The strong weak convergence of the quasi EA

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Peluchetti, Stefano, Roberts, Gareth O. and Casella, Bruno. (2013) The strong weak

convergence of the quasi-EA. Queueing Systems, Volume 73 (Number 4). pp. 447-460.

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DOI 10.1007/s11134-013-9351-0

The strong weak convergence of the quasi-EA

Stefano Peluchetti · Gareth O. Roberts · Bruno Casella

Received: 10 March 2011 / Revised: 14 August 2012 / Published online: 17 March 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance.

Keywords Simulation of SDEs·Biased Brownian Motion·Exact simulation

Mathematics Subject Classification 65C05·65C35

1 Introduction

In this paper, we shall present two convergence results about a novel simulation scheme to the target diffusion process. This scheme is closely related to the Exact Algorithm (EA) for the simulation of diffusion process which was introduced in [1]. In particular, the simulation scheme we consider is essentially EA without the acceptance–rejection correction.

This scheme (which we call the Quasi-EA) is studied for two reasons. Firstly, we are interested in the properties of Quasi-EA as a simulation scheme in its own right. Secondly, a thorough understanding of Quasi-EA contributes to a fuller understanding of EA scheme itself.

S. Peluchetti·G. O. Roberts (

B

The main appeal of EA is that it allows for the exact simulation (i.e., free from any time discretisation error) of any skeleton of the diffusion sample path. Moreover, it is possible to simulate exactly from some classes of path-dependent functionals of the diffusion process. Because EA plays a central rule in our work, we briefly introduce the main ideas behind EA. For a more exhaustive exposition of EA, we refer to [2] and [3].

We consider the diffusion process Y a unique strong solution of the Stochastic Differential Equation (SDE)

dYt =b(Yt)dt+σ (Yt)dBt 0≤t ≤T (1) Y0=y,

where B is scalar Brownian Motion (BM) on the bounded time interval.

We make the mild assumptions thatσ is continuously differentiable and strictly positive, that b is continuously differentiable, and that the diffusion is non-explosive. These assumptions are more than enough to guarantee the existence of a unique strong solution to the SDE (see for example Sect. V.10 of [10]). These assumptions also guar-antee the existence and uniqueness of a bijective functionηsuch that the transformed diffusion process Xt :=η (Yt)satisfies the SDE

d Xt =α (Xt)dt+d Bt 0≤t≤T (2) X0=x :=η (y) .

The drift coefficientαtakes the form:

α(x)= b(η

−1_(x))

σ(η−1_(x)) −σ(η− 1_{(x)) ,}

and inherits continuous differentiability from b andσ. From now on the SDE (2) will be our starting point.

LetQx_TandWx_Tdenote the law of the diffusion X and the law of a BM, respectively, on [0,T ] both started at x. From now on the following hypotheses are assumed to hold:

• ∀x ∈ X Qx_T Wx_T and the Radon–Nikodym derivative is given by Girsanov’s formula:

dQx T

dWx T

(ω)=exp

⎧ ⎨ ⎩

T

0

α (ωs)dωs−

1 2

T

0 α2_(ω

s)ds

⎫ ⎬

⎭ (3)

• αis continuously differentiable onX;

• α2_{+αis bounded below onX.}

We introduce Biased Brownian Motion (BBM) Z and its law Zx_T. This process is defined as a BM on[0,T]started at x and conditioned on having its terminal value

hx,T(u):=ηx,T ×exp A(u)−( u−x)2

2T

. (4)

Here A(u) := _cuα (r)dr for some c ∈ Xand the normalising constant ηx,T is

assumed to be finite. Hence conditionally on the value of ZTthe process Z is distributed

as a Brownian Bridge (BB). Given the hypothesis, it is possible to prove that

dQx T

dZx_T (ω)=ηx,Texp{−A(x)}exp

⎧ ⎨

⎩−

T

0

α2_+α 2 (ωs)ds

⎫ ⎬

⎭ (5)

∝exp

⎧ ⎨

⎩−

T

0

φ (ωs)ds

⎫ ⎬

⎭≤1, (6)

whereφ (u):=α2(u)+α(u)/2−infr∈X

α2_{(r)+α(r)/2. Equation (6)} sug-gests the use of a rejection sampling algorithm to generate realisations from Qx_T. However, it is not possible to generate a sample from Z , Z being an infinite dimen-sional variate, and moreover it is not possible to compute analytically the value of the integral in (6). For ease of exposition, we only consider the case of EA1, whereα2_+α is assumed to be bounded. It should be noted that this hypothesis can be weakened or even removed; see [2,3].

