued random variable defined on some probablity space Ω, P We also use the conventions = 1, and = 0 2 Some deterministic approximation models 21 Cut-off type models We suppose that E n are topological spaces with σ-fields E n that contains the Borel σ-field on E n Our next objective is to find conditions under which we can reduce the bacward functional equation 12 to a sequence of compact sets Ê n To this end, we further assume that the initial measure η 0 and the Marov transition M n of the chain X n satisfy the following tightness property: For every sequence of positive numbers ǫ n [0, 1[, there exists a collection of compact subsets Ên E n st T η 0 Êc 0 ǫ 0 and n 0 sup x n b E n M n+1 x n, Êc n+1 ǫ n+1 For instance, this condition is clearly met for regular gaussian type transitions on the euclidian space, for some collection of increasing compact balls In this situation, a natural cut off consists in considering the Marov transitions M restricted to the compact sets Ê x Ê 1 M x, dy := M x, dy 1 be M 1 be x These transitions are well defined as soon as M x, Ê > 0, for any x Ê 1 Using the decomposition [ M M ]u = M u M 1 be u M 1 be cu 1 = 1 M u 1 be M 1 be cu M 1 be = M 1 be c M 1 be M u 1 be M 1 be c u

13 On the Robustness of the Snell envelope 10 Then using Lemma 11 yields u û be := sup u x û x x E b n M l 1 be c l M l u l 1 M l 1 bel bel bel 1 + M l u l 1 be c bel 1 l bel 1 l=+1 n l=+1 [ ] ǫl M l u l bel 1 + M l u 2 l 1 ǫ 1/2 ǫ 1/2 l be l 1 l We summarize the above discussion with the following result Theorem 21 We assume that the tightness condition T is met, for every sequence of positive numbers ǫ n [0, 1[, and for some collection of compact subsets Ên E n In this situation, for any 0 n, we have that u û be n l=+1 ǫ 1/2 l 1 ǫ 1/2 l M l u 2 l 1/2 be l 1 We notice that n u M,l f l and therefore l= M u 2 be 1 n +1 n M 1,l f l 2 be 1 l= Consequently, one can find sets Êl <l n so that u û be is as small as one wants as soon as M,l f l 2 be <, for any 0 < l n 22 Euler approximation models In several application model areas, the discrete time Marov chain X 0 is often given in terms of an IR d -valued and continuous time process X t t 0 given by a stochastic differential equation of the following form dx t = ax t dt + bx t dw t, lawx 0 = η 0, 21 where η 0 is a nown distribution on IR d, and a, b are nown functions, and W is a d-dimensional Wiener process Except in some particular instances, the time homogeneous Marov transitions M = M are usually unnown, and we need to resort to an Euler approximation scheme The discrete time approximation model with a fixed time step 1/m is defined by the following recursive formulae ξ 0 x = x ξi+1 x = ξ 1 i x + a ξ i x ξ m m m m + b i x m 1 m ǫ i where the ǫ i s are iid centered and IR d -valued Gaussian vectors with unit covariance matrix The chain ξ 0 is an homogeneous Marov with a transition ernel which we denote by M

