Importance Sampling and Statistical Romberg method

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1 Importance Sampling and Statistical Romberg method Mohamed Ben Alaya, Kaouther Hajji, Ahmed Kebaier o cite this version: Mohamed Ben Alaya, Kaouther Hajji, Ahmed Kebaier Importance Sampling and Statistical Romberg method 2013 <hal > HAL Id: hal Submitted on 12 Apr 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not he 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 Importance Sampling and Statistical Romberg method Mohamed Ben Alaya 1, Kaouther Hajji 1, and Ahmed Kebaier 1 1 Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS UMR 7539 mba@mathuniv-paris13fr hajji@mathuniv-paris13fr kebaier@mathuniv-paris13fr April 12, 2013 Abstract he efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods Among these methods we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter θ he optimal choice of θ is approximated using Robbins-Monro procedures, provided that a non explosion condition is satisfied Otherwise, one can use either a constrained Robbins-Monro algorithm see eg Arouna 2 and Lelong 17 or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pagès in 18 In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique he statistical Romberg method introduced by Kebaier in 13 is known for reducing efficiently the complexity compared to the classical Monte Carlo one In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift θ that minimizes the variance associated to the statistical Romberg method hen, we prove a central limit theorem for the new algorithm that we called adaptative statistical Romberg method Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model AMS 2000 Mathematics Subject Classification 60F05, 62F12, 65C05, 60H35 Key Words and Phrases Stochastic algorithm, Robbins-Monro, variance reduction, Central limit theorem, Statistical Romberg method, Euler scheme, Heston model 1 Introduction Monte Carlo methods have proved to be a useful tool for many of numerical computations in modern finance hese includes the pricing and hedging of complex financial products he general problem is to estimate a real quantity EψX, where X t 0 t is a given diffusion, his research benefited from the support of the chair Risques Financiers, Fondation du Risque 1

3 defined on B := Ω, F, F t t 0, P, taking values in R d and ψ a given function such that ψx is square integrable Since the efficiency of the Monte Carlo simulation considerably depends on the smallness of the variance in the estimation, many variance reduction techniques were developed in the recent years Among these methods appears the technique of importance sampling very popular for its efficiency he working of this method is quite intuitive, if we can produce a parametric transformation such that for all θ R q we have EψX = Egθ, X hen it is natural, to implement a Monte Carlo procedure using the optimal θ solution to the problem θ = arg min θ R q Eg2 θ, X, since the quantity Eg 2 θ, X denotes the main term of the limit variance in the central limit theorem associated to the Monte Carlo method But how to compute θ? o solve this problem, one can use the so-called Robbins-Monro algorithm to construct recursively a sequence of random variables θ i i N that approximate accurately θ Convergence results of this procedure requires a quite restrictive condition known as the non explosion condition see eg 5, 9, 15 given by NEC E g 2 θ, X C1 + θ 2, for all θ R q o avoid this restrictive condition, two improved versions of this routine are proposed in the literature he first one, based on a truncation procedure called Projection à la Chen, is introduced by Chen in 8, 7 and investigated later by several authors see, eg Andrieu, Moulines and Priouret in 1 and Lelong in 17 he use of this procedure in the context of importance sampling is initially proposed by Arouna in 2 and investigated afterward by Lapeyre and Lelong in 16 he second alternative, is more recent and introduced by Lemaire and Pagès in 18 In fact, they proposed an unconstrained procedure by using extensively the regularity of the involved density and they prove the convergence of this algorithm In what follows, these two methods will be called respectively constrained and unconstrained algorithms In view of this, a Monte Carlo method that integrates this importance sampling recursion is recommended in practice he aim of this paper is to study a new algorithm based on an original combination of the statistical Romberg method and the importance sampling technique he statistical Romberg method is known for improving the Monte Carlo efficiency when used with discretization schemes and was introduced by Kebaier in 13 However, the main term of the limit variance in the central limit theorem associated to the statistical Romberg method is quite different from that of the crude Monte Carlo method It turns out that the optimal θ, in this case, is solution to the problem θ = arg min θ R q Ẽ g 2 θ, X + x gθ, X U 2, where U t t 0, is a given diffusion associated to the process X t t 0, defined on an extension B = Ω, F, F t t 0, P of the initial space B see further on Moreover, we intend to study the 2

4 discretized version of this problem More precisely, we denote X n resp Un the Euler scheme, with time step /n, associated to X resp U and we consider the optimal θn given by θ n = arg min θ R q Ẽ g 2 θ, X n + xgθ, X n Un 2 he convergence of θn towards θ as n tends to infinity is proved in the next section In section 3 we study the problem of estimating θn using the Robbins-Monro algorithm More preciously, we construct recursively a sequence of random variables θi n i,n N using either the constrained or the unconstrained procedure he aim is to prove that lim i,n θn i = lim lim θi n = lim lim i n n i θn i = θ, P-as his assertion is slightly complicated to achieve for the unconstrained procedure In fact, for fixed i, n N, the term θi+1 n constructed with this latter procedure involves Xn, θn i,i+1, U n, θn i,i+1, a new pair of diffusion, with drift terms containing θi n o overcome this technical difficulty we make use of the θ-sensitivity process given by θ Xn, θ, θ Un, θ and we obtain the announced convergence result see heorem 32 and 33 and Corollary 34 In section 4, we first introduce the new adaptative algorithm obtained by combining together the importance sampling procedure and the statistical Romberg method hen, we prove central limit theorems for both adaptative Monte Carlo method see heorem 42 and the remark below, and adaptative statistical Romberg method see heorem 43 using the Lindeberg-Feller central limit theorem for martingale array In Section 5 we proceed to numerical simulations to illustrate the efficiency of this new method with some applications in finance he last section is devoted to discuss some future openings 2 General Framework Let X := X t 0 t be the process with values in R d, solution to dx t = bx t dt + q j=1 σ j X t dw j t, X 0 = x R d 1 where W = W 1,,W q is a q-dimensional Brownian motion on some given filtered probability space B = Ω, F, F t t 0, P and F t t 0 is the standard Brownian filtration he functions b : R d R d and σ j : R d R d, 1 j q, satisfy condition H b,σ x, y R d bx by + q σ j x σ j y C b,σ x y, with C b,σ > 0, j=1 where denotes the Euclidean norm his ensures strong existence and uniqueness of solution of 1 In many applications, in particular for the pricing of financial securities, we are interested in the effective computation by Monte Carlo methods of the quantity EψX, where ψ is a given function From a practical point of view, we have to approximate the process X by a 3

