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Quasi-Monte Carlo Algorithms with Applications in Numerical Analysis and Finance Reinhold E. Kainhofer Rigorosumsvortrag, Hnstitut f¨ ur Mathematik Fraz, 16. Mai 2003

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Quasi-Monte Carlo Algorithms withApplications in Numerical Analysis and

FinanceReinhold F. Kainhofer

Rigorosumsvortrag, Institut fur Mathematik

Graz, 16. Mai 2003

Inhalt

1. Quasi-Monte Carlo Methoden

2. Sublineare Dividendenschranken in der Risikotheorie

3. Runge-Kutte QMC Methoden fur retardierte Differentialgleichungen

(a) Losungsskizze des Algorithmus

(b) RKQMC Losungsmethoden (Hermite Interpolation, QMC Methoden)

(c) Konvergenzbeweis

(d) Numerische Beispiele

4. QMC Methoden fur singulare gewichtete Integration

(a) Konvergenztheorem und -beweis

(b) Hlawka-Muck Konstruktion

(c) Numerisches Beispiel (Bewertung von asiatischen Optionen)

Quasi-Monte Carlo methods

Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s

f (x)dx =1

N

N∑i=1

f (xi)

Quasi-Monte Carlo methods

Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s

f (x)dx =1

N

N∑i=1

f (xi)

MC methods: xi random points

QMC methods: xi low-discrepancy sequences

Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)

s, good uniform distribution

Quasi-Monte Carlo methods

Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s

f (x)dx =1

N

N∑i=1

f (xi)

MC methods: xi random points

QMC methods: xi low-discrepancy sequences

Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)

s, good uniform distribution

discrepancy of the point set S

DN(S) = supJ⊆[0,1]s

∣∣∣∣A(J , S)

N− λs(J )

∣∣∣∣

Quasi-Monte Carlo methods

Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s

f (x)dx =1

N

N∑i=1

f (xi)

MC methods: xi random points

QMC methods: xi low-discrepancy sequences

Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)

s, good uniform distribution

discrepancy of the point set S

DN(S) = supJ⊆[0,1]s

∣∣∣∣A(J , S)

N− λs(J )

∣∣∣∣Koksma-Hlawka inequality (f of bounded Variation):∣∣∣∣∣ 1

N

N∑n=1

f(xn)−∫

[0,1)s

f(u)du

∣∣∣∣∣ ≤ V ([0, 1)s, f)D∗N (x1, . . . , xN ) .

Low-Discrepancy sequences

D∗N (S) ≤ O

((log N)s

N

)

Low-Discrepancy sequences

D∗N (S) ≤ O

((log N)s

N

)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the

comma

• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals

Low-Discrepancy sequences

D∗N (S) ≤ O

((log N)s

N

)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the

comma

• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Monte Carlo, 3000 points

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Halton sequence, 3000 points

Low-Discrepancy sequences

D∗N (S) ≤ O

((log N)s

N

)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the

comma

• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Monte Carlo, 3000 points

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Halton sequence, 3000 points

Problem: correlations between elements

Risk Model with constant interest forceand non-linear dividend barrier

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timet

xb

dividend barrier breserve R

dividends

ruin

φ(u, b) . . . probability of survival

W (u, b) . . . expected value of discounted dividend payments

Integro-differential equation for W (u, b):

(c+i u) ∂W∂u + 1

α m bm−1∂W∂b − (i+λ) W +λ

u∫0

W (u−z,b)dF (z)=0

with boundary condition ∂W∂u

∣∣u=b

= 1.

Solution is fixed point of integral operator ⇒ apply it iteratively orrecursively

Ag(u, b) =

∫ t∗

0λe−(λ+i)t

∫ (c′+u)eit−c′

0g

((c′ + u)eit − c′ − z,

(bm +

t

α

)1/m)

dF (z)dt

+

∫ ∞

t∗λe−(λ+i)t

∫ (bm+ tα)

1/m

0g

((bm +

t

α

)1/m

− z,

(bm +

t

α

)1/m)

dF (z)dt

+

∫ ∞

t∗λe−λt

∫ t

t∗e−is

((c + i u)eis − 1

mα(bm + s

α

)1−1/m

)ds dt,

=⇒ Solution is just a high-dimensional integration problem.

