Central Limit Theorem for the Multilevel Monte Carlo Euler Method

1 1 Mohamed Ben Alaya & Ahmed Kebaier∗

1 UniversitéParis 13 Sorbonne Paris Cité, LAGA, CNRS (UMR 7539) 99, av. J.B. Clément 93430 Villetaneuse, France [email protected] [email protected]

September 16, 2013

Abstract This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [8] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [16], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out. AMS 2000 Mathematics Subject Classification. 60F05, 62F12, 65C05, 60H35. Key Words and Phrases. Central limit theorem, Multilevel Monte Carlo methods, Euler scheme, finance.

1 Introduction

In many applications, in particular in the pricing of financial securities, we are interested in the effective computation by Monte Carlo methods of the quantity Ef(XT ), where X := (Xt)0≤t≤T is a diffusion process and f a given function. The Monte Carlo Euler method consists of n two steps. First, approximate the diffusion process (Xt)0≤t≤T by the Euler scheme (Xt )0≤t≤T E n 1 N n n with time step T/n. Then, approximate f (XT ) by N i=1 f(XT,i), where f(XT,i)1≤i≤N is ∗This research benefited from the support of the chair ”Risques FinP anciers”, Fondation du Risque.

1 n a sample of N independent copies of f(XT ). This approximation is aﬀected respectively by a discretization error and a statistical error

1 N ε := E (f(Xn) f(X )) and f(Xn ) Ef(Xn). n T − T N T,i − T i=1 X On one hand, Talay and Tubaro [21] prove that if f is sufficiently smooth, then εn c/n with c a given constant and in a more general context, Kebaier [17] proves that the rate of∼ convergence of the discretization error ε can be 1/nα for all values of α [1/2, 1] (see e.g. Kloeden and n ∈ Platen [18] for more details on discretization schemes). On the other hand, the statistical error is controlled by the central limit theorem with order 1/√N. Further, the optimal choice of the sample size N in the classical Monte Carlo method mainly depends on the order of the α discretization error. More precisely, it turns out that for εn =1/n the optimal choice of N is 2α 2α+1 n . This leads to a total complexity in the Monte Carlo method of order CMC = n (see Duffie and Glynn [5] for related results). Let us recall that the complexity of an algorithm is proportional to the maximum number of basic computations performed by this one. Hence, −2−1/α expressing this complexity in terms of the discretization error εn we get CMC = εn . In order to improve the performance of this method, Kebaier introduced a two-level Monte Carlo method [17] (called the statistical Romberg method) reducing the complexity CMC while maintaining the convergence of the algorithm. This method uses two Euler schemes with time steps T/n and T/nβ, β (0, 1) and approximates E f(X ) by ∈ T N1 N2 1 β 1 β f(Xˆ n )+ f(Xn ) f(Xn ), N T,i N T,i − T,i 1 i=1 2 i=1 X X ˆ nβ β where XT is a second Euler scheme with time step T/n and such that the Brownian paths n nβ ˆ nβ used for XT and XT has to be independent of the Brownian paths used to simulate XT . It α turns out that for a given discretization error εn = 1/n (α [1/2, 1]), the optimal choice is 2α 2α−(1/2) ∈ obtained for β = 1/2, N1 = n and N2 = n . With this choice, the complexity of the 2α+(1/2) −2−1/(2α) statistical Romberg method is of order CSR = n = εn which is lower than the classical complexity in the Monte Carlo method. More recently, Giles [8] generalized the statistical Romberg method of Kebaier [17] and proposed the multilevel Monte Carlo algorithm, in a similar approach to Heinrich’s multilevel method for parametric integration [12] (see also Creutzig, Dereich, Müller-Gronbach and Ritter [3], Dereich [4], Giles [7], Giles, Higham and Mao [9], Giles and Szpruch [10], Heinrich [11], Heinrich and Sindambiwe [13] and Hutzenthaler, Jentzen and Kloeden [14] for related results). The multilevel Monte Carlo method uses information from a sequence of computations with decreasing step sizes and approximates the quantity Ef(XT ) by

N0 L Nℓ 1 1 1 1 ℓ,mℓ ℓ,mℓ− N Qn = f(XT,k)+ f(XT,k ) f(XT,k ) , m 0, 1 , N0 Nℓ − ∈ \{ } Xk=1 Xℓ=1 Xk=1 log n where the fine discretization step is equal to T/n thereby L = log m . For ℓ 1, , L , pro- ℓ ℓ−1 ∈ {ℓ ℓ−1} cesses (Xℓ,m ,Xℓ,m ) , k 1, , N , are independent copies of (Xℓ,m ,Xℓ,m ) t,k t,k 0≤t≤T ∈ { ℓ} t t 0≤t≤T 2 whose components denote the Euler schemes with time steps m−ℓT and m−(ℓ−1)T . However, ℓ,mℓ ℓ,mℓ−1 for fixed ℓ, the simulation of (Xt )0≤t≤T and (Xt )0≤t≤T has to be based on the same 1 Brownian path. Concerning the first empirical mean, processes (Xt,k)0≤t≤T , k 1, , N0 , 1 ∈ { } are independent copies of (Xt )0≤t≤T which denotes the Euler scheme with time step T . Here, it is important to point out that all these L + 1 Monte Carlo estimators have to be based on different independent samples. Due to the above independence assumption for the paths, the variance of the multilevel estimator is given by

