THE STRONG WEAK CONVERGENCE OF THE QUASI-EA
STEFANO PELUCHETTI, GARETH O. ROBERTS, BRUNO CASELLA
In this paper we investigate the onvergene of a novel simulation
sheme to the target diusion proess. This sheme, the Quasi-EA, is losely
related to the Exat Algorithm (EA) for diusion proesses, as it is obtained by
negleting the rejetion step in EA. We prove the existene of a myopi oupling
between the Quasi-EA and the diusion. Moreover, an upper bound for the
oupling probability is given. Consequently we establish the onvergene of
the Quasi-EA to the diusion with respet to the total variation distane.
Abstrat.
1.
Introdution
In this paper, we shall present two onvergene results about a novel simulation
sheme to the target diusion proess. This sheme is losely related to the Exat
Algorithm (EA) for the simulation of diusion proess whih was introdued in
Beskos and Roberts [2005℄. In partiular, the simulation sheme we onsider is
essentially EA without the aeptane-rejetion orretion.
This sheme (whih we all the Quasi-EA) is studied for two reasons. Firstly
we are interested in the properties of Quasi-EA as a simulation sheme in its own
right. Seondly a thorough understanding of Quasi-EA ontributes to a fuller understanding of EA sheme itself.
Its main appeal is that it allows for the exat simulation (i.e. free from any time
disretisation error) of any skeleton of the diusion sample path. Moreover, it is
possible to simulate exatly from some lasses of path-dependent funtionals of the
diusion proess. Beause EA plays a entral rule in our work, we briey introdue
the main ideas behind EA. For a more exhaustive exposition of EA we refer to
Beskos et al. [2006a℄ and Beskos et al. [2006b℄.
We onsider the diusion proess Y a unique strong solution of the Stohasti
Dierential Equation (SDE)
(1)
dYt = b (Yt ) dt + σ (Yt ) dBt
Y0 = y
0≤t≤T
B is the salar Brownian Motion (BM) on the bounded time interval [0, T ] and y
is the initial ondition. The drift oeient b and the diusion oeient σ are
impliitly assumed to satisfy the proper onditions that imply the existene and
uniqueness of suh diusion Y .
The mild assumption that σ is ontinuously dierentiable and stritly positive
guarantees the existene and uniqueness of a bjetive funtion η suh that the
Date : January 2008.
1
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
2
transformed diusion proess Xt := η (Yt ) satises the SDE
(2)
0≤t≤T
dXt = α (Xt ) dt + dBt
X0 = x := η (y)
The drift oeient α depends on the funtional form of b and σ and the diusion
oeient is the unitary onstant funtion. The SDE (2) is assumed to admit a
unique strong and non-explosive solution and we denote with X the state spae of
X . From now on this SDE will be our starting point.
Let QxT and WxT denote the law of the diusion X and the law of a BM respetively on [0, T ] both started at x. From now on the following hypotheses are
assumed to hold:
(3)
• ∀x ∈ X QxT ≪ WxT and the Radon-Nikodym derivative is given by the
Girsanov's formula:
(
)
T
dQxT
1 T 2
(ω) = exp
α (ωs ) dωs −
α (ωs ) ds
dWxT
2 0
0
• α is ontinuously dierentiable on X;
• α2 + α′ is bounded below on X.
We introdue the Biased Brownian Motion (BBM) Z and its law ZxT . This proess
is dened as a BM on [0, T ] started at x and onditioned on having its terminal
value ZT distributed aording to the density
(
)
(u − x)2
hx,T (u) := ηx,T × exp A (u) −
(4)
2T
u
Here A (u) := c α (r) dr for some c ∈ X and the normalising onstant ηx,T is
assumed to be nite. Hene, onditionally on the value of ZT the proess Z is
distributed as a Brownian Bridge (BB). Given the hypothesis it is possible to prove
that
( )
T
dQxT
α2 + α′
(5)
(ω) = ηx,T exp {−A (x)} exp −
(ωs ) ds
dZxT
2
0
( )
(6)
T
∝ exp −
φ (ωs ) ds
0
≤1
where φ (u) := α2 (u) + α′ (u) /2 − inf r∈X α2 (r) + α′ (r) /2. Equation (6) suggests the use of a rejetion sampling algorithm to generate realisations from QxT .
