Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

Transcription

1 Approximation of bacward stochastic differential equations using Malliavin weights and least-squares regression Emmanuel Gobet, Plamen Turedjiev To cite this version: Emmanuel Gobet, Plamen Turedjiev. Approximation of bacward stochastic differential equations using Malliavin weights and least-squares regression <hal v1> HAL Id: hal Submitted on 30 Aug 013 (v1), last revised 5 Jan 014 (v) 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 Approximation of bacward stochastic differential equations using Malliavin weights and least-squares regression E. Gobet and P. Turedjiev Centre de Mathématiques Appliquées Ecole Polytechnique and CNRS Route de Saclay 9118 Palaiseau cedex, France August 9, 013 Abstract: We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of bacward stochastic differential equations with the integration by parts-representation of the Z-component by [MZ0]. When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates tae into account the regularity of the terminal function. Keywords: Bacward stochastic differential equations, Malliavin calculus, dynamic programming equation, empirical regressions, non-asymptotic error estimates. MSC 010: 49L0, 60H07, 6Jxx, 65C30, 93E4. 1 Introduction 1.1 The problem Let T > 0 be a fixed terminal time and let (Ω, F, (F t ) 0 t T, P) be a filtered probability space whose filtration is augmented with the P-null sets. Let π = {0 =: t 0 < t 1 <... < t < t N := T } be a given time-grid on [0, T ] and i := t i+1 t i. Additionally, for a fixed q N\{0}, we are given a set {H (i) j : 0 i < j N} of (R q ) -valued random variables in L (F T, P) (i.e. square integrable and This research is part of the Chair Financial Riss of the Ris Foundation and of the FiME Laboratory. emmanuel.gobet@polytechnique.edu Corresponding Author. A significant part of the second author s research has been done while at Humboldt University. turedjiev@cmap.polytechnique.fr 1

3 F T -measurable) that we call Malliavin weights (whose significance is explained below). Here stands for the transpose. In this paper, we introduce a numerical algorithm, named MWLS, to approximate discrete time stochastic processes (Y, Z) defined by Y i = E i [ξ + f j (Y j+1, Z j ) j ], 0 i N, Z i = E i [ξh (i) N j=i + f j (Y j+1, Z j )H (i) j j ], 0 i N 1, where E i [ ] = E[ F ti ], ξ is a R-valued random variable in L (F T, P), and (ω, y, z) f j (ω, y, z) is F tj B(R) B((R q ) )-measurable. This system is solved bacward in time in the order Y N, Z, Y... and it taes the form of a dynamic programming equation with Malliavin weights: we call it Malliavin Weights Dynamic Programming equation (MWDP for short). The main application of (1.1) is to approximate continuous-time, decoupled Forward-Bacward SDEs (FBSDEs) of the form (1.1) T T Y t = ξ + f(s, X s, Y s, Z s )ds Z s dw s (1.) t t where (W s ) s 0 is a Brownian motion in R q, (X s ) s 0 is a diffusion in R d and ξ is of the form Φ(X T ). Indeed, according to [MZ0, Theorem 4.], there is a version of the process (Z t ) 0 t<t given by Z t = E t [ξh (t) T + T where the processes (H s (t) ) 0 t<s T are Malliavin weights defined by t t f(s, X s, Y s, Z s )H (t) s ds] (1.3) H s (t) = 1 ( s (σ 1 (r, X r )D t X r ) ), dw r 0 t < s T, (1.4) s t where (D t X r ) t is the Malliavin derivative of X r and σ(.) is the diffusion coefficient of X. The representation (1.3) is obtained via a Malliavin calculus integration by parts formula, see [Nua06] for a general account on the subject. A discretization procedure to approximate the FBSDE (1.-1.3) with (1.1), including explicit definitions of the random variables H (i) j based on (1.4), is given in [Tur13], where the author also computes the discretization error in terms of N. In honour of the connection between (1.1) and (1.-1.3), call the random variables H (i) j Malliavin weights, ξ the terminal condition, and (i, ω, y, z) f i (y, z) the driver. We say that the pair (Y, Z) satisfying (1.1) solves a discrete BSDE, or a MWDP, with terminal condition ξ and driver f i (x, y). Contributions. In this paper, we are not concerned with the discretization procedure, rather with the analysis of the MWDP equation (1.1) and its numerical resolution via what we call the MWLS algorithm, in which one uses empirical least-squares regressions (approximations on finite basis of functions using simulations) to compute conditional expectations. Since the system (1.1) may be relevant to problems beyond the FBSDE system (1.-1.3), we allow the framewor and assumptions

4 to accomodate as much generality as possible. However, MWLS is, to the best of our nowledge, the first direct implementation of formula (1.3) in a fully implementable numerical scheme. For other applications of Malliavin calculus in numerical simulations, with rather different perspectives and results to ours, see for instance [FLL + 99][KHP0][BT04][GM05][BCZ05][HNS11][BL13]. We adapt the recent theoretical analysis of [GT13] for discrete BSDEs (without Malliavin weights) to the setting of MWDP: actually, the Malliavin weights lead to significantly differences. As in the aforementioned reference, we consider locally Lipschitz driver f i (y, z), to allow the case of some quadratic BSDEs or some proxy/variance reduction methods - see Section 1.3. We prove stability results on the MWDP in Section. These results are instrumental throughout the paper. The stability estimates on Z are at the individual time points (coherently with the representation theorem of [MZ0]) rather than the time-averaged estimates of [GT13, Proposition 3.]: this allows for finer and more precise computations. The time-dependency in our estimates also taes into better account the regularity of the terminal condition, similarly to the continuous-time case [DG06]. Section 3 is the core of the paper: it is dedicated to the MWLS algorithm in the Marovian context Y i = y i (X i ) and Z i = z i (X i ) for some Marov chain X i in R d and unnown functions (y i ( ), z i ( )). In MWLS, the conditional expectations in (1.1) are replaced by Monte-Carlo least-squares regressions: to each point of the time-grid, we use a cloud of independent paths of the explanatory process X and the Malliavin weights H, and some approximation spaces for representing the value functions (y i ( ), z i ( )). The algorithm is detailed in Section 3. and a full error analysis is performed in Sections 3.3 and 3.4 in terms of the number of simulations and the approximation spaces. The final error estimates (Theorem 3.10) are similar to [GT13, Theorem 4.11] in that they are the time-averaged regression errors of the discrete BSDE, but the results are in a stronger norm and the time-dependency is better (in particular we avoid the 1/ i -factor). These error estimates appear to be optimal regarding the convergence rates (up to logarithmic factors) and are valid under rather great generality regarding the distribution of the stochastic model for X and H (model-free estimates). Taing into account the time-dependency as mentioned above is important for the complexity analysis (Section 3.5) and the derivation of optimal convergence rates. Regarding the curse of dimensionality, the rates depend on a dimensionality parameter which is that of the Marov chain X (i.e. d) and (hopefully) not that of the Malliavin weights. This paper is theoretically oriented, and is aimed at paving the way for such new numerical approaches. Further wors will be devoted to a deeper investigation about the numerical performance of the MWLS algorithm compared to other nown numerical schemes. 1. Notation used throughout the paper x stands for the Euclidean norm of the vector x, denotes the transpose operator. U p := (E U p ) 1 p stands for the Lp (P)-norm (p 1) of a random variable U. The F t -conditional version is denoted by U p, := (E U p ) 1 p. To indicate that U is additionally measurable w.r.t. the σ-algebra Q, we may write U L p (Q, P). For a multidimensional process U = (U i ) 0 i N, its l-th component is denoted by U l = (U l,i ) 0 i N. For any finite L > 0 and x = (x 1,..., x n ) R n, define the truncation function T L (x) := ( L x 1 L,..., L x n L). 3