L 2 −1 1 −1 2 σ := V ar(Qn)= N0 V ar(f(XT )) + Nℓ σℓ , Xℓ=1 ℓ ℓ−1 where σ2 = V ar f(Xℓ,m ) f(Xℓ,m ) . Assuming that the diffusion coefficients of X and ℓ T − T the function f are Lipschitz continuous then it is easy to check, using properties of the Euler scheme, that L σ2 c N −1m−ℓ ≤ 2 ℓ Xℓ=0 for some positive constant c2 (see Proposition 1 for more details). Giles [8] uses this computation in order to find the optimal choice of the multilevel Monte Carlo parameters. More precisely, to obtain a desired root mean squared error (RMSE), say of order 1/nα, for the multilevel estimator, Giles [8] uses the above computation on σ2 to minimize the total complexity of the algorithm. It turns out that the optimal choice is obtained for (see Theorem 3.1 of [8])

log n T log n N =2c n2α +1 , for ℓ 0, , L and L = . (1) ℓ 2 log m mℓ ∈{ } log m α Hence, for an error εn = 1/n , this optimal choice leads to a complexity for the multilevel 2α 2 −2 2 Monte Carlo Euler method proportional to n (log n) = εn (log εn) . Interesting numerical tests, comparing three methods (crude Monte Carlo, statistical Romberg and the multilevel Monte Carlo), were processed in Korn, Korn and Kroisandt [19]. In the present paper, we focus on central limit theorems for the inferred error; a question which has not been addressed in previous research. To do so, we use techniques adapted to this setting, based on a central limit theorem for triangular array (see Theorem 2) together with Toeplitz lemma. It is worth to note that our approach improves techniques developed by Kebaier [17] in his study of the statistical Romberg method (see Remark 2 for more details). Hence, our main result is a Lindeberg Feller central limit theorem for the multilevel Monte Carlo Euler algorithm (see Theorem 4). Further, this allows us to prove a Berry-Essen type bound on our central limit theorem. In order to show this central limit theorem, we ﬁrst prove a stable law convergence theorem, for the Euler scheme error on two consecutive levels mℓ−1 and mℓ, of the type obtained in Jacod and Protter [16]. Indeed, we prove the following functional result (see Theorem 3)

ℓ m ℓ ℓ−1 (Xℓ,m Xℓ,m ) stably U, as ℓ , (m 1)T − ⇒ →∞ s − 3 where U is the same limit process given in Theorem 3.2 of Jacod and Protter [16]. Our result uses standard tools developed in their paper but it can not be deduced without a speciﬁc and laborious study. Further, their result, namely

ℓ m ℓ (Xℓ,m X) stably U, as ℓ , r T − ⇒ →∞ is neither suﬃcient nor appropriate to prove our Theorem 4, since the multilevel Monte Carlo ℓ ℓ−1 ℓ Euler method involves the error process Xℓ,m Xℓ,m rather than Xℓ,m X. Thanks to Theorem 4 we obtain a precise− description for the choice of− the parameters to run the multilevel Monte Carlo Euler method. Afterward, by a complexity analysis we obtain the optimal choice for the multilevel Monte Carlo Euler method. It turns out that for a total α error of order εn =1/n the optimal parameters are given by (m 1)T log n N = − n2α log n, for ℓ 0, , L and L = . (2) ℓ mℓ log m ∈{ } log m

2α 2 −2 2 This leads us to a complexity proportional to n (log n) = εn (log εn) which is the same order obtained by Giles [8]. By comparing relations (1) and (2), we note that our optimal sequence of sample sizes (Nℓ)0≤ℓ≤L does not depend on any given constant, since our approach is based on proving a central limit theorem and not on obtaining an upper bound for the variance of the algorithm. However, some numerical tests comparing the runtime with respect to the root mean square error, show that we are in line with the original work of Giles [8]. Nevertheless, the major advantage of our central limit theorem is that it ﬁlls the gap in the literature for the multilevel Monte Carlo Euler method and allows to construct a more accurate conﬁdence interval compared to the one obtained using Chebyshev’s inequality. All these results are stated and proved in section 3. The next section is devoted to recall some useful stochastic limit theorems and to introduce our notations.

2 General framework

2.1 Preliminaries

Let (Xn) be a sequence of random variables with values in a Polish space E defined on a probability space (Ω, , P). Let (Ω˜, ˜, P˜) be an extension of (Ω, , P), and let X be an E- F F F valued random variable on the extension. We say that (Xn) converges in law to X stably and stably write Xn X, if ⇒ E(Uh(X )) E˜(Uh(X)) n → for all h : E R bounded continuous and all bounded random variable U on (Ω, ) . This → F convergence is obviously stronger than convergence in law that we will denote here by “ ”. According to section 2 of Jacod [15] and Lemma 2.1 of Jacod and Protter [16], we have⇒ the following result. Lemma 1 let V and V be defined on (Ω, ) with values in another metric space E′. n F P if V V, X stably X then (V ,X ) stably (V,X). n → n ⇒ n n ⇒ 4 Conversely, if (V,X ) (V,X) and V generates the σ-field , we can realize this limit as n ⇒ F (V,X) with X defined on an extension of (Ω, , P) and X stably X. F n ⇒ Now, we recall a result on the convergence of stochastic integrals formulated from Theorem 2.3 in Jacod and Protter [16]. This is a simplified version but it is sufficient for our study. Let Xn = n,i d (X )1≤i≤d be a sequence of R -valued continuous semimartingales with the decomposition

Xn,i = Xn,i + An,i + M n,i, 0 t T t 0 t t ≤ ≤ where, for each n N and 1 i d, An,i is a predictable process with ﬁnite variation, null at 0 and M n,i is a martingale∈ null≤ at≤ 0.

Theorem 1 Assume that the sequence (Xn) is such that

T M n,i + dAn,i T s Z0 is tight. Let Hn and H be a sequence of adapted, right-continuous and left-hand limited processes all defined on the same filtered probability space. If (Hn,Xn) (H,X) then X is a semimartingale with respect to the filtration generated by the limit process⇒ (H,X), and we have (Hn,Xn, HndXn) (H,X, HdX). ⇒ We recallR also the following LindebergR Feller central limit theorem that will be used in the sequel (see for instance Theorem 7.2 and 7.3 in [1]).

