Presentation.pdf

MLMC Combined with Numerical Smoothing:
Applications in Probabilities/Densities Computation
and Option Pricing
Chiheb Ben Hammouda
Christian Bayer Raúl Tempone
Center for Uncertainty
Quantification
Center for Uncertainty
Quantification
Center for Uncertainty Quantification Logo Lock-up
MCM23, mini-symposuim: MLMC for discontinuous functionals
Paris, France, June 28, 2023
0

Outline
1 Framework and Motivation
2 The Numerical Smoothing Idea
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Numerical Experiments and Results
5 Conclusions and Extensions

0

Framework
Goal: Approximate efficiently E[g(X(T))]
Setting:
▸ Given a (smooth) φ ∶ Rd
→ R, the function g ∶ Rd
→ R:
☀ Indicator functions: g(x) = 1(φ(x)≥0) (probabilities, pricing
digital/barrier options, . . . )
☀ Dirac Delta functions: g(x) = δ(φ(x)=0) (densities, . . . )
▸ X: solution process of a d-dimensional system of SDEs,
approximated by X (via a discretization scheme with N time steps),
E.g., stochastic volatility model: E.g., the Heston model
dXt = µXtdt +
√
vtXtdWX
t
dvt = κ(θ − vt)dt + ξ
√
vtdWv
t ,
(WX
t ,Wv
t ): correlated Wiener processes with correlation ρ.
Challenge: High-dimensional, non-smooth integration problem
E[g(X
∆t
(T))] = ∫
Rd×N
G(z)ρd×N (z)dz
(1)
1 ...dz
(1)
N ...dz
(d)
1 ...dz
(d)
N ,
with G(⋅) maps N × d random inputs to g(X
∆t
(T)); and ρd×N (z):
joint density function of z. 1

Motivation
Table 1: Complexity comparison of the different methods for approximating
E[g(X(T))] within a pre-selected error tolerance, TOL. Given the same initial
problem, and using a weak order one scheme, E.g, the Euler-Maruyama scheme.
Method General Complexity Optimal Complexity
MC O (TOL−3
) O (TOL−3
)
MLMC O (TOL−3+β
), 1
2
≤ β ≤ 1
O (TOL−2
)
Quasi-MC (QMC) O (TOL−1− 2
1+2δ ), 0 ≤ δ ≤ 1
2
O (TOL−2
)
Adaptive sparse
grids quad (ASGQ)
O (TOL−1− 2
p ), p > 0
O (TOL−1
)
Sufficient Regularity Conditions for Optimal Complexity:
▸ MLMC (Cliffe et al. 2011; Giles 2015):
g is Lipschtiz ⇒ (sub) canonical complexity: O (TOL−2
) up to log terms.
▸ QMC (Dick, Kuo, and Sloan 2013):
1 g belongs to the d-dimensional weighted Sobolev space of functions with
square-integrable mixed (partial) first derivatives.
2 High anisotropy between the different dimensions.
▸ ASGQ (Chen 2018; Ernst, Sprungk, and Tamellini 2018):
p is related to the order of bounded weighted mixed (partial) derivatives of g
and the anisotropy between the different dimensions.
⇒ ASGQ Complexity: O (TOL−1− 2
p ) (O (TOL−1
) when p ≫ 1).
2

Our Proposed Strategy to
Recover Optimal Complexities
1 For QMC/ASGQ:
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Numerical smoothing with hierarchical adaptive sparse grids and
quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance 23.2 (2023), pp. 209–227.
2 For MLMC (Topic of the Talk)
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Multilevel Monte Carlo with Numerical Smoothing for Robust
and Efficient Computation of Probabilities and Densities”. In:
arXiv preprint arXiv:2003.05708 (2022).
" The numerical smoothing idea in (Bayer, Ben Hammouda, and
Tempone 2023) and (Bayer, Ben Hammouda, and Tempone 2022) is
similar. However, the analysis is different.
" For a survey on the different smoothing/adaptivity techniques for
MLMC: see (Giles 2023).
3

3

Numerical Smoothing Steps
Motivating Example:
E[g(X
∆t
T )] =?
g ∶ Rd
→ R nonsmooth function: (E.g., g(x) = 1(φ(x)≥0))
X
∆t
T (∆t = T
N ) Euler discretization of d-dimensional SDE , E.g.,
dX
(i)
t = ai(Xt)dt + ∑d
j=1 bij(Xt)dW
(j)
t ,
where {W(j)
}d
j=1 are standard Brownian motions.
X
∆t
T = X
∆t
T (∆W
(1)
1 ,...,∆W
(1)
N ,...,∆W
(d)
1 ,...,∆W
(d)
N )
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=∆W
≡ X
∆t
T (Z), Z = (Zi)dN
i=1 ∼ N(0,IdN ).
The discontinuity is in (N ×d)-dimensional space characterised by
φ(X
∆t
T (Z)) = 0.