14 On the Robustness of the Snell envelope 11 We further assume that the functions a and b are twice differentiable, with bounded partial derivatives of orders 1 and 2, and the matrix bb x is uniformly non-degenerate In this situation, the integral operators M and M admit densities, denoted by p and p According to Bally and Talay [?], we have that [p p] c q and m p p c q 22 with the Gaussian density qx, x := 1 2πσ e 1 2σ 2 x x 2, and a pair of constants c, σ depending only on the pair of functions a, b Let Q, be the Marov integral operator on IR d with density q We consider a sequence of functions f 0 n on IR d We let u 0 n and û 0 n be the Snell envelopes on IR d associated to the pair M, f and M, f Using Lemma 11, we readily obtain the following estimate u û n 1 l= M l M Mu l+1 c m n 1 l= M l Q u l+1 Rather crude upper bounds that do not depend on the approximation ernels M can be derived using the first inequality in 22 u û 1 n c l Q l u l+ m Recalling that u l+ l+ l n Ml l+ f l, we also have that u û 1 n c l Q l m 1 m l=1 n l=1 l+ l n l+ l n l=1 c l l+ Q l l+ f l c l Q l f l = 1 m 1 l n We summarize the above discussion with the following theorem l c l Q l f +l Theorem 22 Suppose the functions f 0 n on IR d are chosen such that Q l f +l x <, for any x IR d, and 1 + l n Then, for any 0 l n, we have the inequalities u û c m n 1 l= M l Q u l+1 1 m 1 l n l c l Q l f +l 23 Interpolation type models Most algorithms proposed to approximate the Snell envelope provide discrete approximations û i at some discrete potentially random points ξi However, for several purposes, it can be interesting to consider approximations û of functions u on the whole space E One motivation to do so is, for instance, to be able to define a new low biased estimator, Ū, using a Monte Carlo approximation

15 On the Robustness of the Snell envelope 12 of 11, with a stopping rule ˆτ associated with the approximate Snell envelope û, by replacing u by û in the characterization of the optimal stopping time τ 11, ie Ū = 1 M fˆτ i M X iˆτ with ˆτ i i = min { l n : û lxl i = f lxl i } i=1 23 where X i = X1 i,, Xi n are iid path according to the reference Marov chain dynamic In this section, we analyse non asymptotic errors of some specific approximation schemes providing such estimators û of u on the whole state E Let M +1 = I M+1 be the composition of approximation Marov transition M +1 from a finite set S into the whole state space E +1, with an auxiliary interpolation type and Marov operator I from E into S, so that x S I x, ds = δ x ds and such that the integrals x E I ϕ x = I x, ds ϕ s S of any function ϕ on S are easily computed starting from any point x in E We further assume that the finite state spaces S are chosen so that f I f E ǫ f, S 0 as S 24 for continuous functions f on E An example of interpolation transition I is provided hereafter We let M = I 1 M be the composition operator on the state spaces Ê = E The approximation models M are non necessarily deterministic In [?], we examined the situation where s S M s, dx = 1 1 i δ X i sdx where X i s stands for a collection of independent random variables with common law M s, dx Theorem 23 We suppose that the Marov transitions M are Feller, in the sense that M CE CE 1, where CE stands for the space of all continuous functions on the E We let u 0 n, and respectively û 0 n be the Snell envelope associated with the functions f = f, and the Marov transitions M, and respectively M = I 1 M on the state spaces Ê = E u û E n 1 [ǫ l M l+1 u l+1, S l + M l+1 M ] l+1 u l+1 Sl l= The proof of the theorem is a direct consequence of Lemma 11 combined with the following decomposition u û E n 1 [ Id I l M l+1 u l+1 El + I l M l+1 M ] l+1 u l+1 El 25 l=

16 On the Robustness of the Snell envelope 13 We illustrate these results in the typical situation where the space E are the convex hull generated by the finite sets S Firstly, we present the definition of the interpolation operators We let P = {P 1,, P m } be a partition of a convex and compact space E into simplexes with disjoint non empty interiors, so that E = 1 i m P i We denote by δp the refinement degree of the partition P δp := sup sup x y 1 i m x,y P i We let S = VP be the set of vertices of these simplexes We denote by I be interpolation operator defined by Ifs = fs, if s S, and if x belongs to some simplex P j with vertices {x j 1,, xj d j } If λ i x i j = λ i fx j i 1 i d j 1 i d j where the barycenters λ i 1 i dj are the unique solution of x = λ i x j i with λ i 1 i dj [0, 1] dj and λ i = 1 1 i d j 1 i d j The Marovian interpretation is that starting from x, one choses the closest simplex and then one chooses one of its vertices x i with probability λ i For any δ > 0, we let ωf, δ be the δ-modulus of continuity of a function f CE ωf, δ := sup x,y E: x y δ fx fy The following technical Lemma provides a simple way to chec condition 24 for interpolation ernels Lemma 24 Then for any f, g CE, sup x E fx Igx max fx gx + ωf, δp + ωg, δp 26 x S In particular, we have that sup fx Ifx ωf, δp x E Proof: Suppose x belongs to some simplex P j with vertices {x j 1,, xj d j }, and let λ i 1 i dj be the barycenter parameters x = 1 i d j λ i x i j Since we have Igx j i = gxj i, and Igxj i = gxj i for any i {1,,d j}, it follows that fx Igx = d j d j λ i fx fx j i + λ i fx j i Igxj i i=1 i=1 d j d j + λ i Igx j i gx i=1 d j λ i fx fx j i + λ i fx j i gxj i i=1 i=1 d j + λ i gx j i gx i=1