However it is not possible to generate a sample from Z , being Z an innite dimensional variate, and moreover it is not possible to ompute analytially the value of
the integral in (6). For ease of exposition we only onsider the ase of EA1, where
α2 + α′ is assumed to be bounded. It should be noted that this hypothesis an
be weakened or even removed, leading to EA2 (Beskos et al. [2006a℄) and to EA3
(Beskos et al. [2006b℄) respetively.
We denote by m the nite supremum of φ. Let Φ = (X , Ψ) be a unit rate Poisson
Point Proess (PPP) on [0, T ]× [0, m] and let N be the number of points of Φ below
φ (ωs ) onditionally on a path ω . If the event Γ is dened as Γ (ω, Φ) := {N = 0}
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
it follows that
(7)
( Pr [Γ|ω] = exp −
T
φ (ωs ) ds
0
3
)
This equivalene is an immediate onsequene of the properties of PPPs. The lhs
of (7) is the probability that no points of Φ fall in the region of the plane bounded
by 0 and φ (ωs ), a region whose area is given by the integral on the rhs of (7). Thus
there is no need to alulate the integral analytially. We have impliitly assumed
that it was possible to generate the innite dimensional variate ω ∼ ZxT . However
to ompute the the aeptane event Γ we only need to know the value of Z on a
nite set of random times only, this set orresponding to the time oordinates of
the realisations of Φ. It is then possible to exhange the order in whih Z and Φ
are generated, obtaining Algorithm 1.
Algorithm 1 Exat Algorithm 1 (EA1)
1.
2.
3.
4.
SAMPLE ZT ∼ hx,T
SAMPLE Φ
= (X , Ψ)
SAMPLE Zχj ; 1 ≤ j ≤ |X | ∼ ZxT | ZT
IF Γ RETURN S := Zχj ; 1 ≤ j ≤ |X | , ZT
ELSE GOTO 1
As already observed, step 3 of Algorithm 1 results in the generation of independent BBs. EA1 returns a skeleton S distributed aording to the true law of the
diusion X . Furthermore, from Beskos et al. [2006a℄ it is known that the onditional law ZxT | S an be expressed as the produt measure of independent BBs.
Using this result it is possible to omplete S on any nite arbitrary number of times
if required and even to generate (exatly) ertain funtionals of the path of X .
The remaining part of this paper is organised as follows. In Setion 2 the QuasiEA is motivated and introdued and the onnetion with EA is examined. In
Setion 3 the two main theorems are proved. We start by establishing a loal
onvergene result that is further extended by means of the maximal oupling inequality. We show that the Quasi-EA is an aurate ontinuous time approximation
of the law of the diusion proess. In fat we prove the existene of a myopi (sequential) oupling between the diusion and the simulation sheme. The existene
of suh a oupling implies the onvergene with respet to the total variation distane, a strong form of weak onvergene aording to Jaka and Roberts [1997℄.
Setion 4 onludes the paper with some remarks about possible future researh on
the topi and pratial onsiderations about our sheme.
The Euler sheme does not generally onverge with respet to the total variation
distane, see Peter Glynn and Goodman [2006℄. However, under mild tehnial
onditions, the Euler sheme does onverge with respet to total variation distane
if the diusion proess has a onstant diusion term (see Jaod and Shiryaev [1987℄).
Genon-Catalot [2007℄ extended this result to prove that the rate of onvergene is
of order ∆1/2 , where ∆ is the length of the (equally spaed) disretisation interval.
Here we fous on the existene of a sequential (or myopi) oupling between
the diusion and the simulation sheme. A sequential oupling is a oupling in
whih, for eah time inrement in turn, we try to maximise the probability that
the two proesses stay together. It is therefore a natural and pratial oupling.
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
4
Whilst total variation onvergene is equivalent to the existene of a oupling, and
thus, the existene of a sequential oupling implies total variation onvergene, the
onverse is false, as we shall see for the Euler sheme. To illustrate this, onsider
(8)
(9)
dYt
dXt
= αdt + dBt
= dBt
where α is a onstant and Bt is a salar BM on [0, T ]. Both Xt and Yt start at
the same value. Let [0, T ] be partitioned in intervals of length ∆. Then by Taylor
expansion arguments it is possible
to show that the total variation distane between
Y∆ and X∆ is of order O ∆1/2 , hene the probability of the oupling sueeds is
1/∆
1 − ∆1/2
and this last quantity tends to 0 and ∆ ↓ 0.