Numerical solution of delayed differen-tial equations using QMC methods

1. Sketch of the numerical solution

2. The RKQMC solution methods (Hermite Interpolation, QMC meth-ods)

3. Convergence proofs

4. Numerical examples

The problem

Heavily varying delay differential equations (DDE) or DDE with heavilyvarying solutions.

y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) , for t ≥ t0, k ≥ 1,

y(t) = φ(t), for t ≤ t0 ,

with

The problem

Heavily varying delay differential equations (DDE) or DDE with heavilyvarying solutions.

y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) , for t ≥ t0, k ≥ 1,

y(t) = φ(t), for t ≤ t0 ,

with

f (t, y(t), yret(t)) . . .piecewise smooth in y and yret,

bounded and Borel measurable in t

y(t) . . .solution, d-dimensional real-valued function

τ1(t), ..., τk(t) . . .cont. delay functions, bounded from below by τ0 > 0,

satisfy t1 − τj(t1) ≤ t2 − τj(t2) for t1 ≤ t2

φ(t) . . .initial function, cont. on

[inf

t0≤t,1≤j≤k(t− τj(t)) , t0

].

Sketch of the numerical solution

For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.

• Hermite interpolation for retarded argument ⇒ ODE

Sketch of the numerical solution

For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.

• Hermite interpolation for retarded argument ⇒ ODE

• use RKQMC methods for ODE:Large Runge-Kutta error for heavily oscillating DEIdea (Stengle, Lecot): integrate over whole step size

Sketch of the numerical solution

For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.

• Hermite interpolation for retarded argument ⇒ ODE

• use RKQMC methods for ODE:Large Runge-Kutta error for heavily oscillating DEIdea (Stengle, Lecot): integrate over whole step size

– Runge Kutta: Integration over y and t discretized

– RK(Q)MC: Integration over y discretized, numerical Integrationin t (using MC or QMC integration to minimize the error)

Hermite interpolation

Use hermite interpolation for the retarded arguments:

Hermite interpolation

Use hermite interpolation for the retarded arguments:

zj(t) := y(t− τj(t)) =

{φ(t− τj(t)), if t− τj(t) ≤ t0

Pq(t− τj(t); (yi); (y′i)) otherwise

Hermite interpolation

Use hermite interpolation for the retarded arguments:

zj(t) := y(t− τj(t)) =

{φ(t− τj(t)), if t− τj(t) ≤ t0

Pq(t− τj(t); (yi); (y′i)) otherwise

DDE transforms to a ODE:

y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .

Hermite interpolation

Use hermite interpolation for the retarded arguments:

zj(t) := y(t− τj(t)) =

{φ(t− τj(t)), if t− τj(t) ≤ t0

Pq(t− τj(t); (yi); (y′i)) otherwise

DDE transforms to a ODE:

y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .

Solution has to be piecewise r/2-times continuously differentiable in t.

Hermite interpolation

Use hermite interpolation for the retarded arguments:

zj(t) := y(t− τj(t)) =

{φ(t− τj(t)), if t− τj(t) ≤ t0

Pq(t− τj(t); (yi); (y′i)) otherwise

DDE transforms to a ODE:

y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .

Solution has to be piecewise r/2-times continuously differentiable in t.Resulting ODE has to fulfill requirements for RKQMC methods (Borel-measurable in t, continously differentiable in y(t)).

Runge Kutta QMC methods for ODE

G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty

y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0

f smooth in y, bounded and Borel measurable in t.

Runge Kutta QMC methods for ODE

G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty

y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0

f smooth in y, bounded and Borel measurable in t.

f is Taylor-expanded only in y ⇒ integral equation in t.

Runge Kutta QMC methods for ODE

G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty

y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0

f smooth in y, bounded and Borel measurable in t.

f is Taylor-expanded only in y ⇒ integral equation in t.

yn+1 = yn +hn

s!N

∑0≤j<N

Gs (tj,n; y)

Gs (tj,n; y) . . . differential increment function of scheme

Runge Kutta QMC methods for ODE

G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty

y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0

f smooth in y, bounded and Borel measurable in t.

f is Taylor-expanded only in y ⇒ integral equation in t.

yn+1 = yn +hn

s!N

∑0≤j<N

Gs (tj,n; y)

Gs (tj,n; y) . . . differential increment function of scheme

G1 (u; y) = f (u, y)

G2 (u; y) = f (u1, y) +1

βf (u2, y)) +

1

αf (u2, y + αhnf (u1, y)))

G3 (u; y) = a1f(u1, y) +L2∑l=1

a2,lf(u2, y + b2,lhnf(u1, y)

)+

+L3∑l=1

a3,l

(u3, y + b

(1)3,l hnf (u1, y) + b

(2)3,l hnf

(u2, yn + c3,lhn(u1, yn)

))

RKQMC for Volterra functional equations

y′(t) = f (t, y(t), z(t)) t ≥ t0

z(t) = (Fy)(t)

y(s) = φ(s) s ≤ t0

RKQMC for Volterra functional equations

y′(t) = f (t, y(t), z(t)) t ≥ t0

z(t) = (Fy)(t)

y(s) = φ(s) s ≤ t0

• (Fy)(t) contains dependence on retarded values.