1 Identify hierarchical representation of integration variables ⇒ locate
the discontinuity in a smaller dimensional space
(a) X
∆t
T (∆W) ≡ X
∆t
T (Z), Z = (Zi)dN
i=1 ∼ N(0,IdN ):
s.t. “Z1 ∶= (Z
(1)
1 ,...,Z
(d)
1 ) (coarse rdvs) substantially contribute even
for ∆t → 0”, through hierarchical path generation (Brownian bridges
/ Haar wavelet construction)
⇒ Discontinuity in d-dimensional space instead of
(N × d)-dimensions.
Haar wavelet construction in one dimension
For i.i.d. standard normal rdvs Z1, Zn,k, n ∈ N0, k = 0,...,2n
− 1, we define the
(truncated) standard Brownian motion
WN
t ∶= Z1Ψ−1(t) +
N
∑
n=0
2n
−1
∑
k=0
Zn,kΨn,k(t).
with Ψ−1(⋅) and Ψn,k(⋅) are the antiderivatives of the Haar basis functions.
" Our approach is different from previous MLMC techniques which
uses conditional expectation at the final step w.r.t ∆W ⇒ smoothing
effect vanishes as ∆t → 0. 5

1 Identify hierarchical representation of integration variables ⇒
locate the discontinuity in a smaller dimensional space
(b) If d > 1, introduce a linear mapping using A: rotation matrix whose
structure depends on the function g.
Y = AZ1.
E.g., for an observable g(x) = 1{(∑d
i=1 cixi(T )−K)≥0}, a suitable A is a
rotation matrix, with the first row leading to Y1 = ∑
d
i=1 Z
(i)
1 up to
rescaling without any constraint for the remaining rows
(Gram-Schmidt procedure).
⇒ Discontinuity in 1-dimensional space instead of d-dimensions.
⇒ y∗
1 (y−1,z
(1)
−1 ,...,z
(d)
−1 ): the exact discontinuity location s.t
φ(X
∆t
T ) = φ(X
∆t
T (y∗
1 ;y−1,z
(1)
−1 ,...,z
(d)
−1 )) = 0. (1)
Notation
▸ x−j: vector of length N − 1 denoting all the variables other than xj
in x ∈ RN
. 6

2
E[g(X(T))] ≈ E[g(X
∆t
(T))]
= ∫
Rd×N
G(z)ρd×N (z)dz
(1)
1 ...dz
(1)
N ...dz
(d)
1 ...dz
(d)
N
= ∫
RdN−1
I(y−1,z
(1)
−1 ,...,z
(d)
−1 )ρd−1(y−1)dy−1ρdN−d(z
(1)
−1 ,...,z
(d)
−1 )dz
(1)
−1 ...dz
(d)
−1
= E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] ≈ E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )], (2)
3
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
= ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
1
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
≈ I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
Mlag
∑
k=0
ηkG(ζk (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 ),
4 Compute the remaining (dN − 1)-integral (expectation) in (2) by MLMC.
Notation
G maps N × d Gaussian random inputs to g(X
∆t
(T));
y∗
1 (y−1,z
(1)
−1 ,...,z
(d)
−1 ): the exact discontinuity location (see (1))
y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding.
MLag: number of Laguerre quadrature points ζk ∈ R, and weights ηk;
ρd×N (z) = 1
(2π)d×N/2 e−1
2
zT z
.
7

Some Remarks
" In (Bayer, Ben Hammouda, and Tempone 2023), we show that
I(⋅) in (2) is C∞
⇒ optimal complexity for ASGQ and QMC.
" Here, for MLMC we need different analysis/arguments to show
that we get the optimal complexity of MLMC (see next slides).
The numerical smoothing can be extended to the case of finitely
many roots.
8