Theorem 2 (central limit theorem for triangular array) Let (kn)n∈N be a sequence such that k as n . For each n, let X , ,X be k independent random variables n −→ ∞ −→ ∞ n,1 n,kn n with ﬁnite variance such that E(Xn,k)=0 for all k 1, ,kn . Suppose that the following conditions hold. ∈ { }

A1. lim kn E X 2 = σ2,σ> 0. n→∞ k=1 | n,k| A2. Lindeberg’sP condition: for all ε> 0, lim kn E X 21 =0. Then n→∞ k=1 | n,k| {|Xn,k|>ε}

kn P X (0, σ2) as n . n,k ⇒N →∞ Xk=1

Moreover, if the Xn,k have moments of order p > 2, then the Lindeberg’s condition can be obtained by the following one

A3 Lyapunov’s condition: lim kn E X p =0. n→∞ k=1 | n,k| P

5 2.2 The Euler scheme Rd Let X := (Xt)0≤t≤T be the process with values in , solution to dX = b(X )dt + σ(X )dW , X = x Rd (3) t t t t 0 ∈ where W =(W 1,...,W q) is a q-dimensional Brownian motion on some given ﬁltered probability space = (Ω, , ( t)t≥0,P ) with ( t)t≥0 is the standard Brownian ﬁltration, b and σ are respectivelyB Rd andF RFd×q valued functions.F We consider the continuous Euler approximation Xn with step δ = T/n given by:

n dXt = b(Xηn(t))dt + σ(Xηn(t))dWt, ηn(t) = [t/δ]δ. It is well known that under the global Lipschitz condition ( ) C > 0, such that, b(x) b(y) + σ(x) σ(y) C y x , x,y Rd, Hb,σ ∃ T | − | | − | ≤ T | − | ∈ the Euler scheme satisﬁes the following property (see e.g. Bouleau and L´epingle [2]) K (T ) ) p 1, X,Xn Lp and E sup X Xn p p , with K (T ) > 0. P ∀ ≥ ∈ | t − t | ≤ np/2 p 0≤t≤T Note that according to Theorem 3.1 of Jacod and Protter [16], under the weaker condition ( ˜ ) b and σ are locally Lipschitz with linear growth, Hb,σ we have only the uniform convergence in probability, namely the property

˜ n P ) sup Xt Xt 0. P 0≤t≤T | − | −→ Following the notation of Jacod and Protter [16], we rewrite diﬀusion (3) as follows

q j dXt = ϕ(Xt)dYt = ϕj(Xt)dYt j=0 X 1 q ′ where ϕj is the j-th column of the matrix σ, for 1 j q, ϕ0 = b and Yt := (t, Wt , , Wt ) . Then, the continuous Euler approximation Xn with≤ time≤ step δ = T/n becomes

q n n n j dXt = ϕ(Xηn(t))dYt = ϕj(Xηn(t))dYt , ηn(t) = [t/δ]δ. (4) j=0 X 3 The Multilevel Monte Carlo Euler method

mℓ −ℓ Let (Xt )0≤t≤T denotes the Euler scheme with time step m T for ℓ 0, , L , where L = log n/ log m. Noting that ∈ { }

L ℓ ℓ−1 Ef(Xn)= Ef(X1 )+ E f(Xm ) f(Xm ) , (5) T T T − T Xℓ=1 6 the multilevel method is to estimate independently by the Monte Carlo method each of the E n expectations on the right-hand side of the above relation. Hence, we approximate f(XT ) by

N0 L Nℓ 1 1 1 1 ℓ,mℓ ℓ,mℓ− Qn = f(XT,k)+ f(XT,k ) f(XT,k ) . (6) N0 Nℓ − Xk=1 Xℓ=1 Xk=1 Here, it is important to point out that all these L + 1 Monte Carlo estimators have to be based on different, independent samples. More precisely, for each ℓ 1, , L the samples ℓ,mℓ ℓ,mℓ−1 ℓ,mℓ ℓ,mℓ−1 ∈ { } (XT,k ,XT,k )1≤k≤Nℓ are independent copies of (XT ,XT ) whose components are the Eu- ler schemes with time steps m−ℓT respectively m−(ℓ−1)T and simulated with the same Brownian 1 path. Concerning the first empirical mean, the samples (XT,k)1≤k≤N0 are independent copies of 1 XT . The following result gives us a first description of the asymptotic behavior of the variance in the multilevel Monte Carlo Euler method.

Proposition 1 Assume that b and σ are functions satisfying condition ( b,σ). For a Lipschitz continuous function f : Rd R we have H −→ L −1 −ℓ V ar(Qn)= O Nℓ m . (7) ℓ=0 X Proof : We have

L ℓ ℓ−1 V ar(Q )= N −1V ar f(X1 ) + N −1V ar f(Xℓ,m ) f(Xℓ,m ) n 0 T ℓ T − T ℓ=1 L X ℓ ℓ−1 N −1V ar f(X1 ) +2 N −1 V ar(f(Xm ) f(X )) + V ar(f(Xm ) f(X )) ≤ 0 T ℓ T − T T − T ℓ=1 X L 2 2 −1 1 −1E mℓ mℓ−1 N0 V ar f(XT ) + 2[f]lip Nℓ sup Xt Xt + sup Xt Xt , ≤ 0≤t≤T − 0≤t≤T − ℓ=1 X where [f] := sup |f(u)−f(v)| . We complete the proof by using ) on the strong convergence lip u=v |u−v| P of the Euler scheme.

Inequality (7) indicates the dependence of the variance of Qn on the choice of the parameters n N0,...,NL. This variance can be smaller than the variance of f(XT ), so that Qn appears as a good candidate for the variance reduction. The main result of this section is a Lindeberg Feller central limit theorem (see Theorem 4 below). In order to prove this result, we need to prove ﬁrst a new stable law convergence theorem for the Euler scheme error adapted to the setting of multilevel Monte Carlo algorithm. This is crucial and is the aim of the following subsection.