We denote by m the finite supremum ofφ. Let=(X, Ψ )be a unit rate Poisson Point Process (PPP) on[0,T] ×[0,m], and let N be the number of points ofΦbelow φ (ωs)conditionally on a pathω. If the eventis defined as (ω, Φ):= {N =0}it

follows that

Pr [|ω]=exp

⎧ ⎨

⎩−

T

0

φ (ωs)ds

⎫ ⎬

⎭. (7)

This equivalence is an immediate consequence of the properties of PPPs. The lhs of (7) is the probability that no points ofΦfall in the region of the plane bounded by 0 andφ (ωs), a region whose area is given by the integral on the rhs of (7). Thus there is

no need to calculate the integral analytically. We have implicitly assumed that it was possible to generate the infinite dimensional variateω∼Z_Tx. However to compute the acceptance eventΓ, we only need to know the value of Z on a finite set of random times only, this set corresponding to the time coordinates of the realisations ofΦ. It is then possible to exchange the order in which Z andΦ are generated, obtaining Algorithm1.

Algorithm 1 Exact Algorithm 1 (EA1)

1. SAMPLE ZT∼hx,T

2. SAMPLE =(X, ) 3. SAMPLE

Z_χj;1≤j≤ |X|∼Zx_T|ZT

4. IF RETURN S:=

Z_χj;1≤j≤ |X|,ZT

As already observed, step 3 of Algorithm1results in the generation of independent BBs. EA1 returns a skeletonSdistributed according to the true law of the diffusion X . Furthermore, from [3] it is known that the conditional lawZx_T |Scan be expressed as the product measure of independent BBs. Using this result it is possible to complete S on any finite arbitrary number of times if required and even to generate (exactly) certain functionals of the path of X .

The remaining part of this paper is organised as follows. In Sect.2, the Quasi-EA is motivated and introduced and the connection with EA is examined. In Sect.3, the two main theorems are proved. We start by establishing a local convergence result that is further extended by means of the maximal coupling inequality. We show that the Quasi-EA is an accurate continuous time approximation of the law of the diffusion process. In fact, we prove the existence of a myopic (sequential) coupling between the diffusion and the simulation scheme. The existence of such a coupling implies the convergence with respect to the total variation distance, a strong form of weak convergence according to [5]. Section4concludes the paper with some remarks about possible future research on the topic and practical considerations about our scheme.

The Euler scheme does not generally converge with respect to the total variation distance; see [9]. However, under mild technical conditions, the Euler scheme does converge with respect to total variation distance if the diffusion process has a con-stant diffusion term (see [6]). Genon-Catalot [4] extended this result to prove that the rate of convergence is of orderΔ1/2, whereΔis the length of the (equally spaced) discretisation interval.

One of the main reasons for the interest in total variation distance is the well-known connection it shares with coupling. Essentially the total variation distance between two stochastic process laws is small if and only if there exists a probability space in which the probability that the two processes are identical is close to unity. However in the context of Markov processes, there is particular interest in jointly Markov (or sequential or myopic) couplings, which explicitly utilise Markov structure. A sequential coupling is a coupling in which, for each time increment in turn, we try to maximise the probability that the two processes stay together. We therefore focus on the existence of a sequential (or myopic) coupling between the diffusion and the simulation scheme.

Although total variation convergence is equivalent to the existence of a coupling, there is no guarantee that such a coupling is sequential. The existence of a sequential coupling clearly implies total variation convergence, however the converse is false in general. To illustrate this in the context of diffusion sample path approximation, consider

dYt =αdt+d Bt (8)

d Xt =d Bt, (9)

whereαis a constant, Btis a scalar BM on[0,T], and both Xtand Ytstart at the same

the probability of the coupling succeeds is1−Δ1/21/Δand this last quantity tends to 0 asΔ↓0.