17 On the Robustness of the Snell envelope 14 This implies that with sup fx Igx max fx gx + ωf, δp j + ωg, δp j x P j x P j ωf, δp j = The end of the proof is now clear sup fx fy and δp j := sup x y x y δp j x,y P j Combining 25 and 26, we obtain the following result Proposition 25 We let P = {P 1,,Pm } be a partition of a convex and compact space E into simplexes with disjoint non empty interiors, so that E = 1 i m P i We let S = VP be the set of vertices of these simplexes We let û 0 n, be the Snell envelope associated with the functions f = f and the Marov transitions M = I 1 M on the state spaces E = Ê u û E n 1 [ωm l+1 u l+1, δp l + M l+1 M ] l+1 u l+1 Sl l= 24 Quantization tree models Quantization tree models belongs to the class of deterministic grid approximation methods The basic idea consists in chosing finite space grids Ê = { x 1,, x m } E = R d and some neighborhoods measurable partitions A i 1 m of the whole space E such that the random state variable X is suitably approximated, as m, by discrete random variables of the following form X := x i 1 A i X X 1 i m The numerical efficiency of these quantization methods heavily depends on the choice of these grids There exists various criteria to choose judiciously these objects, including minimal L p -quantization errors, that ensures that the corresponding Voronoi type quantized variable X minimizes the L p distance to the real state variable X For futher details on this subject, we refer the interested reader to the pioneering article of G Pagès [?], and the series of articles of V Bally, G Pagès, and J Printemps [?], G Pagès and J Printems [?], as well as G Pagès, H Pham and J Printems [?], and references therein The second approximation step of these quantization model consists in defining the coupled distribution of any pair of variables X 1, X by setting P X = x j, X 1 = x i 1 = P X A j, X A j 1 for any 1 i m 1, and 1 j m This allows to interpret the quantized variables X 0 n as a Marov chain taing values in the states spaces Ê 0 n with Marov transitions M x i 1, xj := P X = x j X 1 = x i 1 = P X A j X A i 1

18 On the Robustness of the Snell envelope 15 Using the decompositions M fx i 1 = = m j=1 m j=1 + A j A j fy PX dy X 1 = x i 1 fy PX dy X 1 A i 1 [Mfx i 1 Mfx] PX 1 dx X 1 A i 1 and m M fx i 1 = we find that j=1 A j fx j PX dy X 1 A i 1 [M M ]fx i 1 m = [fy fx j j=1 A j ] PX dy X 1 A i 1 [Mfx i + 1 Mfx ] PX 1 dx X 1 A i 1 We let LipR d be the set of alll Lipschitz functions f on R d, and we set Lf = fx fy sup x,y R d,x y x y for any f LipR d We further assume that M LipR d LipR d From previous considerations, we find that [M M [ ]fx i 1 Lf E X X ] 1 p X 1 = x i p 1 This clearly implies that +LM f E X 1 X 1 p X 1 = x i 1 1 p M,l M l+1 M l+1 f x i Lf [ E X l+1 X l+1 p X = x i ] 1 p +LM l+1 f E X l X l p X = x i 1 p otice that the above inequality can be refined using the fact that a b c d a c b d we observe that f and u +1 LipR d u LipR d with Lu Lf LM +1 u +1 Using Lemma 11, we readily arrive at the following theorem similar to Theorem 2 in [?]

