On the Monte Carlo simulation of BSDE’s: an
improvement on the Malliavin weights
D. Crisan
∗, K. Manolarakis
†, N. Touzi
‡Abstract
Monte Carlo simulation of backward stochastic differential equations offers a probabilistic numerical method for the approximation of solutions of semilinear PDEs. We propose a generic framework for the analysis of such numerical approx-imations. The general results are used to re-visit the convergence of the algorithm suggested by Bouchard and Touzi [6]. By dropping the higher order terms in the Skorohod integrals resulting from the Malliavin integration by parts in [6], we intro-duce a variant of the latter algorithm which allows for a significant reduction of the numerical complexity. We prove the convergence of this improved Malliavin-based algorithm, and we derive a bound on the induced error. In particular, we show that the price to pay for our simplification is to use a more accurate localizing function. Key words: BSDEs, weak approximations, Monte Carlo methods, Malliavin calculus
Mathematics subject classification:65C30; 65C05; 60H07; 62G08
∗Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK. †Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK.Research partially supported by EPSRC and the Greek state scholarship’s foundation
‡Centre de Math´ematiques Appliques Ecole Polytechnique UMR CNRS 7641 91128 Palaiseau Cedex FRANCE
1
Introduction
Let (Ω, F , P ) be a probability space on which we have defined the decoupled forward-backward stochastic differential equation (BSDE henceforth)
dXt = b(Xt)dt + σ(Xt)dWt
−dYt = f (t, Xt, Yt, Zt)dt − ZtdWt
(1.1)
with the initial condition X0 = x for the forward component and the final condition
Y1 = Φ(X1) for the backward component. In (1.1), W = {Wt, t ≥ 0} is a d-dimensional
Brownian motion. The coefficients of the system (1.1) are given by the function b : Rd→ Rd, σ : Rd → Rd×d for the forward component and f : [0, 1] × Rd× R × Rd → R for the backward part. Also Φ : Rd → R is a real valued function. A solution to the forward
backward system (1.1) is the triplet Θt = (Xt, Yt, Zt). The approximation of the forward
part of the solution X is well understood (see, for example [14] and the references therein). The focus of this paper is on obtaining approximations for the process Y . Discrete-time schemes based on the approximation of the Brownian motion have been analysed by Chevance [7], Coquet, Macquevicius and M´emin [11], Briand, Delyon and M´emin [17], Antonelli and Kohatsu-Higa [1], Ma, Protter, San Martin and Torres [13], Bally and Pag`es [2].
In contrast with the above literature, we concenrate on approximations of solutions of backward stochastic differential equations based on the Monte Carlo simulation of the underlying Brownian motion, thus continuing the studies of Bouchard and Touzi [6], Zhang [18] and Gobet, Lemor and Warin [9]. See also Bouchard and Elie [5] for the case of jump-diffusions.
As a first contribution, we propose a generic framework for the analysis of such ap-proximations. By isolating the one-step approximation operator of the algorithm, we provide a transparent set of sufficient conditions in order to study the rate of convergence of the approximation. The main tool is an expansion of the error as a Trotter product. This generic framework applies to various numerical methods. For instance, the above
methodology is applied in [8] in order to prove convergence of a probabilistic numerical algorithm where the regressions are estimated by using the cubature method of Lyons and Victoir [15] to approximate the distribution of (X, W ).
This methodology is applied to the Malliavin intergration-by-parts based algorithm suggested in [6] whose main idea is to exploit the representation of the regression function r(x) := E[Y |X = x] = E[Y ε{x}(X)]/E[ε{x}(X)], where ε{x} denotes the Dirac mass at x,
as r(x) = E h Y 1Rd +(X − x)S h(φ(X − x))i E h 1Rd +(X − x)S h(φ(X − x))i
for smooth random variables X and Y valued respectively in Rdand Rk, for some integers
d, k ≥ 1. Here Sh is an iterated Skorohod integral, and φ is a smooth function with
φ(0) = 1, called localizing function.
The Malliavin integration-by-parts algorithm of [6] turns out to have high numerical complexity mainly due to the two following facts:
• the Heaviside function 1Rd
+ obtained by integrating up the Dirac measure; this
numerical difficulty can be dealt with by appealing to advanced sorting algorithms from the operations research literature, see e.g. [3].
• the weights obtained from the Malliavin integration-by-parts consist of iterated Sko-rohod integrals with exploding number of terms when converted into Itˆo stochastic integrals (which is necessary for the simulation step).
The second main contribution of this paper is to suggest an improvement of the Malliavin integration-by-parts based algorithm by reducing the numerical complexity of the above weights. To do this, we first re-visit completely the convergence results of [6]. In particular our methodology allows to clarify some points in the proofs of [6]. We next suggest to drop higher order terms of the weights by a judicious expansion. This leads to a reduction of the number of terms at least from 4d to 2d, where d is the dimension of the forward
state variable. Hence the number of elementary calculations is at least cut by 2d, for
instance with d = 5 we reduce the effort by more than 96% !
The price to pay for such a reduction in the numerical complexity is a change of scale in the localizing function φh(x) := φ (xh−α) of [4] and [6], where h is the time step. While h = 1/2 in [6] is the optimal scale, we find in the present setting that α must be chosen strictly larger than 1/2.
The paper is organised as follows. After collecting some preliminaries in Section 2, we report our one-step operator approach for the discretisation of BSDEs in Section 3. We provide a review of the Monte Carlo method of [6] in Section 4. Section 5 introduces our main modification by truncating the weights, and provides the corresponding asymptotic results for the error analysis.
2
Preliminaries
We start first by giving some notation and the assumptions that we will be using through-out the paper. We are working on a complete probability space (Ω, F , P, {Ft)}) and on
finite time interval [0, 1] where a d dimensional Brownian motion is defined Wt, t ∈ [0, 1].
The filtration of the space is the augmented Brownian filtration. We denote by Mn the
space of n × n matrices equipped with the norm |A| := Pn
i,j=1|aij|2
1/2
. We denote by A∗ the transpose of an element of Mn. The Euclidean spaces are equipped with the usual
Euclidean norm. The inner product of two elements in Rd is denoted by x · y.
We will be using the following assumptions on the coefficients of system (1.1)
(H1a) The functions b : Rd→ Rd, σ : Rd → Md are Lipschitz continuous
|b(x) − b(y)| + |σ(x) − σ(y)| ≤ K|x − y| and σ is uniformly positive definite.
(H1b) The functions b, σ, σ−1 ∈ C∞ b .
(H2) The driver of the BSDE f : [0, 1] × Rd× R × Rd → R is Lipschitz in the spatial
variables
|f (t, x, y, z) − f (t, x0, y0, z0)| ≤ K(|x − x0| + |y − y0| + |z − z0|)
The terminal condition is a Lipschitz function of the state process ξ = Φ(X1) with
the same Lipschitz constant as above.