5 discretization scheme So, let us consider the Euler continuous approximation X n with time step δ = /n given by dx n t = bx ηntdt + q σ j X ηntdw t, η n t = t/δδ 2 j=1 It is well known that under condition H b,σ we have the almost sure convergence of X n towards X together with the following property see eg Bouleau and Lépingle 6 P p 1, X, X n L p and E sup X t Xt n p K p 0 t n, with K p > 0 p/2 he weak error is firstly studied by alay and ubaro in 20 and now it is well known that if ψ, b and σ j 1 j q are in CP 4, they are four times differentiable and together with their derivatives at most polynomially growing, then we have see heorem 1451 in Kloeden and Platen in 14 ε n := EψX n EψX = O1/n he same result was extended in Bally and alay in 3 for a measurable function ψ but with a non degeneracy condition of Hörmander type on the diffusion In the context of possibly degenerate diffusions, when ψ satisfies ψx ψy C1 + x p + y p x y for C > 0, p 0, the estimate ε n c n follows easily from P Moreover, Kebaier in 13 proved that in addition of assumption H b,σ, if b and σ j 1 j q are C 1 and ψ satisfies condition H f PX / D f = 0, where D f := {x R d f is differentiable at x} then, lim n nεn = 0 Conversely, under the same assumptions, he shows that the rate of convergence can be 1/n α, for any α 1/2, 1 So, it is worth to introduce assumption H εn for α 1/2, 1 n α ε n C ψ, α, C ψ, α R In order to compute the quantity EψX n, one may use the so-called statistical Romberg method, considered by 13 and which is conceptually related to the alay-ubaro extrapolation his method reduces efficiently the computational complexity of the combination of Monte Carlo method and the Euler discretization scheme In fact, the complexity in the Monte Carlo method is equal to n 2α+1 and is reduced to n 2α+1/2 in the statistical Romberg method More precisely, for two numbers of discretionary time step n and m such that m << n, the idea of the statistical Romberg method is to use many sample paths with a coarse time discretization step and few additional sample paths with a fine time discretization step he statistical m n Romberg routine approximates our initial quantity of interest EψX using two empirical means N 1 1 ψ ˆX,i m + 1 N 2 ψx,i n N ψxm,i 2 N 1 he random variables of the first empirical mean are independent copies of ψx m and the random variables in the second empirical mean are also independent copies of ψx n ψxm he associated Brownian paths Ŵ and W are independent Under assumptions H f and 4

6 H εn, this method is tamed by a central limit theorem with a rate of convergence equal to n α More precisely, for N 1 = n 2α, N 2 = n 2α 1/2 and m = n the global error normalized by n α converges in law to a Gaussian random variable with bias equal to C ψ, α and a limit variance equal to VarψX + Var ψx U, where U is the weak limit process of the error nx n X defined on B an extension of the initial space B see heorem 32 in Kebaier 13 More precisely, the process U is solution to du t = ḃx tu t dt + q j=1 σ j X t U t dw j t 1 2 q j,l=1 σ j X t σ l X t d W lj t, 3 where W is a q 2 -dimensional standard Brownian motion, defined on the extension B, independent of W, and ḃ respectively σ j 1 j q is the Jacobian matrix of b respectively σ j 1 j q In view to use importance sampling routine, based on the Girsanov transform, we define the family of P θ, as all the equivalent probability measures with respect to P such that L θ t = dp θ dp F t = exp θ W t 1 2 θ 2 t Hence, Bt θ := W t θt is a Brownian motion under P θ his leads to EψX = E θ ψx e θ Bθ 1 2 θ 2 Let us introduce the process Xt θ solution, under P, to q dxt θ = bxt θ + θ j σ j Xt θ dt + j=1 q j=1 σ j X θ t dw j t, so that the process B θ t, X t 0 t under P θ has the same law as W t, X θ t 0 t under P Henceforth, we get EψX = Egθ, X θ, W, with gθ, x, y = ψxe θ y 1 2 θ 2, x R d and y R q 4 We also introduce the Euler continuous approximation X n,θ of the process X θ solution, under P, to q q dx n,θ t = bx n,θ η + nt θ j σ j X n,θ η nt dt + σ j Xη θ dw j nt t, j=1 Our target now is to use the statistical Romberg method introduced above to approximate EψX = Egθ, X θ, W by j=1 1 N 1 N 1 gθ, ˆX m,θ,i, Ŵ,i + 1 N 2 N 2 gθ, X n,θ,i, W,i gθ, X m,θ,i, W,i 5

7 Of course the Brownian paths generated by Ŵ and W have to be independent According to heorem 32 of Kebaier 13 mentioned above, we have a central limit theorem with limit variance Var gθ, X, θ W + Var x gθ, X, θ W U θ where U θ is the weak limit process of the error nx n,θ X θ defined on the extension B and solution to q q dut θ = ḃxt θ + θ j σ j Xt θ Ut θ dt + σ j Xt θ Ut θ dw j t 1 q σ j Xt θ σ l X θ lj t d W t 5 2 j=1 j=1 herefore, it is natural to choose the optimal θ minimizing the associated variance As Egθ, X θ, W = EψX and Ẽ xgθ, X θ, W U θ = 0 see Proposition 21 in Kebaier 13, it boils down to choose ψx θ = argmin vθ with vθ := Ẽ θ 2 + ψx θ U θ 2 e 2θW θ 2 6 θ R q Note that from a practical point of view the quantity vθ is not explicit, we use the Euler scheme to discretize X θ, U θ and we choose the associated θn := argmin v n θ with v n θ := Ẽ ψx n,θ 2 + ψx n,θ Un,θ 2 e 2θW θ 2 7 θ R q with U n,θ is the Euler discretization scheme of U θ, solution to q du n,θ t = ḃx n,θ η + nt θ j σ j X n,θ η nt U n,θ η dt nt j=1 + q j=1 σ j X n,θ η nt Un,θ η dw j nt t 1 q 2 j,l=1 j,l=1 σ j X n,θ η σ nt lx n,θ lj η nt d W t 8 heorem 21 Suppose σ and b are in C 1 with bounded derivatives hen for any θ R the following property holds P p 1, U θ, U n,θ L p and Ẽ sup Ut θ U n,θ t p K p 0 t n, with K p > 0 p/2 In particular, for θ = 0 the above property holds for the processes U and U n Proof: Following the steps of the proof of the strong rate convergence of the Euler scheme, especially the helpful Gronwall inequality, we obtain the announced results by tedious but standard evaluations he existence and uniqueness of θ is ensured by the following result Proposition 21 Suppose σ and b are in C 1 with bounded derivatives and let ψ satisfying assumption H f such that PψX 0 > 0 If there exists a > 1 such that E ψ 2a X and E ψx 2a are finite, then the function θ vθ is C 2 and strictly convex with vθ = ẼHθ, X, U, W where Hθ, X, U, W := θ W ψx 2 + ψx U 2 e θ W θ 2 9 Moreover, there exists a unique θ R q such that min θ R qvθ = vθ 6