19 On the Robustness of the Snell envelope 16 Theorem 26 Assume that f 0 n LipR d n+1, and M LipR d LipR d, for any 1 n In this case, we have u 0 n LipR d n+1, and for any 0 n, we have the almost sure estimate u û X LM +1 u +1 X X + n 1 l=+1 +Lf n On the other hand, we also have that Using the decomposition we conclude that Lu l + LM l+1 u l+1 E X l X l p X 1 p [ E X n X ] 1 n p p X u X u X Lu X X û X u X = [û X u X ] + [u X u X ], û ξ u X Lf n n 1 + [ E X n X ] 1 n p p X Lu l + LM l+1 u l+1 E X l X l p X 1 p l= 3 Monte Carlo approximation models 31 Path space models The choice of non homogeneous state spaces E n is not innocent In several application areas the underlying Marov model is a path-space Marov chain X n = X 0,,X n E n = E 0 E n 31 The elementary prime variables X n represent an elementary Marov chain with Marov transitions M x 1, dx from E 1 into E In this situation, the historical process X n can be seen as a Marov chain with transitions given for any x 1 = x 0,,x 1 E 1 and y = y 0,, y E by the following formula M x 1, dy = δ x 1 dy 1 M y 1, dy This path space framewor is, for instance, welle suited when dealing with path dependent options as Asian options Besides, this path space framewor is also well suited for the analysis of Snell envelopes under different probability measures To fix the ideas, we associate with the latter a canonical Marov chain Ω, F, X n n 0, P η with initial dis- 0 tribution η 0 on E 0, and Marov transitions M n from E n 1 into E n We use the notation E P to denote the expectations with respect to P η 0 η We further 0 assume that there exists a sequence of measures η 0 n on the state spaces E 0 n such that η 0 η 0 and M x 1, η 32

20 On the Robustness of the Snell envelope 17 for any x 1 E 1, and 1 n We let Ω, F, X n n 0, P η0 be the canonical space associated with a sequence of independent random variables X with distribution η on the state space E, with 1 Under the probability measure P η0, the historical process X n = X 0,, X n can be seen as a Marov chain with transitions M x 1, dy = δ x 1 dy 1 η dy By construction, for any integrable function f on E, we have E P f nx n = E η 0 Pη0 f n X n with the collection of functions f on E given for any x = x 0,,x E by f x = f x dp dp x with dp dp x n = dη 0 dη 0 x 0 1 l dm l x l 1, dη l x l 33 Proposition 31 The Snell envelopes u and u associated with the pairs f, M and f, M are given for any 0 < n by the bacward recursions u = f M +1 u +1 and u = f M +1 u +1 with u n, u n = f n, f n These functions are connected by the following formulae 0 n x = x 0,, x E u x = u x dp dp x 34 Proof: The first assertion is a simple consequence of the definition of a Snell envelope, and formula 34 is easily derived using the fact that u x = f x E +1 This ends the proof of the proposition η +1 dx +1 dm +1 x, x dη +1 u +1x Under condition 32, the above proposition shows that the calculation of the Snell envelope associated with a given pair of functions and Marov transitions f, M reduces to that of the path space models associated with sequence of independent random variables with distributions η n More formally, the restriction P η0,n of reference measure P η0 to the σ-field F n generated by the canonical random sequence X 0 n is given by the the tensor product measure P η0,n = n =0 η evertheless, under these reference distributions the numerical solving of the bacward recursion stated in the above proposition still involves integrations wrt the measures η These equations can be solved if we replace these measures by some sequence of possibly random measures η with finite support on some reduced measurable subset Ê E, with a reasonably large and finite cardinality We extend η to the whole space E by setting η E Ê = 0

