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, IRIA 2010, pp41 <inria v4> HAL Id: inria Submitted on 15 Jan 2011 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 the discrete time optimal stopping problem We consider a series of approximation schemes, 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 We deduce these estimates from a single and general robustness property of Snell envelope semigroups 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 second part of the article, we propose a new approach using a genealogical tree 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 Some simulations results are provided and confirm the interest of this new algorithm 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 (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 : ous analysons les propriètès de robustesse de l équation de l évolution bacward de l enveloppe de Snell pour le problème d arrêt optimal au temps discet ous considèrons une série de schémas d approximation, y compris les approximations de type cut-off, schémas de discrétisation d Euler, des modèles d interpolation, les modèles de quantification, et la méthode de Broadie- Glasserman Dans chaque situation, nous fournissons des estimations de convergence non-asymptotique, y compris les bornes d erreur L p et les inégalités de concentration exponentielle ous en déduisons de ces estimations à partir d une seule propriété générale de robustesse générale de semigroupes enveloppe de Snell En particulier, cette analyse nous permet de retrouver des résultats de convergence existants pour la méthode de quantification et d améliorer significativement la vitesse de convergence obtenue pour l estimateur de Broadie- Glasserman Dans la deuxième partie de l article, nous proposons une nouvelle approche en utilisant une approximation d arbre généalogique du processus de Marov en termes de référence d un modèle de type génétique neutre Contrairement à Broadie-Glasserman Monte Carlo modèles, le coût de calcul de cette nouvelle approximation particulaire stochastique est linéaire du nombre de points échantillonnés Certains résultats des simulations sont fournies et de confirmer l intérêt de ce nouvel algorithme 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 of 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 options pricing and hedging The present paper is restricted to the case of discrete time optimal stopping problem corresponding in finance to the case of Bermudan options It is well nown that the price of the Bermudan option giving the opportunity to exercise a payoff f at discrete dates = 0,, n, can be calculated by a bacward dynamic programming formula This recursion consists in comparing at each time step the immediate payoff f and the expectation of the future gain (or the so-called continuation value, which precisely involves the Marov transition M +1 of the underlying assets process (X One first goal of this paper is to provide a simple framewor to analyze in unison most of the numerical schemes currently used in practice to approximate the Snell envelope, which are precisely based on the approximation of the dynamic programming recursion The idea is to analyze the related approximation error in terms of robustness properties of the Snell envelope with respect to (wrt the pair parameters (f, M Hence, we include in our analysis approximation schemes which are defined in terms of some approximate pairs of functions and transitions ( f, M 0 Then, we deduce from the robustness Lemma 21, stated in the preliminary Section 2, non asymptotic convergence theorems, including L p -mean error bounds and related exponential inequalities for the deviations of Monte Carlo type approximation models In Section 3, this approach allows us to derive non asymptotic error bounds for deterministic approximation schemes such as cut-off techniques, Euler type discrete time approximations, quantization tree models, interpolation type approximations, then recovering or improving some existing results or in some cases providing new bounds We emphasize that this non asymptotic robustness analysis also allows to combine in a natural way several approximation models For instance, under appropriate tightness conditions, cut-off techniques can be used to reduce the numerical analysis of the Snell envelope to compact state spaces 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 In Section 4, we focus on two ind of Monte Carlo importance sampling approximation schemes The first one is the Stochastic Mesh method introduced by M Broadie and P Glasserman in their seminal paper [5] (see also [22], for some recent refinements The principle idea of this method is to operate a change of measure to replace conditional expectations by simple expectations involving the Marov transition densities wrt some reference measures The number of sampled points wrt the reference measures η n required by this model can be constant in every exercise date This technique avoids the explosion issue of the naive Monte Carlo method As any full Monte Carlo type technique, the main advantage of their approach is that it applies to high dimensional Bermudan options with a finite possibly large, number of exercise dates In [5], the authors provide a set of conditions under which the Monte Carlo importance scheme converges as the computational effort increases However, the