A solution to the system (1.1) is an adapted triplet of processes (X, Y, Z) taking values in Rd, R, Rd, respectively, such that
E sup 0≤t≤1 |Yt|2+ Z 1 0 |Zs|2ds < ∞ (2.1)
Following for example, theorem 2.1 of [10], such a solution exists when conditions (H1a) and (H2) are satisfied. In what follows we will be primarily working with these two hy-pothesis. The enhanced condition (H1b) will be required to show additional smoothness, in the Malliavin sense, for the Euler scheme associated with the forward component of (3.2).
3
Discretizing the BSDE
Consider a partition of the time interval π = {0 = t0 < t1 < . . . < tn = 1} along which,
we aim to descretize the system (1.1). Denote by πi = ti − ti−1 and |π| := max1≤i≤nπi
the partition mesh. For the discretization of the Brownian motion we write ∆Wi+1 =
Wti+1 − Wti and ∆W
j
i+1 for its j-th entry, where j = 1, 2, . . . , d. The forward component
of system (1.1) is approximated by an Euler scheme. In particular if
Xtπ(s, x) := x + b(x)(t − s) + σ(x)(Wt− Ws)
then the Euler scheme corresponding to the partition π is defined as
The following estimate is standard (see theorem 10.2.2 [14]) max 0≤i≤n−1E sup ti≤t≤ti+1 |Xt− Xtπ| 2 ≤ C(1 + |x|2)|π| (3.1) For the discretization of the backward components of the solution we define recursively on π the random variables Ytπi, Ztπi, i = 0, . . . , n, as follows:
Ytπ n = Φ(X π tn), Z π tn = 0 Ztπ i−1 = π −1 i Ei−1Ytπi∆Wti (3.2) Ytπi−1 = Ei−1Ytπi + πif (ti−1, X
π ti−1, Y π ti−1, Z π ti−1)
Above Ei stands for conditional expectation with respect to the σ –field Ftπi, the natural
filtration of the Euler scheme, Ftπi = σ
Xtπj : j ≤ i
. When the terminal condition ξ is not path dependent, the random variables Ytπi, Ztπi are (deterministic) functions of the underlying Xπ
ti and the above conditional expectations are actually regressions
Ei−1Ytπi∆Wti = E h Ytπi∆Wti|X π ti−1 i , Ei−1Ytπi = E h Ytπi|Xtπi−1i
The proof of the following theorem may be found in [6] or in a slight different formulation in [18] for the path dependent case.
Theorem 3.1. Associated with the backward scheme (3.2) we define the step processes Yπ
t =
Pn−1
i=0 Ytπi 1[ti,ti+1)(t)+Y
π tn1{tn}(t) and respectively Z π t = Pn−1 i=0 Ztπi1[ti,ti+1)(t)+Z π tn1{tn}(t) . Then E sup 0≤t≤T |Yt− Ytπ| 2+ Z 1 0 |Zt− Ztπ| 2dt ≤ C|π| (3.3) for a constant C independent of the partition.
The above scheme is not directly implementable. Special care needs to be taken for the computation of the involved conditional expectations . However it does highlight the important fact that, to obtain a fully implementable scheme we need to approximate the regression functions as well as the underlying process. In this spirit we start by defining the following operators
Definition 3.2. Let C1
b = {g| g : Rd → R, Lipschitz}. Then define the non linear operators
Ri : Cb1 → Cb1 i = n, . . . , 0 as follows Rng(x) = g(x) Rig(x) = E h g(Xtπi+1(ti, x)) i + πi+1f ti, x, Rig(x), π−1i+1E h g(Xtπi+1(ti, x))∆Wi+1 i (3.4)
Remark 3.3. In this paper we will work with forward backward systems where the termi-nal condition is a Lipschitz function of X1. When an operator from the above family is
applied to a function which is C1
b and the driver f of the BSDE is one time continuously
differentiable then, the function Rig is also differentiable with bounded first order
deriva-tives. This is just an easy consequence of the chain rule and the implicit function theorem (see [18] for details). In fact, there exists a uniform bound on the first derivative of the iteration of the family {Ri}i
max
0≤i≤n−1k∇Ri. . . Rn−1Φ(x)k∞ < ∞
Note that the operator Ri is well defined for a larger class of functions g, not necessarily
in C1
b. For instance, Rig could be defined for functions g of polynomial growth.
Using a backward induction argument we have the following representation
Ytπ
i = RiRi+1. . . Rn−1Φ(X
π
ti) (3.5)
for i = 0, 1, . . . , n. The next result uses the notation:
kf kLp(Px ti,ti+1) := E[|f (Xtπj)| p|Xπ ti = x] 1/p
Lemma 3.4. For every i = 0, . . . , n−1 the operator Ri enjoys the following Lipschitz-type
property : |Rig1− Rig2| (x) ≤ 1 + Cπi+1 1 − Kπi+1 kg1− g2kLp(Px ti,ti+1) (3.6)
for some p > 1, where K is the Lipschitz constant of f and C some generic constant which depends on d, p, and K.
Proof. Let us fix a value for i. Starting from the definition of Ri we have that Rig1(x) − Rig2(x) =E(g1− g2)(Xti+1(ti, x)) + πi+1 f ti, x, Rig1(x), 1 πi+1E g1(Xti+1(ti, x))∆Wi+1 −f ti, x, Rig2(x), 1 πi+1E g2(Xti+1(ti, x))∆Wi+1
By the mean value theorem and the K -Lipschitz property of f , we can find deterministic functions ν(x) : Rd→ R, ζ(x) : Rd → Rd , which are bounded by K, such that
fti, x, Rig1(x), 1 πi+1E g1(Xti+1(ti, x))∆Wi+1 − f ti, x, Rig2(x), 1 πi+1E g2(Xti+1(ti, x))∆Wi+1 = πi+1ν(x) (Rig1(x) − Rig2(x)) + ζ(x) · E(g1− g2)(Xti+1(ti, x))∆Wi+1
and we have that
(1 − πi+1ν(x))(Rig1(x) − Rig2(x)) = (3.7)
E(g1− g2)(Xti+1(ti, x)) + ζ(x) · E g1(Xti+1(ti, x)) − g2(Xti+1(ti, x)) ∆Wi+1
From (3.7) we deduce that
(1 − πi+1K)|(Rig1(x) − Rig2(x))| ≤ E(g1− g2)(Xti+1(ti, x)) (1 + ζ(x) · ∆Wi+1) ≤ kg1− g2kLp(Pti,ti+1x )E [|1 + ζ(x) · ∆Wi+1|q] 1 q ≤ kg1− g2kLp(Pti,ti+1x )E h (1 + ζ(x) · ∆Wi+1)2k i2k1
bounded by the Lipschitz constant of f, K, we have that E(1 + ζ(x) · ∆Wi+1)2k = 2k X j=0 2k j E(ζ(x) · ∆Wi+1)j = 2k X j=0 2k j E(ζ(x) · ∆Wi+1)2j ≤ 1 + Cπi+1
which completes the proof of the lemma.