21 On the Robustness of the Snell envelope 18 We let P bη 0 be the distribution of a sequence of independent random variables ξ with distribution η on the state space Ê, with 1 Under the probability measure P bη 0, the historical process X n = X 0,, X n can now be seen as a Marov chain taing values in the path spaces Ê := Ê 0 Ê with Marov transitions given for any x 1 = x 0,,x 1 Ê 1 and y = y 0,, y Ê by the following formula M x 1, dy = δ x 1 dy 1 η dy otice that the restriction P bη 0,n of these approximated reference measure P bη 0 to the σ-field F n generated by the canonical random sequence X 0 n is now given by the the tensor product measure P bη 0,n = n =0 η We let û be the Snell envelope on the path space Ê, associated with the pair f, M, with the sequence of functions f = f given in 33 By construction, for any 0 n, and any path x = x 0,, x Ê, we have û x = û x dp dp x with the collection of functions û 0 n on the state spaces E 0 n given by the bacward recursions û x = f x M +1 x, dx +1 û +1 x E b +1 with the random integral operator M from E into Ê+1 defined below M +1 x, dx +1 = η +1dx +1 R +1x, x +1 with the Radon iodym derivatives R +1 x, x +1 = dm +1 x, dη +1 x Broadie-Glasserman models We consider the path space models associated to the changes of measures presented in Sub-section 31 We use the same notation as in there We further assume that η = 1 i=1 δ ξ i is the occupation measure associated with a sequence of independent random variables ξ := ξ i 1 i with common distribution η on Ê = E We further assume that ξ 0 n are independent This Monte Carlo type model has been introduced in 1997 by M Broadie, and P Glasserman see for instance [?], and references therein We let Ê be the expectation operator associated with this additional level of randomness, and we set ÊP η0 := Ê E P η0 In this situation, we observe that M +1 M +1x, dx +1 = 1 V +1 dx +1 R +1 x, x +1

22 On the Robustness of the Snell envelope 19 with the random fields V +1 := [ η +1 η +1 ] From these observations, we readily prove that the approximation operators M +1 are unbias, in the sense that 0 l x l E l Ê Pη0 M,l fx l F = M,l fx l 36 for any bounded function f on E l+1 Furthermore, for any even integer p 1, we have [ Ê Pη0 M l+1 M 1 l+1 ]fx p l p 2 ap η l+1 [R l+1 x l,fp ] 1 p The above estimate is valid as soon as the rhs in the above inequality is well defined We are now in position to state and prove the following theorem Theorem 32 For any integer p 1, we denote by p the smallest even integer greater than p Then for any time horizon 0 n, and any x E, we have Ê Pη0 u x û x p 1 p 2ap { M,l x, dx l η l+1 [R 1 l+1 x l,u l+1 p ]} p l<n 37 otice that, as stated in the introduction, this result implies exponential rate of convergence in probability Hence, this allows to improve noticeably existing convergence results stated in [?], with no rate of convergence, and in [?] with a polynomial rate of convergence in probability Proof: For any even integers p 1, any 0 l, any measurable function f on E l+1, and any x E, using the generalized Minowsi inequality we find that Ê Pη0 [ M,l M l+1 M ] l+1 f x p 1 p F 2ap M,l x, dx l η l+1 [R l+1 x l,fp ] 1 p By the unbias property 36, we conclude that Ê Pη0 [ M,l M l+1 M ] l+1 f x p 1 p { 2ap M,lx, dx l η l+1 [R l+1 x l,f p ]} 1/p For odd integers p = 2q + 1, with q 0, we use the fact that EY 2q+1 2 EY 2q EY 2q+1 and EY 2q EY 2q+1 q q+1 for any non negative random variable Y and 2q + 1 q+1 = 2 2q + 1 q+1 and 2q q = 2q + 1 q+1 /2q + 1 so that a2q 2q a2q + 1 2q+1 2 2q+1 2q q+1 /q + 1/2 = a2q + 1 2q+1 2 ÊP η0 M,l [M l+1 M ] l+1 f x 2q q+1 a2q + 1 2q+1 2 M,l x, dx l η [ l+1 Rl+1 x l,f2q+1] q q+1 M,l x, dx l η [ l+1 Rl+1 x l,f2q+1]