The major finding of this paper in Theorem 1 is to show that, unlike the Euler scheme, the Quasi-EA algorithm can be shown to exibit a sequential coupling with the true diffusion which is successful with probability converging to 1 asΔ→0, on the whole interval [0,T ] with rateΔ1/2_.

2 The quasi-EA

The main idea behind the construction of Quasi-EA is very simple. In step 2 of Algo-rithm1, we sample a PPP with unit rate on [0,T ]×[0,m]. The acceptance rate in EA

increases as T ↓0, as the likelihood that no points fromΦare sampled increases too. Clearly in this eventuality, the proposed path is accepted.

As the acceptance rate can be interpreted as a measure of the quality of the proposal measure, we develop a scheme that always accept the proposal variate distributed as Zx

T. Given the previous considerations, this approximation is accurate only if T is quite

small. Hence the time interval[0,T]is partitioned into smaller intervals on which the scheme is applied sequentially.

We now describe the scheme more precisely. The time interval[0,T]is divided into

n smaller intervals having the same lengthΔ=T/n. The continuous time scheme Y

is defined by the following equations

Y0=x (10)

YiΔ∼hYiΔ,Δ (i =1, . . . ,n) (11)

Ys ∼BB

YiΔ,Y(i+1)Δ, Δ

(iΔ <s< (i+1)Δ) (12)

where BB(x,dy,t)is the measure of a BB starting at(0,x)and ending at(t,dy).

It can be easily seen the process Y thus defined consists of n sequential BBMs. The simulation of the Quasi-EA involves the sampling of the sequence of random variables in (11) only. However we are going to prove a stronger result than the convergence of the discretized process{YiΔ;i =0, . . . ,n}to the diffusion process. We shall prove

that the law of the continuous time process Y defined by Eqs. (10)–(12) converges with respect to the total variation distance to the law of the diffusion process X as

n↑ ∞. This result suggests that the BBs are good process to fill-in the gaps between the simulated values of the discretized process.

We denote byY₀x_,,n_T the probability measure induced by the Quasi-EA scheme Y started at x and consisting of n steps on[0,T]. Consistently with the previous notation Yy

0,Δdenotes the probability measure induced by this scheme on the single step [0, Δ] when Y0=y. To sum up

{Ys;0≤s≤T |Y0=x} ∼Yx0,,nT (13)

3 Two convergence results

The two convergence theorems require the two following lemmas.

Lemma 1 Let f : R → Rbe a continuous function, such that for all sufficiently smallΔ >0

1

√

2πΔ

R

|f (y)|e−(y−x)

2

2Δ _dy<∞ ₍₁₅₎

then (for any fixed x ∈R)

lim

Δ↓0 1

√

2πΔ

R

f(y)e−(y−x)

2

2Δ dy = f(x). (16)

This is proved by elementary dominated convergence arguments. We omit the proof. For ease of exposition, we assume that the state space of X isR, although this is easily generalised.

The following conditions, to be used selectively in the following results, are now introduced

• (E1)∃k ∈ (0,1) ,∃c ∈ R₊ : α (u) ≤ c+ _Tku (u≥0)andα (u) ≥ c+ k

Tu (u <0)

• (E2)∃k∈(0,1) ,∃c∈R₊: |α (u)| ≤c+_Tk |u|,u ∈R

• (S1) αis twice continuously differentiable onR.

Lemma 2 If condition(E1)holds, then∀r>0

sup 0≤s≤T

EX∼hx,s[e

r|X|_]<∞.

(17)

Proof From the definition of A, we get

A(u)≤r|u| + k

2Tu 2₍

u ∈R)

and so

hx,s(u)≤exp −( u−x)2

2s +r|u| +

k

2Tu 2

(u ∈R,0≤s≤T) .

If u≥0

hx,s(u)≤exp −(

u−μ₊)2

2σ2 +

whereμ₊=

x+2sr 1−sk_T

, σ+= s

1−sk_T,r+=exp

−x2+x+2sr 1−sk_T

. Similarly if u<0

hx,s(u)≤exp −(

u−μ₋)2

2σ2 −

r₋,

whereμ₋=

x−2sr 1−sk_T

, σ+=σ−,r₋=exp

−x2_+x−2sr

1−sk_T

.