Moreover, the bound given by the sequential oupling on the total variation
distane is not very strit. A trivial example is the Ornstein-Uhlenbek proess.
Simple omputations show that its Euler disretisation onverges with respet to the
total variation distane on the single interval [0, ∆] with rate ∆. Similarly QuasiEA an be shown to onverge with order ∆3/2 on the same interval (implying that
the bound (20) in Theorem 1 is quite sharp). We already stated that the Euler
disretisation onverges on the whole interval [0, T ] with rate ∆1/2 . However, the
oupling inequality of Theorem (2) implies that Quasi-EA onverges with the same
order ∆1/2 .
Finally, it should be observed that the hypothesis used in the derivation of the
two main onvergene theorems inlude the onditions that permits the simulation
of X using EA3. Beause of this we gain more insight into the role of the proposal
measure ZxT in the ontext of EA.
2.
The quasi-EA
The main idea behind the onstrution of Quasi-EA is very simple. In step 2
of Algorithm 1 we sample a PPP with unit rate on [0, T ] × [0, m]. The aeptane
rate in EA inreases as T ↓ 0, as the likelihood that no points from Φ are sampled
inreases too. Clearly in this eventuality the proposed path is aepted. Similar
onsiderations an be made in the ase of EA3 too. As the aeptane rate an
be interpreted as a measure of the quality of the proposal measure, we develop
a sheme that always aept the proposal variate distributed as ZxT . Given the
previous onsiderations, this approximation is aurate only if T is quite small.
Hene the time interval [0, T ] is partitioned into smaller intervals on whih the
sheme is applied sequentially.
We now desribe the sheme more preisely. The time interval [0, T ] is divided
into n smaller intervals having the same length ∆ = T /n. The ontinuous time
sheme Y is dened by the following equations
(10)
Y0 = x
(11)
Yi∆ ∼ hYi∆ ,∆
(12)
Ys ∼ BB Yi∆ , Y(i+1)∆ , ∆
(i = 1, · · · , n)
(i∆ < s < (i + 1) ∆)
where BB (x, dy, t) is the measure of a BB starting at (0, x) and ending at (t, dy).
It an be easily seen the proess Y thus dened onsists of n sequential BBMs.
The simulation of the Quasi-EA involves the sampling of the sequene of random
variables in (11) only. However we are going to prove a stronger result than the
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
5
onvergene of the disretized proess {Yi∆ ; i = 0, · · · , n} to the diusion proess.
We shall prove that the law of the ontinuous time proess Y dened by equations
(10)-(12) onverges with respet to the total variation distane to the law of the
diusion proess X as n ↑ ∞. This result suggests that the BBs are good proess
to ll-in the gaps between the simulated values of the disretized proess.
We denote by Yx,n
0,T the probability measure indued by the Quasi-EA sheme
Y started at x and onsisting of n steps on [0, T ]. Consistently with the previous
notation Yy0,∆ denotes the probability measure indued by this sheme on the single
step [0, ∆] when Y0 = y . To sum up
{Ys ; 0 ≤ s ≤ T | Y0 = x} ∼ Yx,n
0,T
(13)
{Ys ; 0 ≤ s ≤ ∆ | Y0 = y} ∼ Yy0,∆ = Zy∆
(14)
3.
Two onvergene results
The two onvergene theorems require the two following lemmas.
Lemma 1. Let f
∆>0
:R→R
be a ontinuous funtion, suh that for suiently small
1
√
2π∆
(15)
then (for any xed x ∈ R)
R
1
lim √
∆↓0
2π∆
(16)
|f (y)| e−
f (y) e−
(y−x)2
2∆
(y−x)2
2∆
dy < ∞
dy = f (x)
R
Proof. Let ε > 0 , f (y) = f (y) 1{y∈Bε (x)} + f (y) 1{y∈Bcε (x)} where B ε (x) is the
c
losed ball entred in x with radius ε and B ε (x) is the omplementary set of B ε (x).