7 3.1 Stable convergence In what follows, we prove a stable law convergence theorem, for the Euler scheme error on two consecutive levels mℓ−1 and mℓ, of the type obtained in Jacod and Protter [16]. Our result in Theorem 3 below is an innovative contribution on the Euler scheme error that is diﬀerent and more tricky than the original work by Jacod and Protter [16] since it involves the error process ℓ ℓ−1 ℓ ℓ ℓ−1 Xℓ,m Xℓ,m rather than Xℓ,m X. Note that the study of the error Xℓ,m Xℓ,m as ℓ −can be reduced to the study− of the error Xmn Xn as n where Xmn and− Xn stand for→∞ two Euler schemes with time steps T/(mn) and−T/n constructed→∞ on the same Brownian path. Theorem 3 Assume that b and σ are 1 with linear growth then the following result holds. C mn For all m N 0, 1 , (Xmn Xn) stably U, as n , ∈ \{ } (m 1)T − ⇒ →∞ r − with (Ut)0≤t≤T the d-dimensional process satisfying q 1 t U = Z Hi,jdBij, t [0, T ], (8) t √ t s s ∈ 2 i,j=1 0 X Z where Hi,j =(Z )−1ϕ˙ ϕ¯ , with ϕ˙ := ϕ (X ) and ϕ¯ := ϕ (X ), (9) s s s,j s,i s,j ∇ j s s,i i s d×d and (Zt)0≤t≤T is the R valued process solution of the linear equation

q t Z = I + ϕ˙ dY jZ , t [0, T ]. t d s,j s s ∈ j=0 0 X Z Here, ϕj is a d d matrix with ( ϕj)ik is the partial derivative of ϕij with respect to the k-th ∇ ×ij ∇ 2 coordinate, and (B )1≤i,j≤q is a standard q -dimensional Brownian motion independent of W . This process is deﬁned on an extension (Ω˜, ˜, ( ˜ ) , P˜) of the space (Ω, , ( ) , P). F Ft t≥0 F Ft t≥0 Note that by letting formally m tend to inﬁnity we recover the Jacod and Protter’s result [16].

mn,n mn,n Proof: Consider the error process U =(Ut )0≤t≤T , deﬁned by U mn,n := Xmn Xn, t [0, T ]. t t − t ∈ Combining relation (4), for both processes Xmn and Xn, together with a Taylor expansion q dU mn,n = ϕ˙ n (Xmn Xn ) dY j, t t,j ηmn(t) − ηn(t) t j=0 X n whereϕ ˙ t,j is the d d matrix whose i-th row is the gradient of the real-valued function ϕij at ×n mn n a point between Xηn(t) and Xηmn(t). Therefore, the equation satisﬁed by U can be written as

t q mn,n n mn,n j mn,n Ut = ϕ˙ s,j Us dYs + Gt , 0 j=0 Z X 8 with t q t q Gmn,n = ϕ˙ n (Xn Xn ) dY j ϕ˙ n (Xmn Xmn ) dY j. t s,j s − ηn(s) s − s,j s − ηmn(s) s 0 j=0 0 j=0 Z X Z X mn,n Rd×d In the following, let (Zt )0≤t≤T be the valued solution of

t q mn,n n j mn,n Zt = Id + ϕ˙ s,j dYs Zs . 0 j=0 ! Z X mn,n −1 Theorem 48 p.326 in [20], ensures existence of the process ((Zt ) )0≤t≤T deﬁned as the solution of

t q t q (Zmn,n)−1 = I + (Zmn,n)−1 (ϕ ˙ n )2ds (Zmn,n)−1 ϕ˙ n dY j. t d s s,j − s s,j s 0 j=1 0 j=0 Z X Z X Thanks to Theorem 56 p.333 in the same reference [20], we get

t t q U mn,n = Zmn,n (Zmn,n)−1dGmn,n (Zmn,n)−1 (ϕ ˙ n )2(Xn Xn ) ds t t s s − s s,j s − ηn(s) 0 0 j=1 nZ Z X t q + (Zmn,n)−1 (ϕ ˙ n )2(Xmn Xmn ) ds . s s,j s − ηmn(s) 0 j=1 Z X o Since the increments of the Euler scheme satisfy

q q Xn Xn = ϕ¯n (Y i Y i ) and Xmn Xmn = ϕ¯mn(Y i Y i ), s − ηn(s) s,i s − ηn(s) s − ηmn(s) s,i s − ηmn(s) i=0 i=0 X X n n mn mn withϕ ¯s,i = ϕi(Xηn(s)) andϕ ¯s,i = ϕi(Xηmn(s)), it is easy to check that

q t U mn,n = Zmn,n Hi,j,mn,n(Y i Y i ) dY j + Rmn,n + Rmn,n t t s s − ηn(s) s t,1 t,2 i,j=1 0 X Z q t Zmn,n H˜ i,j,mn,n(Y i Y i ) dY j R˜mn,n R˜mn,n (10) − t s s − ηmn(s) s − t,1 − t,2 i,j=1 0 X Z with

q t q t Rmn,n = Zmn,n Ki,mn,n(Y i Y i ) ds, Rmn,n = Zmn,n H0,j,mn,n(s η (s)) dY j, t,1 t s s − ηn(s) t,2 t s − n s i=0 0 j=1 0 X Z X Z and

q t q t R˜mn,n = Zmn,n K˜ i,mn,n(Y i Y i ) ds, R˜mn,n = Zmn,n H˜ 0,j,mn,n(s η (s)) dY j. t,1 t s s − ηmn(s) t,2 t s − mn s i=0 0 j=1 0 X Z X Z 9 where, for (i, j) 0, , q 1, , q , ∈{ }×{ } q Ki,mn,n =(Zmn,n)−1 ϕ˙ n ϕ¯n (ϕ ˙ n )2ϕ¯n , Hi,j,mn,n =(Zmn,n)−1ϕ˙ n ϕ¯n , s s s,0 s,i − s,j s,i s s s,j s,i j=1 ! X and

