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

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

regression

Emmanuel Gobet, Plamen Turkedjiev

To cite this version:

Emmanuel Gobet, Plamen Turkedjiev. Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression. 2014. �hal-00855760v2�

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

E. Gobet^† and P. Turkedjiev^‡ Centre de Math´ematiques Appliqu´ees

Ecole Polytechnique and CNRS Route de Saclay

91128 Palaiseau cedex, France March 25, 2014

Abstract: We design a numerical scheme for solving a Dynamic Programming equation with Malli- avin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the Z-component by [18]. 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 take into account the regularity of the terminal function.

Keywords: Backward stochastic differential equations, Malliavin calculus, dynamic programming equation, empirical regressions, non-asymptotic error estimates.

MSC 2010: 49L20, 60H07, 62Jxx, 65C30, 93E24.

1 Introduction

1.1 Setting

LetT >0 be a fixed terminal time and let (Ω,F,(Ft)_0≤t≤T,P) be a filtered probability space whose filtration is augmented with theP-null sets. Let π ={0 =:t₀ < t₁ < . . . < t_N−1 < 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_j⁽ⁱ⁾ : 0≤i < j ≤N} of (R^q)^>-valued random variables in L2(FT,P) (i.e. square integrable and FT-measurable) that we call Malliavin weights. Here^> stands for the transpose.

∗This research benefited from the support of the Chair Finance and Sustainable Development, under the aegis of Louis Bachelier Finance and Sustainable Growth laboratory, a joint initiative with Ecole polytechnique.

†The author’s research is part of the Chair Financial Risks of the Risk Foundation and of the FiME Laboratory.

Email: emmanuel.gobet@polytechnique.edu

‡Corresponding Author. A significant part of the author’s research has been done while at Humboldt University.

Email: turkedjiev@cmap.polytechnique.fr

In this paper, we introduce the Malliavin Weights Least Squares algorithm, abreviated MWLS, to approximate discrete time stochastic processes (Y, Z) defined by















Yi=Ei[ξ+

N−1

X

j=i

fj(Yj+1, Zj)∆j], 0≤i≤N, Z_i=Ei[ξH_N⁽ⁱ⁾+

N−1

X

j=i+1

f_j(Y_j+1, Z_j)H_j⁽ⁱ⁾∆_j], 0≤i≤N−1,

(1.1)

whereEi[·] :=E[·|F_ti],ξis aR-valued random variable inL₂(F_T,P), and (ω, y, z)7→f_j(ω, y, z) isF_tj⊗ B(R)⊗ B((R^q)^>)-measurable. This system is solved backward in time in the orderY_N, Z_N−1, Y_N−1. . . and it takes the form of a dynamic programming equation with Malliavin weights. We call it the Malliavin Weights Dynamic Programming equation (MWDP for short).

The main application of (1.1) is to approximate continuous-time, decoupled Forward-Backward SDEs (FBSDEs) of the form

Yt=ξ+ Z T

t

f(s, Xs, Ys, Zs)ds− Z T

t

ZsdWs (1.2)

where (Ws)_s≥0 is a Brownian motion inR^q, (Xs)_s≥0 is a diffusion inR^d andξ is of the form Φ(XT).

Indeed, the MWDP (1.1) was inspired by [18, Theorem 4.2], which states that there is a version of the continuous-time process (Zt)_0≤t<T given by

Z_t=Et[ξH_T^(t)+ Z T

t

f(s, X_s, Y_s, Z_s)H_s^(t)ds] (1.3) where the processes (Hs^(t))_0≤t<s≤T are Malliavin weights defined by

H_s^(t)= 1 s−t

Z s t

(σ⁻¹(r, X_r)D_tX_r)^>dW_r>

, 0≤t < s≤T, (1.4) for (DtXr)t being the Malliavin derivative of Xr and σ(·) is the diffusion coefficient of X. The representation (1.3) is obtained via a Malliavin calculus integration by parts formula, see [19] for a general account on the subject. A discretization procedure to approximate the FBSDE (1.2-1.3) with (1.1), including explicit definitions of the random variables H_j⁽ⁱ⁾ based on (1.4), is given in [21][22], where the author also computes the discretization error in terms of N. In honour of the connection between (1.1) and (1.2-1.3), we call the random variables H_j⁽ⁱ⁾ Malliavin weights, ξ the terminal condition, and (i, ω, y, z)7→f_i(y, z) the driver. We say that the pair (Y, Z) satisfying (1.1) solves a MWDP with terminal conditionξand driverf_i(y, z).