Remember that N (x, ∆) onverges weakly to the Dira delta funtion δx as it is
easily proved via harateristi funtion arguments. Additionally, f (y) 1{y∈Bε (x)}
is bounded and the measure with respet to δx of the set of its points of disontinuity
is zero. Using the Skorohod representation theorem and the bounded onvergene
theorem, it follows that
(y−x)2
1
lim √
f (y) 1{y∈B ε (x)} e− 2∆ dy = f (x)
∆↓0
2π∆ R
(y−x)2
|f (y)| −
e 2∆ 1{y∈Bc (x)} is dereasing in ∆ (and integrable for suiently small
As √
2π∆
ε
∆), it is possible to apply the dominated onvergene theorem obtaining
1
(y−x)2
lim √
f (y) 1{y∈B c (x)} e− 2∆ dy ε
∆↓0
2π∆
R
(y−x)2
1
≤ lim √
|f (y)| 1{y∈B c (x)} e− 2∆ dy = 0
ε
∆↓0
2π∆ R
The linearity of the integral ends the proof.
For ease of exposition we assume that the state spae of X is R. This assumption
does not have any impat on our results. The following onditions, to be used
seletively in the following results, are now introdued
• (E1) ∃k ∈ (0, 1) , ∃c ∈ R+ : α (u) ≤ c + Tk u (u ≥ 0) and α (u) ≥ c +
k
T u (u < 0)
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britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
6
• (E2) ∃k ∈ (0, 1) , ∃c ∈ R+ : |α (u)| ≤ c + Tk |u| , u ∈ R
• (S1) α is twie ontinuously dierentiable on R
Lemma 2. If ondition (E1) holds, then ∀r > 0
(17)
sup EX∼hx,s [er|X| ] < ∞
0≤s≤T
Proof. From the denition of A we get
A (u) ≤ r |u| +
and so
(
k 2
u (u ∈ R)
2T
2
(u − x)
k 2
+ r |u| +
u
hx,s (u) ≤ exp −
2s
2T
If u ≥ 0
)
(u ∈ R, 0 ≤ s ≤ T )
(
)
2
(u − µ+ )
hx,s (u) ≤ exp −
r+
2
2σ+
n 2
o
s
x +x+2sr
,
σ
=
where µ+ = x+2sr
,
r
=
exp
−
. Similarly if u < 0
+
+
sk
sk
sk
1− T
1− T
(1− T )
)
(
2
(u − µ− )
hx,s (u) ≤ exp −
r−
2
2σ−
n 2
o
x +x−2sr
,
σ
=
σ
,
r
=
exp
−
where µ− = x−2sr
.
+
−
−
sk
sk
1−
1−
T
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 measures M, N on a measurable spae (E, E), let ||M − N||
be their total variation metri, that is
(18)
||M − N|| := sup |M (A) − N (A)|
A∈E
We are now ready to state the following loalised result:
Theorem 1. If ondition (S1) hold, then for any xed x the law of the BBM
onverges towards the law of the diusion proess Qx∆ with respet to the total
variation metri as ∆ ↓ 0 :
Zx∆
(19)
lim ||Zx∆ − Qx∆ || = 0
∆↓0
and for ∆ suiently small, ε > 0
(20)
||Zx∆ − Qx∆ || ≤ kx ∆3/2−ε
where the leading order onstant kx is a ontinuous funtion of x (i.e. the rate of
onvergene is at least Ox ∆3/2−ε for any ε > 0).
Proof. To ease the notation let W = Wx∆ and Q = Qx∆ in the sope of this proof.
We introdue a new probability measure QT r via the following Radon-Nikodym
derivative
dQT r
= 1T r /qx,∆
dQ
qx,∆ = Q [T r]
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britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
7
where the measurable event T r is dened as T r = sup0≤s≤∆ |ωs − ω0 | ≤ ∆1/2−ε .