7 On the Robustness of the Snell envelope 4 computing time grows quadratically with the number of sampled points in the stochastic mesh In this context, in Section 42, we provide new non asymptotic estimates, including L p -mean error bounds and exponential concentration inequalities Our analysis allows us to derive Theorem 47 improving significantly existing convergence results (see [5]or [1] The second type of Monte Carlo importance sampling scheme discussed in this article is another version of the Broadie-Glasserman model, called average density in the original article The main advantage of this 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, even if the Radon iodym derivatives, R n (x, y = dmn(x, dη n (y is not nown explicitly We only assume that the Marov transitions M n (x, are absolutely continuous with respect to some measures λ n We can then approximate this function with empirical measure In this situation, we can recover a similar approximation to the original stochastic mesh method, except that the Radon iodym derivatives, R +1 (ξ i, ξj +1 is replaced by an approximation The stochastic analysis of this particle model is provided in the second part of Section 42 and follows essentially the same line of arguments as the one of the Broadie-Glasserman model In the final part of the article, Section 5, we present a new Monte Carlo approach 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 model is a Marov chain with a selection/mutation transition During the mutation phase, the particles explore the state space independently according to the Marov transitions while the selection step induces interactions between the various particles This type of model is frequently used in biology, and genetic algorithms literature (see for instance [14], and references therein An important observation concerns the genealogical tree structure of the genetic particle model that we consider 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 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 We can then construct the approximation û as the Snell envelope associated with this -approximated Marov chain Several estimates of convergence are provided in Section 5 Finally, some numerical simulations are performed and show the interest of our new algorithm 2 Preliminary In a discrete time setting, the problem is related to pricing of Bermuda options and is defined in terms of a 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

8 On the Robustness of the Snell envelope 5 of (Z 0 n, is the stochastic process (U 0 n defined for any 0 < n by the following bacward equation U = Z E(U +1 F, with the terminal condition U n = Z n The main property of this stochastic process is that U = sup τ T E(Z τ F = E(Z τ F (21 with τ = min { l n : U l = Z l } T 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 is 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 = Law(X 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 nonnegative measurable functions f n on E n In this situation, the computation of the Snell envelope amounts to solve the following bacward functional equation u = H +1 (u +1 = max(f, M +1 (u +1 = f M +1 (u +1, (22 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 E +1 = E(u +1 (X +1 X = x We let H,l = H +1 H +1,l, with l n, be the nonlinear semigroups associated with the bacward equations (22 We use the convention H, = Id, the identity operator, so that u = H,l (u l, for any l n Given a sequence of bounded integral operator M from some state space E 1 into another E, let us denote by M,l the composition operator such that M,l := M +1 M +2 M l, for any l, with the convention M, = Id, the identity operator With this notation, 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, 1 n = f u M,l f l, 1 n l n (23 From the readily proved Lipschitz property H (u H (v M +1 ( u v, for any functions u, v on E, we also have that H,l (u H,l (v M,l ( u v (24

9 On the Robustness of the Snell envelope 6 for any functions u, v on E l, and any l n Even if it loos innocent, the numerical solving of the recursion (22 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 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 associated with the functions f and the sequence of integral operators M from Ê 1 into Ê As in (22, the computation of the Snell envelope û amounts to solve the following bacward functional equation û = Ĥ+1(û +1 = f M +1 (û +1 (25 We let Ĥ,l = Ĥ+1 Ĥ+1,l, with l n, be the nonlinear semigroups associated with the bacward equations (25, so that û = Ĥ,l(û l, for any l n Using the elementary inequality (a a (b b a b + a b,, which is valid for any a, a, b, b R, for any 0 < n and for any functions u on E +1 one readily obtains the local approximation inequality H +1 (u Ĥ+1(u f f + (M +1 M +1 (u (26 To transfer these local estimates to the semigroups H,l and Ĥ,l we use the same perturbation analysis as the one presented [10, 12, 21, 28] in the context of nonlinear filtering semigroups and particle approximation models The difference between the approximate and the exact Snell envelope can be written as a telescoping sum u û = n [Ĥ,l (H l+1 (u l+1 Ĥ,l(Ĥl+1(u l+1 ] l= setting for simplicity H n+1 (u n+1 = u n and Ĥn+1(u n+1 = û n, for l = n Combining the Lipschitz property (24 of the semigroup Ĥ,l with the local estimate (26, one finally gets the following robustness lemma, which is a natural and fundamental tool for the analysis of the Snell envelope approximations Lemma 21 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= M,l (M l+1 M l+1 u l+1 The perturbation analysis of nonlinear semigroups described above and the resulting robustness lemma are not really new As we mentioned above, it is a rather standard tool in approximation theory and numerical probability More precisely, these Lipschitz type estimates are often used by induction or as