1.2 Contributions

In this paper, we are not concerned with the discretization procedure, rather with the analysis of the MWDP equation (1.1) itself and its numerical resolution via 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 (1.2-1.3), we allow the framework and assumptions to accomodate as much general- ity as possible. However, MWLS is, to the best of our knowledge, the first direct implementation of formula (1.3) in a fully implementable numerical scheme. For other applications of Malliavin calcu- lus in numerical simulations, with rather different perspectives and results to ours, see for instance [8][17][3][12][1][16][4].

We adapt the recent theoretical analysis of the Least Squares Multi-step Forward Dynamical Pro- gramming algorithm (LSMDP) of [13] for discrete BSDEs (without Malliavin weights) to the setting of MWDP. As in the aforementioned reference, we consider locally Lipschitz driver fi(y, z) that is locally bounded at (y, z) = (0,0) - see Section 1.4. This allows the algorithm of the current paper to be applied for the approximation of some quadratic BSDEs and for some proxy/variance reduction methods. See [13, Section 2.2] for more details on these applications. Moreover, we make use of analo- gous stability results and conditioning arguments in the proof of the main result, Theorem 3.10, as in the proof of [13, Theorem 4.11]. However, the Malliavin weights lead to significantly differences in the main theorem and stability results, both in the technical elements of the proofs and the results. We develop seemingly novel Gronwall type inequalities to handle the technical differences; these results are outlined in Section 2.1 and proved in Appendix A.1. Furthermore, the stability results are rather more powerful and the complexity of the MWLS is rather better than the LSMDP, as will be discussed in what follows.

We would like to mention that the class of quadratic problems we can treat with these assumptions is quite different to the recent [5]. Here we are treating the non-degenerate setting where the terminal condition may be H¨older continuous, whereas the other reference allows degeneracy at the expense of requiring locally Lipschitz terminal conditions.

We prove stability results on the MWDP in Section 2. Much effort is made to keep the constants explicit. These results are instrumental throughout the paper. The stability estimates on Z are at the individual time points (coherently with the representation theorem of [18]) rather than the time- averaged estimates of [13, Proposition 3.2]. This allows for finer and more precise computations.

The time-dependency in our estimates also takes better into account the regularity of the terminal condition, similarly to the continuous-time case [7].

Section 3 is the core of the paper: it is dedicated to the MWLS algorithm in the Markovian context Yi =yi(Xi) andZi=zi(Xi) for some Markov chainXi in R^d and unknown functions (yi(·), zi(·)). In MWLS, the conditional expectations in (1.1) are replaced by Monte-Carlo least-squares regressions.

For each point of the time-grid, we use a cloud of independent paths of the Markov chainX and the Malliavin weightsH, and some approximation spaces for representing the value functions (y_i(·), zi(·)).

The algorithm is detailed in Section 3.2, and a full error analysis in terms of the number of simulations and the approximation spaces is performed in Sections 3.3 and 3.4. The final error estimates (Theorem 3.10) are similar to [13, Theorem 4.11] in that they are the time-averaged local regression errors of the discrete BSDE, but the results are in a stronger norm and the time-dependency is better. The constants are completely explicit. Although the norms are stronger than in to [13], the estimates do not deteriorate; rather, they are significantly improved. This is intrinsically due to the MWDP representation, which avoids the usual 1/∆i-factor in front of all controls on Z. This improvement can be tracked by inspecting the a.s. bounds (compare (2.10) and [13, Eq. (14)]) and the statistical error bounds (compare the ^K_M^Z,k

k -terms in (3.10) of Theorem 3.10 and the _∆^KZ,k

kM_k-terms of [13, Theorem 4.11]). These error estimates are optimal with respect to the convergence rates (up to logarithmic factors) under rather great generality regarding the distribution of the stochastic model for X and H, even if the constants may be rather conservative. This is because the local regression errors are optimal under model-free estimates (Proposition 3.9).