5 For finite x > 0, log(x) is the natural logarithm of x. 1.3 Assumptions First set of hypotheses. The following assumptions hold throughout the entirety of the paper. Let R π > 0 be a fixed parameter: this constant determines which time-grid can be used. The larger R π, the larger the class of admissible time-grids. All subsequent error estimates depend on R π. (A ξ ) ξ is in L (F T, P), (A F ) i) (ω, y, z) f i (y, z) is F ti B(R) B((R q ) )-measurable for every i < N, and there exist deterministic parameters θ L (0, 1] and L f [0, + ) such that f i (y, z) f i (y, z ) for any (y, y, z, z ) R R (R q ) (R q ). L f (T t i ) (1 θ L)/ ( y y + z z ), ii) There exist deterministic parameters θ c (0, 1] and C f [0, + ) such that f i (0, 0) C f, 0 i < N. 1 θc (T t i ) iii) The time-grid π := {0 = t 0 <... < t N = T } satisfies i+1 max R π. 0 i N i (A H ) For all 0 i < j N, the Malliavin weights satisfy E[H (i) j F ti ] = 0, [ E[ H (i) j F ti ] ] 1/ C M (t j t i ) 1/ for a finite constant C M 0. Comments. We remar that assumptions (A ξ ) and (A F -i-ii) are the same as their equivalents in [GT13, Section ]: the usual case of Lipschitz BSDE is covered by θ L = θ c = 1. As explained in [GT13], the case of locally Lipschitz driver (θ L < 1 or/and θ c < 1) is interesting since it allows a large variety of applications, such as solving BSDEs using proxy methods or variance reduction methods, and solving quadratic BSDEs. We refer the reader to [GT13, Section ] for details. Assumption (A F -iii) is much simpler compared to [GT13]. If R π 1, (A F -iii) is satisfied by any time grid with non-increasing time-step, such as the grids of [GM10, Ric11, GGG1], which may be valuable for future wor on time-grid optimization. Assumption (A H ) is specific to the dynamic programming equation with Malliavin weights. It is satisfied for the weights derived in [MZ0], and this can remain true after discretization (see [Tur13] or [GM05]). Second set of hypotheses: Marovian assumptions. The following assumptions will mostly be used in Section 3. They give us a Marov representation for solutions of the discrete BSDEs (see 4

6 Equation (3.1) later). We also include additional assumptions on the terminal condition to obtain tighter estimates on Z i (see Corollary.6 and subsequent remars). (A X ) X is a Marov chain in R d (1 d < + ) adapted to (F ti ) i. For every i < N and j > i, there exist G i B(R d )-measurable functions V (i) j : Ω R d R d where G i F T is independent of F ti, such that X j = V (i) j (X i ). (A ξ ) i) ξ is a bounded F T -measurable random variable: C ξ := P ess sup ω ξ(ω) < +. ii) ξ is of form ξ = Φ(X N ) for a bounded, measurable function Φ. (A ξ ) In addition to (A ξ ), for some θ Φ [0, 1] and a finite constant C Φ 0, we have ξ E i ξ,i C Φ (T t i ) θφ/ for any i {0,..., N}. (A F ) For every i < N, the driver is of the form f i(ω, y, z) = f i (X i (ω), y, z), and (x, y, z) f i (x, y, z) is B(R d ) B(R) B((R q ) )-measurable and (A F ) is satisfied. (A H ) In addition to (A H), for every i < N and j > i, there is a function h (i) j : Ω R d (R q ) that is G i B(R d )-measurable, where G i F T is independent of F ti, such that H (i) j = h (i) j (X i ). Comments. (A X ) is usually satisfied when X is the solution of SDE or its Euler scheme built on the time grid π. (A ξ ), which is inspired by the fractional smoothness condition of [GM10] although somewhat stronger, is satisfied, for instance, if the terminal function Φ is θ Φ -Hölder and E i [ X N X i ] C X (T t i ), which is satisfied by a diffusion process (possibly including bounded jumps) with bounded coefficients and by the Euler scheme for such a diffussion. Indeed, we have ξ E i ξ,i Φ(X N ) Φ(X i ),i C Φ (C X (T t i )) θ Φ. Regarding (A H ), the Malliavin weights (1.4) satisfy this assumption (under the conditions that the drift coefficient b, the diffusion coefficient σ, and the inverse of the diffusion coefficient, as well as their first space-derivatives, are all uniformly bounded) because of the flow property of the diffusion X and since D t X r = X r X 1 t σ(t, X t )1 t r = x X t,x x=xt σ(t, X t )1 t r where X t,x denotes the SDE solution starting from x at time t, and X t := x X 0,x t. Stability Suppose that (Y 1, Z 1 ) (resp. (Y, Z )) solves a MWDP with terminal condition/driver (ξ 1, f 1,i ) (resp. (ξ, f,i )). We are mainly interested in studying the differences (Y 1 Y, Z 1 Z ). The driver f 1,i (y, z) is not assumed to be Lipschitz continuous, but we assume that each f 1,i (Y 1,i+1, Z 1,i ) is in L (F T ) so that Y 1,i and Z 1,i are also square integrable for any i (thans to (A H )). The driver f,i (y, z) is locally Lipschitz continuous w.r.t. (y, z) as in (A F -i), which is crucial for the validity of the a priori estimates. Additionally, we do not insist that the drivers be adapted, which will be needed in the setting of sample-dependant drivers. r 5

7 .1 Gronwall type inequalities Here we gather deterministic inequalities frequently used throughout the paper. They show how linear inequalities with singular coefficients propagate. They tae the form of unusual Gronwall type inequalities. Their proofs are postponed to Appendix A.1. We assume that π is in the class of timegrids satisfying (A F -iii). Lemma.1. Let α, β > 0 be finite. There exists a finite constant B α,β 0 depending only on R π, α and β (but not on the time-grid) such that, for any 0 i < N, 1 (t t j ) α 1 j B α,1 (t t i ) α, j=i 1 (t t j ) α 1 (t j t i ) β 1 j B α,β (t t i ) α+β 1. Lemma. (exponent improvement in recursive equations). Let α 0, β (0, 1 ] and {0,..., N 1}. Suppose that, for a finite constant C u 0, the finite non-negative real-valued sequences {u l } l and {w l } l satisfy u j w j + C u u l l, j N. (.1) (T t l ) 1 β (t l t j ) 1 α Then, for two finite constants C (.a) 0 and C (.b) 0 that depend only on C u, T, α, β and R π, u j C (.a) w j + C (.a) (T t l ) 1 β (t l t j ) + C 1 α (.b) u l l, j N. (T t l ) 1 β Lemma.3 (intermediate a priori estimates). Let α 0, β ]0, 1 ] and {0,..., N 1}. Assume that the finite non-negative real-valued sequences {u l } l and {w l } l satisfy (.) for finite constants C (.a) 0 and C (.b) 0. Then, for any finite γ > 0, there is a finite constant C (γ) (.3) 0 (depending only on C (.a), C (.b), T, α, β, R π and γ) such that u l l (T t l ) 1 β C(γ) (t l t j ) 1 γ (.3) Plugging (.3) with γ = 1 + α into (.1) gives a ready-to-use result. (.), j N. (.3) (T t l ) 1 β (t l t j ) 1 γ Proposition.4 (final a priori estimates). Under the assumptions of Lemma., (.1) implies u j w j + C ( 1 +α) (.3) C u, j N. (T t l ) 1 β (t l t j ) 1 α. Stability of discrete BSDEs with Malliavin weights Define: Y = Y 1 Y, Z = Z 1 Z, ξ = ξ 1 ξ, 6