23 On the Robustness of the Snell envelope 20 using the fact that EY q q+1 EY q q+1, we prove that the rhs term in the above display is upper bounded by 2 2q+1 a2q + 1 2q+1 { 2 M,lx, dx l η l+1 [R l+1 x l,f 2q+1]} 2 1 2q+1 1 from which we conclude that [ Ê Pη0 M M l+1 M ] l+1 f x 2q+1 1,l 2q+1 2a2q + 1 { M,l x, dx l η [ 1 l+1 Rl+1 x l,f2q+1]} 2q+1 This ends the proof of the theorem The L p -mean error estimates stated in Theorem 32 are expressed in terms of L p norms of Snell envelope functions and Radon iodym derivatives The terms in rhs of 37 have the following interpretation: [ M,l x, dx l η l+1 R l+1 x l,u l+1 p ] = E [R l+1 X l, ξ1 l+1 u l+1ξl+1 1 /X ]p = x In the above display, E stands for the expectation wrt some reference probability measure under which X l is a Marov chain with transitions M l, and ξl+1 1 is a independent random variable with distribution η l+1 Loosely speaing, the above quantities can be very large when the sampling distributions η l+1 are far from the distribution of the random states X l+1 of the reference Marov chain at time l + 1 ext we provide an original strategy that allows to tae η l+1 = LawX l+1 as the sampling distribution In what follows, we let η = 1 i=1 δ ξ i be the occupation measure associated with independent copies ξ = ξ i 1 i of the Marov chain X, from the origin = 0 up to the final time horizon = n In what follows, we let F be the sigma field generated by the random sequence ξ l 0 l We also assume that the Marov transitions M nx n 1, dx n are absolutely continuous with respect to some measures λ n dx n on E n and we have H 0 x n 1, x n E n 1 E n H n x n 1, x n = dm n x n 1, dλ n x n > 0 In this situation, we have η +1 λ +1, with the Radon iodym derivative given below η +1 dx +1 = η M +1 dx +1 = η H+1, x +1 λ +1 dx +1 Also notice that the bacward recursion of the Snell envelope u can be rewritten as follows u x = f x η +1 dx +1 dm +1 x, x +1 E +1 dη u +1 x = f x E +1 η +1 dx +1 H +1 x, x +1 η H +1, x +1 u +1 x +1

24 On the Robustness of the Snell envelope 21 Arguing as in 35, we define the approximated Snell envelope û 0 n on the state spaces E 0 n by setting û x = f x b E +1 M +1 x, dx +1 û +1 x +1 with the random integral operator M from E into Ê+1 defined below M +1x, dx +1 = η +1 dx +1 dm +1 x, d η M +1 x +1 = η +1 dx H +1 x +1, x +1 η H +1, x +1 By construction, these random approximation operators M +1 satisfy the unbias property stated in 36, and we have M +1 M +1 x, dx +1 = 1 V +1 dx +1 R +1 x, x +1 with the random fields V +1 and the F -measurable random functions R +1 defined below V +1 := [ η +1 η M +1] and R+1 x, x +1 := H +1x, x +1 η H +1, x +1 Furthermore, for any even integer p 1, and any measurable function f on E l we have Ê Pη0 [ M l+1 M l+1 ]fx l p 1 [ p Fl 2 ap η l M l+1 R 1 l+1 x l,fp] p The above estimate is valid as soon as the rhs in the above inequality is well defined For instance, assuming that H 1 M l+1 H l+1 x u2p l l+1 < and sup, x l+1 x H l,y l E l l+1 y l, x l+1 h l+1x l+1 with M l+1 h2p l+1 < we find that [ E M l+1 M ] l+1 u l+1x l p 1 p F l 2 ap M l+1h 2p 1 l+1 M l+1u l+1 2p 2p Rephrasing the proof of Theorem 32, we prove the following result Theorem 33 Under the conditions H 0 and H 1 stated above, for any even integer p > 1, any 0 n, and x E, we have E u x û x p 1 p 2ap l<n M l+1h 2p 1 l+1 M l+1u l+1 2p 2p 38