8 Proof: First of all, note that according to Girsanov theorem, the process B, X, U under P θ has the same law as W, X θ, U θ under P So, using a change of probability, we get vθ := Ẽ ψx 2 + ψx U 2 e θw θ 2 he function θ ψx 2 + ψx U 2 e θw θ 2 is infinitely continuously differentiable with a first derivative equal to Hθ, X, U, W Note that, for c > 0 we have sup Hθ, X, U, W c + W ψx 2 + ψx U 2 e c W c2 θ c Using Hölder s inequality, it is easy to check that Ẽ sup θ c Hθ, X, U, W is bounded by e 1 2 c2 ψ 2 X a e c W c + W a + ψx 2 a 1 a U 2 2a e c W c + W 2a a 1 a 1 Since Eψ 2a X and E ψx 2a are finite we conclude, thanks to property P, the boundedness of Ẽ sup θ c Hθ, X, U, W According to Lebesgue s theorem we deduce that v is C 1 in R q and vθ = ẼHθ, X, U, W In the same way, we prove that v is of class C 2 in R q So, we have Hessvθ = Ẽ θ W θ W + I q ψ 2 X + ψx U 2 e θw θ 2 Since PψX 0 > 0, we get for all u R q \{0} u Hessvθ u = Ẽ u 2 + uθ W 2 ψ 2 X + ψx U 2 e θw θ 2 > 0 Hence, v is strictly convex Consequently, to prove that the unique minimum is attained for a finite value of θ, it will be sufficient to prove that lim θ vθ = + Recall that vθ = Ẽ ψx 2 + ψx U 2 e θw θ 2 Using Fatou s lemma, we get + = Ẽ lim inf his completes the proof ψx 2 + ψx U 2 e θw 1 + θ 2 θ 2 he same results can be obtained for the Euler scheme X n lim inf Ẽ ψx 2 + ψx U 2 e θw θ 2 θ + Proposition 22 Suppose σ and b are in C 1 with bounded derivatives For a given n N, let ψ satisfying assumption H f and such that PψX n 0 > 0 If there exists a > 1 such that E ψ 2a X n and E ψxn 2a are finite, then the function θ v n θ is C 2 and strictly convex with v n θ = ẼHθ, Xn, Un, W Moreover, there exists a unique θn R q such that min θ R qv n θ = v n θn Further, we prove the convergence of θ n towards θ as n tends to infinity heorem 22 Suppose σ and b are in C 1 with bounded derivatives Let ψ satisfying assumption H f such that PψX 0 > 0 and for all n N, PψX n 0 > 0 If there exists a > 1 such that E ψ 2a X, sup n N E ψ 2a X n, E ψx 2a and sup n N E ψx n 2a are finite hen, θn θ, as n 7

9 Proof: For the sake of clearness, we give the proof for d = q = 1 First of all, we know that v θ = 0 and also v n θ n = 0 hen, using the mean value s theorem, there exists ρn θ between θ and θn such that v θ v nθ n = v θ v nθ + v nρ n θ θ θ n = 0 Hence, as v n is strictly convex satisfying v n ρn θ > 0, we have θ θn = v n θ v θ v n ρn θ he numerator is equal to Ẽ θ W ψ 2 X n + ψ XU n n 2 ψ 2 X ψ X U 2 e θ W θ 2 Now, let 1 < ã < a, applying Hölder s inequality several times it is easy to check that there exists Cã, > 0 such that, v n θ v θ is bounded by } Cã, { ψ 2 X n ψ2 X ã + ψ 2 X n ψ 2 X ã U n 2ã + ψ 2 X ã U n U 2ã ã 1 ã 1 Since conditions H b,σ and H f are satisfied, we have the almost sure convergence of ψ 2 X n towards ψ 2 X and ψ 2 X n towards ψ 2 X hese both convergences hold also in Lã thanks to the uniform boundedness in L a of both quantities ψ 2 X n and ψ 2 X n Consequently, thanks to property P the error v n θ v θ vanishes as n tends to infinity Now, it remains to bound from below the denominator uniformly with respect to n and we have v n ρn θ = Ẽ + ρn θ W 2 ψ 2 X n + ψ X n Un 2 e ρn θ W ρn θ 2 v n ρ n θ v nθ n Let us assume for a while that lim inf v n θ n = 0 then by Fatou s lemma we get lim inf n ψ2 X n + ψ X n Un 2 e θn W θn 2 = 0 P-as So, on the event {ψx 0} we have lim inf n e θn W θn 2 = 0 which is impossible his yields inf n N v n θn > 0, which completes the proof 3 Robbins-Monro Algorithms he aim now is to construct for fixed n some sequences θ n i i N such that lim i θ n i = θ n almost surely It is well known that stochastic algorithms can be used to answer this issue and find an accurate approximation of θ n = arg min θ R v n θ Indeed, using the Robbins-Monro algorithm, we construct recursively the sequence of random variables θ n i i N in R q given by θ n i+1 = θn i γ i+1 Hθ n i, Xn,i+1, Un,i+1, W,i+1, i 0, θ n 0 Rq, 10 where H is given by relation 9, the gain sequence γ i i 1 is a decreasing sequence of positive real numbers satisfying γ i = and γi 2 < 11 8

10 Here X,i n, Un,i i 1 is a sequence of independent copies of the Euler scheme associated to X n, Un adapted to the filtration F,i = σw t,l, W t,l, l i, t, where W i, W i i 1 are independent copies of the pair W, W introduced before in equation 3 o obtain the almost sure convergence of the above algorithm to θn = arg min θ R v n θ, we need to check a first condition: θ θn, v nθ, θ θn > 0, which is satisfied in our context thanks to the convexity property of v n Secondly we need also a sub-quadratic assumption known as the non explosion condition NEC Ẽ Hθ, X n, Un, W 2 C1 + θ 2, for all θ R q Unfortunately, this condition is not satisfied in our context and we will study two different stochastic algorithms using the Robbins-Monro procedure and avoiding the above restriction 31 Constrained stochastic algorithm he idea of the Projection à la Chen is to kill the classic Robbins-Monro procedure when it goes close to explosion and to restart it with a smaller step sequence his can be described as some repeated truncations when the algorithm leaves a slowly growing compact set waiting for stabilization hen, the algorithm behaves like the Robbins-Monro algorithm Formally, for a fixed number of discretization time step n 1, the repeated truncations can be written in our context as follows Let K i i N denote an increasing sequence of compact sets satisfying i=0 K i = R d and K i K i+1, i N For θ n 0 K 0, α n 0 = 0 and a gain sequence γ i i N satisfying 11, we define the sequence θ n i, αn i i N recursively by if θi n γ i+1 Hθi n, X,i+1 n, Un,i+1, W,i+1 K α n i, then θi+1 n = θn i γ i+1hθi n, Xn,i+1, Un,i+1, W,i+1, and αi+1 n = αn i 12 else θi+1 n = θn 0 and αn i+1 = αn i + 1, where the function H is given above in relation 9 For i N, αi n represents the number of truncations of the first i iterations In fact, as we can see, if the i + 1 th iteration of the Robbins-Monro is in the compact set K α n i, then the algorithm will behave like a regular Robbins-Monro However, if the i + 1 th iteration outside the compact set K α n i, it will be reinitialized hen, we increase the domain of projection, so we consider the new compact set K α n i +1 heorem 31 Suppose σ and b are C 1 with bounded derivatives and ψ satisfying assumption H f Assume that for all n N, PψX n 0 > 0 and there exists a > 1 such that E ψ 4a X n and E ψxn 4a are finite, then the sequence θi n i 0 given by routine 12, satisfies 1 For all n N, we have θ n i i θ n, almost surely where θ n is given by relation 7 2 Reversely, for all i N, we have θ n i n θ i, almost surely, where the sequence θ i i 0 is obtained by replacing in routine 12, X n,i, Un,i by their limit X,i, U,i, i 1 9