With the error estimates of Theorem 3.10 in hand, we perform a complexity analysis in Section 3.5. We propose a choice of basis functions and use it to calibrate the number of simulations in order to achieve a specified error level. This then allows us to compute the complexity of the algorithm for that error level. The methodology for doing this is analogous to [13, Section 4.4] in that we use the same basis functions - which also enable us to study the benefit of smoothness properties of the underlying Markov functions (yi(·), zi(·)) - and also in that we set the ensure the global error level by calibrating the local regression errors. However, the conclusion of this section is that MWLS yields

better performance in terms of complexity than LSMDP. The main reason for this is the improved time-dependancy of the error estimates, which is a systemic improvement that allows one to make generate fewer simulations to obtain certain error levels. Unfortunately, the complexity reduction does not reduce the dependence on the dimension compared to the LSMDP. The curse of dimensionality still appears, and the rates depend on the dimension of the Markov chainX (i.e. d). Nevertheless, the reduction of complexity is substantial and, since one is able to make fewer simulations to obtain the same error level, will help alleviate the pressure on memory resources that is typical with least-squares Monte Carlo algorithms.

This paper is theoretically oriented, and is aimed at paving the way for new such numerical ap- proaches based on Malliavin calculus. Future works will be devoted to a deeper investigation about the numerical performance of the MWLS algorithm compared to other known numerical schemes.

1.3 Notation used throughout the paper

• |x|stands for the Euclidean norm of the vectorx,^> denotes the transpose operator.

• |U|p := (E[|U|^p])^p1 stands for the Lp(P)-norm (p ≥ 1) of a random variable U. The Ft_k- conditional version is denoted by |U|p,k := (Ek[|U|^p])^1p. To indicate that U is additionally measurable w.r.t. the σ-algebraQ, we may writeU ∈Lp(Q,P).

• For a multidimensional processU = (Ui)_0≤i≤N, itsl-th component is denoted byUl= (Ul,i)_0≤i≤N.

• For any finite L > 0 and x = (x1, . . . , xn) ∈ Rⁿ, define the truncation function TL(x) :=

(−L∨x1∧L, . . . ,−L∨xn∧L).

• For finitex >0, log(x) is the natural logarithm ofx.

1.4 Assumptions

First set of hypotheses. The following assumptions hold throughout the entirety of the paper. Let Rπ >0 be a fixed parameter: this constant determines which time-grid can be used. The larger Rπ, the larger the class of admissible time-grids. All subsequent error estimates depend onRπ.

(Aξ) ξis in L2(FT,P),

(AF) i) (ω, y, z)7→ fi(y, z) is Fti⊗ B(R)⊗ B((R^q)^>)-measurable for everyi < N, and there exist deterministic parametersθ_L∈(0,1] andL_f ∈[0,+∞) such that

|fi(y, z)−f_i(y⁰, z⁰)| ≤ L_f

(T−ti)^(1−θL)/2(|y−y⁰|+|z−z⁰|), for any (y, y⁰, z, z⁰)∈R×R×(R^q)^>×(R^q)^>.

ii) There exist deterministic parameters θc∈(0,1] andCf ∈[0,+∞) such that

|fi(0,0)| ≤ Cf

(T−ti)^1−θc, ∀0≤i < N.

iii) The time-grid π:={0 =t₀< . . . < t_N =T} satisfies

0≤i≤N−2max

∆i+1

∆_i ≤Rπ.

(A_H) For all 0≤i < j≤N, the Malliavin weights satisfy E[H_j⁽ⁱ⁾|Ft_i] = 0,

E[|H_j⁽ⁱ⁾|²|Ft_i]1/2

≤ CM

(t_j−t_i)^1/2 for a finite constantC_M ≥0.

Comments. We remark that assumptions (A_ξ) and (A_F-i-ii) are the same as their equivalents in [13, Section 2]. The usual case of “Lipschitz” BSDE is covered byθL =θc = 1. As explained in [13], the case of locally Lipschitz driver (θL<1 and/orθc<1) is interesting because 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 [13, Section 2.2] for details.

Assumption (AF-iii) is much simpler compared to [13]. IfRπ≥1, (AF-iii) is satisfied by any time grid with non-increasing time-step, such as the grids of [11, 20, 9]. This may be valuable for future work on time-grid optimization.

Assumption (AH) is specific to the dynamic programming equation with Malliavin weights. It is satisfied for the weights derived in [18], and this can remain true after discretization (see [21] or [12]). It is also satisfied ifX takes the formX_t=g(t, W_t) (like multi-dimensional geometric Brownian motion), by a simple change of variables one can use the Malliavin weightsHs^(t)= ^(Ws_s−t^−Wt)> (note the processX may be degenerate).

Second set of hypotheses: Markovian assumptions. The following assumptions will mostly be used in Section 3. (A_X), (A⁰_F), and (A⁰_H) give us a Markov representation for solutions of the discrete BSDEs (see Equation (3.1) later). (A⁰_ξ) is used for obtaining (model free) estimates on regression errors. We also include additional optional assumptions, (A⁰⁰_ξ), on the terminal condition to obtain tighter estimates onZ_i (see Corollary 2.6 and subsequent remarks).