8 f i = f 1,i (Y 1,i+1, Z 1,i ) f,i (Y 1,i+1, Z 1,i ). Let {0,..., N 1} be fixed: throughout this subsection, F t -conditional L -norms are considered and we recall the notation U, := E [ U ] for any square integrable random variable U. For j, define Θ j, = Y j+1, + Z j,. Using (A H ), we obtain E i [ ξh (i) N ] = E i[( ξ E i ξ)h (i) N ] and C M E i [ ξh (i) N ] E i [ ξ E i ξ ] (t N t i ), E i[ f j H (i) j ] C M E i[ f j ] j i + 1. (.4) t j t i Combining this ind of estimates with (A F -i), our stability equations (for i) are Y i, ξ, + j=i Z i, C M ξ E i ξ, T ti + f j, j + j=i L f Θ j, (T t j ) (1 θ L)/ j, (.5) C M f j, tj t i j + L f C M Θ j, (T t j ) (1 θ L)/ t j t i j. (.6) Proposition.5. Taing α = 0, β = θ L / and C u = L f (C M + T ) in Lemmas. and.3, recall the constant C (γ) (.3). Assume that ξ j is in L (F T ). Moreover, for each i {0,..., N 1}, assume that f 1,i (Y 1,i+1, Z 1,i ) is in L (F T ) and f,i (y, z) is locally Lipschitz continuous w.r.t. y and z as in (A F -i), with a constant L f. Then, under (A H ), we have Y i, C y (1) ξ, + C y () f j, j, 0 i N, j=i Z i, C z (1) ξ E i ξ, + C T z () ti where the above constants can be written explicitly: f j, tj t i j + C (3) z ξ, (T t i ) θ L, 0 i < N, C y (1) := 1 + L f C (1) (.3) (C M B θ L,1 + B θ L 1 + θ L,1 T )T, C () y := 1 + L f C (1) (.3) (C M + T )B θ L,1 T θ L, C (1) z := C M (1 + L f C ( 1 ) (.3) C M B θ L, 1 T θ L ), C () z := C M (1 + L f C ( 1 ) (.3) (C M + T )B θ L, 1 T θ L ), C z (3) := C M L f C ( 1 ) (.3) B 1 + θ L., 1 Proof. Using (.5) and (.6), we obtain ξ E j ξ, Θ j, C M + ξ, + (C M + T ) T tj f l, l tl t j 7

9 + (C M + T ) L f Θ l, l (T t l ) (1 θ L)/ t l t j, j. (.7) Upper bound for (.7). We apply Lemmas. and.3 under the setting u j = Θ j,, w j = ξ E C j ξ, M + ξ, + (C M + T ) f l, l, α = 0, β = θ L, C u = L f (C M + T ). To T tj tl t j mae results fully explicit, we first need to upper bound quantities of the form (γ > 0) I (γ) j+1 := (T t l ) 1 θ L (t l t j ) 1 γ. Using that ξ E l ξ, is non-increasing in l and Lemma.1, we obtain I (γ) j+1 = C M ξ E l ξ, T tl ξ E j+1 ξ, C M B θ L,γ (T t j ) 1 θ L γ + (C M + T )B θ L,γ + ξ, + (C M + T ) f r, tr t r r=l+1 l (T t l ) 1 θ L l (t l t j ) 1 γ l=j+ ξ, + B 1 + θ L,γ (T t j ) 1 θ L γ f l, l (t l t j ) 1 θ L γ. (.8) Upper bound for Y i,. Starting from (.5) and applying Lemma.3, we get Y i, ξ, + f j, j + L f C (1) (.3) I(1) i ; j=i then using the estimate (.8) and ξ E i ξ, ξ,, we obtain the announced inequality. Upper bound of Z i,. Starting from (.6) and applying Lemma.3, we have Z i, C M ξ E i ξ, T ti + C M f j, tj t i j + L f C M C ( 1 ) (.3) I( 1 ) i+1 ; therefore using the estimate (.8), we derive the advertised upper bound on Z i,..3 Almost sure bounds The following bounds are needed for the Monte-Carlo scheme. Corollary.6. Assume (A ξ -i), (A F) and (A H ) and recall the constants C y ( ) and C z ( ) from Proposition.5 where L f is replaced by L f. Then, we have Y i C y,i := C (1) y C ξ + C () y C f B θc,1(t t i ) θc, (.9) Z i C z,i := C z (1) ess sup ω ξ E i ξ,i T ti + C() z C f B θc, 1 + C (3) (T t i ) 1 θc z C ξ (T t i ) θ L. (.10) 8

10 The above upper bounds are able to handle the rather general terminal values ξ admitted by (A ξ -i). Without any further information on ξ, we can derive the simple bounds Y i + T t i Z i C y,z (.11) for an explicit, time uniform constant C y,z. It may, however, be useful to tae advantage of additional information on ξ. In Section 3.5, we tune the parameters of the MWLS method, and here finer estimates on C y,i and C z,i are useful. Two situations are of particular interest. For zero terminal condition, Y and Z get smaller and smaller as t i goes to T as expected: Y i + T t i Z i C(T t i ) θc for a constant C depending only on C y (), C z (), C f, θ c and R π. This result is useful for variance reduction methods lie the proxy method of [GT13, Section.], the method of Martingale basis [BS1], and the multilevel method of [Tur13]. Under (A ξ ), we have ξ E iξ,i C Φ (T t i ) θφ/, which leads to an improved estimate for Z: Z i C(T t i ) 1 +θc θ Φ for some constant C depending only on C z (1), C z (), C z (3), C f, θ c, R π, T, C ξ and C Φ. This is why in the subsequent analysis, we eep trac on the general dependence on i of the constants C y,i and C z,i. Proof of Corollary.6. (0, 0) is the solution of the MWDP with data (ξ 1 0, f 1,i 0). Applying Proposition.5 with (Y 1, Z 1 ) = (0, 0) and (Y, Z ) = (Y, Z) yields Y i, C y (1) ξ, + C y () f j (0, 0), j, j=i Z i, C(1) z ξ E i ξ, + C () T ti z f j (0, 0), tj t i j + C (3) z ξ, (T t i ) θ L, for i = 0,..., N 1. Taing = i, plugging in the almost sure bounds on ξ from (A ξ -i)and f j(0, 0) from (A F -ii), and using Lemma.1 then yields the result. 3 Monte-Carlo regression scheme Throughout this section, the Marovian assumptions (A X ), (A ξ ), (A F ) and (A H ) are in force. The notation and preliminary results used in this section overlap with [GT13, Section 4], and we recall and adapt them to the setting of MWLS in Section 3.1 for completeness. 3.1 Preliminaries Due to the Marovian assumptions, there are measurable, deterministic (but unnown) functions y i ( ) : R d R and z i ( ) : R d (R q ) for each i {0,..., N 1} such that the solution (Y i, Z i ) 0 i of the discrete BSDE (1.1) is given by (Y i, Z i ) := ( y i (X i ), z i (X i ) ). (3.1) 9