11 Proof: At the beginning, note that for n N the existence of θ n is ensured by Proposition 22 Concerning, the first assertion, we have to check both assumptions of heorem 31 in 16 he first one given by θ θ n, v n θ, θ θ n > 0, is satisfied in our context thanks to the convexity property of v n So, it remains to check the second assumption given by c > 0, sup Ẽ Hθ, X n, Un, W 2 < 13 θ c his assumption relaxes the usual NEC condition on function H used to run the Robbins- Monro algorithm Let c > 0, we have sup Hθ, X n, Un, W 2 2c + W 2 ψx n 4 + ψx n Un 4 e 2c W +c 2 θ c Using several times Hölder s inequality together wit property P, it is easy to check assumption 13, since Eψ 4a X n and E ψxn 4a are finite he second assertion follows easily by induction on θi n, αn i, using that for all i 1, the pair X,i n, Un,i converges almost surely to X,i, U,i combined with the continuity of ψ Now, by replacing X n, Un by their limit X, U in the proof of the first assertion above, we easily get the following result Corollary 31 Suppose σ and b are C 1 with bounded derivatives and ψ satisfying assumption H f Assume that PψX 0 > 0 and there exists a > 1 such that E ψ 4a X and E ψx 4a are finite, then the sequence θ i i 0 introduced in the above theorem satisfies where θ is given by relation 6 θ i θ i as, he following corollary follows immediately thanks to theorems 22 and 31 and Corollary 31 Corollary 32 Under assumptions of heorem 31 and Corollary 31, the constrained algorithm given respectively by routine 12 satisfies lim i,n θn i = lim lim θi n = lim lim i n n i θn i = θ, P-as, where θ is given by relation 6 32 Unconstrained stochastic algorithm In their recent paper 18, Lemaire and Pagès proposed a new procedure using Robbins-Monro algorithm that satisfies the classical non explosion condition NEC In fact, a new expression of the gradient is obtained by a third change of probability Recall that by Proposition 22 we have v n θ = Ẽ θ W ψx n 2 + ψx n Un 2 e θ W θ 2 10

12 he aim now is to use their idea in our context o do so, we apply Girsanov theorem, with the shift parameter θ Let B θ t := W t + θt and L θ t := dp θ dp F t = e θwt 1 2 θ 2t, we obtain v n θ = Ẽ θ 2θ B θ ψx n 2 + ψx n Un 2 e θ 2 As B θ, X n, U n under P θ has the same law as W, X n, θ, U n, θ under P, we write v n θ = Ẽ 2θ W ψx n, θ 2 + ψx n, θ U n, θ 2 e θ 2 We need in our context to control the growth of ψ So, let us assume that function ψ satisfies for a given λ > 0 ψx C ψ 1 + x λ for all x R d Miming the algorithm proposed by 18, we introduce for a given η > 0, a new function H η θ, X n, θ, U n, θ, W = e η θ 2 2θ W ψx n, θ 2 + ψx n, θ U n, θ 2 hen, we introduce for a gain sequence γ i i N satisfying 11, the algorithm θ n i+1 = θn i γ i+1h η θ n i, Xn, θn i,i+1, U n, θn i,i+1, W,i+1, θ 0 R 14 his algorithm would behave like a classical Robbins-Monro one and does not suffer from the violation of NEC Our aim now is to establish the same results satisfied by the constrained routine 12 and given by heorem 31 his is splitted into two different theorems heorem 32 Suppose σ and b are C 1 with bounded derivatives Let ψ satisfying assumption H f and such that for and for all n N, PψX n 0 > 0 In addition, assume that for λ > 0 we have ψx C ψ 1 + x λ for all x R d and C ψ > 0 hen, the sequence θ n i i 0 given by routine 14, satisfies where θ n is given by relation 7 n N, θ n i i θ n, Proof: o prove the almost sure convergence we will use the classical Robbins-Monro theorem see heorem 2212 page 52 in 9 Let n N, under our assumptions the existence of θ n is ensured by Proposition 22 and we have to check first that θ θn h n θ, θ θn > 0, where h nθ = ẼH ηθ, X n, θ, U n, θ, W his is immediate since h n θ = K η θ v n θ with K η > 0 and v n is a strictly convex function Now it remains to prove that sup Ẽ θ R q H η θ, X n, θ, U n, θ, W 2 <, which guaranties the NEC condition By Cauchy-Schwartz inequality we obtain Ẽ H η θ, X n, θ, U n, θ, W 2 e 2η θ 2 2θ W 2 2 ψx n, θ ψx n, θ U n, θ 2 2 as 11

13 Using both assumptions on ψ, the second and third term in the right side of the above inequality can be bounded respectively up to a standard positive constant by 1 + X n, θ 2λ+1 2 and 1 + X n, θ 4λ 2 + U n, θ 4 2 In the following proof, C denotes, a positive standard constant that may change Let λ 1 = 4λ 2λ + 1, using the identity 1 + x ρ C1 + x ρ for x 0 and ρ 1, then we have Ẽ H η θ, X n, θ, U n, θ, W 2 Ce 2η θ θ X n, θ λ 1 + U n, θ Recall that according to properties P and P, we have the boundedness in L p, p > 1, of the processes X n and U n independently of n So, it follows that Ẽ H η θ, X n, θ, U n, θ, W 2 Ce 2η θ θ X n, θ X n λ 1 + U n, θ U n Using the same techniques as in the proof of existence and uniqueness for stochastic differential equations with Lipschitz coefficients ie Gronwall inequality, we obtain that and 0 Ẽ X n, θ Ẽ X n 2λ 1 C θ 2λ 1 U n, θ q Ẽ j=1 U n 8 C θ 8 Ẽ σ j X n, θ s U n, θ s 0 ds 8 ds As B θ, X n, U n under P θ has the same law as W, X n, θ, U n, θ under P, we write 2λ Ẽ σ j Xs n, θ ds = 2λ1 Ẽ θ σ j Xs n ds = 2λ1 Ẽ 1 σ j Xs n ds e θ W 1 2 θ 2 2λ 1 and Ẽ 0 U n, θ s ds 8 = Ẽ θ 0 U n s ds 8 = Ẽ 0 8 Us n ds e θ W 1 2 θ 2 Now using Hölder s inequality, with = 1, the linear growth of σ r r j 1 j q, properties P and P 1 rθ W and Ẽe r 2 θ 2 r = e r 1 2 θ 2, we obtain Ẽ H η θ, X n, θ, U n, θ, W 2 Ce 2η θ θ 2 We complete the proof by choosing r 1, 1 + 8η 1 + θ λ 1 + θ 4 e r 1 4 θ 2 In the same way as in the constrained case, we deduce the following result if we replace X n, Un by their limit X, U in the above proof 12