Extending Numerical Smoothing for
Density Estimation
Goal: Approximate the density ρX at u, for a stochastic process X
ρX(u) = E[δ(X − u)], δ is the Dirac delta function.
" Without any smoothing techniques (regularization, KDE,. . . )
MC/MLMC fail due to the infinite variance caused by the Dirac
distribution function, δ(⋅).
Strategy in (Bayer, Ben Hammouda, and Tempone 2022):
Conditioning with respect to Z−1 (randomness related to the
Brownian bridge)
ρX(u) =
1
√
2π
E [exp(−(Y ∗
1 (u))
2
/2)∣
dY ∗
1
dx
(u)∣]
≈
1
√
2π
E
⎡
⎢
⎢
⎢
⎣
exp(−(Y
∗
1(u))
2
/2)
R
R
R
R
R
R
R
R
R
R
R
dY
∗
1
dx
(u)
R
R
R
R
R
R
R
R
R
R
R
⎤
⎥
⎥
⎥
⎦
,
Y ∗
1 (x;Z−1): the exact singularity; Y
∗
1(x;Z−1): the approximated
singularity obtained by solving X
∆t
(T;Y
∗
(x),Z−1) = x.

Why not Kernel Density Estimator (KDE)
in Multiple Dimensions?
Similar to approaches based on MLMC with parametric regularization (Giles,
Nagapetyan, and Ritter 2015) or QMC with KDE techniques (Ben Abdellah et al.
2021).
This class of approaches has a pointwise error that increases exponentially with
respect to the dimension of the state vector X.
For a d-dimensional problem, a KDE with a bandwidth matrix, H = diag(h,...,h)
MSE ≈ c1M−1
h−d
+ c2h4
. (3)
M is the number of samples, and c1 and c2 are constants.
Our approach in high dimension: For u ∈ Rd
ρX(u) = E[δ(X − u)] = E[ρd (Y∗
(u))∣det(J(u))∣]
≈ E[ρd (Y
∗
(u))∣det(J(u))∣], (4)
▸ Y∗
(u;⋅): the exact discontinuity; Y
∗
(u;⋅): the approximated discontinuity.
▸ J is the Jacobian matrix, with Jij =
∂y∗
i
∂uj
; ρd(⋅) is the multivariate Gaussian density.
Exact conditioning with respect to the remaining Brownian bridge noise ⇒ the
smoothing error in our approach is insensitive to the dimension of the problem.
10

10

Multilevel Monte Carlo (MLMC)
(Heinrich 2001; Kebaier 2005; Giles 2008)
Setting
▸ A hierarchy of nested meshes of [0,T] (sequence of finer discretizations).
▸ ∆t` ∶= K−`
∆t0: the time steps size for levels ` ≥ 0; K>1, K ∈ N. (∆t0 > ... > ∆tL)
▸ X` ∶= X
∆t`
: The approximate process generated using a step size of ∆t`.
MLMC idea
E[g(X(T))] ≈ E[g(XL(T))] = E[g(X0(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
+
L
∑
`=1
E[g(X`(T)) − g(X`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(5)
Var[g(X0(T))] ≫ Var[g(X`(T)) − g(X`−1(T))] ↘ as ` ↗
M0 ≫ M` ↘ as ` ↗
MLMC estimator: ̂
QMLMC
∶=
L
∑
`=0
̂
Q`, (sample independently each term of (5) with MC)
̂
Q0 ∶=
1
M0
M0
∑
m0=1
g(X0(T;ωm0 )); ̂
Q` ∶=
1
M`
M`
∑
m`=1
(g(X`(T;ωm`
)) − g(X`−1(T;ωm`
))), 1 ≤ ` ≤ L
Compared to MC: MLMC reduces the variance of the deepest level using samples on
coarser (less expensive) levels.
11

Multilevel Monte Carlo with Numerical Smoothing:
Estimator and Notation
Recall
▸ E[g(X(T))] ≈ E[g(X
∆t
(T))] = E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] ≈ E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
▸
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
= ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
1
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
≈ I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
Mlag
∑
k=0
ηkG(ζk (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 ),
where y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding.
I`:= I`(y`
−1,z
(1),`
−1 ,...,z
(d),`
−1 ): level ` approximation of I in ̂
QMLMC
, computed with
step size ∆t`; MLag,` Laguerre points; TOLNewton,` as the Newton tolerance at level `.
̂
QMLMC
∶=
L
∑
`=L0
̂
Q`,
with
̂
QL0 ∶=
1
ML0
ML0
∑
mL0
=1
IL0,[mL0
]; ̂
Q` ∶=
1
M`
M`
∑
m`=1
(I`,[m`] − I`−1,[m`]), L0 + 1 ≤ ` ≤ L, (6)