(AX) X is a Markov chain in R^d (1≤d <+∞) adapted to (Ft_i)i. For every i < N and j > i, there existGi⊗ B(R^d)-measurable functionsV_j⁽ⁱ⁾: Ω×R^d→R^dwhereGi⊂ FT is independent ofFt_i, such thatX_j=V_j⁽ⁱ⁾(X_i).

(A⁰_ξ) i) ξis a boundedFT-measurable random variable: Cξ :=P−ess sup_ω|ξ(ω)|<+∞.

ii) ξis of form ξ:= Φ(XN) for a bounded, measurable function Φ.

(A⁰⁰_ξ) In addition to (A⁰_ξ), for some θΦ ∈[0,1] and a finite constantCΦ ≥0, we have |ξ−Eiξ|2,i ≤ CΦ(T−ti)^θΦ/2 for anyi∈ {0, . . . , N}.

(A⁰_F) For everyi < N, the driver is of the form fi(ω, y, z) =fi(Xi(ω), y, z), and (x, y, z)7→fi(x, y, z) isB(R^d)⊗ B(R)⊗ B((R^q)^>)-measurable and (AF) is satisfied.

(A⁰_H) In addition to (AH), for everyi < N andj > i, there is a functionh_j⁽ⁱ⁾: Ω×R^d→(R^q)^> that isGi⊗ B(R^d)-measurable, where Gi ⊂ FT is independent ofFt_i, such thatH_j⁽ⁱ⁾:=hj⁽ⁱ⁾(Xi).

Comments. (AX) is usually satisfied when X is the solution of SDE or its Euler scheme built on the time gridπ.

The statement of (A⁰⁰_ξ) is inspired by the fractional smoothness condition of [11], although somewhat stronger. It is satisfied, for instance, if the terminal function Φ is θΦ-H¨older continuous and if the Markov chain satisfiesEi[|XN −X_i|²]≤C_X(T−t_i). This is a reasonable assumption on the Markov

chain, since it is satisfied by a diffusion process (possibly including bounded jumps) with bounded coefficients and also by its Euler approximation. Indeed, we have

|ξ−Eiξ|2,i≤ |Φ(XN)−Φ(Xi)|2,i≤CΦ(CX(T −ti))^θ2Φ.

(A⁰_H) is satisfied by the Malliavin weights (1.4) under various conditions. It is valid for the example Xt = g(t, Wt) mentioned before. Consider now the more complex case of the SDE with coefficients b(t, x) for the drift and (σ1(t, x), . . . , σd(t, x)) for the diffusion (q=d) both having first space-derivatives that are uniformly bounded. Recall the relation for the Malliavin derivative of a SDE given by

DtXr=∇Xr∇X_t⁻¹σ(t, Xt)1t≤r=∇xX_r^t,x|x=X_tσ(t, Xt)1t≤r

where X^t,x denotes the SDE solution starting from x at time t, and ∇Xt := ∇xX_t^0,x for ∇xX_t^0,x solving the (d×d)-dimensional, matrix valued linear SDE

∇_xX_r^t,x=I_d+ Z r

t

∇_xb(u, X_u^(t,x))∇_xX_u^t,xdu+

d

X

j=1

Z r t

∇_xσ_j(u, X_u^(t,x))∇_xX_u^t,xdW_j,u.

Then, it is an easy exercice to prove that if σ and its inverse are uniformly bounded, then (A⁰_H) is fulfilled.

2 Stability

2.1 Gronwall type inequalities

Here we gather deterministic inequalities frequently used throughout the paper. These inequalities are crucial due to novel technical problems caused by the Malliavin weights. They show how linear inequalities with singular coefficients propagate. They take the form of unusual Gronwall type inequal- ities. Their proofs are postponed to Appendix A.1. We assume that π is in the class of time-grids satisfying (A_F-iii).