We need to clarify how the above result iterates on the family {Ri}i, a point which
will be used in the sequel. We apply Lemma 3.4 to |RkRk+1g1− RkRk+1g2| (x) and to
|Rk+1h(x) − Rk+1g(x)| to get |RkRk+1h(x) − RkRk+1(x)| ≤ 1 + Cπk+1 1 − Kπk+1 kRk+1h − Rk+1gkLp(Px tk,tk+1) and |Rk+1h(x) − Rk+1g(x)| ≤ 1 + Cπk+2 1 − Kπk+2 kh − gkLp(Px tk+1,tk+2). Then |RkRk+1h(x) − RkRk+1g(x)| ≤ 1 + Cπk+1 1 − Kπk+1 1 + Cπk+2 1 − Kπk+2 kh − gk Lp(Ptk,tk+2x )
by the semi group property. In general it holds that
|R0. . . Rig(x) − R0. . . Rih(x)| ≤ i+1 Y j=1 1 + Cπj 1 − Kπj kg − hk Lp(Pt0,ti+1x ) for all i = 0, . . . , n − 1.
As mentioned above, the conditional expectations involved in (3.2) are not computable in most cases. Hence, the next step would be to substitute the expectation operator E[·] in (3.4), with a simulation-based approximating operator ˆE[·] that is explicitly computable.
The simulation could rely on a Monte-Carlo method as is the case in [6] and [12], or on an evaluation on a tree as in [8]. Given, such an operator ˆE[·] we define for i = n, . . . , 0
ˆ Rng(x) = g(x) ˆ Rig(x) =ˆE h g(Xtπ i+1(ti, x)) i (3.8) + πi+1f ti, x, ˆRig(x), 1 πi+1 ˆ E h g(Xtπi+1(ti, x))∆Wi+1 i
Given the above family of empirical operators, we define as approximating value of Yti
ˆ
Ytπi = ˆRiRˆi+1. . . ˆRn−1Φ(Xtπi)
We are now able to express the error between the Backward Euler scheme Yπ and the
empirical value ˆYπ, as an error between the operators E[·], ˆE[·]. Let Ri,n−1g := Ri. . . Rn−1g Rˆi,n−1g := ˆRi. . . Rn−1g
Then the global error at each step i = 0, . . . , n − 1 is given by
Ytπi − ˆYtπi = Ri,n−1Φ(x) − ˆRi,n−1Φ(x). (3.9)
We also set by convention ∆Wi+10 := 1, and we introduce the error
V[g; l](x) := E −Eˆh
g(Xtπi+1(ti, x))∆Wi+1l
i
and denote the local p- errors at time ti as
Eip[g; l] := V[g; l](X π ti) p (3.10) where i = 0, 1 . . . , n − 1, l = 0, 1, . . . , d and p > 1.
Theorem 3.5. Assume that the partition is equidistant (ti = 1/n) and that (H1a) and
(H2) are satisfied. Then, for any p > 1 and i = 0, . . . , n − 1, the following holds
|Ytπi − ˆYtπi| ≤ C
|π| 0≤l≤dmax 0≤i≤n−1
Eiph ˆRi, nΦ; l
i
Proof. In this proof C will be a constant which may vary from line to line. It is clearly sufficient to prove the result for i = 0.
We start from the definition of the regression and the empirical regression operators. For any function g, such that the corresponding expectation is well defined, we have that
Rig(x) − ˆRig(x) = (E − ˆE)[g(Xtπi+1(ti, x))] + πi+1 f ti, x, Rig(x), 1 πi+1E h g(Xtπi+1(ti, x))∆Wi+1 i − f ti, x, Rig(x), 1 πi+1 ˆ E h g(Xtπi+1(ti, x))∆Wi+1 i
Using the Lipschitz property of f we may argue as in the proof of Lemma 3.4 to deduce that |Rig(x) − ˆRig(x)| ≤ K 1 − Kπi+1 n E −Eˆ [g(Xtπ i+1(ti, x))] + E −Eˆ [g(Xtπ i+1(ti, x))∆Wi+1] o (3.11)
We now go back to (3.9) and develop the error as a Trotter product, that is we add and subtract n terms where every term is a combination of n − i empirical backward projections and i perfect ones R0, iRˆi+1,n−1Φ(Xtπ0).
|Yπ t0 − ˆY π t0| = |R0,nΦ(X π t0) − ˆR0,nΦ(X π t0)| ≤ n−1 X i=0 |R0, i−1Rˆi, n−1Φ(Xtπ0) − R0, i ˆ Ri+1,nΦ(Xtπ0)| (3.12)
We may apply the Remark following lemma 3.6, to every term in the sum to deduce that
R0,iRˆi+1,n−1Φ(x) − R0,i−1Rˆi,n−1Φ(x)
≤ i Y j=1 1 + Cπj+1 1 − Kπj+1 (Ri− ˆRi) ˆRi+1,nΦ(X π ti(0, x)) p
Using (3.11) we conclude that Ri− ˆRi) ˆRi+1. . . ˆRn−1Φ(Xtπ i(0, x) p (3.13) ≤ K(d + 1) 1 − Kπi+1 max 0≤l≤dE p i h ˆRi,nΦ; l i
Plugging these estimates in (3.12) we have that |Ytπ0 − ˆYtπ0| ≤ K(d + 1) 1 − Kπi+1 n−1 X i=0 i Y j=1 1 + Cπj+1 1 − Kπj+1 max 0≤l≤dE p i h ˆRi,nΦ; l i ≤ K(d + 1) 1 − Kπi+1 max 0≤l≤d 0≤i≤n−1 Eiph ˆRi,nΦ; l iXn−1 i=0 i Y j=1 1 + Cπj+1 1 − Kπj+1
and the proof completes by noting thatPn−1
i=0 Qi j=1 1+Cπ j+1 1−Kπj+1 ≤ eC+K |π| .
4
The Malliavin calculus approach
We will now depart from the abstract level of the previous section to analyze the method presented in [6]. Throughout this section the stronger condition (H1b) is assumed to hold. Also, the partition is assumed to be equidistant so that πi = |π|, ∀i. The choice for the
simulation-based approximation ˆE[·] of E[·] is based on a representation of the conditional expectation obtained by means of the (Malliavin) integration by parts formula. The exact formula of this representation involves some weights, that are in fact iterated Skorohod integrals. So, we first introduce the related notation.
We denote by Jk the subset of Nk with elements I = (i1, . . . , ik) that satisfy 1 ≤ i1 <
. . . < ik ≤ d. Trivially, we set J0 = ∅. Given two elements I ∈ Jk, J ∈ Jq we define their
concatenation I ∗ J := (r1, . . . , rl) with k ∧ q ≤ l ≤ d ∧ (k + q), ri ∈ I ∪ J for every i ≤ l,
and 1 ≤ r1 < . . . < rp ≤ d. Finally, for any J ∈ Jk, k = 0, 1 . . . , d we write Jc for its
complementary set, that is , the unique set such that J ∗ Jc = {1, 2, . . . , d} .