10 On the Robustness of the Snell envelope 7 an intermediate technical step in the proof of a convergence theorem of some particular approximation scheme In the context of optimal stopping problems and numerical quantization schemes, these techniques are used for instance in the papers of Egloff [16] and Gobet, Lemor and Warin [19] or Pagès [24] To the best of our nowledge, the general and abstract formulation given above and its direct application to different approximation models seems to be the first result of this type for this class of models Besides the fact that the convergence of many Snell approximation schemes result from a single robustness property, the lemma 21 can be used sequentially and without further wor to obtain non asymptotic estimates for models combining several levels of approximations In the same vein, and whenever it is possible, lemma 21 can also be used as a technical tool to reduce the analysis of Snell approximation models on compact state spaces or even on finite but possibly large quantization trees or Monte Carlo type grids To interpret better the L p -mean error bounds appearing in this article, we end this section with the following lemma Lemma 22 Suppose the estimates have the following form: sup E( u (x û (x p 1 p x E a(pb (n, where b (n are some finite constants whose values do not depend on the parameter p and a(p is a collection of constants such that for all nonnegative integer r: a(2r 2r = (2r r 2 r and a(2r + 1 2r+1 = (2r + 1 r+1 r + 1/2 2 (r+1/2, (27 with the notation (q p = q!/(q p!, for any 1 p q Then we deduce the following exponential concentration inequality ( sup P u (x û (x > b (n + ǫ exp ( ǫ 2 /(2b (n 2 (28 x E Proof: This result is a direct consequence from the fact that, for any nonnegative random variable U, if there exists a bounded positive real b such that where a(r is defined by (27, then r 1 E(U r 1 r a(rb, P (U b + ǫ exp ( ǫ 2 /(2b 2 To chec this implication, we first notice that P (U b + ǫ inf t 0 {e t(b+ǫ E[e tu ]} Then developing the exponential and using the moments boundedness assumption implies that for all t 0 E ( e tu ( (bt 2 exp + bt 2

11 On the Robustness of the Snell envelope 8 Finally ( P (U b + ǫ exp sup (ǫt (bt2 t 0 2 Hence, for any approximation model whose L p -mean error bound has the form listed in the above lemma, we can interpret that the probability that the approximation model maes some level of error is exponentially small 3 Some deterministic approximation models In this section, we analyze the robustness of the Snell envelope wrt some deterministic approximation schemes that are parts of many algorithms proposed to approximate the Snell envelope Hence, the non asymptotic error bounds provided in this section can be applied and combined to derive convergence rates for such algorithms We recover or improve previous results and in some cases, state new error bounds 31 Cut-off type models It is often useful, when computing the Snell envelope, to approximate the state space by a compact set Indeed, Glasserman and Yu (2004 [18] showed that for standard (unbounded models (lie Blac-Scholes, the Monte Carlo estimation requires samples of exponential size in the number of variables of the value function, whereas the bounded state space assumption enables to estimate the Snell envelope from samples of polynomial size in the number of variables For instance, in [17], the authors propose a new algorithm that first requires a cut off step which consists of replacing the price process by another process illed at first exit from a given bounded set However, no bound is provided for the error induced by this cut off approximation In this section, we formalize a general cut-off model and provide some bounds on the error induced on the Snell envelope We suppose that E n are topological spaces with σ-fields E n that contain the Borel σ-field on E n Our next objective is to find conditions under which we can reduce the bacward functional equation (22 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 Euclidean 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