11 The proof of this is analogous to that of [GT13, Equation (4.1)]: one needs to apply Lemma 3.1 combined with G = G i defined in the assumptions (A X ) and (A H ) U = X i, and F (x) := Φ(V (i) N (x)) + =i and F (x) := Φ(V (i) N (x))h(i) N (x) + ( (i) f V (x), y +1(V (i) +1 (x)), z (V (i) (x))) for y i ( ), =i+1 ( (i) f V (x), y +1(V (i) +1 (x)), z (V (i) (x))) h (i) (x) for z i ( ). Lemma 3.1 ([GT13, Lemma 4.1]). Suppose that G and H are independent sub-σ-algebras of F. For l 1, let F : Ω R d R l be bounded and G B(R d )-measurable, and U : Ω R d be H-measurable. Then, E[F (U) H] = j(u) where j(h) = E[F (h)] for all h R d. We recall the general notation of [GT13, Section 4.1] for ordinary least-squares regression problems: Definition 3. (Linear least-squares regression). For l, l 1 and for probability spaces ( Ω, F, P) and (R l, B(R l ), ν), let S be a F B(R l )-measurable R l -valued function such that S(ω, ) L (B(R l ), ν) for P-a.e. ω Ω, and K a linear vector subspace of L (B(R l ), ν) spanned by deterministic R l -valued functions {p (.), 1}. The least-squares approximation of S in the space K with respect to ν is the ( P ν-a.e.) unique, F B(R l )-measurable function S given by S (ω, ) := arg inf φ(x) S(ω, x) ν(dx). (3.) φ K We say that S solves OLS(S, K, ν). On the other hand, suppose that ν M = 1 M M m=1 δ X (m) is a discrete probability measure on (Rl, B(R l )), where δ x is the Dirac measure on x and X (1),..., X (M) : Ω R l are i.i.d. random variables. For an F B(R l )-measurable R l -valued function S such that S ( ω, X (m) (ω) ) < for any m and P-a.e. ω Ω, the least-squares approximation of S in the space K with respect to ν M is the ( P-a.e.) unique, F B(R l ) measurable function S given by S 1 (ω, ) := arg inf φ K M We say that S solves OLS(S, K, ν M ). M φ ( X (m) (ω) ) S ( ω, X (m) (ω) ). (3.3) m=1 From (3.1), the MWDP (1.1) can be reformulated in terms of Definition 3.: taing for K (l ) i any dense subset in the R l -valued functions belonging to L (B(R d ), P (X i ) 1 ), for each i {0,..., N 1}, y i ( ) solves OLS( S Y,i (x (i) ), K (1) i, ν i ), for S Y,i (x (i) ( ) := Φ(x N ) + f x, y +1 (x +1 ), z (x ) ), =i z i ( ) solves OLS( S Z,i (h (i), x (i) ), K (q) i, ν i ), for S Z,i (h (i), x (i) ( ) := Φ(x N )h N + f x, y +1 (x +1 ), z (x ) ) h =i+1 ν i := P (H (i) i+1,..., H(i) N, X i,..., X N ) 1, 10 (3.4)

12 h (i) := (h i+1,..., h N ) ((R q ) ) N i, x (i) := (x i,..., x N ) (R d ) N i+1. (3.5) However, the above least-squares regressions encounter two computational problems: (CP1) L (B(R d ), P (X i ) 1 ) is usually infinite dimensional; (CP) (3.) is computed using the presumably untractable law of (H (i) i+1,..., H(i) N, X i,..., X N ). Therefore, the functions y i ( ) and z i ( ) are to be approximated on finite dimensional function spaces with the sample-based empirical version of the law, as described below. 3. Algorithm The first computational problem (CP1) is handled using a predetermined finite dimensional vector spaces. Definition 3.3 (Finite dimensional approximation spaces). For i {0,..., N 1}, we are given two finite functional linear spaces of dimension K Y,i and K Z,i K Y,i := span{p (1) Y,i,..., p(k Y,i) Y,i K Z,i := span{p (1) Z,i,..., p(k Z,i) Z,i }, for p () Y,i : Rd R s.t. E[ p () Y,i (X i) ] < +, }, for p () Z,i : Rd (R q ) s.t. E[ p () Z,i (X i) ] < +. The function y i ( ) (resp. z i ( )) will be approximated in the linear space K Y,i (resp. K Z,i ). The best approximation errors are defined by ] ] EApp.,i Y := inf E [ φ(x i ) y i (X i ), EApp.,i Z := inf E [ φ(x i ) z i (X i ). φ K Y,i φ K Z,i The second computational problem (CP) is solved using the empirical measure built from independent simulations with distribution ν i. The number of simulations is large enough to avoid having under-determined systems of equations to solve. Definition 3.4 (Simulations and empirical measures). For i {0,..., N 1}, generate M i K Y,i K Z,i independent copies C i := {(H (i,m), X (i,m) ) : m = 1,..., M i } of (H (i), X (i) ) := (H (i) i+1,..., H(i) N, X i,..., X N ): C i forms a cloud of simulations used for the regression at time i. Denote by ν i,m the empirical probability measure of the C i -simulations, i.e. ν i,m := 1 M i M i δ (i,m) (H i+1,...,h(i,m) N,X (i,m) i,...,x (i,m) m=1 N ). (3.6) Furthermore, we assume that the clouds of simulations (C i : 0 i < N) are independently generated. All these random variables are defined on a probability space (Ω (M), F (M), P (M) ). Observe that allowing time-dependency in the number of simulations M i and in the vector spaces K Y,i and K Z,i is coherent with our setting of time-dependent local Lipschitz driver. Denoting by (Ω, F, P) the probability space supporting (H (0),..., H (), X), which serves as a generic element for the clouds of simulations, the full probability space used to analyze our algorithm is the product space ( Ω, F, P) = (Ω, F, P) (Ω (M), F (M), P (M) ). By a slight abuse of notation, we 11