14 Corollary 33 Suppose σ and b are C 1 with bounded derivatives Let ψ satisfying assumption H f such that PψX 0 > 0 and ψx C ψ 1 + x λ for all x R d and C ψ, λ > 0 hen, the sequence θ i i 0, obtained when replacing in routine 14 X,i n, Un,i i 1 by their limit X,i, U,i i 1, satisfies θ i θ, as i where θ is given by relation 6 he aim now is to prove that the same property 2 in heorem 31, is satisfied by the unconstrained algorithm 14 his task looks more complicated to achieve, since for a fixed i 0 the stochastic term θi n also appears in the drift part of the pair X n, θn i,i+1, U n, θn i,i+1 o overcome this technical difficulty we firstly strengthen our hypothesis on ψ and secondly make use of the so called θ-sensitivity process given by θ Xn, θ, θ Un, θ heorem 33 Suppose b and σ are C 2 with bounded first and second derivatives Let ψ satisfying assumption H f and such that for all n N, PψX n 0 > 0 Assume also that for λ > 1, function ψ satisfies Hess ψx C ψ 1 + x λ 1 for all x R d and C ψ > 0 hen, for all p 1, there exists C > 0 such that i N, n N, Ẽ θ n i+1 θ i+1 2p C n p Moreover, i N, θi n θ i, as n where the sequence θ i i 0 is introduced in the above corollary Proof We give the proof in the case of dimension one for simplicity of notations only he below proof is constructed so that it works for any dimension We first proceed by induction on i N to prove the first assertion he case when i = 0 is trivial since θ 0 R We now assume the assertion holds for a fixed integer i and show that it also holds for i + 1 In the following proof, C denotes, a positive standard constant that may change from line to line In this proof the constant C depends essentially on p, λ and For all p 1 relation 14 yields Ẽθi+1 n θ i+1 2p CẼθn i θ i 2p + Cγi+1Ẽ 2p H η θi n, X n, θn i,i+1, U n, θn i 2p,i+1, W,i+1 H η θ i, X θ i,i+1, U θ i,i+1,i+1, W 15 Using the induction assumption we only need to control the second term in the right hand side of the above inequality Ẽ H η θi n, Xn, θn i,i+1, U n, θn i 2p,i+1, W,i+1 H η θ i, X θ i,i+1, U θ i,i+1, W,i+1 C ẼH 2p 1 + ẼH2p

15 where H 1 := e η θn i 2 2θ n i W,i+1 ψx n, θn i,i+1 2 ψx θn i,i H 2 := e η θn i 2 2θi n W,i+1 ψx θn i,i e η θ i 2 2θ i W,i+1 ψ X n, θn i,i+1 U n, θn i 2,i+1 ψ X θn i,i+1 U θn i 2,i+1 ψ X θn i,i+1 U θn i,i ψx θ i,i ψ X θ i,i+1 U θ i,i+1 erm H 1 : Using that θi n is F,i -measurable, W,i+1 F,i we write ẼH2p 1 = ẼAθn i where for all θ R Aθ := e 2pη θ 2 Ẽ 2θ W 2p { ψx n, θ 2 ψx θ 2 + ψ X n, θ U n, θ 2 ψ X θ } 2 2p U θ Since B θ, X n, U n, X, U under P θ has the same law as W, X n, θ, U n, θ, X θ, U θ under P for all θ R, we obtain by a change of probability measure Aθ = e 2pη 1 2 θ 2 Ẽ θ W 2p e θw {ψx n 2 ψx 2 + ψ X nu n 2 ψ X U 2} 2p By Hölder s inequality, we obtain r 1, r 2 and r 3 1, st 1 r r r 3 = 1, Aθ Ce 2pη 1 2 θ 2 Ee r 1θW 1 r 1 Eθ W 2pr 2 1 r 2 ψx n 2 ψx 2 2p r3 + ψ X n Un 2 ψ X U 2 2p r3 Using assumptions on ψ, ψ and ψ together with assumptions P and P we get by standard evaluations, for all θ R Aθ C n p1 + θ2p e r 1 2 2pη 1 2 θ 2 C n p, since one can choose r 1 1, 1 + 4pη So, we deduce easily that p 1, we have ẼH 2p 1 C np 17 erm H 2 : Here, we need first to introduce, for all u R, the couple of u-sensitivity processes Y u t, Z u t t 0, given by Y u t := X u t u and Z u t 14 := U u t, t 0, u

16 solutions to the following system of SDEs dy u t = σx u t + Y u t b X u t uy u t σ X u t dt + Y u t dz u t = σ X u t + Y u t b X u t uy u t σ X u t U u t dt + b X u t uσ X u t Z u t dt + Y u t U u t σ X u t 1 2 σ X u t σx u t + σ X u t 2 Y u t d W t σ X u t dw t, + Z u t σ X u t dw t Our assumptions on b and σ ensure existence and uniqueness of X u, U u, Y u, Z u Note that all theoretical results known for the tangent process of a given SDE, that is the differentiation of the flow of that SDE with respect to its initial value, can be extended to any parameter hus, following heorem 103 of section I in 4, we have the differentiability of θ X θ t, U θ t and it follows that for all θ, θ R 2 X θ t X θ t = θ,θ Y u t du and U θ t U θ t = θ,θ Z u t du, P-as herefore, since θ n i, θ i are both F,i -measurable and W,i+1 F,i, we write ẼH2p 2 = ẼBθn i, θ i where for all θ, θ R 2 Bθ, θ = Ẽ θ,θ { e η u 2 2u W ψx u 2 + u ψ X u } 2 2p U u du Now, we can make explicit the derivative within the above integral and get easily that Bθ, θ C 4 B iθ, θ where B 1 θ, θ = Ẽ θ,θ B 2 θ, θ = Ẽ B 3 θ, θ = Ẽ B 4 θ, θ = Ẽ { 2 2ηu2u W e η u 2 θ,θ θ,θ θ,θ { { { 22u W e η u 2 U u 22u W e η u 2 U u 22u W e η u 2 Y u ψx u ψ X u ψ X u ψ X u Y u U u ψ X u } 2p 2 Z u du } 2 2p du } 2p ψ X u ψx u du } 2p du In the following, we choose to treat only the term B 3 θ, θ, since the others are totally similar By means of the Jensen inequality we get for all θ, θ R 2 B 3 θ, θ C θ θ 2p 1 Ẽ 2u W 2p e 2pη u 2 U u 2p ψ X u 4p Z u du 2p θ,θ Note that the same probability change leading to cancel the u-term in the drift part of X u operates in the same way for the other processes U u, Y u and Z u Let Y, Z be solution 15