Given a matrix valued process h with columns denoted by hi and a random variable F we denote by
Sih[F ] := Z ∞
0
F (hit)∗dWt
and for a multi index I = (i1, . . . , ik)
where the integrals are understood in the Skorohod sense. We extend the definition to I = ∅ by setting Sh
∅[F ] = F. Let φ be a bounded and continuous real valued function with
φ(0) = 1. We will say that φ is a smooth localizing function if
∂Iφ(x) ∈ Cb0(Rd), for any k = 0, 1, . . . , d, I ∈ Jk
trivially we set ∂∅φ = φ. We denote the collection of all smooth localizing functions by L.
For integrability reasons, we will follow [6] by requiring the localizing function to satisfy the condition: X I∈∪dk=1Jk Z Rd |x|m∂ Iφ(x)2dx < ∞. (4.1)
By L0, we denote the collection of all localizing functions satisfying (4.1). For i =
0, 1, . . . , n, we consider the matrix valued process
hi(t) = 1 |π|σ −1 (Xtπi−1) 1[ti−1,ti)(t)− 1 |π|σ −1 (Xtπi) Id+|π|∇b(Xtπi)+ d X j=1 ∇σj(Xtπi)∆Wi+1j 1[ti,ti+1)(t)
Under (H1b), the Euler scheme is infinitely many times differentiable in the Malliavin sense, and the derivative is given by
DtXtπ1 = σ(X π t0)1t≤t1, DtXtπi = DtX π ti−1+ |π|∇b(X π ti−1)DtX π ti−1 + d X j=1 ∇σj(Xπ ti−1)DtX π ti−1∆W j i + σ(X π ti−1)1(ti−1,ti](t)
Observe that the process hi(t) satisfies the identities
Z 1 0 DtXtπihi(t)dt = Id, Z 1 0 DtXtπi+1hi(t)dt = 0 (4.2)
Also, for all I ∈ ∪dk=0Jk, l = 0, . . . , d, and φ ∈ L0, the iterated integrals
Shi I ∆W l i+1φ(X π ti− x) = X J ⊆I (−1)|J |∂Jφ(Xtπi− x)S hi Jc[a(∆Wi+1)] (4.3)
is well defined, and belong to D1,2, the space of one time Malliavin differentiable random
variables. The latter equality follows from the first property of (4.2). The following representation is reported from [4] and [6].
Theorem 4.1. Let ρ is a real valued function, 0 ≤ l ≤ d, and assume that the random variable F = ρ(Xπ
ti+1)∆W
l
i+1∈ L2. Then for all localizing functions φ1, φF ∈ L0:
EF |Xtπi = x = EQF[hi, φF](x) E [Q1[hi, φ1](x)] , (4.4) where QF[hi, φ](x) := Hx(Xtπi)ρ(X π ti+1)S hi (1,...,d)[∆W l i+1φR(Xtπi − x)],
and Hx(y) :=Qdi=11xi≤yi is the Heaviside function in d-dimensions.
Observe that, qi1(x) = E [Q1[hi, φ1](x)] is the density of Xtπi. The above representation
paves the way for a Monte Carlo approach to the problem of conditioning. In particular it suggests to substitute the regression function EF |Xπ
ti = x with a ratio of empirical
means. However we should note that in the simulation, there could be integrability issues. If we consider N independent random variables {Bk(x)}Nk=1 with E[Bk(x)] =
qi(x), ∀k then the estimator of the denominator ˆqi1(x) = N−1
PN
k=1Bk(x) has a gaussian
asymptotic distribution. For this reason, in [6]( Lemma 3.3) the authors take advantage of preliminary bounds available for Yπ
ti, Ei−1[Y
π ti∆W
l
i+1]. In particular, there exists functions
Ψi(x), Ψi(x), ζi(x), ζi(x) of polynomial growth, such that
Ψi(Xtπi) ≤ Y π ti ≤ Ψi(X π ti) (4.5) ζ i−1(X π ti−1) ≤ Ei−1[Y π ti∆W l i] ≤ ζi−1(X π ti−1)
for i = 0, . . . , n and l = 0, . . . , d. With the above bounds at hand let us now build the backward algorithm. Let us denote by (Ω0, F0, P0) the original probability space. Consider nN copies of the Euler scheme approximation of the forward diffusion, that is, will be making use of N copies at every point tkon the grid. We define n probability spaces,
{(Ωi, Fi, Pi)}n
i=1, where in each of these spaces N copies live, i.e.
n Xπjo
j∈Ni
{N i+1, . . . , N (i+1)}, live on the space Ωi. Further on, we consider the product probability space ¯ Ω, ¯F , ¯P = ⊗n i=0Ω i, ⊗n i=0F i, ⊗n i=0P i (4.6)
and we lift all processes in this space. Integration with respect to the measure ¯P is denoted by ¯E, while integration with marginal measures, Pi, Pi× Pk, is denoted by Ei, Ei,k etc. The operator ˆE suggested from Theorem 4.4 is defined as follows for i = n − 1, . . . , 1:
ˆ Ei h g(Xtπ i+1(ti, x))∆W l i+1 i = Tiζh ˜Ei h g(Xtπ i+1(ti, x))∆W l i+1 ii (4.7) with Tiζ(x) = ζi(Xtπi) ∧ x ∨ ζ i(X π ti) for any x ∈ R d and ˜ Ei h g(Xtπ i+1(ti, x))∆W l i+1 i = ˆ QF[h i, φF](x) ˆ Q1[h i, φ1](x) ,
for possibility different localizing functions φ1 and φF ∈ L0, and
ˆ QFi [hi, φF](x) = 1 N X j∈Ni+1 Hx(Xπ l ti )g(X πl ti+1)S hl i h φR Xtπil− x∆Wi+1l,k i ˆ Q1i[hi, φ1](x) = 1 N X l∈Ni+1 Hx(Xπ l ti )S hl i h φ1(Xπ l ti − x) i
Here, ∆Wi+1l,0 = 1 and for 1 ≤ k ≤ d, ∆Wi+1l,k is the k-th component of the l-th copy in the Ni+1 bundle of copies of the increment ∆Wi+1. Moreover, hli stands for the evaluation of
the function hi using the l-th copy of the Euler scheme. At time t0, ˜E is just the empirical
mean ˜ E0g(Xt1(0, x)∆W l 1 = 1 N X j∈N1 g(Xtπ1l)a(∆W1j,l)
1. In the current set up (3.8) becomes ˜ Rng(x) = g(x) ˜ Rig(x) = ˆE h g(Xtπ i+1(ti, x)) i + |π|f ti, x, ˜Rig(x), |π|−1Eˆ h g(Xtπi+1(ti, x))∆Wi+1 i (4.8) ˆ Rig(x) = TiΨ( ˜Rig(x)) i = 0, 1, . . . , n where TΨ i (F ) = Ψi∨ F ∧ Ψi.
Remark 4.2. The conclusions of theorem 4.1 remain valid when applied to a random function %. In particular, if % := %(x, ξ) where ξ is independent of Fπ the representation
(4.4) is true, only now both sides involve random functions instead of deterministic ones. A careful inspection of the algorithm shows that (4.4) needs to be applied to such functions.