13 write P (resp. E) to mean P (resp. Ē) from now on. The subsequent use of conditioning arguments is based on the following definition. Definition 3.5. Define the σ-algebras F ( ) i := σ(c i+1,..., C ), F (M) i := F ( ) i σ(x (i,m) i : 1 m M i ). For every i {0,..., N 1}, let E M i [ ] (resp. P M i ) with respect to F (M) i. We now come to the definition of the MWLS algorithm: this is merely the finite-dimensional version of (3.4) plus a soft truncation of the solutions using the truncation function T. (.) (defined in Section 1.). Definition 3.6 (MWLS algorithm). Set y (M) N ( ) := Φ( ). For each i = N 1, N,..., 0, set the random functions y (M) i ( ) and z (M) i ( ) recursively as follows. (a) First, define z (M) i where ( ) = T Cz,i ( ψ (M) Z,i ( )) where C z,i is the almost sure bound of Corollary.6 and ψ (M) Z,i ( ) solves OLS( S(M) Z,i (h(i), x (i) ), K Z,i, ν i,m ) for S (M) Z,i (h(i), x (i) ) := Φ(x N )h N + =i+1 where h (i), x (i), ν i,m are defined in (3.5) and (3.6). f ( x, y (M) +1 (x +1), z (M) (x ) ) h, (3.7) (b) Second and similarly, define y (M) ( (M) i ( ) := T Cy,i ψ Y,i ( )) where ψ (M) Y,i ( ) solves OLS(S(M) Y,i (x(i) ), K Y,i, ν i,m ) for S (M) Y,i (x(i) ) := Φ(x N ) + =i f ( x, y (M) +1 (x +1), z (M) (x ) ). (3.8) Before performing the error analysis, we state the following uniform (resp. conditional variance) bounds on the functions S (M) Y,i ( ) (resp. the l-th coordinate of S(M) Z,i (H(i,m), X (i,m) ) for each m and l). These bounds are used repeatedly in Section 3.3. The proof is postponed to Appendix A.. Lemma 3.7. For all i {0,..., N 1}, there are finite constants C y,i 0 and C z,i 0 such that S (M) Y,i (x(i) ) C y,i, x (i), q [ Var S (M) l,z,i (H(i,m), X (i,m) ) ] (M) F i C z,i, m {1,..., M i }. l=1 We can write a precise time-dependency of the constants C y,i and C z,i : C y,i := c 1 C ξ + c C f (T t i ) θc, Cz,i := c 3 C ξ (T t i ) 1/ + c 4 C f (T t i ) θc 1, (3.9) where (c j ) 1 j 4 depend only on (L f, C M, q, C y (1), C y (), C z (1), C z (), C z (3), T, R π, θ L, θ c ) (computed explicitely in the proof). 1

14 The above time-dependency is to be used to derive convergence rates for the complexity analysis. 3.3 Main result: error analysis We precise the random norms used to quantify the error of MWLS. Definition 3.8. Let ϕ : Ω (M) R d R or (R q ) be F (M) B(R d )-measurable. For each i {0,..., N 1}, define the random norms ϕ i, := R d ϕ(x) P X 1 i (dx), ϕ i,m := 1 M i The accuracy of the MWLS algorithm is measured as follows: [ Ē(Y, M, i) := E [ E(Y, M, i) := E y (M) i y (M) i M i m=1 ] [ ( ) y i ( ) i,, Ē(Z, M, i) := E ] [ ( ) y i ( ) i,m, E(Z, M, i) := E ϕ(x (i,m) i ). z (M) i z (M) i ] ( ) z i ( ) i,, ( ) z i ( ) i,m In our analysis, we will have to switch from errors in true measure Ē(... ) to errors in empirical measure E(... ), and vice-versa: this is not trivial since (y (M) i (.), z (M) i (.)) and the empirical norm. i,m depend on the same sample. However, the switch can be performed using concentration-of-measure estimates uniformly on a class of functions [GKKW0, Chapter 9]. We directly state the ready-to-use result, which is a straightforward adaptation of [GT13, Proposition 4.10] to our context. Proposition 3.9. Recall the constants C y,i (resp. C z,i ) from Corollary.6, and define the interdependence errors EDep.,i Y 08(K Y,i + 1) log(3m i ) := C y,i, E Z 08(K Z,i + 1)q log(3m i ) Dep.,i := C z,i. M i M i For each i {0,..., N 1}, we have ]. Ē(Y, M, i) E(Y, M, i) + E Y Dep.,i, Ē(Z, M, i) E(Z, M, i) + E Z Dep.,i. The aim is to determine a rate of convergence for E(Y, M, ) = (E y y M,M ) 1 and E(Z, M, ) = (E z z M,M ) 1 using the local error terms (E()) defined below. Theorem 3.10 (global error of the MWLS algorithm). For 0 N 1, define E() := EApp.,+1+ Y C K Y,+1 y,+1 + EApp., Z + M C KZ, ( z, + L f E Y +1 M Dep.,+1 + E Z ) Dep.,. (3.10) For every {0,..., N 1}, [ ] 1/ E( y y M,M ) E Y App., + C KY, y, M 13 + C y (M) j= E(j) j (T t j ) (1 θ L)/, (3.11)

15 [ ] 1/ E( z z M,M ) E Z App., + C KZ, z, M + C (M) z j=+1 E(j) j (T t j ) (1 θ L)/ t j t, (3.1) where, recalling the constant C (γ) (.3) from Lemma.3 (with α = 0, β = θ L, γ { 1, 1} and C u = L f ( C M + 4 T )), C (M) y := + 4L f C (1) (.3) (1 + B θ L,1 T θ L (CM + T )), C (M) z := C M + C M L f C ( 1 ) (.3) (1 + B θ L, 1 T θ L (CM + T )). Discussion. Observe that owing to Proposition 3.9, similar estimates (with modified constants) are valid for Ē(Y, M, ) = (E y y M, ) 1 and Ē(Z, M, ) = (E z z M, ) 1. The global error ( ) is a weighted time-average of three different errors. 1) The contributions E Ȧpp.,. are the best approximation errors using the vector spaces of functions: this accuracy is achieved asymptotically with an infinite number of simulations (tae M + in our estimates). K.,. ) The contributions M. are the usual statistical error terms: the larger the number of simulations or the smaller the dimensions of the vector spaces, the better the estimation error. 3) The contributions E Ḋep.,. are related to the interdependencies between regressions at different times: this is intrinsic to the dynamic programming equation with N nested empirical regressions. However, due to Proposition 3.9, the latter contributions are of same magnitude as statistical error terms (up to logarithmic factors). Therefore roughly speaing, the global error is of order of the best approximation errors plus statistical errors, as if there were a single regression problem [GKKW0, Theorem 11.1]. In this sense, these error bounds are optimal: it is not possible to improve the above estimates with respect to the convergence rates (but only possibly with respect to the constants). An optimal tuning of parameters is proposed in Section 3.5. In comparison to [GT13], where a different Monte-Carlo regression scheme is analyzed, the upper bound for the global error has a similar shape, but with two important differences. Norm on Z. In [GT13] it is a time average of L -norms, whereas here the norm used is time-wise: it currently leads to more informative error bounds. This is an advantage of the discrete BSDE with Malliavin weights against the MDP of [GT13], and we expect a better estimation of the Z-component. Time-dependency. The MWDP yields better estimates on y(.) and z(.) w.r.t. time, which allows better parameters tuning, and finally better convergence rates (see Section 3.5). 3.4 Proof of Theorem Preliminary results The following proposition is a ey tool: the two first properties are of deterministic nature, the two last are probabilistic. 14