• F and f are Lipschitz continuous in all arguments except t.

RKQMC for Volterra functional equations

y′(t) = f (t, y(t), z(t)) t ≥ t0

z(t) = (Fy)(t)

y(s) = φ(s) s ≤ t0

• (Fy)(t) contains dependence on retarded values.

• F and f are Lipschitz continuous in all arguments except t.

RKQMC method (n ≥ 0):

yn+1 = yn + hn

N∑i=1

Gs (tn,i; yn, z(t))

z(t) = (F yj)(t)

Convergence: RKQMC for DDE

Theorem[K., 2002] If

(i) RKQMC method converges for ordinary differential equations withorder p

(ii) the increment function Gs of the method and F are Lipschitz

(iii) the interpolation fulfills a Lipschitz condition

(iv) Hermite interpolation is used with order r

(v) the initial error ‖e0‖ vanishes

then the method converges.

Convergence: RKQMC for DDE

Theorem[K., 2002] If

(i) RKQMC method converges for ordinary differential equations withorder p

(ii) the increment function Gs of the method and F are Lipschitz

(iii) the interpolation fulfills a Lipschitz condition

(iv) Hermite interpolation is used with order r

(v) the initial error ‖e0‖ vanishes

then the method converges. If at least ii and iii hold, the error is boundedby

‖ej+1‖ ≤ ‖e0‖ eLtj +(etjL − 1)

L(∥∥EODE

G

∥∥ + L1L2

∥∥E interpolr

∥∥) .

Special case: one retarded argument

Choose the RKQMC method:

• Gs Lipschitz (L2) in 2nd and 3rd argument, bounded variation (in the sense of Hardy andKrause).

• ∃ c1, c2, c3 such that for a p > 0

loc. trunc. error ‖εn‖ ≤ c1(hn)hpn

RK error ‖δn‖ ≤ c2(hn) ‖en‖QMC error ‖dn‖ ≤ c3(hn)D∗

N (S)

• interpolation order p, fulfils a certain Lipschitz condition

Special case: one retarded argument

Choose the RKQMC method:

• Gs Lipschitz (L2) in 2nd and 3rd argument, bounded variation (in the sense of Hardy andKrause).

• ∃ c1, c2, c3 such that for a p > 0

loc. trunc. error ‖εn‖ ≤ c1(hn)hpn

RK error ‖δn‖ ≤ c2(hn) ‖en‖QMC error ‖dn‖ ≤ c3(hn)D∗

N (S)

• interpolation order p, fulfils a certain Lipschitz condition

Then the error ‖en‖ = ‖yn − y(tn)‖ of the method is bounded fromabove by:

‖en‖ ≤ ‖e0‖ etn(c2+L2s! ) +

etn(c2+L2s! ) − 1

c2 + L2

s!

·

·{

c3D∗N(X) +

L2

s!MHq + c1H

p

}

Numerical examples

y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0

y(t) = 1, t ≤ 0 ,

Numerical examples

y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0

y(t) = 1, t ≤ 0 ,

0 5 10 15 20Log2HΛL

-12

-10

-8

-6

-4

Log2HSHmethL,ΛL

RKQMC3, N=10RKQMC3, N=100RKQMC3, N=1000RKQMC2, N=1000RKQMC1, N=1000RungeHeunButcher

Fig: Error of the RK and RKQMC methods, hn = 0.001 for RK, hn = 0.01 for RKQMC

Numerical examples

y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0

y(t) = 1, t ≤ 0 ,

0 5 10 15 20Log2HΛL

-12

-10

-8

-6

-4

Log2HSHmethL,ΛL

RKQMC3, N=10RKQMC3, N=100RKQMC3, N=1000RKQMC2, N=1000RKQMC1, N=1000RungeHeunButcher