The key point is Theorem 5.1 in [6]. This gives the description for the local error.
Theorem 4.3. For 0 ≤ i ≤ n − 1 and 0 ≤ l ≤ d, consider the random variable F = %(Xtπi+1, ω)∆Wi+1l , where %(·, ω) is a random function with ω ∈ Ωi+2× . . . Ωn.
For φF ∈ L0, let qiF(x) := E0QF[hi, φF](x) be defined as in Theorem 4.1 so that ri(x) :=
qF
i (x)/qi1(x) = Ei+1,...,n[F |Xtπi = x], and consider the regression estimator
ˆ ri(x) := Tiζ(˜ri(x)), with ˜ri(x) := ˆ QF i [hi, φ](x) ˆ Q1 i[hi, ψ](x) .
Then, for any p > 1, we have the error estimate:
kri(Xtπi) − ˆri(X π ti)kp ≤ 1 N1/2p(2Γ(F )) 1/p (4.9) where Γ(F ) := Z Rd γ(x)kQF[hi, φ](x)kL2(¯ P)+ (|r(x)| + γ(x))kQ 1 [hi, ψ](x)kL2(¯ P) dx and γ(x) := |Ψi(x) − Ψi(x)| ∨ |ζi(x) − ζi(x)|.
For the sake of completeness, we include the proof of Theorem 4.3 in the Appendix. In particular, the factorisation (4.6) of the probability space Ω, ¯¯ F , ¯P clarifies some points in the proof of [6].
The above result provides us with a control on the error of the simulation of the conditional expectation. The error is controlled by means of the paremeter N , which is the number of simulated paths used at time ti for the Monte Carlo method. However,
there is a further negative influence of the partition mesh |π| hidden in the functional Γ. In particular , the second norm of the involved random variables explodes when the partition mesh shrinks. The exact rate of explosion is given in the following lemma, reported from [6].
Lemma 4.4. Let µ : Rd→ R be a function of polynomial growth sup
x∈Rd
|µ(x)|
1 + |x|m < ∞ for some m ≥ 1
For φ ∈ L0, set φπ(x) := φ(|π|−1/2x) and let F be defined as in Theorem 4.3. Assume
further that F ∈ L2+ for some > 0. Then
max 1≤i≤n Z Rd µ(x)kQF[hi, φπ](x)kL2(¯P)dx ≤ C |π|d/4.
We are now able to conclude the estimate for the global error of the algorithm.
Theorem 4.5. Assume that (H1b) and (H2) hold true. Then for the Malliavin calculus algorithm the following error estimate holds
|Yπ t0 − ˆY π t0| ≤ C |π|1+d/4pN1/2p
Proof. An easy induction argument shows that V := ˆRi,n−1g(x) = %(x, ω) for ω ∈ Ωi+2×
4.3 and this result gives us for every 0 ≤ l ≤ d: kE −Eˆh%(Xti+1(ti, x), ω)∆W l i+1 i x=Xπ ti kp ≤ 1 N1/2p Z Rd γ(x) kQV[h i, φπ](x)kL2(¯ P) + (|r(x)| + γ(x))kQ1[hi, ψπ](x)kL2(¯ P) dx 1/p
The above combined with theorem 3.5 gives
|Yπ t0 − ˆY π t0| ≤ C |π| 1 N1/4 Z Rd γ(x)hkQV[h i, φπ](x)kL2(¯P) + (|r(x)| + γ(x))kQ1[hi, ψπ](x)kL2(¯ P) i dx 1/p
The functions γ(x), r(x) are of linear growth (see [6]) so lemma 4.4 finishes the proof.
5
An improvement on the Malliavin weights
The Malliavin calculus algorithm provides us with an efficient method for the numerical solution of a BSDE. However one serious drawback of this method is that it can become quite heavy from the implementation viewpoint when the dimension is high. Our aim in this section is to present a variation of the algorithm presented in [6] that reduces the computational effort. We will show that one can consider the function
¯ hi(t) = 1 |π| σ−1(Xtπi−1)1{[ti−1,ti)}(t) − σ −1 (Xtπi)1{[ti,ti+1)}(t) (5.1)
in place of hi and in effect S ¯
hi in place of Shi, and form a backward induction scheme
based on these new weights. Let us try to appreciate the gain from this truncation. To simplify things, in the definition of h, we treat 1
|π|σ −1(Xπ ti) Pd j=1∇σ j(Xπ ti)∆W j i+1 as
being one term (when in fact there are d). Under this assumption, we may deduce that the iterated Skorohod integral Shi[φ(Xπ
ti − x)∆W
l
i+1] constitutes of 4d terms. The same
algorithm, we have to put in only (1/2)d of the effort required for the original algorithm,
when computing the weights. For example, in five dimensions, we make approximately 4% of the original effort. To add to this, observe that the above assumption is quite moderate. The following should also be considered:
• By excluding the last part of h from our computations we are able to avoid d + 1 terms for every integral instead of two.
• Avoiding the last part of h, we avoid the computation of the derivatives of the matrix valued function σ(·).
• Lastly, the last part of h is also the anticipating part of it. Hence, it is the one contributing most to the Malliavin correction terms in the Skorohod integrals, which are now excluded from the computations.
The price that has to be paid for the reduction in the computational effort, is that more copies of the Euler scheme need to be used, to obtain error estimates equivalent to the previous section. This will be made precise in Theorem 5.6.
Remark 5.1. The truncation of the h process does not affect the denominators in the representation of Theorem 4.1. Indeed, a repetitive application of Lemma 3.2.1 of [16] shows that q1(x) = EHx(Xtπi) EiS hi[φ(Xπ ti − x)] = E Hx(Xtπi) Ei[S hi −I[φ(Xtπi− x)] 1 |π|(−σ(X π ti−1)) 1· ∆W i] (5.2) = E h Hx(Xtπi) S ¯ hi 1 Ei[S−Ih [φ(X π ti− x)] i = . . . = EhHx(Xtπi) S ¯ hi[φ(Xπ ti− x)] i
In other words, in the denominator the truncation is exact and no precision is lost. Given this, our effort will focus on the truncation of the numerator.
Observing that R DtXtπi
¯
hi(t)dt = Id, we see that expansion (4.3) holds for ¯hi as well:
S¯hi I [∆W l i+1ϕ(X π ti− x)] = X J ⊆I (−1)|J|∂Jϕ(Xtπi − x)S ¯ hi Jc[∆Wi+1l ] (5.3)
for every 0 ≤ l ≤ d. Hence, in order to analyze the error due to the truncation of hi, we
will concentrate on the difference Shi
Jc[∆Wi+1l ]−S
¯ hi
Jc[∆Wi+1l ] for every J ⊆ I and 0 ≤ l ≤ d.