17 to the following system of SDEs dy t = σx t + Y t b X t dt + Y t σ X t dw t, dz t = σ X t + Y t b X t U t dt + b X t Z t dt + Y t U t σ X t + Z t σ X t dw t 1 σ X t σx t + σ X t 2 Y t d W t 2 18 hat is by applying Girsanov theorem and using that B u, X, U, Y, Z under P u has the same law as W, X u, U u, Y u, Z u under P for all u R, we get B 3 θ, θ C θ θ 2p 1 Ẽ u W 2p e 2pη u 2 U 2p ψ X 4p Z 2p e uw 1 2 u 2 du θ,θ By Hölder s inequality, we obtain for all q 1, q 2 and q 3 1, such that 1 q q q 3 = 1 B 3 θ, θ C θ θ 2p u 2p e q pηu2 U 2p ψ X 4p Z 2p q2 du θ,θ By choosing q 1 1, 1 + 4pη and using our assumption on ψ, it follows that B 3 θ, θ C θ θ 2p, this is immediate, since X and U satisfy properties P and P and Z is a diffusion process with enough smooth coefficients satisfying likewise the same type of properties Now, as mentioned above, similar arguments hold true to get the same upper bound for B 1 θ, θ, B 2 θ, θ, and B 4 θ, θ So that, we obtain for all p > 1 Bθ, θ C θ θ 2p Now, since ẼH2p 2 = ẼBθn i, θ i, it follows for all p > 1 ẼH 2p 2 CE θ n i θ i 2p C n p, By the induction hypothesis Combining this inequality together with relations 15, 16 and 17 give us the first assertion of the theorem he almost sure convergence, for all i N, of θ n i towards θ i as n tends to is a classical and immediate consequence of the first assertion shown above, based on Borel-Cantelli lemma he following corollary follows immediately thanks to theorems 22, 32 and 33 and Corollary 33 Corollary 34 Under assumptions of heorem 33 and PψX 0 > 0, the unconstrained algorithm given respectively by routine 14 satisfies lim i,n θn i = lim lim θi n = lim lim θi n = θ, i n n i P-as, where θ is given by relation 6 16

18 4 Central limit theorem for the adaptative procedure In this section we prove a central limit theorem for both adaptative Monte Carlo and adaptative statistical Romberg methods Let us recall that the adaptative importance sampling algorithm for the statistical Romberg method approximates our initial quantity of interest EψX = E ψx θe θ W 1 2 θ 2 by 1 N 1 N 1 gˆθ m i, ˆX m,ˆθ m i,i+1, Ŵ,i N 2 N 2 gθi n, Xn,θn i,i+1, W,i+1 gθi n, Xm,θn i,i+1, W,i+1, 19 where for all x R d and y R q, gθ, x, y = ψxe θ y 1 2 θ 2 Here the paths generated by W and Ŵ are of course independent In order to prove a central limit theorem for this algorithm, we need to study independently each of the above empirical means his is the aim of subsections 42 and 43 We need first to recall some useful results 41 echnical results Let us recall the Central Limit heorem for martingales array see eg 9 heorem 41 Suppose that Ω, F, P is a probability space and that for each n, we have a filtration F n = F n k k 0, a sequence k n as n and a real square integrable vector martingale M n = M n k k 0 which is adapted to F n and has quadratic variation denoted by M n k k 0 We make the following two assumptions A1 here exists a deterministic symmetric positive semi-definite matrix Γ, such that M n k n = k n k=1 E M n k M n k 1 2 F n k 1 P Γ n A2 Lindeberg s condition holds: that is, for all ε > 0, k n k=1 E Mk n Mn k { M n k Mk 1 n >ε} Fk 1 n P 0 n hen M n k n L N0, Γ as n Remark he following assumption known as the Lyapunov condition, implies the Lindeberg s condition A2, A3 here exists a real number a > 1, such that k n k=1 E M n k Mn k 1 2a F n k 1 P 0 n 17

19 As a prelude to the results of this subsection, we give a double indexed version of the oeplitz lemma that will be very helpful in the sequel Lemma 41 Let a i 1 i kn a sequence of real positive numbers, where k n as n tends to infinity, and x n i i 1,n 1 a double indexed sequence such that i lim n 1 i k n a i = ii lim i,n xn i = lim i lim n x n i = lim n lim i xn i = x < hen kn lim a ix n i n + kn a i Proof For all ε > 0, there exists N 1 ε such that for all n N 1 ε and i N 1 ε, we have that: x n i x ε 2 As k n goes to infinity, there exists N 2 ε such that for all n N 2 ε, we have k n N 1 ε herefore, for all n supn 1 ε, N 2 ε = Nε, we can write: k n a i x n i x = N 1 ε 1 = x a i x n i x + For the second term of the expression above, we have: k n i=n 1 ε a i x n i x ε 2 k n i=n 1 ε k n i=n 1 ε a i ε 2 a i x n i x k n a i On the other hand, by assumptions i and ii there exists Ñε such that for all n Ñε sup 1 i N1 ε 1 sup n 1 x n i x 1 i N 1 ε 1 a i 1 i k n a i ε 2 herefore, for all n Ñε his completes the proof kn a ix n i kn a i x ε Let B = Ω, F, F t t 0, P be the extension probability space introduced in Section 2 endowed with the filtration F,i = σw t,l, W t,l, l i, t given in the very beginning of Section 3 In what follows, let θi n i 0, n N and θ i i 0 a family of sequences satisfying For each n N, θi n i 0 and θ i i 0 are F,i i 0 -adapted H θ with deterministic limits θ and θ n lim lim i n θn i = lim θ i = lim lim θi n = lim θn = θ, i n i n 18 P-as,

20 42 he adaptative Monte Carlo method Let us recall that the statistical Romberg algorithm 19 runs successively two independent empirical means he first one is a crude Monte Carlo simply depending on the Euler scheme with the coarse time step /m However, the second empirical mean involves the functional difference between the fine Euler scheme with time step /n and the coarse one constructed from the same Brownian path he task now is to prove a central limit theorem for the first empirical mean heorem 42 Let θ n i i 0, n N and θ i i 0 be a family of sequences satisfying H θ Moreover, assume that b and σ satisfy the global Lipschitz condition H b,σ and the function ψ is a real valued function satisfying assumption H εn, with α 1/2, 1 and C ψ R, such that ψx ψy C1 + x p + y p x y, for some C, p > 0, then the following convergence holds n α 1 n 2α n 2α gθi n, X n,θn i,i+1, W,i+1 EψX L N C ψ, σ 2 where σ 2 := E ψx 2 e θ W 1 2 θ 2 EψX 2 and for all x R d and y R q, gθ, x, y = ψxe θ y 1 2 θ 2 Furthermore, we have also for all α, β > 0 n α 1 n 2α n 2α gθi nβ, X nβ,θ n β i,i+1, W,i+1 EψX nβ L N 0, σ 2 Proof At first, we rewrite the total error as follows 1 n 2α n 2α gθ n i, Xn,θn i,i+1, W,i+1 EψX = 1 n 2α n 2α gθi n, Xn,θn i,i+1, W,i+1 Egθi n, Xn,θn i,i+1, W,i+1 + EψX n EψX Note that Ẽgθn i, Xn,θn i,i+1, W,i+1 = ẼẼ gθi n, Xn,θn i,i+1, W,i+1 F,i = EψX n Assumption H εn ensures that n α EψX n EψX C ψ as n Consequently, it remains to study the asymptotic behavior of the martingale array Mk n k 1 given by M n k := 1 n α k gθi n, Xn,θn i,i+1, W,i+1 Egθi n, Xn,θn i,i+1, W,i+1 o do so, we split the proof into two steps devoted to apply the central limit theorem for martingales array see heorem 41 and comments their 19