12 On the Robustness of the Snell envelope 9 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 Then using Lemma 21 yields = M (1 be c M (1 be M (u 1 be M (1 be c u 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 1/2 ǫ 1/2 1 ǫ l be l 1 l We summarize the above discussion with the following result Theorem 31 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 We notice that and therefore u û be u n l=+1 n M,l (f l l= ǫ 1/2 l 1 ǫ 1/2 l M (u 2 be 1 (n + 1 M l (u 2 l 1/2 be l 1 n M 1,l (f l 2 be 1 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 A similar cut-off approach was intoduced and analyzed in Bouchard and Touzi [6], but the cut-off was operated on some regression functions and not on the transition ernels 32 Euler approximation models l= 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 = a(x t dt + b(x t dw t, law(x 0 = η 0, (31

13 On the Robustness of the Snell envelope 10 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 In this situation, any approximation or the Snell envelope, which is based on simulations of the price process will be impacted by the error induced by the Euler scheme used in simulations We propose here to provide bounds for this error otice that in this setting, the exercise dates are discrete and fixed, so that our results are not comparable with those from Dupuis and Wang (2004 [15] who analyzed the convergence of the discrete time optimal stopping problem to a continuous time optimal stopping when the frequency of exercise dates increases to infinity Similarly, for numerical approximations of Bacward Stochastic Differential Equations (BSDE, [6] and [19] also analysed the case where the number exercise opportunities grows to infinity The discrete time approximation model with a fixed time step 1/m is defined by the following recursive formula ξ 0 (x = x ξ(i+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 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 [4], we have that [p p] c q and m p p c q, (32 with the Gaussian density q(x, 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 21, 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 ( 32 u û 1 n c l Q l u l+ m l=1

14 On the Robustness of the Snell envelope 11 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 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 32 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 33 Interpolation type models Most algorithms proposed to approximate the Snell envelope provide discrete approximations û i at some discrete (potentially random points ξi of E 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 of (21, 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 (21, ie Ū = 1 M fˆτ i M (X iˆτ with ˆτ i i = min { l n : û l (Xl i = f l (Xl} i i=1 (33 where X i = (X1 i,, Xi n are iid path according to the reference Marov chain dynamic In this section, we analyze non asymptotic errors of some specific approximation schemes providing such interpolated estimators û of u on the whole state E Let M +1 = I M+1 be the composition of the 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, (34

15 On the Robustness of the Snell envelope 12 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 [13], we examined the situation where s S M (s, dx = 1 1 i δ X i (s(dx, where X i (s stands for a collection of independent random variables with common law M (s, dx Theorem 33 We suppose that the Marov transitions M are Feller, in the sense that M (C(E C(E 1, where C(E stands for the space of 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 21 combined with the following decomposition u û E (35 n 1 [ (Id I l M l+1 u l+1 El + I l (M l+1 M ] l+1 u l+1 El l= 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 δ(p the refinement degree of the partition P δ(p := sup sup x y 1 i m x,y P i We let S = V(P be the set of vertices of these simplexes We denote by I be the interpolation operator defined by I(f(s = f(s, if s S, and if x belongs to some simplex P j with vertices {x j 1,,xj d j } I(f( 1 i d j λ i x i j = 1 i d j λ i f(x j i, where the barycenters (λ i 1 i dj are the unique solution of x = 1 i d j λ i x j i with (λ i 1 i dj [0, 1] dj and 1 i d j λ i = 1 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