In the rest of this section, we analyse the error due to the truncation in order to better understand the price to be paid for the above gain in complexity. For this purpose we need to redefine the regression function according to the new “truncated” weights. In the same spirit with Theorem 4.1 we introduce an approximating expectation operator ¯E[·] as follows: For R = %(Xπ
ti+1)∆W
l
i+1, 1 ≤ i ≤ d and 0 ≤ l ≤ d, we write
¯
qR(x) := EQR[¯hi, φπR](x) , ¯q 1
(x) := EQ1[¯hi, φπ1](x) = q 1(x)
where given any φ1, φR ∈ L0, we denote φπ(x) := φ(x/|π|α) and set
¯ E h %(Xtπ i+1(ti, x))∆W l i+1 i = Tζi(¯qR(x)/¯q1(x)), a(x) = x l, l = 1, . . . , d TΨi(¯qR(x)/¯q1(x)), a(x) = 1 (5.4)
Observe that, here we have introduced a normalization in the localizing function, dividing by |π|α. In the previous section we had α = 1/2, whereas here we need to consider α > 1/2. This is necessary for the control of the error and it is exactly what will lead us to worse variance estimates in comparison to the previous section, or in other words, it will lead us to require the generation of more copies of the forward Euler scheme (larger N ) to obtain a rate of convergence of the same order with the previous section.
The next step is, using this new regression function ¯E[·] to redefine the operators Ri
of (3.4) as ˇ ¯ Rig(x) = ¯E h g(Xtπi+1(ti, x)) i + |π|fti, x, ˇR¯ig(x), |π|−1E¯ h g(Xtπi+1(ti, x))∆Wi+1 i ¯ Rig(x) = TΨi( ˇR¯ig(x)) (5.5)
for i = n − 1, . . . , 0 and any function g in a suitable class. The iteration of the above family will give us the values
¯ Ytπ
i := ¯Ri. . . ¯Rn−1Φ(x)
for i = n − 1, . . . , 0 where Φ(·) is the function of the terminal condition of the BSDE. Finally we introduce the family of operators { ˆR¯i}ni=1 that corresponds to the
simulation-based estimation of the family { ¯Ri}ni=1 in a Monte Carlo fashion similar to the previous
section. That is, the definition for the family { ˆR¯i}ni=1is similar to (4.8), with the difference
that we use ¯hi(t) in place of hi(t) for every step i.
To produce meaningful error estimates, we first need to look at the Lp norms of the iterated Skorohod integrals. We first introduce some additional notations. Let
ˆ hi(t) := hi(t) − ¯hi(t) = − 1 |π|σ −1 (Xtπ i) |π|∇b(X π ti) + d X j=1 ∇σj(Xπ ti)∆W j i+1 ! 1[ti,ti+1)(t).
For γ ∈ {±1}|I|, I ∈ ∪dk=1Jk, and i = 1, . . . , d denote
SI,γ[F ] := Si1,γ1 ◦ . . . ◦ Si|I|l,γ|I|[F ],
where
Si,+1[F ] := δ(F ˆhi) and Si,−1[F ] := δ(F ¯hi).
Then, by an easy induction argument, we see that for any Im ⊆ I:
Shi Im[∆W l i+1] − S ¯ hi Im[∆W l i+1] = X γ∈{±1}m\{−1,−1,...,−1} SIm, γ[∆W l i+1]. (5.6)
When these integrals are of length one, the following estimate follows directly from the properties of Skorohod integration:
S¯hi j [∆W l i+1] = ∆W l i+1¯h j i(ti−1) ∆Wi+ ∆Wi+1l ¯h j i(ti)∆Wi+1− Z ti+1 ti Tr[Ds(∆Wi+1l )¯h j i(ti))]ds,
which implies that kS¯hi j [∆W l i+1]kp ≤ C p|π|, kDs1,...,sρS ¯ hi j [∆W l i+1]kp ≤ C |π|. We also note that,
Sˆhi j [∆W l i+1] = ∆W l i+1ˆh j i(ti−1) ∆Wi+ ∆Wi+1l ˆh j i(ti)∆Wi+1− Z ti+1 ti Tr[Ds(∆Wi+1l )ˆh j i(ti))]ds,
which implies that
kSˆhi j kp ≤ C, kDtS ˆ hi j kp ≤ C p|π|, but kDs1,...,sρS ˆ hi j kp ≤ C |π|.
For the general case, we have the following estimate on the explosion rate. The proof is in the spirit of Lemma 6.1 of [6].
Lemma 5.2. For 1 ≤ k, m ≤ d, consider a vector of times τ = (s1, s2, . . . , sk) ∈
[ti−1, ti+1]k, a multi index Im ∈ Jm. Then for every γ ∈ {±1}m\{−1, −1, . . . , −1}:
kDτSIm, γ[∆W l i+1]kp ≤ C|π|(1−k−m)/2, 0 ≤ l ≤ d. (5.7) Moreover kShi[∆Wl i+1] − S ¯ hi[∆Wl i+1]kp ≤ C |π|(d−1)/2 and, for p > 1, kShi[∆Wl i+1]kp, kS ¯ hi[∆Wl i+1]kp = O(|π|−d/2).
Proof. We only prove (5.7), as the remaining estimates follows the same line of argument as in [6]. For simplicity we write SI, γ[∆Wi+1l ] = SI, γ. The proof is done with induction
Consider Ik = (j1, . . . , jk) and γ ∈ {±1}k and we assume for simplicity that γ1 = −1. Then kDτSIk, γkp = Dτ SIk−1, γ−Sj1, γ1(1) − Z ti+1 ti−1 DsSIk−1, γ−¯h j1(s)ds p ≤ X Λ⊆{t1,...,tl} kDΛSIk−1, γk2pkDΛcδ(¯h j1)k 2p + Z ti+1 ti−1 kDΛ,sSIk−1, γ−k2pkDΛc¯h j1 γ(s)k2pds,
where we have used the elementary rule δ(F ut) = F δ(ut)−
R1
0 DtF utdt and Exercise 1.2.14
pg 34 of [16]. The term involving the Riemann integral is
Z ti+1 ti−1 kDΛ,sSIk−1, γ−k2pkDΛc¯h j1 γ (s)k2pds ≤ 2C|π||π|(1−|Λ|−1−k+1)/2|π|−1 = |π|(1−|Λ|−k)/2.
As for the other terms, if Λ = τ by the induction hypothesis and the remark
kDτSIk−1, γ−k2p kδ(¯h j1)k 2p ≤ C|π|(1−l−k+1)/2|π|−1/2 = C|π|(1−l−k)/2, and if Λ ⊂ τ kDΛSIk−1γ−k2p kDΛcδ(¯h j1)k 2p≤ C|π|(1−|Λ|−k+1)/2|π|−1 = |π|(−|Λ|−k)/2 ≤ |π|(1−l−k)/2
as |Λ| ≤ l − 1. Putting all the estimations together we have the result.
Remark 5.3. As in the previous section, it is an easy induction argument to show that for any i = 0, 1, . . . , n − 1, the iteration of the family { ˆR¯i}ni=1 produces random functions
that may be written as ˆR¯i. . . ˆR¯nΦ(x, χ) with χ a random variable. Observe that, with
the notation of the previous section, χ can be shown to be a functional of the copies of the Euler scheme {Xπl, l = i + 1, . . . , n}.