MLMC with Numerical Smoothing: Analysis
Let g(x) = 1(φ(x)≥0) or δ (φ(x) = 0)
Theorem 3.1 (Variance Decay (Bayer, Ben Hammouda, and Tempone 2022))
Under some regularity assumptions for the drift and diffusion, using Euler–Maruyama,
V` ∶= Var[I` − I`−1] = O (∆t1
` ), compared with O (∆t
1/2
` ) for MLMC without smoothing.
" General MLMC Complexity: O (TOL
−2−max(0,γ−β
α
)
log (TOL)2×1{β=γ}
),
where α: weak rate; β: variance decay rate; γ : work growth rate.
Corollary 3.2 (Complexity (Bayer, Ben Hammouda, and Tempone 2022))
Under some regularity assumptions for the drift and diffusion, the complexity of MLMC
combined with numerical smoothing is O (TOL−2
) up to log terms, compared with
O (TOL−2.5
) for MLMC without smoothing.
" Milstein scheme: we show that we obtain the canonical complexity (O (TOL−2
)).
Corollary 3.3 (Robustness (Bayer, Ben Hammouda, and Tempone 2022))
Let κ` be the kurtosis of the r.d.v I` − I`−1, then under some regularity assumptions of the
drift & diffusion, we get κ` = O (1) compared to O (∆t
−1/2
` ) for MLMC without smoothing.
" The assumptions in Theorem 3.1 and Corollary 3.2 are sufficient but not necessary.
13

Sketch of the Proof of Theorem 3.1:
Goal and Notations
Goal: We want to show V` ∶= Var[I` − I`−1] ≤ E[(I` − I`−1)
2
] = O (∆t`).
Notations
X`,X`−1: the coupled paths of the approximate process X, simulated with time
step sizes ∆t` and ∆t`−1, respectively.
W` and B`: coupling Wiener and related Brownian bridge processes at levels `
and ` − 1, respectively.
For t ∈ [0,T], e`(t;Y,B`) is defined as
(X` − X`−1)(t) = ∫
t
0
(a(X`(s)) − a(X`−1(s)))ds + ∫
t
0
(b(X`(s)) − b(X`−1(s)))dW`(s)
= ∫
t
0
(a(X`(s)) − a(X`−1(s)))ds + ∫
t
0
(b(X`(s)) − b(X`−1(s)))
Y
√
T
ds
+ ∫
t
0
(b(X`(s)) − b(X`−1(s)))dB`(s)
=∶ e`(t;Y,B`),
where a(X(s)) = a(X(tn)), b(X(s)) = b(X(tn)), for tn ≤ s < tn+1, on the time
grid 0 = t0 < t1 < ... < tN = T.

Sketch of the Proof of Theorem 3.1: Step 1
For Euler–Maruyama scheme and p ≥ 1,
Under global Lipschitzity of drift and diffusion coefficients
Assumption, we have (Kloeden and Platen 1992)
E[e2p
` (T)] = O (∆tp
` ). (7)
In (Bayer, Ben Hammouda, and Tempone 2022), assuming further
regularity assumptions of the drift and diffusion, we prove that
E[(∂ye`)2p
(T)] = O (∆tp
` ). (8)
" The proof is based on the Grönwall, Hölder, Jensen and
Burkholder-Davis-Gundy inequalities.
15

Using (i) integration by parts, and (ii) the mean value, Fubini, and dominated convergence
theorems, we show that
∆I`(B`) ∶= (I` − I`−1)(B`) ∶= ∫
R
(g(X`(T;y,B`)) − g(X`−1(T;y,B`)))ρ1(y)dy
= −∫
1
0
[∫
R
e`(T;y,B`)g(z(θ;y,B`))(∂y ((∂yz(θ;y,B`))
−1
) − y (∂yz(θ;y,B`))
−1
)ρ1(y)dy]dθ
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(I)
−∫
1
0
[∫
R
∂ye`(T;y,B`)g(z(θ;y,B`))(∂yz(θ;y,B`))
−1
ρ1(y)dy]dθ
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(II)
, (9)
with
z(θ;y,B`) ∶= X`−1(T;y,B`) + θe`(T;y,B`), θ ∈ (0,1)
= (1 − θ)X`−1(T;y,B`) + θX`(T;y,B`)
16