Consequently,
2
′
dQT r
− ∆ α +α
(ωs )−inf ω
2
= 1T r e 0
dZ
“
α2 +α′
2
”
ds
2
′
(ωs )−inf ω α +α
2
”
ds
“
−
∆
α2 +α′
2
= 1T r e 0
“ 2
′
− ∆ α +α
(ωs )−inf ω
2
= EZ 1T r e 0
dqx,∆
/dα,∆ × qx,∆
/dqα,∆
” α2 +α′
2
ds
The triangular inequality gives
sup |Z (A) − Q (A)| ≤ sup Z (A) − QT r (A) + sup QT r (A) − Q (A)
A∈C
A∈C
A∈C
We now proeed by establishing the onvergene of the two terms on the rhs of the
inequality. Conerning the rst term we have that
Z (A) =
1T r + 1T rc dZ (ω)
A
Q
Tr
1
(A) =
dqx,∆
1T r e
−
t “ α2 +α′
2
0
(ωs )−inf ω
α2 +α′
2
α2 +α′
2
Regarding the seond term it sue to observe that
1
dx,∆
QT r (A) =
e
−
A
1
dx,∆ qx,∆
ds
dZ (ω)
A
and trivial omputations gives
sup Z (A) − QT r (A)
A∈C
“ 2
′
1
− ∆ α +α
(ωs )−inf ω
2
≤Z [T rc ] +
2 1T r 1 − e 0
dqx,∆
Q (A) =
”
∆ “ α2 +α′
2
0
1T r e
(ωs )−inf ω
−
α2 +α′
2
∆ “ α2 +α′
0
2
”
ds
” ds c
dZ
(ω)
+
Z
[T
r
]
dZ (ω)
(ωs )−inf ω
α2 +α′
2
”
ds
dZ (ω)
A
and therefore similar omputations leads to
sup QT r (A) − Q (A)
A∈C
“ 2
′
1
1
− ∆ α +α
(ωs )−inf ω
c
2
≤
Z [T r ] +
EZ 1T rc e 0
dqx,∆
dx,∆
α2 +α′
2
”
ds
Combining the two results yields
sup |Q (A) − Z (A)|
“ 2
′
1
1
− 0∆ α +α
(ωs )−inf ω
c
2
(2 + dqx,∆ ) Z [T r ] + 2 +
EZ 1T rc e
≤
dqx,∆
dx,∆
A∈C
−
∆ “ α2 +α′
(ω )−inf
α2 +α′
α2 +α′
2
” ds
”
ds
s
ω
2
2
Moreover dqx,∆ ∈ (0, 1) and the funtions 1T r e 0
and 1T r
are both positive and inreasing as ∆ ↓ 0. By two appliations of the monotone
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
onvergene theorem it follows that
“ 2
′
(ωs )−inf ω
− ∆ α +α
2
lim EZ 1T r e 0
α2 +α′
2
∆↓0
= lim Z [T r]
8
” ds
∆↓0
As we will shortly show Z [T rc ] ↓ 0 as ∆ ↓ 0, so dqx,∆ ↑ 1. Similarly an appliation
of the monotone onvergene theorem shows that dx,∆ ↑ 1 as ∆ ↓ 0. As a onsequene we an nd two positive onstants c1 , c2 (independent of x) suh at that
for ∆ suiently small
sup |Q (A) − Z (A)|
“ 2
′
− 0∆ α +α
(ωs )−inf ω
c
2
≤c1 Z [T r ] + c2 EZ 1T rc e
A∈C
α2 +α′
2
” ds
All that remains to be heked is the rate of onvergene of the two terms of the
inequality's rhs. The rst one is
h
i
1
EW T rc eA(ωt )
Z [T rc ] =
EW eA(ωt )
and an appliation of the Cauhy-Shwartz inequality to EW T rc eA(ωt ) shows that
2A(ω ) 21
∆
c
c 12 EW e
EW [T r ] ≤ (W [T r ])
EW eA(ω∆ )
1
1
From Lemma 1 EW eA(ω∆ ) → eA(x) as ∆ ↓ 0 and EW e2A(ω∆ ) 2 → e2A(x) 2 =
eA(x) . Hene we an nd a positive onstant c3 (independent of x) so that for ∆
suiently small
1
EW [T rc ] ≤ c3 (W [T rc ]) 2
By writing W0 for the law of the BM started at zero on [0, ∆], by knowing that if
B is a BM , −B is a BM too, and by using the reetion priniple, we obtain
1
W sup |ωs − ω0 | > ∆ 2 −ε
0≤s≤∆
h
i
1
≤4W0 ω∆ > ∆ 2 −ε
By hanging the variable and by applying the bound
∞
e−
y2
2
x
it follows that
1
dy ≤
1 − x2
e 2
x2
(W [T rc ]) 2 ≤ c4 ∆ε e
−c
1
4∆
that dereases exponentially
as ∆ ↓ 0“ . 2 ′
− ∆ α +α
(ωs )−inf ω
2
We now onsider EZ 1T r 1 − e 0
Let
2ε
α2 +α′
2
”
ds
.