As k ∈ (0,1)it follows that r₊,r₋, μ₊, μ₋, σ₊, σ₋are bounded for s ∈ (0,T]

and the result follows.

For any two probability measuresM,Non a measurable space(E,E), let||M−N||

be their total variation metric, that is

||M−N|| := sup

A_∈E

|M(A)−N(A)|. (18)

Finally we require the following proposition concerning total variation distance. Again no proof will be provided for these elementary facts.

Proposition 1 (i) Suppose thatP1is a probability measure and B aP1-measurable set. DefineP2(A)=P1(A∩B)/P1(B). Then

P1−P2 =P1(Bc) .

(ii) LetP3be a probability measure with

dP3 dP1 ≥e

−δ

for some non-negative constantδ. Then

P1−P3 ≤1−e−δ≤δ .

We are now ready to state the following localised result:

Theorem 1 If condition(S1)holds and one of conditions(E1)and(E2)holds, then for any fixed x the law of the BBMZx_Δ converges towards the law of the diffusion processQ_Δx with respect to the total variation metric asΔ↓0:

lim

Δ↓0

_Zx

Δ−QΔx=0. (19)

Moreover, for allΔsufficiently small,ε >0

_Zx

Δ−QxΔ≤kxΔ3/2−ε, (20)

where the leading order constant kx is a continuous function of x (i.e., the rate of convergence is at leastOx

Proof To ease the notation letW=Wx_ΔandQ=Qx_Δin the scope of this proof, and in fact generally suppress any unneccessary dependence on the initial state x. We set

B to be the event

B= sup 0≤s≤Δ

|ωs−ω0| ≤Δ1/2−ε

.

We introduce a new probability measureQBvia the following Radon–Nikodym deriv-ative

dQB

dQ =1B/qx,Δ qx,Δ =Q[B] .

Similarly defineZB_{to beZconditioned on the event B. The triangle inequality gives}

Z−Q ≤ZB_−QB_+QB_−Q+ZB_−Z

=T1+T2+T3,

and we shall deal separately with the three terms.

First note that from (S1), there exists a constant k1such that

α2+₂ α(z)−α

2_+α 2 (x)

≤k1|x−z| ≤k1Δ1/2−

at least for z such that|z−x| ≤Δ1/2−. For T1note that ifdQ_dWB(ω)B = f(ω)then

inf

ω f(ω)≥eδsup_ω f(ω),

whereδ =2k1Δ3/2−. Therefore by Proposition1(ii),

T1≤k1Δ3/2−.

By Proposition1(i), the proof will thus be complete if we can demonstrate thatQ(BC)

andZ(BC)can be similarly bounded. From the definition ofZand an application of the Cauchy–Schwarz inequality, we get

Z(BC)=EW[1{BC}e A(w_Δ)_]

EW[eA(w_Δ)_] ≤

W(BC)12E_W[e2 A(wΔ)] 1 2

EW[eA(w_Δ)_] .

Now recall that by assumption(E1)(or(E2)), A is bounded by a quadratic function so

Z(BC)≤k2W(BC)21.

Similar considerations can be applied toQ(BC)yielding:

Q(BC)≤

W(BC)12E_W

e2 A(wΔ)−2₀Δφ(ωs)ds 1

2

E_WeA(wΔ)−0Δφ(ωs)ds

and asφis a positive function by Lemma 1∃k3>0 s.t. Q(BC)≤k3W(BC)12.

On the other hand, we can use the standard fluctuation result for BM:

W(BC)≤2W( sup 0_≤s_≤Δw

s ≥x+Δ1/2−)

=4W(wΔ≥x+Δ1/2−)=O(exp{−Δ−2/2}).

thus completing the proof.

The extension of this localised result to the global case relies on the maximal coupling inequality. The coupling method (see [11]) is already prevalent in the field of SDEs, mainly in the multi-dimensional case. Our approach is very similar to that of [7]. Given the relevance of the coupling method, it is sensible to briefly introduce its basic elements. We recall:

Definition 1 Let (E,E) be a Polish measurable space, and M,N be two proba-bility measures on (E,E). We state that a probability measure Pˆ on E2,E2 is a coupling of (M,N) if its marginals are M and N. We also say that a random object,F,P,X,Y, where,F,Pis a probability space andX,Y

is a F/E2-measurable function, is a coupling of (M,N) if the image measure P_X,Y−1

is a coupling of(M,N).