For term (I), taking expectation w.r.t the Brownian bridge and using
Hölder’s inequality (p,q,p1,q1 ∈ (1,+∞), 1
p + 1
q = 1 and 1
p1
+ 1
q1
= 1), result in
E [(I)
2
] ≤ (E [∣∣g(z(⋅;⋅,B`))(∂y ((∂yz(⋅;⋅,B`))
−1
) − Y (∂yz(⋅;⋅,B`))
−1
)∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
× (E [∣∣e`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
= O (∆t`). (10)
Choosing p and p1 such that 2p1
p ≤ 1, and applying Jensen’s inequality:
(E [∣∣e`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
=
⎛
⎝
E
⎡
⎢
⎢
⎢
⎢
⎣
(∫
R
∣ep
` (T;y,B`)∣ρ1dy)
2p1
p
⎤
⎥
⎥
⎥
⎥
⎦
⎞
⎠
1/p1
≤ (E [∫
R
∣ep
` (T;y,B`)∣ρ1dy])
2
p
= O (∆t`) (using Fubini’s theorem and (7)).
We show that
(E [∣∣g(z(⋅;⋅,B`))(∂y ((∂yz(⋅;⋅,B`))−1
) − Y (∂yz(⋅;⋅,B`))−1
)∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
< ∞,
17

For the term (II) in (9), we redo same steps as for term (I)
E [(II)
2
] ≤ (E [∣∣g(z(⋅;⋅,B`))(∂yz(⋅;⋅,B`))
−1
∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
× (E [∣∣∂ye`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
= O (∆t`) (11)
Using (8), we show (EB`
[∣∣∂ye`(T;⋅,B`)∣∣2p1
Lp
ρ1
(R)
])
1/p1
= O (∆t`).
We show that
(E [∣∣g(z(⋅;⋅,B`))(∂yz(⋅;⋅,B`))−1
∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
< ∞,
18

Error Discussion for MLMC
̂
QMLMC
: the MLMC estimator
E[g(X(T)] − ̂
QMLMC
= E[g(X(T))] − E[g(X
∆tL
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆tL)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error II: numerical smoothing error of O(M
−s/2
Lag,L
)+O(TOLNewton,L)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − ̂
QMLMC
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: MLMC statistical error of O
⎛
⎝
√
∑L
`=L0
√
MLag,`+log(TOL−1
Newton,`
)
⎞
⎠
Notations
y∗
1: the approximated location of the non smoothness obtained by Newton
iteration ⇒ ∣y∗
1 − y∗
1∣ = TOLNewton
MLag is the number of points used by the Laguerre quadrature for the one
dimensional pre-integration step.
s > 0: For the parts of the domain separated by the discontinuity location,
derivatives of G with respect to y1 are bounded up to order s. 19

19

MLMC for Probability in the GBM model:
Euler–Maruyama
0 2 4 6 8
-10
-8
-6
-4
-2
0 2 4 6 8
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
102
103
kurtosis
(a) without numerical smoothing.
0 2 4 6 8
-20
-18
-16
-14
-12
-10
-8
0 2 4 6 8
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
3
4
5
6
kurtosis
(b) With numerical smoothing.
Figure 4.1: MLMC for probability computation under the geometric Brownian
motion (GBM): Variance, cost, L1
-distance and kurtosis per level. P`: the
numerical approximation of the QoI at level `. 20

MLMC for Probability in the GBM Model: Milstein
0 2 4 6 8
-15
-10
-5
0 2 4 6 8
-25
-20
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
103
104
105
kurtosis
0 2 4 6 8
-35
-30
-25
-20
0 2 4 6 8
-25
-20
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
101
kurtosis
Figure 4.2: MLMC with Milstein scheme for probability computation under
the geometric Brownian motion (GBM): Variance, cost, L1
-distance and
kurtosis per level. P`: the numerical approximation of the QoI at level `.
21