16 On the Robustness of the Snell envelope 13 For any δ > 0, we let ω(f, δ be the δ-modulus of continuity of a function f C(E ω(f, δ := sup (x,y E: x y δ f(x f(y The following technical Lemma provides a simple way to chec condition (34 for interpolation ernels Lemma 34 Then for any f, g C(E, sup x E f(x Ig(x max f(x g(x + ω(f, δ(p + ω(g, δ(p (36 x S In particular, we have that sup f(x If(x ω(f, δ(p 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 Ig(x j i = g(xj i, and Ig(xj i = g(xj i for any i {1,,d j}, it follows that f(x Ig(x This implies that with = d j d j λ i (f(x f(x j i + λ i f(x j i Ig(xj i i=1 d j d j i=1 + λ i Ig(x j i g(x i=1 d j λ i (f(x f(x j i + λ i f(x j i g(xj i i=1 d j i=1 + λ i g(x j i g(x i=1 sup f(x Ig(x max f(x g(x + ω(f, δ(p j + ω(g, δ(p j, x P j x P j ω(f, δ(p j = The end of the proof is now clear sup f(x f(y and δ(p j := sup x y x y δ(p j x,y P j Combining (35 and (36, we obtain the following result Proposition 35 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 = V(P 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 [ω(m l+1 u l+1, δ(p l + (M l+1 M ] l+1 u l+1 Sl l=

17 On the Robustness of the Snell envelope Quantization tree models Quantization tree models belong to the class of deterministic grid approximation methods The basic idea consists in choosing finite space grids Ê = { x 1, },xm 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 ensure that the corresponding Voronoi type quantized variable X minimizes the L p distance to the real state variable X For further details on this subject, we refer the interested reader to the pioneering article of G Pagès [24], and the series of articles of V Bally, G Pagès, and J Printemps [2], G Pagès and J Printems [25], as well as G Pagès, H Pham and J Printems [26], 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 1 A i 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, x j := P X = x j X 1 = x i 1 = P X A j X A i 1, Using the decompositions M (f(x i 1 = = m j=1 m j=1 A j A j f(y P(X dy X 1 = x i 1 f(y P(X dy X 1 A i 1 + [M(f(x i 1 M(f(x ] P(X 1 dx X 1 A i 1, and m M (f(x i 1 = we find that j=1 A j f(x j P(X dy X 1 A i 1, [M M ](f(x i 1 m = [f(y f(x j j=1 A j ] P(X dy X 1 A i 1 [M(f(x i + 1 M(f(x] P(X 1 dx X 1 A i 1

18 On the Robustness of the Snell envelope 15 We let Lip(R d be the set of all Lipschitz functions f on R d, and we set L(f = f(x f(y sup x,y R d,x y x y for any f Lip(R d We further assume that M (Lip(R d Lip(R d From previous considerations, we find that [M M [ ](f(x i 1 L(f E X X ] 1 p X 1 = x i 1 p This clearly implies that +L(M (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 L(f [ E( X l+1 X l+1 p X = x i ] 1 p, +L(M l+1 (f E( X l X l p X = x i 1 p We also observe that ( f and u +1 Lip(R d ( u Lip(R d with L(u L(f L(M +1 (u +1 Using Lemma 21, we readily arrive at the following Proposition similar to Theorem 2 in [2] Proposition 36 Assume that (f 0 n Lip(R d n+1, and M (Lip(R d Lip(R d, for any 1 n In this case, we have (u 0 n Lip(R d n+1, and for any 0 n, we have the almost sure estimate u û ( X L(M +1 (u +1 X X + Proof: Using the decomposition we have that n 1 l=+1 +L(f n (L(u l + L(M l+1 (u l+1 E( X l X l p X 1 p [ E( X n X ] 1 n p p X û ( X u (X = [û ( X u ( X ] + [u ( X u (X ], then the proof is ended by û ( ξ u (X L(f n u ( X u (X L(u X X n 1 + [ E( X n X ] 1 n p p X (L(u l + L(M l+1 (u l+1 E( X l X l p X 1 p l=

19 On the Robustness of the Snell envelope 16 4 Monte Carlo importance sampling approximation schemes 41 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 (41 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, well 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 (42 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 η 0 n (X n = E P η0 (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 = dη 0 dη 0 (x 0 1 l dm l (x l 1, dη l (x l (43 Proposition 41 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