q K˜ i,mn,n =(Zmn,n)−1 ϕ˙ n ϕ¯mn (ϕ ˙ n )2ϕ¯mn , H˜ i,j,mn,n =(Zmn,n)−1ϕ˙ n ϕ¯mn. s s s,0 s,i − s,j s,i s s s,j s,i j=1 ! X Now, let us introduce

t q Z = I + ϕ˙ dY j Z , withϕ ˙ = ϕ (X ). t d s,j s s t,j ∇ j t Z0 j=0 X −1 Moreover, ((Zt) )0≤t≤T exists and satisﬁes the following explicit linear stochastic diﬀerential equation t q t q (Z )−1 = I + (Z )−1 (ϕ ˙ )2ds (Z )−1 ϕ˙ dY j. t d s s,j − s s,j s 0 j=1 0 j=0 Z X Z X Thanks to the uniform convergence in probability of the Euler scheme and according to Theorem 2.5 in Jacod and Protter [16], we have

mn,n P mn,n −1 −1 P sup Zt Zt 0 and sup (Zt ) (Zt) 0. (11) 0≤t≤T | − | → 0≤t≤T − →

i,j,mn,n ˜ i,j,mn,n Furthermore, in relation (10), one can replace respectively Hs and Hs by their i,j common limit Hs given by relation (9). So that relation (10) becomes

q t U mn,n = Zmn,n Hi,j(Y i Y i ) dY j + Rmn,n, (12) t t s ηmn(s) − ηn(s) s t i,j=1 0 X Z with Rmn,n = Rmn,n + Rmn,n + Rmn,n R˜mn,n R˜mn,n R˜mn,n t t,1 t,2 t,3 − t,1 − t,2 − t,3 where Rmn,n and R˜mn,n, i 1, 2 , are introduced by relation (10) and t,i t,i ∈{ } q t Rmn,n = Zmn,n (Hi,j,mn,n Hi,j)(Y i Y i ) dY j t,3 t s − s s − ηn(s) s i,j=1 0 X Z q t R˜mn,n = Zmn,n (H˜ i,j,mn,n Hi,j)(Y i Y i ) dY j, t,3 t s − s s − ηmn(s) s i,j=1 0 X Z The remainder term process Rmn,n vanishes with rate √n in probability. More precisely, we have the following convergence result.

10 mn,n Lemma 2 The rest term introduced in relation (12) is such that sup0≤t≤T √nRt converges to zero in probability as n tends to inﬁnity.

For the reader’s convenience, the proof of this lemma is postponed to the end of the current subsection. The task is now to study the asymptotic behavior of the process given by relation (12)

q t √nZmn,n Hi,j(Y i Y i ) dY j. t s ηmn(s) − ηn(s) s i,j=1 0 X Z In order to study this process, we introduce the martingale process,

t M n,i,j = (Y i Y i ) dY j, (i, j) 1, , q 2, t ηmn(s) − ηn(s) s ∈{ } Z0 and we proceed to a preliminary calculus of the expectation of its bracket. Let (i, j) and (i′, j′) ∈ 1, , q 2, we have { } for j = j′, the bracket M n,i,j, M n,i′,j′ =0 • ′ for j = j′ and i = i′, E M n,i,j, M n,i ,j =0 • for j = j′ and i = i′, E M n,i,j = t(η (s) η (s)) ds, t [0, T ] and we have • t 0 mn − n ∈ ηn(t) R 1 E( M n,i,j ) = (η (s) η (s))ds + O( ) t mn − n n2 Z0 [t/δ]−1 m−1 (mk+ℓ+1)δ/m 1 = (ηmn(s) ηn(s)) ds + O( 2 ) (mk+ℓ)δ/m − n Xℓ=0 Xk=0 Z [t/δ]−1 m−1 δ2 mk + ℓ 1 (m 1)δ2 1 = k + O( )= − [t/δ]+ O( ) m m − n2 2m n2 Xℓ=0 Xk=0 (m 1)T 1 = − t + O( ). (13) 2mn n2

Having disposed of this preliminary evaluations, we can now study the stable convergence 2mn n,i,j of (m−1)T M . By virtue of Theorem 2-1 in [15], we need to study the asymptotic 1≤i,j≤q n,i,j n,i′,j′ n,i,j j′ behaviorq of both brackets n M , M t and √n M ,Y t, for all t [0, T ] and all (i, j, i′, j′) 1, , q 4. The case j = j′ is obvious and we only proceed to prove∈ that ∈{ } P ′ n,i,j j for j = j , √n M ,Y t 0, for all t [0, T ]. • n−→→∞ ∈ P ′ ′ n,i,j n,i′,j for j = j and i = i , n M , M t 0, for all t [0, T ]. • n−→→∞ ∈ P ′ ′ n,i,j (m−1)T for j = j and i = i , n M t t, for all t [0, T ]. • n−→→∞ 2m ∈

11 For the ﬁrst point, we consider the L2 convergence t 2 E n,i,j j 2 E i i M ,Y t = (Yηmn(s) Yηn(s))ds 0 − t Z t = E (Y i Y i )(Y i Y i ) dsdu ηmn(s) − ηn(s) ηmn(u) − ηn(u) Z0 Z0 = 2 g(s,u)dsdu Z0

Proof of Lemma 2 : At ﬁrst, we prove the uniform convergence in probability toward zero mn,n ˜mn,n of the normalized rest terms √nRt,i for i 1, 2 . The convergence of √nRt,i i 1, 2 is a straightforward consequence of the previous∈{ one.} The main part of these rest terms∈{ can} be represented as integrals with respect to three types of supermartingales that can be classiﬁed through the following three cases

t t t Dn,0,0 = √n (s η (s)) ds, Dn,i,0 = √n (Y i Y i ) ds, M n,0,j = √n (s η (s)) dY j, t − n t s − ηn(s) t − n s Z0 Z0 Z0 13 where (i, j) 1, , q 2 and t [0, T ]. In the ﬁrst case the supermartingale is deterministic ∈{ } ∈ of ﬁnite variation and its total variation on the interval [0, T ] has the following expression