25 On the Robustness of the Snell envelope Genealogical tree based models 331 eutral genetic models In this section, we propose a new model whose purpose is to reduce the number of calculations in selecting trajectories from the large exploding tree, but maintain the precision Using the notation of Sub-section 31, we set X n = X 0,, X n E n = E 0 E n We further assume that the state spaces E n are finite We denote by η the distribution of the path-valued random variable X on E, with 0 n We also set M the Marov transition from X 1 to X, and M the Marov transition from X 1 to X In Sub-section 31, we have seen that M x 0,,x 1, dy 0,, y = δ x 0,,x 1 dy 0,,y 1 M y 1, dy In the further development, we fix the final time horizon n, and for any 0 n, we denote by π the -th coordinate mapping π : x n = x 0,, x n E n = E 0 E n π x n = x E In this notation, for any 0 < n, x E and any function f BE +1, we have η n = LawX 0,, X n and M +1 fx := η n1 x π f π +1 η n 1 x π 39 By construction, it is also readily checed that the flow of measure η 0 n also satisfies the following equation 1 n η := Φ η with the linear mapping Φ η 1 := η 1 M The genealogical tree based particle approximation associated with these recursion is defined in terms of a Marov chain ξ = ξ i, 1 i in the product state spaces E, where = 0 is a given collection of integers P ξ = x 1,,x ξ 1 = Φ 1 δ ξ i 1 1 x i 1 i 1 i 1 The initial particle system ξ 0 = ξ i, 0 311, is a sequence of 0 iid 0 i 0 random copies of X 0 We let F be the sigma-field generated by the particle approximation model from the origin, up to time To simplify the presentation, when there is no confusion we suppress the population size parameter, and we write ξ and ξ i instead of ξ and ξ i, By construction, ξ is a genetic type model with a neutral selection transition and a mutation type exploration ξ E Selection ξ := ξi E b 1 i b Mutation ξ +1 E

26 On the Robustness of the Snell envelope 23 with := +1 During the selection transition, we select randomly +1 path valued particles ξ := ξi among the path valued particles ξ = ξ i 1 i 1 i +1 Sometimes, this elementary transition is called a neutral selection transition in the literature on genetic population models During the mutation transition ξ ξ, every selected path valued individual ξ i evolves randomly to a new path valued individual ξ+1 i = x randomly chosen with the distribution M +1 ξ i, x, with 1 i By construction, every particle is a path-valued random variable defined by ξ i := ξ0, i, ξi 1,,,ξi, ξ i := ξi 0,, ξ 1,, i, ξ, i E := E 0 E By definition of the transition in path space, we also have that ξ i +1 = = ξ0,+1 i, ξi 1,+1,,ξi,+1, ξ i +1,+1 }{{} {}}{ ξi 0,, ξi 1,,, ξi,, ξ+1,+1 i = ξi, ξ i +1,+1 where ξ+1,+1 i is a random variable with distribution M +1 ξ, i, In other words, the mutation transition ξ i ξi +1 simply consists in extending the selected path ξ i with an elementary move ξ, i ξi +1,+1 of the end point of the selected path From these observations, it is easy to chec that the terminal random population model ξ, = ξ, i and ξ, = ξi, is again defined 1 i 1 i +1 as a genetic type Marov chain defined as above by replacing the pair E, M by the pair E, M, with 1 n The latter coincides with the mean field particle model associated with the time evolution of the -th time marginals η of the measures η on E Furthermore, the above path-valued genetic model coincide with the genealogical tree evolution model associated with the terminal state random variables We let η and η be the occupation measures of the genealogical tree model after the mutation and the selection steps; that is, we have that η := 1 δ ξ i and η := δ bξ i 1 i 1 1 i b In this notation, the selection transition ξ, ξ consists in choosing conditionally independent and identically distributed random paths ξ i with common distribution η In other words, η is the empirical measure associated with conditionally independent and identically distributed random paths ξ i with common distribution η Also observe η is the empirical measure associated with conditionally independent and identically distributed random paths ξ i with common distribution η 1 M