α2 + α′
(ωs ) > −∞
ω∈T r s∈[0,∆]
2
r = arg inf
inf
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britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
For ∆ suiently small
kx :=
sup
|y|≤|x|+1
≥
2
2 ′
α +α
2 (y) −
sup
|x−y|≤∆1/2−ε
′
α2 +α′
2
|y − x|
2 ′
α +α
2 (y) −
9
(x)
α2 +α′
2
|y − x|
(x)
and as (S1) ⇒ α +α
is loally Lipshitz it follows that kx is ontinuous in x and
2
for ∆ suiently small
2
α + α′
α2 + α′
(ωs ) −
(r) ≤ kx |ωs − r|
2
2
≤ 2kx ∆1/2−ε
by the denition of T r . Finally
“ 2
′
− 0∆ α +α
(ωs )−inf ω
2
EZ 1T r 1 − e
≤1 − e2kx ∆
α2 +α′
2
3/2−ε
”
ds
≤2kx ∆3/2−ε
for suiently small ∆.
The extension of this loalised result to the global ase relies on the maximal
oupling inequality. The oupling method (see Thorisson [2000℄) is already prevalent in the eld of SDEs, mainly in the multi-dimensional ase. Our approah is
very similar to that of Odasso [2005℄. Given the relevane of the oupling method
it is sensible to briey introdue its basi elements. We reall:
Denition 1. Let (E, E) be a Polish measurable spae, and M, N be two proba
bility measures on (E, E). We state that a probability measure P̂ on E 2 , E 2 is
a oupling of (M, N) if its marginals are M and N. We also say that a random
objet (Ω′ , F ′ , P′ , (X ′ , Y ′ )), where (Ω′ , F ′ , P′ ) is a probability spae and (X ′ , Y ′ )
is a F ′ /E 2 -measurable funtion, is a oupling of (M, N) if the image measure
−1
P′ (X ′ , Y ′ ) is a oupling of (M, N).
The power of the oupling argument omes from the following Lemma
Lemma 3. Let ||M − N|| be the total variation metri, that is
(21)
(Coupling
(22)
(Maximal
s.t.
(23)
||M − N|| := sup |M (A) − N (A)|
A∈E
Inequality) For any oupling (Ω′, F ′, P′, (X ′, Y ′ )) of (M, N)
||M − N|| ≤ P′ [X ′ 6= Y ′ ]
Coupling Equality) There is a oupling
h
i
||M − N|| = P̂ X̂ 6= Ŷ
Ω̂, F̂, P̂, X̂, Ŷ
of (M, N)
This oupling is alled the maximal oupling of (M, N).
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
10
Theorem 2. (Main Convergene Theorem) If in addition to ondition (S1)
onditions (E2) and (S2) also hold, where
• (S2) α′ is sub-quadrati , that is
(24)
|α′ (u)| ≤ c 1 + u2
or ondition (E1) and (S3) also hold, where
• (S3) α and α′ are sup-exponential , whih are:
(25)
|α (u)| , |α′ (u)| ≤ c 1 + ec|u|
(u ∈ R)
(u ∈ R)
b X̂, Ŷ of Qx , Yx,n suh that
then there exist a (myopi) oupling P,
0,T
0,T
h
i
lim P̂ X̂ 6= Ŷ = 0
(26)
n→∞
and the rate of onvergene is at least O ∆1/2−ε for any ε > 0. As a onsequene
x
of the oupling inequality, Yx,n
0,T onverges towards Q0,T with respet to the total
variation metri with the same rate of onvergene.
Remark 1. If (E2) holds then ondition (S2) is a weak assumption. As α2 + α′
is bounded below, this additional ondition means that the drift oeient annot
osillate too quikly as |u| → ∞. Moreover in most diusion models ondition (E1)
is satised, as otherwise the diusion would exhibit explosive behaviour.
Proof. We build a probability spae Ω̂, F̂, P̂ and two measurable funtions
Ω̂, F̂, P̂ → X̃ (s, x, y)
s ∈ (0, ∆]
→ Ỹ (s, x, y)
s ∈ (0, ∆]
that dene the maximal oupling of Qx0,∆ , Yy0,∆ . A oupling of Qx0,T , Yx,n
0,T that
starts from this maximal oupling is onstruted on the single time interval (0, ∆].