16 Proposition 3.11 ([GT13, Proposition 4.1]). With the notation of Definition 3., suppose that K is finite dimensional and spanned by the functions {p 1 (.),..., p K (.)}. Let S solve OLS(S, K, ν) (resp. OLS(S, K, ν M )), according to (3.) (resp. (3.3)). The following properties are satisfied: (i) linearity: the mapping S S is linear. (ii) contraction property: S L(B(R l ),µ) S L(B(R l ),µ), where µ = ν (resp. µ = ν M ). (iii) conditional expectation solution: in the case of the discrete probability measure ν M, assume additionally that the sub-σ-algebra Q F is such that ( p j (X (1) ),..., p j (X (M) ) ) is Q-measurable for every j {1,..., K}. Setting S Q (X (m) ) := Ẽ[S(X (m) ) Q] for each m {1,..., M}, then Ẽ[S Q] solves OLS ( ) S Q, K, ν M. (iv) bounded conditional variance: in the case of the discrete probability measure ν M, suppose that S(ω, x) is G B(R l )-measurable, for G F independent of σ(x (1),..., X (M) ), there exists a Borel measurable function g : R l E, for some Euclidean space E, such that the random variables {p j (X (m) ) : m = 1,..., M, j = 1,..., K} are H := σ(g(x (m) ) : m = 1,..., M)- measurable, and there is a finite constant σ 0 that uniformly bounds the conditional variances Ẽ [ S(X (m) ) Ẽ(S(X (m) ) G H) G H ] σ P-a.s. and for all m {1,..., M}. Then [ ] Ẽ S ( ) Ẽ[S ( ) G H] L (B(R l ),ν M ) G H σ K M. Intermediate processes and local error terms. For each {0,..., N 1}, recall the functions S Y, (x (i) ) and S Z, (h (i), x (i) ) from (3.4), the linear spaces K Y, and K Z, from Definition 3.3, and the empirical measure ν,m from (3.6), and set ψ Y, ( ) solves OLS( S Y, (x (i) ), K Y,, ν,m ), ψ Z, ( ) solves OLS( S Z, (h (i), x (i) ), K Z,, ν,m ). From Lemma 3.1 and our Marovian assumptions, observe that (E M [S Y,(X (,m) )], E M [S Z,(H (,m), X (,m) )]) = ( y (X (,m) ), z (X (,m) ) ) for each m {1,..., M } where ( y ( ), z ( ) ) are the unnown functions defined in (3.1). Proposition 3.11(iii) implies the first statement of the following lemma. The second statement results from a direct interchange of inf and E, and from the identical distribution of (X (,m) ) for all m. Lemma 3.1. For each {0,..., N 1}, E M [ψ Y,( )] solves OLS( y (.), K Y,, ν,m ), E M [ψ Z,( )] solves OLS( z (.), K Z,, ν,m ). In addition, recalling the local error terms EApp., Y and E App., Z from Definition 3.3, E [ E M [ψ Y, ( )] y ( ) ] [,M = E inf φ( ) y ( ) ],M (E Y App., ), φ K Y, E [ E M [ψ Z, ( )] z ( ) ] [,M = E inf φ( ) z ( ) ],M (E Z App., ). φ K Z, } } 15

17 3.4. Proof of Theorem 3.10 Step 1: decomposition of the error on Y. From T Cy, (y ) = y and the Lipschitz continuity of T Cy,, it follows that y ( ) y (M) ( ),M is less than or equal to y ( ) ψ (M) Y, ( ),M. Using the triangle inequality for the.,m -norm, it follows that Because S (M) Y, y ( ) y (M) ( ),M y ( ) E M [ψ Y, ( )],M + E M [ψ Y, ( )] ψ (M) Y, ( ),M. (3.13) ( ) depends on z(m) ( ) computed with the same cloud of simulations C as that used to ( ), it raises some interdependency issues that we solve by maing a small define the OLS solution ψ (M) Y, perturbation to the intermediate processes as follows: for x () = (x,..., x N ), define S (M) Y, (x() ) := Φ(x N ) + f ( x, y (M) +1 (x +1), z (x ) ) + ψ (M) (M) Y, ( ) solves OLS( S Y, (x() ), K Y,, ν,m ). i=+1 f i ( xi, y (M) i+1 (x i+1), z (M) i (x i ) ) i, This perturbation is not needed for the Z-component, because S (M) Z, (h(), x () ) depends only on the subsequent clouds of simulations {C j, j > }. Applying the L -norm, the triangle inequality in (3.13), and the first part of Lemma 3.1 yields E(Y, M, ) EApp., Y + E M (M) [ ψ Y, ( ) ψ (M) Y,( )],M + ψ Y, ( ) EM (M) [ ψ Y, ( )],M (M) + ψ Y, ( ) ψ(m) Y, ( ),M. (3.14) Let us handle each term in the above inequality separately. Term E M (M) [ ψ Y, ( ) ψ Y,( )],M. Set S (M) ξ Y,(x) (M) := E( S Y, (X() ) S Y, (X () ) X () = x, F (M) ). Recalling that Y, (x() ) S Y, (x () ) is built only using the clouds {C j, j + 1}, it follows from Lemma 3.1 that E M (M) [ S Y, (X(,m) ) S Y, (X (,m) )] is equal to ξ Y, (X(,m) ) for every m {1,..., M }. Then, using Proposition 3.11(i)(iii), E M (M) [ ψ Y, ( ) ψ Y,( )] solves OLS( ξ Y, ( ), K Y,, ν,m ). By Proposition 3.11(ii), E [ E M (M) [ ψ Y, ( ) ψ Y,( )] ] [,M E ξ Y, ( ) ] [,M = E ( ξ Y, (X )) ], where the final equality follows from the fact that ξ Y, ( ) is generated only using the simulations in the clouds {C j : j > } and {X, X (,1),..., X (,M ) } are identically distributed. Defining the triangle inequality yields ξ Y,(x) := E[S (M) Y, (X() ) S Y, (X () ) X () = x, F (M) ], (3.15) ξ Y,(X (M) ) S Y, (X() ) S (M) Y, (X() ) + ξy,(x ) 16

18 (M) Term ψ Y, ( ) EM f (X, y (M) +1 (X +1), z (M) (X )) f (X, y (M) +1 (X +1), z (X )) + ξy,(x ) L f Ē(Z, M, ) + ξ (T t ) 1 θ L Y,(X ). is bounded above by C y, (lie S (M) Y, (M) ψ Y, ( ) EM S (M) (M) [ ψ Y, ( )],M. Since Y, (.) depends only on the clouds {C j, j > } and (.), see Lemma 3.7), it follows from Proposition 3.11(iv) that [ ψ (M) Y, ( )],M is bounded above by C y, KY, /M : this is similar to the statistical error term in usual regression theory. (M) Term ψ Y, ( ) ψ(m) Y, ( ),M. Owing to Proposition 3.11(i)(ii), (M) above by S Y, ( ) S(M) Y, ( ),M, which equals M M m=1 f (X (,m), y (M) L f z ( ) z (M) (T t ) 1 θ L +1 (X(,m) +1 ( ),M. ), z(m) (X (,m) (M) ψ Y, ( ) ψ(m) Y, ( ),M is bounded )) f (X (,m), y (M) +1 (X(,m) +1 ), z (X (,m) )) Collecting the bounds on the three terms, substituting them into (3.14) and applying Proposition 3.9 yields E(Y, M, ) E Y App., + ξ Y,(X ) + C y, KY, M + L f (T t ) 1 θ L { (1 + } )E(Z, M, ) + EDep., Z. (3.16) Step : decomposition of the error on Z. Analogously to (3.14), one obtains the upper bound E(Z, M, ) E Z App., + E M [ψ (M) Z, ( ) ψ Z,( )],M + ψ (M) Z, ( ) EM [ψ (M) Z, ( )],M. Since S (M) Z, (.) depends only on the clouds {C j, j > } and the F (M) -conditional variance of S (M) Z, (H(,m), X (,m) ) is bounded above by C z, for all m (see Lemma 3.7), it follows from Proposition 3.11(iv) that ψ (M) Z, ( ) EM [ψ(m) Z, ( )],M is bounded above by C z, KZ, /M. Defining ξ Z,(x) := E[S (M) Z, (H(), X () ) S Z, (H (), X () ) X () = x, F (M) ], (3.17) it follows that E M [ψ(m) Z, ( ) ψ Z,( )] solves OLS(ξ Z, ( ), K Z,, ν,m ). Therefore, E(Z, M, ) E Z App., + ξ Z,(X ) + C z, KZ, M. (3.18) Step 3: error propagation and a priori estimates. Observe that (ξy, (X ), ξz, (X )) defined in (3.15,3.17) solves a MWDP with terminal condition 0 and driver f ξ,(y, z) := f (X, y (M) +1 (X +1), z (M) (X )) f (X, y +1 (X +1 ), z (X )). Applying Proposition.5 with L f = 0 and local Lipschitz continuity of 17