Fig: Error of the RK and RKQMC methods, hn = 0.001 for RK, hn = 0.01 for RKQMC

slowly varying (small λ): conventional RK betterrapidly varying (high λ): RKQMC outperform higher order RK

Advantage of RKQMC for unstable DDE

y′(t) = πλ

2

(y

(t− 2− 3

)− y

(t− 2− 1

)), t ≥ 0

y(t) = sin(λtπ), t < 0 ,

Advantage of RKQMC for unstable DDE

y′(t) = πλ

2

(y

(t− 2− 3

)− y

(t− 2− 1

)), t ≥ 0

y(t) = sin(λtπ), t < 0 ,

Exact Solution: y(t) = sin(λtπ)

• No discontinuities in any derivative

• k-th derivative only bounded by λk → ”exploding” error

Advantage of RKQMC for unstable DDE

y′(t) = πλ

2

(y

(t− 2− 3

)− y

(t− 2− 1

)), t ≥ 0

y(t) = sin(λtπ), t < 0 ,

Exact Solution: y(t) = sin(λtπ)

• No discontinuities in any derivative

• k-th derivative only bounded by λk → ”exploding” error

5 10 15 20Log2HΛL5

10

15

20

Log2HSΛ,HmethLL

RKQMC3, N=1000

RKQMC2, N=1000

RKQMC1, N=1000

Runge

Heun

Butcher

RKQMC schemes can delay the instability of the solution for heavilyoscillating delay differential equations.

Time-corrected error

QMC integration is more expensive than 1 evaluation (Runge-Kutta)

Time-corrected error

QMC integration is more expensive than 1 evaluation (Runge-Kutta)→ Compare a time-corrected error

Time-corrected error

QMC integration is more expensive than 1 evaluation (Runge-Kutta)→ Compare a time-corrected error

5 10 15 20Log2HΛL5

10

15

Log2HSTΛ,HmethLL

RKQMC3, N=1000

RKQMC2, N=1000

RKQMC1, N=1000

Runge

Heun

Butcher

Fig: Time-corrected error of the RK and RKQMC methods

Result: RKQMC methods loose some advantage, but still better thanRunge-Kutta for heavily oscillating DDE.

Non-uniform QMC integration of singu-lar integrands

1. Convergence theorem

2. Sketch of proof

3. Hlawka-Muck construction for h-distributed low-discrepancy se-quences

4. Numerical example (pricing of Asian options)

H-Diskrepancy

H-discrepancy of a sequence ω = (y1, y2, . . . ):

DN,H(ω) = supJ⊆L

∣∣∣∣ 1

NAN(J, ω)−H(J)

∣∣∣∣ ,with L . . . support of H ,

J . . . intervals[~a,~b], AN . . . number of elements of ω lying in J .

H-Diskrepancy

H-discrepancy of a sequence ω = (y1, y2, . . . ):

DN,H(ω) = supJ⊆L

∣∣∣∣ 1

NAN(J, ω)−H(J)

∣∣∣∣ ,with L . . . support of H ,

J . . . intervals[~a,~b], AN . . . number of elements of ω lying in J .

Koksma-Hlawka inequality (arbitrary distribution)

Let f a function of bounded variation (in the sense of Hardy andKrause) on L and ω = (y1, y2, . . . ) a sequence on L. Then∣∣∣∣∣

∫L

f (x)dH(x)− 1

N

N∑n=1

f (yn)

∣∣∣∣∣ ≤ V (f )DN,H(ω).

Convergence of the multidimensional QMC estimator

Convergence of the multidimensional QMC estimator

Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),

Convergence of the multidimensional QMC estimator

Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),

and let furthermore

cN,j = min1≤n≤N

y(j)n ∧ aj < cj ≤ cN,j.

Convergence of the multidimensional QMC estimator

Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),

and let furthermore

cN,j = min1≤n≤N

y(j)n ∧ aj < cj ≤ cN,j.

If the improper integral exists, and if

DN,H (ω) · V[c,b](f ) = o(1),

then the QMC estimator converges to the value of the improper integral:

limN→∞

1

N

N∑n=1

f (yn) =

∫[a,b]

f (x) dH(x).

Hlawka-Muck Transformation (1 dimension)

yk =1

N

N∑r=1

b1 + xk −H(xr)c =1

N

N∑r=1

χ[0,xk](H(xr))

Hlawka-Muck Transformation (1 dimension)

yk =1

N

N∑r=1

b1 + xk −H(xr)c =1

N

N∑r=1

χ[0,xk](H(xr))

DN,H(ω) ≤ (1 + 2sM)DN(ω)

0 might appear among the yk ⇒ adapt the construction as follows(needed for singular integrands): We define the sequence ω for i =1, . . . , N as:

yk =

{yk wenn yk ≥ 1

N,

1N

wenn yk = 0.