21 Step 1 We need first to study the asymptotic behavior of the quadratic variation of the martingale array Mk n k 1 given by M n n = 1 n 2α 2 Ẽ gθ n 2α n 2α i, Xn,θn i,i+1, W,i+1 Ẽgθn i, Xn,θn i,i+1, W,i+1 F,i Since θ n i is F,i -measurable and W,i+1 F,i, we obtain easily that M n n 2α = 1 n 2α n 2α ν n θi n EψX n 2, 20 where for all θ R q ν n θ := E ψx n,θ 2 e 2θW θ 2 = E ψx n 2 e θ W θ 2 It is clear that by assumption H εn, the last term in the right side of the relation 20 converges to EψX 2, as n tends to infinity Concerning the first term, we introduce νθ := E ψx 2 e θ W θ 2 and we get for all θ R ψx n ν n θ νθ E 2 ψx 2 e θ W θ 2 e 3 2 θ 2 ψx n 2 ψx 2 2 Under the condition on ψ together with property P, there exists C > 0 such that ν n θ νθ C n e 3 2 θ 2, θ R q By similar calculations, we check easily the equicontinuity of the family functions ν n n 1 and we deduce thanks to property H θ lim ν nθi n = νθ i,n P-as herefore, Lemma 41 applies and we deduce that M n n σ 2 2α n Step 2 We will check now the Lyapunov condition, that is assumption A3, which implies the Lindeberg condition A2 Let a > 1, we have n 2α Ẽ M ni M ni 1 2a F,i 1 = 1 n 2aα 22a 1 n 2aα n 2α Ẽ n 2α gθ n i, Xn,θn i,i+1, W,i+1 EψX n 2a F,i ν a,n θ n i + 22a 1 n 2αa 1EψXn 2a where for all θ R q, ν a,n θ = E ψx n2a e 2a 1θ W a 3 2 θ 2 Following the same arguments detailed in the first step, we prove that 1 n 2α n 2α ν a,n θ n i n ν aθ P-as where for all θ R q, ν a θ = E ψx 2a e 2a 1θ W a 3 2 θ 2 he second assertion is easily obtained following the above proof with α, β > 0 his completes the proof 20

22 Remark If one have in mind to reduce the variance by using an adaptative crude Monte Carlo method, it appears clear that the natural choice is θ = arg min Ẽ g 2 θ, X and θ θ R q n = arg min Ẽ g 2 θ, X n θ R q for n 1 Under suitable conditions on ψ, b and σ, one can of course construct sequences θ n i i 0, n N and θ i i 0 satisfying H θ by either the constrained or the unconstrained Robbins-Monro algorithm 43 he adaptative statistical Romberg method As we pointed out at the beginning of the above subsection, the statistical Romberg algorithm 19 consists of two empirical means So our task now is to study the asymptotic behavior of the second one in view to establish a central limit theorem for the method heorem 43 Let θ n i i 0, n N and θ i i 0 be a family of sequences satisfying H θ Moreover, assume that b and σ are C 1 functions satisfying the global Lipschitz condition H b,σ and ψ is a real valued function satisfying assumptions H f, H εn, with constants α 1/2, 1 and C ψ R, such that ψx ψy C1 + x p + y p x y, for some C, p > 0 If we choose N 1 = n 2α, N 2 = n 2α β and m = n β, 0 < β < 1 then the statistical Romberg algorithm denoted by V n in 19 satisfies n α V n EψX L N C ψ, σ 2 + σ 2 as n, where σ 2 = E ψx 2 e θ W 1 2 θ 2 EψX 2, σ 2 := Ẽ ψx U 2 e θ W 1 2 θ 2 and U is the process introduced from the beginning by relation 3 Proof First of all, note that we can rewrite the normalized total error as follows n α V n EψX := A n 1 + An 2 with A n 1 := n α V n EψX nβ EψXn ψxnβ, and A n 2 := n α EψX n ψx So, assumption H εn yields the convergence of the second term A n 2 towards the discretization constant C ψ, as n tends to infinity he first term A n 1 can be also rewritten as follows A n 1 := A n 1,1 + An 1,2, where A n 1,1 := 1 n2α n α A n 1,2 := n 1 2α β n α β gˆθ nβ i, ˆX nβ,ˆθ n β i,i+1, W,i+1 EψX nβ, gθi n, X n,θn i,i+1, W,i+1 gθi n, X nβ,θi n,i+1, W,i+1 EψX n ψx nβ 21

23 Using the independence between A n 1,1 and A n 1,2, we study separately their asymptotic behavior Concerning the first term, the second assertion in heorem 42 applies and gives the asymptotic normality of A n 1,1, A n 1,1 L N 0, σ 2, as n 21 Now, concerning the second term A n 1,2 we introduce the martingale arrays Mn k k 1 M n k := 1 n α β k gθi n, Xn,θn i,i+1, W,i+1 gθi n,,θ n Xnβ i,i+1, W,i+1 EψX n ψxnβ, in view to apply heorem 41 o do so, we will verify both assumptions A1 and A3 in the following two steps Step 1 he quadratic variation of M evaluated at n 2α β is given by where θ R q, ξ n θ := E M n n = 1 n 2α β n β ξ 2α β n 2α β n θi n n β 2 2 EψX n EψX nβ, 22 ψx n ψxnβ 2 e θ W θ 2 Now, assumption H εn with 1/2 < α 1 ensures that the second term in the right side of relation 22 vanishes as n tends to infinity We focus now on the asymptotic behavior of n β ξ n θ Under assumption H f, we apply the aylor expansion theorem twice to get for all θ R q where n β 2 ψx n ψx nβ e 1 2 θ W θ 2 = n β 2 ψx X n X nβ e 1 2 θ W θ 2 + R n, R n := n β 2 X n X εx, X n X n β 2 X n β X εx, X nβ X with εx, X n X P-as 0 and εx, X nβ X P-as 0 as n, since the global Lipschitz condition H b,σ is satisfied Further, as b and σ are C 1 functions then according to heorem 32 in 12 we have the tightness of n β 2 X n X and n β 2 X nβ X and we deduce the convergence in probability of the remaining term R n to zero as n tends to infinity Once again, by the same theorem in 12, we get for all θ R q n β 2 ψx n ψx nβ e 1 2 θ W θ 2 stably = ψx U e 1 2 θ W θ 2 23 Otherwise, θ R q and a > 1 we have by Cauchy-Schwarz inequality E n β 2 ψx n ψx nβ e 1 2 θ W θ 2 2a n E βa ψx n ψx nβ 4a 1 2 a e 2a +1 θ 2 2 hanks to the assumption on ψ together with property P, we obtain sup E n β 2 ψx n ψx nβ e 1 2 θ W θ 2 2a < 24 n Hence, by the stable convergence obtained in 23 and the uniform integrability property given by 24 we deduce θ R q lim n nβ ξ n θ = Ẽ ψx U 2 e θ W θ 2 := ξθ 25 22