Probability Computation under the GBM Model:
Numerical Complexity Comparison
10-4
10-3
10-2
10-1
TOL
10-6
10-4
10-2
100
102
104
106
E[W]
MLMC without smoothing (Euler)
TOL-2.5
MLMC without smoothing (Milstein)
MLMC+ Numerical Smoothing (Euler)
TOL-2
log(TOL)2
MLMC+ Numerical Smoothing (Milstein)
TOL-2
Figure 4.3: Probability Computation under GBM: Comparison of the
numerical complexity of the different MLMC estimators.
22

MLMC for Probability under the Heston Model
0 2 4 6
-9
-8
-7
-6
-5
-4
-3
-2
0 2 4 6
-15
-10
-5
0
0 2 4 6
1
2
3
4
5
6
7
8
0 2 4 6
102
kurtosis
0 2 4 6
-16
-14
-12
-10
-8
-6
-4
-2
0 2 4 6
-15
-10
-5
0
0 2 4 6
1
2
3
4
5
6
7
8
0 2 4 6
10
12
14
16
18
20
kurtosis
Figure 4.4: MLMC with FT Euler–Maruyama scheme for probability
computation under the Heston model: Variance, cost, L1
-distance and
kurtosis per level. P`: the numerical approximation of the QoI at level `.
23

Density Estimation under the Heston Model
0 2 4 6 8
-15
-10
-5
0
5
0 2 4 6 8
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
10
15
20
25
30
35
kurtosis
(a) Asset price density
0 2 4 6 8
-10
-5
0
0 2 4 6 8
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
13
14
15
16
17
18
19
kurtosis
(b) Joint density
Figure 4.5: Density of Heston: Convergence plots for MLMC with numerical
smoothing combined with the FT Euler scheme, for computing the asset price
density ρX(T ) at u = 1 and the joint density ρX(T ),v(T ) at u = 1 and v = 0.04.
24

24

Conclusions
1 The numerical smoothing approach is adapted to the MLMC
context for efficient probability computation,
univariate/multivariate density estimation, and option pricing.
2 Compared to the case without smoothing
▸ We significantly reduce the kurtosis at the deep levels of MLMC
(becomes bounded instead of blow-up) which improves the
robustness of the estimator.
▸ We improve the MLMC strong convergence (variance decay) rate ⇒
improvement of MLMC complexity from O (TOL−2.5
) to
O (TOL−2
) (we recover the MLMC complexities obtained for
Lipschitz functionals).
3 When estimating densities: Compared to the smoothing strategies
based on MLMC with parametric regularization as in (Giles,
Nagapetyan, and Ritter 2015) or QMC with kernel density
techniques as in (Ben Abdellah et al. 2021), the error of our
approach does not increase exponentially with respect to the
dimension of state vector
25

Extensions
1 Extend our techniques to efficiently compute
▸ Sensitivities (Financial Greeks): ∂
∂α
E[f(ω,α)].
▸ Risk quantities ⇒ nested expectations problems
E [g (E [f(X,Y )∣X])].
▸ Computing nonsmooth quantities (such as probabilities) of a
functional of a solution arising from a random PDE.
2 Combine the numerical smoothing technique with multilevel QMC
to profit from the good features of QMC and MLMC.
3 Combine the numerical smoothing technique with antithetic
MLMC (Giles and Szpruch 2014) for multi-dimensional SDEs to
recover the optimal complexity.
26

Related References
Thank you for your attention!
[1] C. Bayer, C. Ben Hammouda, R. Tempone. Numerical Smoothing
with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo
Methods for Efficient Option Pricing. Quantitative Finance 23, no.
2 (2023): 209-227.
[2] C. Bayer, C. Ben Hammouda, R. Tempone. Multilevel Monte Carlo
with Numerical Smoothing for Robust and Efficient Computation of
Probabilities and Densities, arXiv:2003.05708 (2022).
27

Notations and Assumptions
Notation
X`(T) ∶= X`(T;(Z`
1,Z`
−1)).
We denote X`(T) by X
N`
T .
X
N`
k are the Euler–Maruyama increments of X
N`
T for 0 ≤ k ≤ N`
with X
N`
T = X
N`
N`
.
Assumption 5.1
For p ∈ N s.t. 1 ≤ p ≤ 4, there are positive rdvs Cp with finite moments
of all orders such that
∀N` ∈ N, ∀k1,...,kp ∈ {0,...,N` − 1} ∶
R
R
R
R
R
R
R
R
R
R
R
R
R
∂p
X
N`
T
∂X
N`
k1
⋯∂X
N`
kp
R
R
R
R
R
R
R
R
R
R
R
R
R
≤ Cp a.s.
Assumption 5.1 is fulfilled if the drift and diffusion coefficients are
smooth.
28