0 might appear among the yk ⇒ adapt the construction as follows(needed for singular integrands): We define the sequence ω for i =1, . . . , N as:

yk =

{yk wenn yk ≥ 1

N,

1N

wenn yk = 0.

The discrepancy of this sequence can be bounded as

DN,H(ω) ≤ (1 + 4M)sDN(ω).

Sample problem: Valuing Asian options

arithmetic mean until expiration time

Pay-Off (discrete Asian option, call)

P (ST ) =

(1

n

n∑i=1

Sti −K

)+

(St)t≥0 ... price process, K ... strike price

St = eXt with Levy process (Xt)t≥0

Increments hi = Xi −Xi−1 with distribution H(e.g. NIG, Variance-Gamma, Hyperbolic, ...)

NIG distribution

Use the NIG distribution for the increments hi ∼ HQ.Advantage: closed under convolution ⇒ dimension reduction, sampleonly weekly instead of daily

Valuation

Using fundamental theorem (Schachermayer):

Ct0 := e−r(tn−t0)EQ

[(1

n

n∑i=1

Sti −K

)+]r ... constant interest rateQ ... equivalent martingale measure (Esscher measure)

NIG distribution

Use the NIG distribution for the increments hi ∼ HQ.Advantage: closed under convolution ⇒ dimension reduction, sampleonly weekly instead of daily

Valuation

Using fundamental theorem (Schachermayer):

Ct0 := e−r(tn−t0)EQ

[(1

n

n∑i=1

Sti −K

)+]r ... constant interest rateQ ... equivalent martingale measure (Esscher measure)

Straigforward simulation (crude Monte Carlo): sample N price pathesand take the mean

Quasi-Monte Carlo schemes

Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)

Quasi-Monte Carlo schemes

Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)

2 Solutions:

1. Hlawka-Muck method for direct creation of (xn)1≤n≤N

i.i.d.∼ NIG⇒ direct QMC calculation of the expectation value

Quasi-Monte Carlo schemes

Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)

2 Solutions:

1. Hlawka-Muck method for direct creation of (xn)1≤n≤N

i.i.d.∼ NIG⇒ direct QMC calculation of the expectation value

2. Transformation of the integral using a suitable density (Ratio ofuniforms, ”Hat”) ⇒ variance reduction (if done right)

Transformation

Using a distribution K(~x) = u:∫Rn

P (~x)fQH (~x)d~x =

∫[0,1]n

P (K−1(u))fQ

H (K−1(u))

k(K−1(u))du

Problem: ”Usual” transformation F (x) = u leads to unbounded varia-tion

Discrepancy for Hlawka-Muck

The discrepancy of (yk)1≤k≤N can be bounded by

DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))

with M(ρ) = sup ρ.

Discrepancy for Hlawka-Muck

The discrepancy of (yk)1≤k≤N can be bounded by

DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))

with M(ρ) = sup ρ.

QMC estimator

1. Creation of low-discrepancy points with densityfQ

H (K−1(x))k(K−1(x))

(Transformation of the integral Rn to [0, 1]n using double-exponential distribution K(x))

Discrepancy for Hlawka-Muck

The discrepancy of (yk)1≤k≤N can be bounded by

DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))

with M(ρ) = sup ρ.

QMC estimator

1. Creation of low-discrepancy points with densityfQ

H (K−1(x))k(K−1(x))

(Transformation of the integral Rn to [0, 1]n using double-exponential distribution K(x))

2. Transformation [0, 1]n

to Rn of the sequence using double-exponential distribution K−1(x).

Estimator similar to crude Monte Carlo

Numerical results (4 dimensions)

210 212 214 216 218N

-14

-12

-10

-8

-6

-4

Log2HerrL 4 dimensions

double exp.

HM IS

Hlawka-Mück

Ratio of unif.

Import. samp.

MC

ROU and Hlawka-Muck are considerably better than Monte Carlo andcontrol variate

Dimension 12

210 212 214 216 218N

-9

-8

-7

-6

-5

-4

Log2HerrL 12 dimensions

double exp.

HM IS

Hlawka-Mück

Ratio of unif.

Import. samp.

MC

• ROU looses performance

• Hlawka-Muck gives best results