Lemma 2.1. Let α, β >0 be finite. There exists a finite constant Bα,β ≥0 depending only on Rπ, αandβ (but not on the time-grid) such that, for any 0≤i < k≤N,

k−1

X

j=i

∆_j

(tk−tj)^1−α ≤Bα,1(tk−ti)^α,

k−1

X

j=i+1

∆_j

(tk−tj)^1−α(tj−ti)^1−β ≤Bα,β(tk−ti)^α+β−1. Lemma 2.2(exponent improvement in recursive equations). Letα≥0, β∈(0,¹₂]andk∈ {0, . . . , N− 1}. Suppose that, for a finite constant Cu ≥0, the finite non-negative real-valued sequences {ul}_l≥k and{wl}_l≥k satisfy

uj≤wj+Cu N−1

X

l=j+1

ul∆l

(T−t_l)^12−β(t_l−t_j)^12−α, k≤j≤N . (2.1) Then, for two finite constantsC_(2.2a)≥0 andC_(2.2b)≥0that depend only onC_u, T, α, β andR_π,

uj≤ C(2.2a)wj+C(2.2a) N−1

X

l=j+1

w_l∆_l

(T−tl)^12−β(tl−tj)^12−α +C(2.2b) N−1

X

l=j+1

u_l∆_l

(T−tl)^12−β, k≤j ≤N . (2.2)

Lemma 2.3 (intermediate a priori estimates). Let α≥0, β ∈(0,¹₂] andk∈ {0, . . . , N−1}. Assume that the finite non-negative real-valued sequences{ul}_l≥k and{wl}_l≥k satisfy (2.2)for finite constants C(2.2a)≥0 andC(2.2b)≥0. Then, for any finite γ >0, there is a finite constant C_(2.3)^(γ) ≥0 (depending only onC_(2.2a),C_(2.2b),T,α,β,Rπ andγ) such that

N−1

X

l=j+1

ul∆l

(T−tl)^12−β(tl−tj)^1−γ ≤ C_(2.3)^(γ)

N−1

X

l=j+1

wl∆l

(T −tl)^12−β(tl−tj)^1−γ, k≤j≤N . (2.3) Plugging (2.3) with γ= ¹₂+αinto (2.1) gives a ready-to-use result.

Proposition 2.4(final a priori estimates). Under the assumptions of Lemma 2.2, (2.1)implies uj≤wj+C_(2.3)^(12+α)Cu

N−1

X

l=j+1

wl∆l

(T −t_l)^12−β(t_l−t_j)^12−α, k≤j ≤N.

2.2 Stability of discrete BSDEs with Malliavin weights

Suppose that (Y₁, Z₁) (resp. (Y₂, Z₂)) solves a MWDP with terminal condition/driver (ξ₁, f_1,i) (resp.

(ξ₂, f_2,i)). We are interested in obtaining estimates on the differences (Y₁−Y₂, Z₁−Z₂). To give a notion of how stability estimates are used, the processes (Y1, Z1) are typically obtained by construction.

For example, in Section 2.3, they are (0,0), whereas in the proof of Theorem 3.10, they are a set of processes determined from a series of arguments based on conditioning w.r.t. the Monte Carlo samples.

One then applies the stability estimates based on a priori knowledge that what stands on the right hand side is beneficial to the computations. In Corollary 2.6, for example, the right hand side yields almost sure bounds for the processes (Y, Z). We note that the assumptions on the drivers in this section are rather weaker than the general assumptions of Section 1.4. The driverf1,i(y, z) does not have to be Lipschitz continuous, but we assume that eachf1,i(Y1,i+1, Z1,i) is in L2(FT) so thatY1,i andZ1,i are also square integrable for anyi(thanks to (AH)). The driverf2,i(y, z) is locally Lipschitz continuous w.r.t. (y, z) as in (AF-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. Define

∆Y :=Y1−Y2, ∆Z :=Z1−Z2, ∆ξ:=ξ1−ξ2,

∆fi:=f1,i(Y1,i+1, Z1,i)−f2,i(Y1,i+1, Z1,i).

Letk∈ {0, . . . , N−1} be fixed: throughout this subsection,F_tk-conditionalL₂-norms are considered and we recall the notation|U|_2,k:=p

Ek[|U|²] for any square integrable random variableU. Forj ≥k, define

|Θj|2,k:=|∆Yj+1|2,k+|∆Zj|2,k. Using (AH), we obtainEi[∆ξH_N⁽ⁱ⁾] =Ei[(∆ξ−Ei∆ξ)H_N⁽ⁱ⁾] and

|Ei[∆ξH_N⁽ⁱ⁾]|²≤Ei[|∆ξ−Ei∆ξ|²] C_M²

(tN −ti), |Ei[∆f_jH_j⁽ⁱ⁾]|²≤C_M² Ei[|∆fj|²] tj−ti

j≥i+ 1. (2.4) Combining this kind of estimates with (AF-i) and the triangle inequality, our stability equations (for k≤i) are