T T T 2 dDn,0,0 = √n (s η (s)) ds . t − n ≤ √n Z0 Z0

So, the process Dn,0,0 converges to 0 and is tight. In the second case, for i 1, , q , the supermartingale is also of ﬁnite variation and its total variation on the interval∈ { [0, T ] has} the following expression T T dDn,i,0 = √n Y i Y i ds. t | s − ηn(s)| Z0 Z0

It is clear that sup E T dDn,i, 0 < which ensures the tightness of the process Dn,i,0. n 0 | s | ∞ n,i,0 2 Therefore, we only needR to establish the convergence of Dt towards 0 in L (Ω), for t [0, T ]. In fact, we have ∈

E (Dn,i,0)2 =2n E (Y i Y i )(Y i Y i ) ds du. t s − ηn(s) u − ηn(u) Z0

t T 2 E (Dn,i,0)2 2T (u η (u))du 2 t. t ≤ − n ≤ n Z0 It follows from all these that Dn,i,0 0. In the last case, for j 1, , q , the process M n,0,j ⇒ ∈{ } t is a square integrable martingale and its bracket has the following expression

T T 3 M n,0,j = n (s η (s))2 ds . T − n ≤ n Z0 E n,0,j n,0,j It is clear that supn M T < , so we deduce the tightness of the process M and the convergence M n,0,j 0. ∞ Now thanks to property⇒ ˜) and relation (11), it is easy to check that the integrand processes i,mn,n 0,j,mn,n P Ks and Hs , introduced in relation (10), converge uniformly in probability to their respective limits Ki = (Z )−1 ϕ˙ ϕ¯ q (ϕ ˙ )2ϕ¯ and H0,j = (Z )−1ϕ˙ ϕ¯ , where s s s,0 s,i − j=1 s,j s,i s s s,j s,i ϕ˙ s,j = ϕj(Xs) andϕ ¯s,i = ϕi(Xs). Therefore, by Theorem 1 we deduce that the integral ∇ P processes given by

t t √n Ki,mn,n(Y i Y i ) ds and √n H0,j,mn,n(s η (s)) dY j s s − ηn(s) s − n s Z0 Z0 vanish. Consequently, we conclude using relation (11) that √nRmn,n 0 for i 1, 2 . i ⇒ ∈{ }

14 We now proceed to prove that Rmn,n 0. The convergence of the process R˜mn,n toward 0 is 3 ⇒ 3 obviously obtained from the previous one. The main part of this rest term can be represented as a stochastic integral with respect to the martingale process given by

t N n,i,j = √n (Y i Y i ) dY j, t s − ηn(s) s Z0 with (i, j) 1, , q 1, , q . It was proven in Jacod and Protter [16] that ∈{ }×{ } n Bij N n,i,j stably , T ⇒ √ r 2 ij 2 where (B )1≤i,j≤q is a standard q -dimensional Brownian motion deﬁned on an extension (Ω˜, ˜, ( ˜t)t≥0, P˜) of the space (Ω, , ( t)t≥0, P), which is independent of W . Thanks to prop- ertyF˜)F and relation(11), the integrandF F process Hi,j,mn,n Hi,j 0 and once again by Theorem 1 weP deduce that the integral processes given by − ⇒

t √n (Hi,j,mn,n Hi,j)(Y i Y i ) dY j s − s s − ηn(s) s Z0 vanish. All this allows us to conclude using relation (11).

3.2 Central limit theorem Let us recall that the multilevel Monte Carlo method uses information from a sequence of computations with decreasing step sizes and approximates the quantity Ef(XT ) by

N0 L Nℓ 1 1 1 1 ℓ,mℓ ℓ,mℓ− N log n Qn = f(XT,k)+ f(XT,k ) f(XT,k ) , m 0, 1 and L = . N0 Nℓ − ∈ \{ } log m Xk=1 Xℓ=1 Xk=1 In the same way as in the case of a crude Monte Carlo estimation, let us assume that the discretization error ε = Ef(Xn) Ef(X ) n T − T is of order 1/nα for any α [1/2, 1]. Taking advantage from the limit theorem proven in the above section, we are now able∈ to establish a central limit theorem of Lindeberg Feller type on the multilevel Monte Carlo Euler method. To do so, we introduce a real sequence (aℓ)ℓ∈N of positive terms such that

L L 1 p/2 ( ) lim aℓ = and lim p/2 aℓ =0, for p> 2. W L→∞ ∞ L→∞ L ℓ=1 a ℓ=1 X ℓ=1 ℓ X P and we assume that the sample size Nℓ depends on the rest of parameters by the relation

n2α(m 1)T L log n Nℓ = ℓ− aℓ, ℓ 0, , L and L = . (20) m aℓ ∈{ } log m Xℓ=1 15 We choose this form for Nℓ because it is a generic form allowing us a straightforward use of Toeplitz lemma that is a crucial tool used in the proof of our central limit theorem. Indeed, property ( ) implies that if (x ) is a sequence converging to x R as ℓ tends to infinity W ℓ ℓ≥1 ∈ then L a x lim ℓ=1 ℓ ℓ = x. L→+∞ L P ℓ=1 aℓ In the sequel, we will denote by E˜ respectivelyP V˜ ar the expectation respectively the variance defined on the probability space (Ω˜, ˜, P˜) introduced in Theorem 3. We can now state the central limit theorem under strengthenedF conditions on the diffusion coefficients.