Notations and Assumptions
Assumption 5.2
For p ∈ N s.t. 1 ≤ p ≤ 4, there are positive rdvs Dp with finite moments
of all orders such that a
⎛
⎝
∂X
N`
T
∂y
(Z`
1,Z`
−1)
⎞
⎠
−p
≤ Cp a.s.
a
y ∶= z−1
In (Bayer, Ben Hammouda, and Tempone 2023), we show
sufficient conditions where this assumption is valid.
For instance, Assumption 5.2 is valid for
▸ one-dimensional SDEs with a linear or constant diffusion.
▸ multivariate SDEs with a linear drift and constant diffusion,
including the multivariate lognormal model (see (Bayer,
Siebenmorgen, and Tempone 2018)).
29

How Does Regularity Affect MLMC Complexity?
Complexity analysis for MLMC
MLMC Complexity (Cliffe et al. 2011)
O (TOL
−2−max(0, γ−β
α
)
log (TOL)2×1{β=γ}
)
(12)
i) Weak rate:
∣E[g (X`(T)) − g (X(T))]∣ ≤ c12−α`
ii) Variance decay rate:
Var[g (X`(T)) − g (X`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=V`
≤ c22−β`
iii) Work growth rate: W` ≤ c32γ`
(W`:
expected cost)
For Euler-Maruyama (γ = 1):
▸ If g is Lipschitz ⇒ V` ≃ ∆t` due to strong rate 1/2, that is β = γ and MLMC complexity
O (TOL−2
) (up to log terms);
▸ Otherwise (without any smoothing or adaptivity techniques):
β < γ ⇒ worst-case complexity, O (TOL− 5
2 ).
Higher order schemes, E.g., the Milstein scheme, may lead to better complexities
even for non-Lipschitz observables (Giles, Debrabant, and Rößler 2013; Giles 2015).
However,
▸ For moderate/high-dimensional SDEs, the scheme becomes computationally expensive.
▸ Deterioration of the robustness of the MLMC estimator because the kurtosis explodes
as ∆t` decreases: O (∆t−1
` ) compared with O (∆t
−1/2
` ) for Euler-Maruyama without
smoothing (Giles, Nagapetyan, and Ritter 2015).
30

How Does Regularity Affect MLMC Robustness?
/ For non-lipschitz payoffs (without any smoothing or adaptivity
techniques):
The Kurtosis, κ` ∶=
E[(Y`−E[Y`])4
]
(Var[Y`])2 is of O(∆t
−1/2
` ) for Euler-Maruyama.
Large kurtosis problem: discussed previously in (Ben Hammouda, Moraes,
and Tempone 2017; Ben Hammouda, Ben Rached, and Tempone 2020) ⇒
/ Expensive cost for reliable/robust estimates of sample statistics.
Why is large kurtosis bad?
σS2(Y`) =
Var[Y`]
√
M`
√
(κ` − 1) +
2
M` − 1
; " M` ≫ κ`.
Why are accurate variance estimates, V` = Var[Y`], important?
M∗
` ∝
√
V`W−1
`
L
∑
`=0
√
V`W`.
Notation
Y` ∶= g(X`(T)) − g(X`−1(T))
σS2(Y`): Standard deviation of the sample variance of Y`;
M∗
` : Optimal number of samples per level; W`: Cost per sample path.
31

Extending Numerical Smoothing for
Multiple Discontinuities
Multiple Discontinuities: Due to the payoff structure/use of Richardson extrapolation.
R different ordered multiple roots, e.g., {y∗
i }R
i=1, the smoothed integrand is
I (y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
+
R−1
∑
i=1
∫
y∗
i+1
y∗
i
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1,
and its approximation I is given by
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
MLag,1
∑
k=0
ηLag
k G(ζLag
k,1 (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 )
+
MLag,R
∑
k=0
ηLag
k G(ζLag
k,R (y∗
R),y−1,z
(1)
−1 ,...,z
(d)
−1 )
+
R−1
∑
i=1
⎛
⎝
MLeg,i
∑
k=0
ηLeg
k G(ζLeg
k,i (y∗
i ,y∗
i+1),y−1,z
(1)
−1 ,...,z
(d)
−1 )
⎞
⎠
,
{y∗
i }R
i=1: the approximated discontinuities locations; MLag,1 and MLag,R: the number
of Laguerre quadrature points ζLag
.,. ∈ R with corresponding weights ηLag
. ; {MLeg,i}R−1
i=1 :
the number of Legendre quadrature points ζLeg
.,. with corresponding weights ηLeg
. .
I can be approximated further depending on (i) the decay of G × ρ1 in the semi-infinite
domains and (ii) how close the roots are to each other.