20 On the Robustness of the Snell envelope 17 These functions are connected by the following formula 0 n x = (x 0,, x E u (x = u (x dp dp (x (44 Proof: The first assertion is a simple consequence of the definition of a Snell envelope, and formula (44 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 +1 dη u +1 (x Under condition (42, 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 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 (43 By construction, for any 0 n, and any path x = (x 0,, x Ê, we have û (x = û (x dp dp (x,

21 On the Robustness of the Snell envelope 18 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 +1, (45 E b +1 with the random integral operator M from E into Ê +1 defined below M +1(x, dx +1 = η +1 (dx +1 R +1 (x, 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 41 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 [5], 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 +1(x, dx +1 = 1 V+1 (dx +1 R +1 (x, x +1, with the random fields V +1 := [η +1 η +1 ] From these observations, we readily prove that the approximation operators M +1 are unbiased, in the sense that 0 l x l E l Ê Pη0 ( M,l (f(x l F = M,l(f(x l, (46 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 ](f(x p l p 2 a(p 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 42 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 (47 2a(p l<n { M,l (x, dx l η l+1 [(R l+1 (x l,u l+1 p ]} 1 p

22 On the Robustness of the Snell envelope 19 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 [5], with no rate of convergence, and in [1] 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 l 2a(p M,l(x, dx l η l+1 [(R l+1 (x l,f p ] 1 p By the unbias property (46, we conclude that ( [ Ê Pη0 M,l M l+1 M ] l+1 (f (x p 1 p { 1/p 2a(p M,l (x, dx l η l+1 [(R l+1 (x l ]},fp For odd integers p = 2q + 1, with q 0, we use the fact that E(Y 2q+1 2 E(Y 2q E(Y 2(q+1 and E(Y 2q E(Y 2(q+1 q q+1, for any nonnegative random variable Y and (2(q + 1 q+1 = 2 (2q + 1 q+1 and (2q q = (2q + 1 q+1 /(2q + 1, so that a(2q 2q a(2(q + 1 2(q+1 2 (2q+1 (2q q+1/(q + 1/2 = ( a(2q + 1 2q+1 2 ( ÊP η0 M,l [M l+1 M ] l+1 (f (x 2q+1 2 ( 2 (2q+1 a(2q + 1 2q+1 2 M,l (x, dx l η [ l+1 (Rl+1 (x l,f2(q+1] q q+1 M,l (x, dx l η [ l+1 (Rl+1 (x l,f2(q+1] Using the fact that E(Y q q+1 E(Y q q+1, we prove that the rhs term in the above display is upper bounded by ( 2 (2q+1 a(2q + 1 2q+1 { 2 2(1 2(q+1 M,l (x, dx l η l+1 [(R 1 l+1 (x l,f2(q+1]}, from which we conclude that Ê Pη0 ( M,l [ M l+1 M ] l+1 (f (x 2q+1 1 2q+1 2a(2q + 1 { M,l (x, dx l η [ 1 l+1 (Rl+1 (x l,f2(q+1]} 2(q+1

23 On the Robustness of the Snell envelope 20 This ends the proof of the theorem The L p -mean error estimates stated in Theorem 42 are expressed in terms of L p norms of Snell envelope functions and Radon iodym derivatives The terms in the rhs of (47 have the following interpretation: M,l (x, dx l η l+1 [(R l+1 (x l,u l+1 p ] (Rl+1 ] = E[ (X l, ξ1 l+1 u l+1(ξl+1 1 p X = 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 an 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 for instance to tae η l+1 = Law(X l+1 as the sampling distribution, even if R l+1 is not nown (ie cannot be evaluated at any point of E l+1 In the sequel, we consider independent copies (ξ0, i ξn i 1 i of the Marov chain (X 0, X 1, X n, from the origin = 0 up to the final time horizon = n Then, for all = 0, n, we define the associated occupation measure η = 1 i=1 δ ξ i For all = 0, n, we let F be the sigma field generated by the random sequence (ξ l 0 l We also assume that the Marov transitions M n (x 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, where H n is supposed to be nown up to a normalizing constant 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 Arguing as in (45, we define the approximated Snell envelope (û 0 n on the state spaces (E 0 n by setting ( û (x = f (x M +1(x, dx +1 û +1(x +1, be +1 with the random integral operator M from E into Ê+1 defined below M +1(x, dx +1 = η +1 (dx +1 dm +1 (x, d η M +1 (x +1 = η +1 (dx +1 H +1 (x, x +1 η (H +1 (, x +1