19 f j (.) yields ξy,(x ) L f j= ξz,(x ) C M L f Ē(Y, M, j + 1) + Ē(Z, M, j) (T t j ) 1 θ L j, j=+1 Ē(Y, M, j + 1) + Ē(Z, M, j) (T t j ) 1 θ L j. tj t Next, introducing the notation Θ j follows that := E(Y, M, j + 1) + E(Z, M, j) and applying Proposition 3.9, it ξy,(x ) L f j= ξz,(x ) C M L f Θ j j (EDep.,j+1 Y + L + E Dep.,j Z ) j (T t j ) 1 θ L f, j= (T t j ) 1 θ L j=+1 Θ j j (T t j ) 1 θ L tj t + C M L f j=+1 (E Y Dep.,j+1 + E Z Dep.,j ) j (T t j ) 1 θ L tj t. Substituting the above into (3.16) and (3.18), and merging together the terms in Z, it follows that E(Y, M, ) E Y App., + C y, KY, M + L f EApp., Y + C KY, y, + M E(Z, M, ) E Z App., + C z, KZ, M + C M L f j=+1 + C M j= j= (EDep.,j+1 Y + E Dep.,j Z ) j + 4L (T t j ) 1 θ L f j= E(j) j + 4L (T t j ) 1 θ L f j= j=+1 E(j) j (T t j ) 1 θ L tj t Θ j j (T t j ) 1 θ L Θ j j (T t j ) 1 θ L, (3.19) Θ j j (T t j ) 1 θ L tj t. (3.0) Step 4: final estimates. Now, summing (3.0) and (3.19), one obtains an estimate for Θ : Θ E() + (C M + T ) + L f ( C M + 4 T ) j=+1 j=+1 E(j) j (T t j ) 1 θ L tj t Θ j j (T t j ) 1 θ L tj t. Thus, using Lemmas. and.3 with α = 0, β = θ L, C u = L f ( C M + 4 T ), w := E() + (C M + T ) E(j) j j=+1 (T t j) 1 θ L, we can control weighted sums involving (Θ ) using weighted sums tj t of (w ), which is exactly what we need to complete the upper bounds ( ) for E(Y, M, ) and 18

20 E(Z, M, ). Namely, let γ > 0: j=+1 j=+1 w j j (T t j ) 1 θ L (t j t ) 1 γ E(j) j (T t j ) 1 θ L (t j t ) 1 γ + (C M + T ) (1 + B θ L,γ T θ L (CM + T )) l=+1 l=+ E(l) l (T t l ) 1 θ L E(l) l (T t l ) 1 θ L (t l t ) 1 γ, l 1 j=+1 j (t l t j ) 1 θ L (t j t ) 1 γ where we have applied Lemma.1. Then, we obtain j=+1 Θ j j (T t j ) 1 θ C(γ) L (.3) (t j t ) (1 + B 1 γ θ L,γ T θ L (CM + T )) l=+1 E(l) l (T t l ) 1 θ L (t l t ) 1 γ. Plug the above inequality into (3.19) and (3.0) to derive (3.11) and (3.1). 3.5 Complexity analysis We discuss the complexity in different cases according to the regularity of the value functions (y i ( ), z i ( )) and the choice of the grid π. In order to have a fair comparison with other numerical schemes, we revisit the setting of [GT13, Section 4.4], which we partly recall for completeness, and extend the analysis to include more general settings. We perform an asymptotic complexity analysis as the number N of grid times goes to +. We are concerned with time-dependent bounds: thus in the following, the order convention, O(.) or o(.), is uniform in t i. The grids under consideration are of the form π (θπ) := {t i = T T (1 i N ) 1 θπ } for θ π (0, 1] (inspired by [GM10, GGG1]). Observe that their time-step i is not-increasing in i, hence they all satisfy (A F -iii) with the same parameter R π = 1. The magnitude of the final accuracy is denoted by N θconv for some parameter θ conv > 0. This is usually related to time-discretization errors between the continuous-time BSDE and the discretetime one, θ conv may range from 0 + (for non smooth data [GM10, Theorem 1.1]) to 1 (in the case of smooth data [GL07, Theorems 7 and 8]). The approximation spaces are given by local polynomials of degree n (n 0) defined on hypercubes with edge length δ > 0, covering the set [ R, R] d (R > 0): we denote it by P n,δ,r loc.. The functions in P n,δ,r loc. tae values in R for the y-component and in (R q ) for z (using local polynomials component-wise), but we omit this in the notation. The best-approximation errors are easily controled (using the Taylor formula): inf ϕ P n,δ,r loc. ϕ(x i ) u(x i ) u (P( X i > R)) 1/ + c n D n+1 u δ n+1 (3.1) 19

21 for any function u that is bounded, n + 1-times continuously differentiable with bounded derivatives, and where the constant c n does not depend on (R, u, δ). The dimension of the vector space P n,δ,r loc. is bounded by c n (R/δ) d where c n depends on d and n. A significant computational advantage of local polynomial basis is that the cost of computing the regression coefficients associated to a sample of size M dim(p n,δ,r loc. ) is O(M) flops [GVL96], and the cost of evaluating pointwise the approximated function is c d,n (a constant that does not depend on the number of hypercubes). On the other hand, the cost of generating the clouds of simulations and computing the simulated functionals (S (M) Y,i (X(i,m) ), S (M) Z,i (H(i,m), X (i,m) )) i,m is O( i=0 NM i), which is clearly dominant in the computational cost C of the MWLS algorithm. To summarize, the computational cost is C = O( NM i ). i=0 Another advantage of the local polynomial basis is that there is substantial potential for parallel computing. To mae the tail contributions (outside [ R, R] d ) small enough, we assume that X i has exponential moments (uniformly in i), i.e. sup N 1 sup 0 i N E(e λ Xi ) < + for some λ > 0, so that the choice R := θ conv λ 1 log(n + 1) is sufficient to ensure (P( X i > R)) 1/ = O(N θconv ). To simplify the discussion, we assume θ L = θ c = 1. Smooth functions. Assume that y i ( ), z i ( ) are respectively of class C l+1 b (R d, R) and Cb l(rd, (R q ) ) (bounded with bounded derivatives) for some l N\{0}: this is similar to the discussion of [GT13, Section 4.4]. In fact, this is usually valid for the continuous-time limit (a priori estimates on the semi-linear PDE, see [DG06, CD1]) provided that the data are smooth enough. In particular, we may assume (A ξ ) with θ Φ = 1. This leads to time-uniform bounds on the quantities C y,i, C z,i, C y,i, T t i Cz,i. Set δ y,i := N θconv l+1, δz,i := N θconv l, M i := (log(n + 1)) d+1 N θconv(+ d l ), tae K Y,i := P l,δy,i,r loc. and K Z,i := P l 1,δz,i,R loc.. From Proposition 3.9, Theorem 3.10 and the inequality (3.1), it is easy to chec that E Y App.,i = O(N θconv ), E Y Dep.,i = o(n θconv ), Cy,i KY,i M i ( = o N θconv / ) log(n + 1), E Z App.,i = O(N θconv ), E Z Dep.,i = O(N θconv ), Cz,i KZ,i M i = (T t i ) 1 O ( N θconv / log(n + 1) Consequently, using Lemma.1, we finally obtain ). [ E( y i y M i i,m )] 1/ = O(N θ conv ), [ 1/ E( z i zi M i,m )] = O(N θ conv ) (1 + (T t i) 1 ). log(n + 1) For any time-grid π = π (θπ), we get sup 0 i N E( y i yi M i,m ) + i=0 ie( z i zi M i,m ) = O(N θconv ). The computational cost is C = O ( log(n + 1)) d+1 N θconv(+ d )+) l. Ignoring the loga- 0