To obtain a rate of convergence for this variation of the algorithm presented in [6], we need to compare Yπ t0 = R0. . . RnΦ(x) with ˆ ¯ Yπ t0 = ˆ ¯
R0. . . ˆR¯nΦ(x) when X0 = x. The first
step is to look at the local error between Ri, ¯Ri, i = 0, . . . , n − 1, when applied to random
functions of the form described in Remark 5.3.
Theorem 5.4. Given a localizing function φ ∈ L0, let φπ(x) := φ (|π|−αx), and consider
the definition of the regression (4.4) and the truncated regression functions (5.4) with localising function φπ. Then, there exists a constant C independent of the partition such
that:
1. If % is a Lipschitz real valued function , then for any i = 0, 1, . . . , n − 1 and p > 1:
Ri%(Xtπi) − ¯Ri%(X π ti) p ≤ C|π| 1 p(α+ 1 2).
2. Let χ be a random variable independent of Fπ
1 and assume that %(Xtπi+1, χ)∆W
l i+1∈
Lq, q > 1, with % a real valued random function and l = 0, 1, . . . , d. Then Ri%(Xtπi) − ¯Ri%(X π ti) p ≤ C|π| α pk%(Xπ ti+1, χ)k 1/p p .
Proof. Let us first assume that we are working with a Lipschitz deterministic function %. Using the notation and arguments presented in Theorem 3.5, where here we consider
¯
E ≡E, we know that there exists a constant C such thatˆ
k(Ri− ¯Ri)%(Xtπi)kp ≤
C
1 − K|π|0≤l≤dmaxE p i [%; l] .
Hence we may focus on the right hand side and for that, we fix a value for l ∈ {0, . . . , d}. Let
νi(x) := ¯qR(x)/¯q1(x).
Using the truncation, we have that
Eip[%; l]p = E − ¯ E h %(Xtπ i+1(ti, X π ti))∆W l i+1 i p p ≤ E νi(Xtπi) − ri(X π ti) p ∧ γ(Xπ ti) p ≤ E νi(Xtπi) − ri(X π ti) γ(Xtπ i) p−1 ,
and E νi(Xtπi) − ri(X π ti) γ(Xtπ i) p−1 = E qR(Xπ ti) − ¯q R(Xπ ti) q1(Xπ ti) γ(Xtπ i) p−1 = Z Rd EHx(Xtπ i)%(X π ti+1) S hi − S¯hi[∆Wl i+1φ π (Xtπi− x)] γ(x)p−1dx .
Using the representation (4.3) we get
E|νi(Xtπi) − ri(X π ti)| γ(X π ti) p−1 = Z Rd E n Hx(Xtπi)%(X π ti+1) X J ∈S k≤d−1Jk (−1)|J|∂Jφπ(Xtπi − x) Shi Jc − S ¯ hi Jc [∆Wi+1l ] o γ(x) p−1 dx. (5.8)
Arguing as in (5.2), with the help of Lemma 3.2.1 of [16], we have
E h Shi Jc− S ¯ hi Jc [∆Wi+1l ] Fti i = 0 .
So we may condition in (5.8) the integrand with respect to Fti to get
E|νi(Xtπi) − ri(X π ti)| γ(X π ti) p−1 = Z Rd E n Hx(Xtπi) %(Xtπi+1) − %(Xtπi) × X J ∈S k≤d−1Jk (−1)|J|∂Jφπ(Xtπi − x) Shi Jc− S ¯ hi Jc [∆Wi+1l ]o γ(x) p−1dx
Our next step is to perform an ω-by- ω change of variables for the Riemann integral by setting u = (Xπ
ti− x)/|π|
α. Observe that the definition of the localizing function φπ yields
∂Jφπ(x) = |π|−α|J|∂Jφ(|π|xα). Hence we have that
E νi(Xtπi) − ri(X π ti) γ(Xtπ i) ≤ X J ∈∪k≤d−1Jk |π|α(d−|J |) Z Rd+ E n (%(Xtπ i+1)−%(X π ti))(S hi Jc−S ¯ hi Jc)[∆Wi+1l ]γ(Xtπ i−|π| αu)p−1o |∂Jφ(u)|du
We now apply a H¨older inequality and using the assertion of Lemma 5.2, the Lipschitz assumption on % and the obvious estimate kXπ
ti+1 − X π tikk ≤ C|π| 1/2 for any k ≥ 1, we have E νi(Xtπi) − ri(X π ti) γ(Xtπ i) p−1 ≤ C X J ∈S k≤d−1Jk |π|12+(α− 1 2)(d−|J |)k%(Xπ ti+1) − %(X π ti)kp Z Rd+ kγ(Xπ ti− |π| αu)p−1k q2|∂Jφ(u)|du ≤ C X J ∈S k≤d−1Jk |π|1+(α−12)(d−|J |) Z Rd+ kγ(Xtπi− |π|αu)p−1kq2|∂Jφ(u)|du (5.9)
for some q2 > 1 and C, a constant independent of the partition. Now, since the function
γ(·) is of polynomial growth we have, with r an integer, that Z Rd+ Eγ(Xtπi− |π| α u)q2(p−1)1/q2|∂ Jφ(u)|du ≤ C Z Rd+ E h 1 + r X k=0 r k |Xπ ti| k||π|αu|r−kq2(p−1)i1/q2|∂ Jφ(u)|du ,
and since |∂Jφ(u)| integrates against polynomials, by the assumptions on the localizing
function, we have the first result. The second part of the theorem follows from (5.9) without using the last step estimate.
As already mentioned, here we need to consider an α > 1/2 and this will have a negative impact, relative to the results of the previous section, on the integrated variance that controls the error of the simulation-based estimation of the family { ¯Ri}n−1i=0. In
particular using the assumptions and the notation of Lemma 4.4, we may show with identical arguments that
lim sup |π|→0 |π|αd/2 max 1≤i≤n Z Rd µ(x)kQR[¯hi, φπ](x)k2dx < ∞ , (5.10)
where φπ(x) := φ(|π|xα) and µ(·) is a function of polynomial growth.
In our main result we will make use of the following version of Gronwall’s whose proof follows from a straightforward induction. lemma.
Lemma 5.5. Let ai, bi, i = 0, 1, . . . , n be two sequences of positive numbers, such that ai ≤ bi+ n X j=i+1 aj, i = 0, . . . , n − 1,
with > 0. Then we have that
n X i=0 ai ≤ n X i=0 (1 + )ibi+ (1 + )nan.
Theorem 5.6. Let φ, φπ be as in Theorem 5.4 and consider the three families of operators {Ri}n−1i=0, { ¯Ri}i=0n−1, { ˆR¯i}n−1i=0 defined with the function φπ, on the partition π. As before, we
have that Yπ
ti = Ri. . . RnΦ(X
π
ti(0, x)), ∀i = 0, . . . , n − 1 and we set
ˆ ¯ Ytπ i := ˆ ¯ Ri. . . ˆR¯nΦ(Xtπi(0, x)).