The power of the coupling argument comes from the following Lemma

Lemma 3 Let||M−N||be the total variation metric, that is

||M−N|| := sup

A∈E

|M(A)−N(A)| (21)

(Coupling Inequality) For any coupling,F,P,X,Yof(M,N)

||M−N|| ≤P_{X=Y (22)}

(Maximal Coupling Equality) There is a coupling,ˆ F,ˆ P,ˆ Xˆ,Yˆof(M,N)s.t.

||M−N|| = ˆPXˆ = ˆY

(23)

Theorem 2 (Main Convergence Theorem) If in addition to condition(S1)conditions

(E2)and(S2)also hold, where

• (S2) αis sub-quadratic, that is

_α(u)≤c1+u2 ₍

u ∈R) (24)

or condition(E1)and(S3)also hold, where

• (S3) αandαare sub-exponential, that is that:

|α (u)|,α(u)≤c1+ec|u| (u ∈R) (25)

then there exist a (myopic) coupling

_P,Xˆ,Yˆ

of

Qx

0,T,Y x,n

0,T

such that

lim

n_→∞Pˆ

ˆ

X = ˆY

=0 (26)

and the rate of convergence is at leastOΔ1/2−εfor anyε >0. As a consequence

of the coupling inequality, Yx₀,_,n_T converges towards Qx_0,T with respect to the total variation metric with the same rate of convergence.

Remark 1 If (E2) holds then condition (S2) is a weak assumption. As α2+α is bounded below, this additional condition means that the drift coefficient cannot oscil-late too quickly as|u| → ∞. Moreover in most diffusion models condition(E1)is satisfied, as otherwise the diffusion would exhibit explosive behaviour.

Proof We build a probability space

ˆ

,F,ˆ Pˆand two measurable functions

ˆ

,F,ˆ Pˆ→ ˜X(s,x,y) s∈(0, Δ]

→ ˜Y(s,x,y) s∈(0, Δ]

that define the maximal coupling of

Qx

0,Δ,Y

y

0,Δ

. A coupling of

Qx

0,T,Y x,n

0,T

that starts from this maximal coupling is constructed on the single time interval(0, Δ]. The initial step is defined by settingXˆ0= ˆY0=x andXˆs = ˜X(s,x,x) ,Yˆs = ˜Y(s,x,x)

for s∈(0, Δ]. Now suppose that

ˆ

X,Yˆ

is a coupling of

Qx

0,iΔ,Y x,n

0,iΔ

. Let

ˆ

Xi_Δ+s := ˜X

s,Xˆi_Δ,Yˆi_Δ

s∈(0, Δ]

ˆ

YiΔ+s := ˜Y

s,XˆiΔ,YˆiΔ

s∈(0, Δ]

independent ofXˆ,Yˆon [0,iΔ]. From the time homogeneity and Markov property of

the processes X,Y (limited to the set of times{iΔ;i=1, . . . ,n−1}) it follows that

ˆ

X,Yˆ

s∈(iΔ,(i+1)Δ]is a coupling of

QXˆiΔ

iΔ,(i+1)Δ,Y ˆ

YiΔ

iΔ,(i+1)Δ

process

ˆ

X,Yˆ

s∈[0,(i+1)Δ] is a coupling of

Qx

0_,(i₊1_)Δ,Y

x,n

0_,(i₊1_)Δ

. The induction

step is thus satisfied. No measurability problem arise in the definition of X˜,Y and˜

they can be chosen to be jointly measurable in(x,y), see [7] for the technical details. The coupling inequality yields

Yx0,n

0_,T −Q x0

0_,T

≤ ˆP∃s∈[0,T ]: ˆXs = ˆYs

≤ ˆPXˆs = ˆYson [0, Δ]

n−2

i₌0

ˆ

PXˆ = ˆY on((i+1) Δ,iΔ] | ˆXiΔ= ˆYiΔ

from the structure of the coupling

ˆ

X,Yˆ

, and the generic term of this last quantity

is exactly the maximal coupling of

QXˆiΔ

iΔ,(i+1)Δ,Y ˆ

YiΔ

iΔ,(i+1)Δ

. Hence, it is possible to use the maximal coupling equality. By defining a family of sub-probability measures onRbySi(A):= ˆP