|∆Yi|2,k≤ |∆ξ|2,k+

N−1

X

j=i

|∆fj|2,k∆j+

N−1

X

j=i

Lf₂|Θj|2,k

(T −t_j)^(1−θL)/2∆j, (2.5)

|∆Zi|2,k≤C_M|∆ξ−Ei∆ξ|2,k

√T−ti

+

N−1

X

j=i+1

C_M|∆fj|2,k

√tj−ti

∆j+

N−1

X

j=i+1

L_f2C_M|Θj|2,k

(T−tj)^(1−θL)/2√ tj−ti

∆j. (2.6) Proposition 2.5. Taking α= 0,β =θL/2 andCu=Lf2(CM+√

T)in Lemmas 2.2 and 2.3, recall the constantC_(2.3)^(γ) . Assume thatξj is inL2(FT). Moreover, for each i∈ {0, . . . , N−1}, assume that f_1,i(Y_1,i+1, Z_1,i)is inL₂(F_T)andf_2,i(y, z)is locally Lipschitz continuous w.r.t. y andz as in(A_F-i), with a constantL_f2. Then, under (A_H), we have

|∆Yi|2,k≤C_y⁽¹⁾|∆ξ|2,k+C_y⁽²⁾

N−1

X

j=i

|∆fj|2,k∆j, 0≤k≤i≤N,

|∆Z_i|_2,k≤C_z⁽¹⁾|∆ξ−Eⁱ∆ξ|2,k

√T−t_i +C_z⁽²⁾

N−1

X

j=i+1

|∆fj|2,k

√t_j−t_i∆_j+C_z⁽³⁾|∆ξ|_2,k(T−t_i)^θL2 , 0≤k≤i < N,

where the above constants can be written explicitly:

C_y⁽¹⁾:= 1 +Lf₂C_(2.3)⁽¹⁾ (CMBθL 2 ,1+B1

2+^θL₂ ,1

√ T)T^θL2 , C_y⁽²⁾:= 1 +Lf₂C_(2.3)⁽¹⁾ (CM+

√ T)BθL

2 ,1T^θL2 , C_z⁽¹⁾:=CM(1 +Lf₂C_(2.3)⁽¹²⁾ CMBθL

2 ,¹₂T^θL2 ), C_z⁽²⁾:=CM(1 +Lf₂C_(2.3)⁽¹²⁾ (CM+√

T)BθL

2 ,¹₂T^θL2 ), C_z⁽³⁾:=CMLf₂C_(2.3)⁽¹²⁾ B1

2+^θL₂ ,¹₂. Proof. Using (2.5) and (2.6), we obtain

|Θj|2,k≤CM

|∆ξ−Ej∆ξ|2,k

pT−t_j +|∆ξ|2,k+ (CM +

√ T)

N−1

X

l=j+1

|∆fl|2,k∆l

√tl−tj

+ (CM+

√ T)

N−1

X

l=j+1

L_f2|Θl|2,k∆_l (T−tl)^(1−θL)/2√

tl−tj

, j ≥k. (2.7)

Upper bound for (2.7). We apply Lemmas 2.2 and 2.3 under the setting u_j = |Θ_j|_2,k, w_j = C_M^|∆ξ−√^Ej∆ξ|2,k

T−tj

+|∆ξ|2,k+ (C_M +√

T)PN−1 l=j+1

|∆f√_l|_2,k∆_l t_l−tj

, α= 0, β = ^θ₂^L,C_u =L_f2(C_M +√ T). To make results fully explicit, we first need to upper bound quantities of the form (γ >0)

I_j+1^(γ) :=

N−1

X

l=j+1

wl∆l

(T−tl)^12−θL2 (tl−tj)^1−γ .

Using that|∆ξ−El∆ξ|2,k is non-increasing inl and Lemma 2.1, we obtain I_j+1^(γ) =

N−1

X

l=j+1

C_M^{|∆ξ−√El∆ξ|2,k}

T−tl +|∆ξ|_2,k+ (C_M+√

T)PN−1 r=l+1

|∆f√r|2,k∆r

t_r−tl

(T−tl)^12−θL2 (tl−tj)^1−γ

∆_l

≤CMBθL 2 ,γ

|∆ξ−Ej+1∆ξ|2,k

(T−tj)^{1−θL2 −γ} +B1

2+^θL₂ ,γ

|∆ξ|2,k

(T−tj)^{12−θL2 −γ}