Then, for any p > 1 and α ≥ (3p − 1)/2, we have
max 0≤i≤n−1kY π ti − ˆ ¯ Ytπ ikp ≤ C |π|1p( 1 2+α)−1)+ 1 |π|1+αd/2pN1/2p
for a constant C independent of the partition.
Remark 5.7. The result above reports on the rate of convergence for the global error. The restriction on α appears because the use of Lemma 5.5, at the end of our proof calls for an = O(1/n). Observe that with an equidistant partition, say |π| = 1/n, this value of α gives us an error accuracy of order 1/√n, as long as we make use of N = n3p+(3p−1)d2
copies of the Euler scheme at every point of the partition.
Proof of Theorem 5.6: We will estimate |Yπ t0− ˆY¯
π
t0| and remark that the estimate for i > 0
follows identical arguments. For ease of presentation we make the conventions that R−1 =
product expansion for the error: |Yπ t0 − ˆ ¯ Ytπ 0| = |R0. . . Rn−1Φ(x) − ˆ ¯ R0. . . ˆR¯n−1Φ(x)| ≤ n−1 X i=0
|R0,iRˆ¯i+1,n−1Φ(x) − R0,i−1Rˆ¯i,n−1Φ(x)|
≤ n−1 X i=0 (Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p ,
where we have used Lemma 3.4, with p > 1. For every i we clearly have
k(Ri− ˆR¯i) ˆR¯i+1,n−1Φ(Xtπi) p ≤ (Ri− ¯Ri) ˆ ¯ Ri+1,n−1Φ(Xtπi) p+ ( ¯Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p (5.11) To estimate the second term above, we may appeal to Theorem 4.3. The proof of the latter comes through in this case to give us,
( ¯Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p ≤ 2 N1/2 Z Rd γ(x)kQR[¯hi, φ](x)kp+ (|r(x)| + γ(x))kQ1[¯hi, ψ](x)kp dx 1/p .
Plugging the estimate (5.10) in the above we get
( ¯Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p ≤ C |π|αd/2pN1/2p. (5.12)
As for the first term in (5.11) we have
k(Ri− ¯Ri) ˆR¯i+1,n−1Φ(Xtπi) p ≤ k(Ri− ¯Ri)Ri+1,n−1Φ(X π ti)kp + k(Ri− ¯Ri)( ˆR¯i+1,n−1− Ri+1,n−1)Φ(Xtπi)kp.
We know from Remark 3.3 that the functions Ri+1,n−1Φ are Lipschitz uniformly in i.
Hence we apply the first assertion of Theorem 5.4 to the first term above, while the second assertion is applied to the other one, to obtain
(Ri− ¯Ri) ˆ ¯ Ri+1,n−1Φ(Xtπi) p ≤ C |π|α+1/2p + |π| α p ( ˆ ¯
Ri+1,n−1− Ri+1,n−1)Φ(Xtπi+1)
1/p p . (5.13)
We would now like to eliminate the 1/p-root above. To this end, observe that for any a > 0 and p > 1, it holds that a1/p ≤ a + 1. Hence
( ˆ
¯
Ri+1,n−1− Ri+1,n−1)Φ(Xtπi+1)
1/p p ≤ |π| 1 2p− 1 2 ( ˆ ¯
Ri+1,n−1− Ri+1,n−1)Φ(Xtπi+1)
p + |π| 1/2p . (5.14) Putting together (5.12), (5.13), (5.14) we have that
(Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p ≤ C |π|α+1/2p + 1 |π|αd/2pN1/2p + |π|αp+ 1 2p− 1 2 ( ˆ ¯
Ri+1,n−1− Ri+1,n−1)Φ(Xtπi+1)
p ,
which provides, by performing a Trotter product expansion to the last term, (Ri− ˆ ¯ Ri) ˆR¯i+1,n−1Φ(Xtπi) p ≤ C |π|α+1/2p + 1 |π|αd/2pN1/2p + |π|αp+ 1 2p− 1 2 n−1 X j=i+1 ( ˆ ¯ Rj− Rj) ˆR¯j+1,n−1Φ(Xtπj) p ,
and the proof is completed by appealing to Lemma 5.5.
Appendix
Proof of Theorem 4.3. Using the bounds on ri we have
E0,i+1 h (ri(Xtπi)−ˆri(X π ti)) pi ≤ E0,i+1|˜r i(Xtπi) − ri(X π ti)| p∧ γ(Xπ ti) p (.15) = E0,i+1 " εF(Xπ ti) − ri(X π ti)ε 1(Xπ ti) ˆ Q1(Xπ ti) p ∧ γ(Xtπi)p # , where εF(x) := ˆQF(x) − qiF(x), and ε1(x) := ˆQ1(x) − q1i(x).
We will be using later the fact that
Ei+1|εF(x)| ≤ kεF(x)kL2(Pi+1) ≤
1 N1/2V
0
where V0[QF] is the variance of the random variable QF[h
i; φ] with respect to the measure
P0. Notice that the last inequality holds in the marginal measure Pi+1. A similar result is true for ε1.
Next, for x ∈ Rd let us consider the event M(x) = ω : | ˆQ1(x, ω) − qi1(x)| ≤ 1 2q 1 i(x) .
Using this set we may split the expectation above
E0,i+1 h εF(Xπ ti) − ri(X π ti)ε 1(Xπ ti) ˆ Q1(Xπ ti) p ∧ γ(Xtπi)p i ≤ E0,i+1 2 εF(Xtπi) − ri(Xtπi)ε 1(Xπ ti) q1 i(Xtπi) p γ(Xtπi)1M(Xπ ti) + E0,i+1 h γ(Xtπi)p 1Mc(Xπ ti) i ,
where we have used the inequality ap∧ bp ≤ abp−1. For the first term on the right hand
side we compute 2 Z Rd Ei+1 h |εF(x)−r i(x)ε1(x)|γ(x) i dx ≤ 2 Z Rd (kεF(x)kL2(Pi+1)+ |ri(x)|kε1(x)kL2(Pi+1))γ(x)p−1dx = 2 N1/2 Z Rd (VR0(x)1/2+ |ri(x)|V10(x) 1/2)γ(x)p−1dx ≤ 2 N1/2 Z Rd (kQFkL2(P0)+ |ri(x)|kQ1kL2(P0))γ(x)p−1dx.
As for the second term we estimate it by means of the Chebychev inequality E0,i+1 h γ(Xtπ i) p 1 Mc(Xπ ti) i = E0Ei+1[γ(Xtπi) p 1 Mc(Xπ ti)] = E0γ(Xtπi) p Pi+1[Mc(Xtπi)] ≤ E0 γ(Xtπ i) p 1 q1 i(Xtπi) Ei+1[2| ˆQ1(Xtπi) − q 1 i(X π ti)|] = Ei+1 γ(x)p Z Rd 2| ˆQ1(x) − qi1(x)|dx ≤ 1 N1/2 Z Rd γ(x)pVQ1(x)1/2dx.
We now have an estimate of the error with respect to the measure dP0 × dPi+1. The
claimed result follows from an application of Fubini’s theorem.
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