22 rithmic factors, we obtain a final accuracy in terms of the computational cost: C 1 (+ d l )+ θconv. It should be compared with the rate C 1 (+ d l )+ θconv 3 which is valid for the LSMDP algorithm [GT13]. This shows a small improvement on the rate. Moreover the controls are stated in stronger norms. The ratio d/l is the usual balance between dimension and smoothness, arising when approximating a multidimensional function. Hölder terminal condition. We investigate the case of non-smooth terminal condition, where nevertheless there is a smoothing effect of the conditional expectation yielding smooth value functions (y i ( ), z i ( )). Namely, assume that Φ is bounded and θ Φ -Hölder continuous (in particular with (A ξ )), and that, for all i, the function y i( ) (resp. z i ( )) is (l + 1)-times (resp. l-times) continuously differentiable with highest derivatives bounded by D l+1 x y i C(T t i ) (θφ l)/, D l xz i C(T t i ) (θφ (l+1))/. (3.) These qualitative assumptions are related to the wors of [DG06, CD1], who have determined similar estimates for the gradients of quasi-linear PDEs under quite general conditions on the driver, terminal condition, and differential operator: their estimates cover the case l = 0 [DG06, Theorem.1] or θ Φ = 0 and l 1 [CD1, Theorem 1.4], but the Hölder continuous setting is not investigated. We therefore extrapolate these results in the assumptions (3.) for the purposes of this discussion. In this setting, we have time-uniform bounds on the quantities C y,i, (T t i ) 1 θ Φ C z,i, C y,i, T t i Cz,i. Set δ y,i := T t i N θconv l+1, δz,i := T t i N θconv l, M i := (log(n + 1)) d+1 N θconv(+ d l ) (T t i ) d/, tae K Y,i := P l,δy,i,r loc. and K Z,i := P l 1,δz,i,R loc.. Similarly to before, using in particular (3.1), we eventually obtain E Y App.,i = O(N θconv ), E Y Dep.,i = o(n θconv ), Cy,i KY,i M i EApp.,i Z = (T t i ) θ Φ 1 O(N θconv ), EDep.,i Z = (T t i ) θ Φ 1 O(N θconv ), KZ,i ( C z,i = (T t i ) 1 O N θconv / ) log(n + 1). M i Consequently, using Lemma.1, we finally obtain [ E( y i y M i i,m )] 1/ = O(N θ conv ), ( = o N θconv / ) log(n + 1), [ 1/ E( z i zi M i,m )] = O(N θ conv ) ((T t i ) θ Φ 1 + (T t i) 1 ). log(n + 1) 1

23 The computation cost is given by (under the assumption π = π (θπ) ) C = O ( i=0 ) ( NM i = O log(n + 1)) d+1 N 1+θconv(+ d )) l (1 i d N ) Up to possibly a log(n)-factor, the last sum is O(N d θπ 1 ), and ignoring the logarithmic factors, we obtain C = O(N 1+ d θπ 1+θconv(+ d l ) ). Equivalently, as a function of the computational cost, the convergence rate of the final accuracy equals C 1 (+ d l )+ θconv 1 (1+ θπ d 1). i=0 θπ. Following [GM10] (under suitable assumptions), two time-grid choices are possible for solving the same BSDE. The uniform grid π = π (1) gives θ conv = θ Φ / (at least). The convergence order becomes 1 + d l + θ (1+ d 1). Φ The grid π = π (θ) (for θ < θ Φ ) gives θ conv = 1/. 1 + d l +(1+ d θ Φ 1). Taing θ θ Φ, the convergence order is The grid π (θ) exhibits a better convergence rate compared to the uniform grid. This corroborates the interest in time grids that are well adapted to the regularity of the data. These features will be investigated in subsequent more experimental wors. A Appendix A.1 Proof of Lemmas.1,. and.3 A.1.1 Proof of Lemmas.1 The first inequality, for α 1, follows by bounding the sum by t t i (t t) α 1 dt, whence B α,1 = 1/α. The case α > 1 is obvious with B α,1 = 1. For the second inequality, there are two main cases: If α 1 and β 1, the advertised inequality is obvious with B α,β = 1. Now, assume α < 1 or β < 1, and first consider the case t i = 0 and t = 1. We set ϕ(s) = ϕ(s)ds (equivalent to the usual beta function with parameters (α, β)) to bound the sum. A simple but useful property is that ϕ is either monotone or has a unique minimum on (0,1), whence (1 s) α 1 s β 1 and we use the integral 1 0 tj tj+1 (1 t j ) α 1 t β 1 j j R π ϕ(s)ds + ϕ(s)ds. t j 1 t j Summing up over j and defining B α,β = (1 + R π ) 1 ϕ(s)ds concludes the proof for the simple case. 0 For general t i and t one can use the bounds on the simple case by rearranging the j-sum which is

24 equal to 1 (t t i ) α+β 1 (1 t j t i ) α 1 ( t j t i ) β 1 B α,β (t t i ) α+β 1. t t i t t i t t i j A.1. Proof of Lemma. If α 1, the result trivially holds with C (.a) = 1 and C (.b) = C u T α 1. Now, assume α < 1 : if (.1) holds, of course we also have u j w j + (T t l ) 1 β (t l t j ) 1 α + C u By substituting (A.1) into the last sum, and using Lemma.1 we observe u l l (T t l ) 1 β (t l t j ) 1 α (T t l ) 1 β (t l t j ) 1 α + + C u u r r r=l+1 (T t r) 1 β (t r t l ) 1 α l (T t l ) 1 β (t l t j ) 1 α (T t l ) 1 β (t l t j ) + B 1 α α+β, 1 +α + C u B α+β, 1 +α r=j+ u l l. (A.1) (T t l ) 1 β (t l t j ) 1 α w r r r=l+1 (T t r) 1 β (t r t l ) 1 α l (T t l ) 1 β (t l t j ) 1 α r=j+ u r r (T t r ) 1 β (t r t j ) 1 α β. w r r (T t r ) 1 β (t r t j ) 1 α β Substituting into (A.1), we observe that we have an equation of similar form to (A.1), except that, in the sum involving u, α α + β and C u CuB α+β, 1 +α, and, in the sum involvung w, w (1 + C u (1 + T α+β B α+β, 1 +α ))w. After κ iterations of the previous step, ) we obtain α κ (α+β) β =: α κ. Hence, for κ sufficiently, we obtain the bound advertised in the Lemma statement. large so that α κ 1, i.e. κ log ( 1 +β α+β 3