Theorem 4 Assume that b and σ are 1 functions satisfying the global Lipschitz condition ( ). Let f be a real valued function satisfyingC Hb,σ p p ( f ) f(x) f(y) C(1 + x + y ) x y , for some C,p > 0. H | − | ≤ | | | | | − | Assume P(X / )=0, where := x Rd; f is diﬀerentiable at x , and that for some T ∈ Df Df { ∈ } α [1/2, 1] we have ∈ α ( εn ) lim n εn = Cf (T,α). H n→∞ Then, for the choice of N , ℓ 0, 1, , L given by equation (20), we have ℓ ∈{ } nα Q E (f(X )) C (T,α), σ2 n − T ⇒N f with σ2 = V˜ ar f(X ).U and (C (T,α), σ2) denotes a normal distribution. ∇ T T N f The global Lipschitz condition ( ) seems to be essential to establish our result, since it Hb,σ ensures property ). Otherwise, Hutzenthaler, Jentzen and Kloeden [14] prove that under weaker conditionsP on b and σ the multilevel Monte Carlo Euler method may diverges whereas the crude Monte Carlo method converges.

Proof: To simplify our notations we give the proof for α = 1, the case α [1/2, 1) is a ∈ straightforward deduction. Combining relations (5) and (6) together we get

Q E (f(X )) = Qˆ1 + Qˆ2 + ε , n − T n n n where

N0 ˆ1 1 1 E 1 Qn = f(XT,k) f(XT ) N0 − Xk=1 L Nℓ 1 1 ˆ2 1 ℓ,mℓ ℓ,mℓ− E ℓ,mℓ ℓ,mℓ− Qn = f(XT,k ) f(XT,k ) f(XT ) f(XT ) . Nℓ − − − Xℓ=1 Xk=1

16 Using assumption ( εn ) we obviously obtain the term Cf (T,α) in the limit. Taking N0 = 2 H P n (m−1)T L a , we can apply the classical central limit theorem to Qˆ1 . Then we have nQˆ1 a0 ℓ=1 ℓ n n → ˆ2 0. Finally,P we have only to study the convergence of nQn and we will conclude by establishing nQˆ2 0, V˜ ar f(X ).U . n ⇒N ∇ T T To do so, we plan to use Theorem 2 with the Lyapunov condition and we set

Nℓ n mℓ,mℓ−1 mℓ,mℓ−1 ℓ,mℓ ℓ,mℓ−1 E ℓ,mℓ ℓ,mℓ−1 Xn,ℓ := ZT,k and ZT,k := f(XT,k ) f(XT,k ) f(XT,K ) f(XT,k ) . Nℓ − − − k=1 X (21) In other words, we will check the following conditions : lim L E(X )2 = V˜ ar f(X ).U • n→∞ ℓ=1 n,ℓ ∇ T T (Lyapunov condition) there exists p> 2 such that lim L E X p =0. P n→∞ ℓ=1 n,ℓ For• the ﬁrst one, we have | | P L L L 2 1 E 2 n mℓ,mℓ− (Xn,ℓ) = V ar(Xn,ℓ)= V ar ZT,1 Nℓ Xℓ=1 Xℓ=1 Xℓ=1 L ℓ 1 m ℓ ℓ−1 = a V ar Zm ,m . (22) L ℓ (m 1)T T,1 ℓ=1 aℓ Xℓ=1 − Otherwise, since P(X / ) = 0, applying theP Taylor expansion theorem twice we get T ∈Df ℓ,mℓ ℓ,mℓ−1 mℓ,mℓ−1 f(XT ) f(XT )= f(XT ).UT + − ℓ∇ ℓ ℓ−1 ℓ−1 (Xℓ,m X )ε(X ,Xℓ,m X ) (Xℓ,m X )ε(X ,Xℓ,m X ). T − T T T − T − T − T T T − T ℓ,mℓ P The function ε is given by the Taylor-Young expansion, so it satisﬁes ε(XT ,XT XT ) 0 ℓ−1 − ℓ ℓ−→→∞ ℓ,m P mℓ ℓ,m and ε(XT ,XT XT ) 0. By property ) we get the tightness of (XT XT ) − ℓ−→→∞ P (m−1)T − ℓ−1 and mℓ (Xℓ,m X ) and we deduce q (m−1)T T − T q ℓ m ℓ,mℓ ℓ,mℓ ℓ,mℓ−1 ℓ,mℓ−1 P (XT XT )ε(XT ,XT XT ) (XT XT )ε(XT ,XT XT ) 0. s(m 1)T − − − − − ℓ−→→∞ − So, according to Lemma 1 and Theorem 3 we conclude that

ℓ 1 m ℓ,mℓ ℓ,mℓ− stably f(XT ) f(XT ) f(XT ).UT , as ℓ . (23) s(m 1)T − ⇒ ∇ →∞ − Using ( f ) it follows from property ) that H P 2+ε ℓ E m ℓ,mℓ ℓ,mℓ−1 ε> 0, sup f(XT ) f(XT ) < . ∀ ℓ s(m 1)T − ∞ −

17 We deduce using relation (23) that

k ℓ k E m ℓ,mℓ ℓ,mℓ−1 E˜ f(XT ) f(XT ) f(XT ).UT < for k 1, 2 . s(m 1)T − ! → ∇ ∞ ∈{ } − Consequently, ℓ m ℓ ℓ−1 V ar(Zm ,m ) V˜ ar ( f(X ).U ) < . (m 1)T T,1 −→ ∇ T T ∞ − Hence combining this result with relation (22), we obtain the ﬁrst condition using Toeplitz lemma. Concerning the second one, by Burkholder’s inequality and elementary computations, we get for p> 2

p Nℓ p 1 p 1 p E p n E mℓ,mℓ− n E mℓ,mℓ− Xn,ℓ = p ZT,1 Cp p/2 ZT,1 , (24) | | Nℓ ≤ N ℓ=1 ℓ X where C is a numerical constant depending only on p. Otherwise, property ) ensures the p P existence of a constant Kp > 0 such that

ℓ ℓ−1 p K E Zm ,m p . T,1 ≤ mpℓ/2

Therefore L L p ˜ L E p ˜ n Cp p/2 Xn,ℓ Cp p/2 p/2 aℓ 0. (25) | | ≤ pℓ/2 ≤ L n−→→∞ ℓ=1 ℓ=1 Nℓ m a ℓ=1 X X ℓ=1 ℓ X This completes the proof. P