27 On the Robustness of the Snell envelope 24 In practice, we can tae 0 = 1 = n = when we do not have any information on the variance of X In the case when we now the approximate variance of X, we can tae a large when the variance of X is large To clarify the presentation, In the further development of the article we further assume that the particle model has a fixed population size =, for any Convergence analysis For general mean field particle interpretation models 311, several estimates can be derived for the above particle approximation model see for instance [?] For instance, for any n 0, r 1, and any f n Osc 1 E n, and any 1, we have the unbias and the mean error estimates: E ηn f n = η n f n = E η n f n and [ ] E η n η n fn r 1 r with the Dobrushin ergodic coefficients 2 ar 313 n βm p,n p=0 βm p,n := sup M p,n x p, M p,n y p, tv x p,y p E p and the collection of constants ap defined in 17 Arguing as in 19, for time homogeneous population sizes n =, for any functions f Osc 1 E n, we conclude that P [ η n η n ] f bn + ǫ exp ǫ2 2bn 2 with bn := 2 n βm p,n 314 For the path space models 39, we have βm p,n = 1 so that the estimates 313 and 314 taes the form p=0 E [ η n η n ] fn r 1 r 2 ar n and [ P η n η ] n f 2n ǫ exp ǫ2 8n In the next lemma we extend these estimates to unbounded functions and non homogeneous population size models Lemma 34 For any p 1, we denote by p the smallest even integer greater than p In this notation, for any 0 and any function f, we have the almost sure estimate E [η n η 1M 1,n ]f p F 1 1 p 2ap n l= [ ] 1 η 1M 1,l M l,n f p p 316 In particular, for any f L p η n, we have the non asymptotic estimates E [η n η n]f p 1/p 2 ap f p,η n n

28 On the Robustness of the Snell envelope 25 Proof : In writing η 1 M 0 = η 0, for any 0 we have the decomposition [η n η 1 M,n] = n [ηl ηl 1 M l]m l,n l= with the semigroup Using the fact that we prove that M,n = M +1 M +2 M n E η l f η l 1 = η l 1 M l f E [η l η l 1 M l]f p F l 1 1 p E [ηl µ l ]f p 1 Fl 1 p where µ l := 1 i=1 δ ζl i stands for a independent copy of η l Kintchine s type inequalities we have given η l 1 Using E [η l µ l ]f p F l 1 1 p f 2 ap E ξ 1 l p Fl 1 [ ] 1 = 2 ap ηl 1M l f p p 1 p Using the unbias property of the particle scheme, we have This implies that l n E η l f F 1 = η 1 M 1,l f E [η l η l 1 M l]f p F 1 1 p 2 ap E η l 1 M l f p 1 F p 1 [ ] 1 = 2 ap η 1 M 1,l f p p The end of the proof of 316 is now a direct application of Minowsi s inequality The proof of 317 is a direct consequence of 316 This ends the proof of the lemma Particle approximations of the Snell envelope In sub-section 331 we have presented a genealogical based algorithm whose occupation measures ηn converge, as, to the distribution η n of the reference Marov chain X 0,, X n from the origin, up to the final time horizon n Mimicing formula 39, we define the particle approximation of the Marov transitions M as follows : M +1fx := η n 1 x π f π +1 η n 1 x π := 1 i 1 xξ i,n fξi +1,n 1 i 1 xξ i,n for every state x in the support Ê,n of the measure ηn π 1 otice that Ê,n coincides with the collection of ancestors ξ,n i at level of the population