References I
[1] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Multilevel Monte Carlo with Numerical Smoothing for Robust
and Efficient Computation of Probabilities and Densities”. In:
arXiv preprint arXiv:2003.05708 (2022).
[2] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Numerical smoothing with hierarchical adaptive sparse grids
and quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance 23.2 (2023), pp. 209–227.
[3] Christian Bayer, Markus Siebenmorgen, and Rául Tempone.
“Smoothing the payoff for efficient computation of basket option
pricing.”. In: Quantitative Finance 18.3 (2018), pp. 491–505.
[4] Amal Ben Abdellah et al. “Density estimation by randomized
quasi-Monte Carlo”. In: SIAM/ASA Journal on Uncertainty
Quantification 9.1 (2021), pp. 280–301.
33

References II
[5] Chiheb Ben Hammouda, Nadhir Ben Rached, and Raúl Tempone.
“Importance sampling for a robust and efficient multilevel Monte
Carlo estimator for stochastic reaction networks”. In: Statistics
and Computing 30.6 (2020), pp. 1665–1689.
[6] Chiheb Ben Hammouda, Alvaro Moraes, and Raúl Tempone.
“Multilevel hybrid split-step implicit tau-leap”. In: Numerical
Algorithms 74.2 (2017), pp. 527–560.
[7] Peng Chen. “Sparse quadrature for high-dimensional integration
with Gaussian measure”. In: ESAIM: Mathematical Modelling
and Numerical Analysis 52.2 (2018), pp. 631–657.
[8] K Andrew Cliffe et al. “Multilevel Monte Carlo methods and
applications to elliptic PDEs with random coefficients”. In:
Computing and Visualization in Science 14.1 (2011), p. 3.
34

References III
[9] Josef Dick, Frances Y Kuo, and Ian H Sloan. “High-dimensional
integration: the quasi-Monte Carlo way”. In: Acta Numerica 22
(2013), pp. 133–288.
[10] Oliver G Ernst, Bjorn Sprungk, and Lorenzo Tamellini.
“Convergence of sparse collocation for functions of countably
many Gaussian random variables (with application to elliptic
PDEs)”. In: SIAM Journal on Numerical Analysis 56.2 (2018),
pp. 877–905.
[11] Michael B Giles. “MLMC techniques for discontinuous
functions”. In: arXiv preprint arXiv:2301.02882 (2023).
[12] Michael B Giles. “Multilevel Monte Carlo methods”. In: Acta
Numerica 24 (2015), pp. 259–328.
[13] Michael B Giles. “Multilevel Monte Carlo path simulation”. In:
Operations Research 56.3 (2008), pp. 607–617.
35

References IV
[14] Michael B Giles, Kristian Debrabant, and Andreas Rößler.
“Numerical analysis of multilevel Monte Carlo path simulation
using the Milstein discretisation”. In: arXiv preprint
arXiv:1302.4676 (2013).
[15] Michael B Giles, Tigran Nagapetyan, and Klaus Ritter.
“Multilevel Monte Carlo approximation of distribution functions
and densities”. In: SIAM/ASA Journal on Uncertainty
Quantification 3.1 (2015), pp. 267–295.
[16] Michael B Giles and Lukasz Szpruch. “Antithetic multilevel
Monte Carlo estimation for multi-dimensional SDEs without
Lévy area simulation”. In: (2014).
[17] Stefan Heinrich. “Multilevel monte carlo methods”. In:
International Conference on Large-Scale Scientific Computing.
Springer. 2001, pp. 58–67.
36

References V
[18] Ahmed Kebaier. “Statistical Romberg extrapolation: a new
variance reduction method and applications to option pricing”.
In: The Annals of Applied Probability 15.4 (2005), pp. 2681–2705.
[19] Peter E Kloeden and Eckhard Platen. “Stochastic differential
equations”. In: Numerical solution of stochastic differential
equations. Springer, 1992, pp. 103–160.
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