ˆ

Xi_Δ= ˆYi_Δ∈A

and by using this last consideration

Yx0,n

0,T −Q x0

0,T

≤Yx0

0,Δ−Q

x0

0,Δ

+

n−2

i=0

_Ys

0,Δ−Qs0,ΔdSi(s)

≤kx0Δ

3/2−ε_+Δ3/2−ε

n−2

i=0

ksdSi(s)

Similarly to the proof of Theorem1

kx:= sup

|y|≤|x|+1

α2+2α (y)−α

2_+α

2 (x)

|y−x| .

We see that the behaviour in the tails as|x| → ∞is determined by

lim_|x|→∞k˜x :=lim|x|→∞

α2+2α (x)

|x|

ksdSi(s)≤k+k

|s|dSi(s)≤k+k

|s|dQx0

i_Δ(s) ,

where the last integral is the absolute momentum of XiΔ:E[|XiΔ| |X0=x0]. It is a known result that(E2)implies

E[|Xi_Δ| |X0=x0]≤E

sup 0≤s≤T

|Xs|2|X0=x0

<∞

∀i ∈N. Therefore

Yx0,n

0,T −Q x0

0,T

≤kx0Δ

3/2−ε_+Δ3/2−ε

n−2

i=0

k 1+E

sup 0≤s≤T

|Xs|2|X0=x0

!

≤dΔ3/2−ε

n= d TΔ

1/2−ε

forΔsufficiently small.

If instead condition (S3) holds, by looking at lim_|x|→∞k˜x, we obtain the

sub-exponential growth condition on kx, that is

kx ≤k

1+ek|x|

(x∈R) .

As a consequence

ksdSi(s)≤k+k

ek|s|dSi(s)≤k+k

ek|s|dZx0

i_Δ(s).

Again, the last integral isEek|YiΔ|_{|Y0=x0. As condition(E1)holds, Lemma2} implies that

sup

i=1,...,n−1

Eek|YiΔ|_|Y0=x0<∞.

Finally

Yx0,n

0,T −Q x0

0,T

≤kx0Δ

3/2−ε_+Δ3/2−ε

n−2

i=0

k 1+ sup

i=1,...,n−1

Eek|YiΔ|_|Y0=x0!

≤dΔ3/2−εn= d TΔ

1/2−ε_.

4 Conclusion

In this paper, we proved two convergence results about the Quasi-EA simulation scheme.

We shown the convergence of the law of the BBM Z to the law of the diffusion process X when both are started at the same value and the time interval[0, Δ]shrinks to zero. The convergence is obtained with respect to the total variation distance and an upper bound for the rate of the convergence is shown to beOx

Δ3/2−ε _{∀ε >0. The} notation underlines that this speed of convergence is not necessarily uniform in x.

We also extend this convergence to the global case of a fixed time interval[0,T]. In this case,[0,T]is uniformly partitioned in n intervals and the convergence is obtained as n ↑ ∞. The main difficulty that the starting point for X and Y on each single interval is not the same anymore is overcome using the coupling method. We are thus able to construct a successful myopic coupling of the Quasi-EA and the diffusion. Consequently, we obtain the convergence with respect to the total variation distance and an upper bound for the rate of convergenceOΔ1/2−ε∀ε >0.

Very efficient algorithms to sample from the parametric family of densities

"

hx,T

#

x∈Xare introduced in [8]. However, while the quasi-EA is more efficient than

the Euler scheme, a brief simulation study suggests that Predictor–Corrector schemes result in a more accurate simulation method.

As already noted, the hypothesis used in the derivation of these results include the conditions that permit the simulation of X using EA3. It is possible to weaken the condition used in our work and prove the convergence of the Quasi-EA even in models where EA3 can not be applied, and this will be the focus of future research. This could be worked out in future research. The main contribute in this paper is to obtain an insight into the role of the BBM Z in the context of EA.

Acknowledgments All three authors would like to thank EPSRC for support through grant EP/D505607/2. Roberts would additionally like to thank EPSRC for support through grants EP/20620/01 and EP/S61577/01.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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