24 Using property P with assumption on ψ, it is easy to check by standard evaluations the equicontinuity of the family functions n β ξ n n 1 So under assumption H θ, we get lim i,n nβ ξ n θi n = ξθ P-as hen, Lemma 41 yields lim n M n n = ξθ, P-as 2α β Step 2 he second step consists on checking Lyapunov assumption A3 Let a > 1, n 2α β Ẽ M ni M ni 1 2a F,i 1 22a 1 n 2α β n βa ξ n a2α β a,n θi n + 22a 1 n βa n 2α βa 1 EψXn EψXnβ 2a where for all θ R q, ξ a,n θ := E ψx n ψxnβ 2a e 2a 1θ W a 3 2 θ 2 Miming the same arguments used in the first step, we prove under assumption H θ using relations 23 and Lemma 41, that n 1 2α β n βa ξ n 2α β a,n θi n ξ a θ := Ẽ ψx U 2a e 2a 1θ W a 3 2 θ 2, P-as n Consequently, since a > 1, we conclude using assumption H εn that A3 holds his gives the asymptotic normality of A n 1,2,2 so that we have A n 1,2 N0, σ 2, as n his completes the proof Remark We recall that for the adaptative statistical Romberg method the optimal choice of θ and θn is given respectively by relations 6 and 7 According to Corollary 32 resp Corollary 34, the sequences θi n i 0, n N and θ i i 0 obtained by the constrained Robbins- Monro algorithm resp the unconstrained Robbins-Monro algorithm satisfy H θ under some regularity conditions on ψ, b and σ Complexity analysis According to the main theorems of this section, we deduce that for a total error of order 1/n α, α 1/2, 1, the minimal computational effort necessary to run the adaptative statistical Romberg algorithm is obtained for N 1 = n 2α, N 2 = n 2α β and m = n β his leads to a time complexity given by C SR = C n 2α+β + n + n β n 2α β, with C > 0 So the time complexity reaches its minimum for the optimal choice of β = 1/2 Hence, the optimal parameters to run the method are given by m = n, N 1 = n 2α and N 2 = n 2α 1/2 hen the optimal complexity of the adaptative statistical Romberg algorithm is given by C SR C n 2α+1 2 However, for the same error of order 1/n α, the optimal complexity of the adaptative Monte Carlo algorithm is given by C MC = C N n = C n 2α+1 We conclude that the adaptative statistical Romberg method is more efficient in terms of time complexity 5 Numerical results for the Heston model Stochastic volatility models are increasingly important in practical derivatives pricing applications In this section we show, throughout the problem of option pricing with a stochastic 23 L

25 volatility model, the efficiency of the importance sampling statistical Romberg method compared to the importance sampling Monte Carlo one he popular stochastic volatility model in finance is the Heston model introduced by Heston in 11 as solution to { dst = rs t dt + V t S t dw 1 t dv t = κ v V t dt + σ V t ρdw 1 t + σ V t 1 ρ2 dw 2 t, where W 1 and W 2 are two independent Brownian motions Parameters κ, σ, v and r are strictly positive constants and ρ 1 In this model, κ is the rate at which V t reverts to v, v is the long run average price variance, σ is the volatility of the variance, r is the interest rate and ρ is a correlation term Our aim is to use the importance sampling method in order to reduce the variance when computing the price of an European call option, with strike K, under the Heston model he payoff of the option is ψs = S K + hen, the price is e r EψS After a density transformation, given by Girsanov theorem, the price will be defined by: e r E gθ, S θ = e r E ψs θ e θw 1 2 θ 2, θ R 2 For more details on definitions of the function g and S θ, see relation 4 and related results given in the same page o approximate S θ, we consider the step /n and we discretize the stochastic process using the Euler scheme For i 0, n 1 and θ = θ 1, θ 2 R 2, S n,θ t i+1 = S n,θ t i 1 + r + θ 1 V n,θ t i n + V n,θ t i n Z 1,i+1, V n,θ t i+1 = V n,θ t i + κ v V n,θ t i + σ V n,θ t i ρθ ρ 2 θ 2 n + σ V n,θ t i n Z 2,i+1, with Z 1,i, Z 2,i 1 i n is a sequence of a standard Gaussian random vectors taking values in R 2 Hence, the price of the European call option is firstly approximated by e r E gθ, S n,θ = e r E ψs n,θ e θw 1 2 θ 2, θ R 2 he choice of θ depends on using the classical Monte Carlo method or the statistical Romberg one he optimal θ for the first method is given by θn = arg mine θ R 2 ψ 2 S n,θ e 2θW θ 2 However, he optimal θ for the second one is θ n = arg mine θ R 2 ψ 2 S n,θ + ψsn,θ Un,θ 2 e 2θW θ 2 where U n,θ denotes the Euler discretization scheme obtained when we replace coefficients b and σ of relation 8 by the corresponding parameters in the Heston model Here, we have also the choice of the algorithm approximating both θ n and θ n We can use either the constrained or the unconstrained stochastic algorithms studied in section 3 above 24,

26 Approximation of θ n by i Constrained algorithm: let K i i N denote an increasing sequence of compact sets satisfying i=0 K i = R d and K i K i+1, i N For θ n 0 K 0, α n 0 = 0 and a gain sequence γ i i N satisfying 11, we define the sequence θ n i, αn i i N recursively by if θi n γ i+1hθi n, Sn,i+1, Un,i+1, W,i+1 K α n i, then θi+1 n = θi n γ i+1 Hθi n, S,i+1 n, Un,i+1, W,i+1, and αi+1 n = αi n 26 else θi+1 n = θn 0 and αn i+1 = αn i + 1, where Hθ n i, Sn,i+1, W,i = θ n i W,i+1 ψ 2 S n,i+1 e θn i W,i θn i 2 ii Unconstrained algorithm : θ n i+1 = θ n i γ i+1 2θ n i W,i+1 ψ 2 S n, θn i,i+1 e η θn i 2, with η > 0 Approximation of θ n by i Constrained algorithm: we use the same routine 26 with Hθ n i, S n,i+1, W,i = θ n i W,i+1 ψ 2 S n,i+1 + ψs n,i+1u n,i+1 2 e θn i W,i θn i 2 ii Unconstrained algorithm: we use the routine θi+1 n = θi n γ i+1 2θi n W,i+1 ψ 2 S n, θn i,i+1 + ψsn, θn i,i+1 Un,i+1 e 2 η θn i 2 o compare these different routines we run a number of iterations M = he parameters in the Heston model are chosen as follows: S 0 = 100, V 0 = 001, K = 100, the free interest rate r = log11, σ = 02, k = 2, v = 001, ρ = 05 and maturity time = 1 able 1 gives the obtained values of the two-dimensional vectors θ n and θ n Constrained algorithm Unconstrained algorithm θn 07906, , θ n 07884, , able 1: Estimation of θ n and θ n In Figure 1, we test the robustness of both routines, for the computation of θ n, using the averaged algorithm à la Ruppert & Poliak see eg 19 known to give optimal rate for convergence We implement this averaged algorithm using both constrained and unconstrained procedures So, we proceed as follows, i first, we choose a slowly decreasing step: γ i = γ 0 /i α, for α 1 2, 1 and γ 0 > 0 ii hen, we compute the empirical mean of all the previous observations, θ n i+1 := 1 i i θ k n k=0