Remark 1 From Theorem 2 page 544 in [6], we prove a Berry-Essen type bound on our central limit theorem. This improves the relevance of the above result. Indeed, take α = 1 as in the proof, for X = nQˆ1 and X given by relation (21), with ℓ 1, , L , put n,0 n n,ℓ ∈{ } L L s2 = E X 2, ρ = E X 3 n | n,ℓ| n | n,ℓ| Xℓ=0 Xℓ=0 and denote by F the distribution function of n(Q Ef(Xn))/s . Then for all x R and n n − T n ∈ n N∗ ∈ ρn Fn(x) G(x) 6 3 , (26) | − | ≤ sn where G is the distribution function of a standard Gaussian random variable. If we interpret the output of the above inequality as sum of independent individual path simulation, we get

L 1 ℓ ℓ−1 s2 = a V ar f(X1 ) + a mℓV ar f(Xℓ,m ) f(Xℓ,m ) . n L 0 T ℓ T − T (m 1)T ℓ=1 aℓ ℓ=1 ! − X P 18 According to the above proof, it is clear that sn behaves like a constant but getting lower bounds for sn seems not to be a common result to our knowledge. Concerning ρn, taking p =3 in both inequalities (24) and (25) gives us an upper bound. In fact, when f is Lipschitz, there exists a positive constant C depending on b, σ, T and f such that

L C 3/2 ρn 3/2 aℓ . ≤ L a ℓ=1 ℓ=1 ℓ X P For the optimal choice aℓ = 1, given in the below subsection, the obtained Berry-Essen type bound is of order 1/√log n.

Remark 2 Note that the above proof diﬀers from the ones in Kebaier [17]. In fact, here our proof is based on the central limit theorem for triangular array which is adapted to the form of the multilevel estimator, whereas Kebaier used a crude approach based on studying the associated characteristic function. Further, this latter approach needs a control on the third moment, whereas we only need to control a moment strictly greater than two. Also, it is worth to note that the limit variance in Theorem 4 is smaller than the limit variance in Theorem 3.2 obtained by Kebaier in [17].

3.3 Complexity analysis From a complexity analysis point of view, we can interpret Theorem 4 as follows. For a total error of order 1/nα the computational eﬀort necessary to run the multilevel Monte Carlo Euler method is given by the sequence of sample sizes speciﬁed by relation (20). The associated time complexity is given by:

L ℓ ℓ−1 CMMC = C N0 + Nℓ(m + m ) with C > 0 × ! Xℓ=1 2α L 2 L L n (m 1)T 2α (m 1)T 1 = C − aℓ + n − aℓ . × a0 m aℓ ! Xℓ=1 Xℓ=1 Xℓ=1 ∗ The minimum of the second term of this complexity is reached for the choice of weights aℓ = 1, ℓ 1, , L , since the Cauchy-Schwartz inequality ensures that L2 L 1 L a , and ℓ=1 aℓ ℓ=1 ℓ the∈{ optimal complexity} for the multilevel Monte Carlo Euler method is≤ given by P P (m 1)T (m2 1)T C = C − n2α log n + − n2α(log n)2 = O n2α(log n)2 . MMC × a log m m(log m)2 0 α It turns out that for a given discretization error εn = 1/n to be achieved the complexity is given by C = O (ε−2(log ε )2). Note that this optimal choice a∗ = 1, ℓ 1, , L , with MMC n n ℓ ∈{ } taking a0 = 1 corresponds to the sample sizes given by (m 1)T N = − n2α log n, ℓ 0, , L . ℓ mℓ log m ∈{ }

19 2 10 Our routine Giles routine

1 10 Runtime

0 10

−1 10 −5 −4 −3 10 10 10 RMSE

Figure 1: Comparison of both routines.

Hence, our optimal choice is consistent with that proposed by Giles [8]. Nevertheless, unlike the parameters obtained by Giles [8] for the same setting (see relation (1)), our optimal choice of the sample sizes N , ℓ 1, , L does not depend on any given constant, since our approach ℓ ∈{ } is based on proving a central limit theorem and not on getting upper bounds for the variance. α Otherwise, for the same error of order εn = 1/n the optimal complexity of a Monte Carlo method is given by 2α+1 −2−1/α CMC = O n = O εn which is clearly larger than CMMC. So we deduce that the multilevel method is more efficient. Note also that the optimal choice of the parameter m is obtained for m∗ = 7. Otherwise, any 2α β choice N0 = n (log n) , 0 <β< 2, leads to the same result. Some numerical tests comparing original Giles work [8] with the one of us show that both error rates are in line. Here in Figure 1, we make a simple log-log scale plot of CPU time with respect to the root mean square error, 2α 1.9 for European call and with N0 = n (log n) . It is worth to note that the advantage of the central limit theorem is to construct a more accurate confidence interval. In fact, for a given root mean square error RMSE, the radius of the 90%-confidence interval constructed by the central limit theorem is 1.64 RMSE. However, × without this latter result one can only use Chebyshev’s inequality which yields a radius equal to 3.16 RMSE. Finally note that, taking α =1/2 still gives the optimal rate and allows us to cancel× the bias in the central limit theorem due to the Euler discretization.

20 4 Conclusion

The multilevel Monte Carlo algorithm is a method that can be used in a general framework: as soon as we use a discretization scheme in order to compute quantities such as Ef (X , 0 t T ), we can implement the statistical multilevel algorithm. And this is worth t ≤ ≤ because it is an efficient method according to the original work by Giles [8]. The central limit theorems derived in this paper fill the gap in literature and confirm superiority of the multilevel method over the classical Monte Carlo approach.

Acknowledgments The authors are greatly indebted to the Associate Editor and the referees for their many suggestions and interesting comments which improve the paper.

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