24 On the Robustness of the Snell envelope 21 By construction, these random approximation operators M +1 satisfy the unbias property stated in (46, 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 M +1 η +1] and R+1 (x, x +1 := H +1(x, 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 (f(x l p 1 [ p F l 2 a(p η l M l+1 ( R l+1 (x l,f p] 1 p The above estimate is valid as soon as the rhs in the above inequality is well defined For instance, assuming that (H 1 and M l+1(u 2p sup x l,y l E l l+1 < H l+1 (x l, x l+1 H l+1 (y l, x l+1 h l+1(x l+1 with M l+1 (h2p l+1 <, we find that ( [ E M l+1 M l+1 ](u l+1 (x l p 1 p Fl ( 1 2 a(p M l+1 (h2p l+1 M l+1 ((u l+1 2p 2p Rephrasing the proof of Theorem 42, we prove the following result Theorem 43 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 (48 2a(p ( M l+1(h 2p 1 l+1 M l+1((u l+1 2p 2p l<n In the end of this subsection, recovering and extending results from [5], it is interesting to point out that both the Broadie-Glasserman estimator and this new BG type adapted estimator have positive bias Proposition 44 For any 0 n and any x E E(û (x u (x (49 Proof: This inequality can be proved easily by a simple bacward induction The terminal condition û n = u n implies directly the inequality on instant n Assuming the inequality holds true in instant, then Jensen s inequality implies that E(û (x f (x E( M +1 (û +1(x f (x M +1u +1 (x = u (x

25 On the Robustness of the Snell envelope 22 This ends the proof of the proposition 5 A genealogical tree based model 51 eutral genetic models Using the notation of Sub-section 41, 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 41, we have seen that M ((x 0,,x 1, d(y 0,, y = δ (x 0,,x 1 (d(y 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 B(E +1, we have η n = Law(X 0,, X n and M +1 (f(x := η n((1 x π (f π +1 η n ((1 x π (51 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 1, (52 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 x i 1 1 i 1 i 1 The initial particle system ξ ( 0 = ( ξ (i, 0 (53, 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,

26 On the Robustness of the Snell envelope 23 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 (54 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 ξ1,, i,ξ, i 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 ξ1,+1, i, ξ,+1 i, 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

27 On the Robustness of the Snell envelope 24 distribution η In other words, η is the empirical measure associated with conditionally independent and identically distributed random paths ξ i with is the empirical measure associated common distribution η Also observe η with conditionally independent and identically distributed random paths ξ i with common distribution η 1 M 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 0 In the sequel, the simulation of the path valued particle system (ξ 0 n will be called the Forward step and is summarized in the following algorithm 511 Forward algorithm Initialization At time step = 0, generate iid random copies of X 0 and set ξ 0 = ( ξ0 i 0 i At each time step = 1,, n 1 Selection: For each i = 1,,, generate independently an indice I i {1,, } with probability P(I i = j = 1/ Then set ˆξ i 1 = ξ Ii 1 2 Mutation: For each i = 1,,, generate independently iid random variables (ξ i, 0 i according to the transition ernel M (ˆξ i 1, 1, Then set ξi = (ˆξ i 1, ξi, 52 Convergence analysis For general mean field particle interpretation models (53, several estimates can be derived for the above particle approximation model (see for instance [11] 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: and E ( ηn (f n = η n (f n = E ( η n (f n (55 ( [ ] E η n η n (fn r n 1 r 2 a(r β(m p,n, p=0 with the Dobrushin ergodic coefficients β(m 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 a(p defined in (27 Arguing as in (28, for time homogeneous population sizes n =, for any functions f Osc 1 (E n,