The initial step is dened by setting X̂0 =
=
Ŷ0 =
x and X̂s = X̃(s, x, x) , Ŷs Ỹ (s, x, x) for s ∈ (0, ∆]. Now suppose that X̂, Ŷ
Let
X̂i∆+s := X̃ s, X̂i∆ , Ŷi∆
Ŷi∆+s := Ỹ s, X̂i∆ , Ŷi∆
is a oupling of Qx0,i∆ , Yx,n
0,i∆ .
s ∈ (0, ∆]
s ∈ (0, ∆]
independent of X̂, Ŷ on [0, i∆]. From the time homogeneity and Markov property
of theproesses
X, Y (limited to the set of times {i∆; i = 1, · · · , n − 1}) it follows
Ŷi∆
i∆
that X̂, Ŷ
is a oupling of QX̂
and that the
i∆,(i+1)∆ , Yi∆,(i+1)∆
s∈(i∆,(i+1)∆]
extended proess X̂, Ŷ
is a oupling of Qx0,(i+1)∆ , Yx,n
0,(i+1)∆ . The
s∈[0,(i+1)∆]
indution step is thus satised. No measurability problem arise in the denition of
X̃, Ỹ and they an be hosen to be jointly measurable in (x, y), see Odasso [2005℄
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
11
for the tehnial details. The oupling inequality yields
x0 ,n
0 Y0,T − Qx0,T
h
i
h
i
≤P̂ X̂ 6= Ŷ = P̂ ∃s ∈ [0, T ] : X̂s 6= Ŷs
h
i
≤P̂ X̂s 6= Ŷs on [0, ∆]
n−2
X
h
i
P̂ X̂ 6= Ŷ on ((i + 1) ∆, i∆] | X̂s = Ŷs on[0, i∆]
n−2
X
h
i
P̂ X̂ 6= Ŷ on ((i + 1) ∆, i∆] | X̂i∆ = Ŷi∆
i=1
h
i
=P̂ X̂s 6= Ŷs on [0, ∆]
i=0
from the struture of the oupling X̂, Ŷ , and the generi term of this last quantity
Ŷi∆
i∆
is exatly the maximal oupling of QX̂
i∆,(i+1)∆ , Yi∆,(i+1)∆ . Hene, it is possible to
use the maximal oupling
equality. By
h
i dening a family of sub-probability measures
on R by Si (A) := P̂ X̂i∆ = Ŷi∆ ∈ A and by using this last onsideration we obtain
by the disintegration of the onditional probability that
x0 ,n
x0 Y0,T − Q0,T
n−2
X 0
0 ≤ Yx0,∆
− Qx0,∆
+
Ysi∆,(i+1)∆ − Qsi∆,(i+1)∆ dSi (s)
i=0
n−2
X 0
0 Ys − Qs dSi (s)
= Yx0,∆
− Qx0,∆
+
0,∆
0,∆
i=0
n−2
X
3/2−ε
≤kx0 ∆3/2−ε + ∆
ks dSi (s)
i=0
As in the proof of Theorem 1
kx :=
sup
|y|≤|x|+1
2 ′
α +α
2 (y) −
α2 +α′
2
|y − x|
(x)
We see that the behaviour in the tails as |x| → ∞ is determined by
2 ′
α +α
2 (x)
lim|x|→∞ k̃x := lim|x|→∞
|x|
If ondition (S2) holds the positive funtion k̃x an diverge at most linearly. So
kx ≤ k (1 + |x|) and
ks dSi (s) ≤ k + k
|s| dSi (s) ≤ k + k
|s| dQxi∆0 (s)
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
12
where the last integral is the absolute momentum of Xi∆ : E [|Xi∆ | | X0 = x0 ]. It
is a known result that (E2) implies
2
E [|Xi∆ | | X0 = x0 ] ≤ E sup |Xs | | X0 = x0 < ∞
0≤s≤T
∀i ∈ N. Therefore
x0 ,n
0 Y0,T − Qx0,T
3/2−ε
≤kx0 ∆
3/2−ε
+∆
n−2
X
i=0
≤d∆3/2−ε n =
2
k 1 + E sup |Xs | | X0 = x0
0≤s≤T
d 1/2−ε
∆
T
for ∆ suiently small.
If instead ondition (S3) holds, by looking at lim|x|→∞ k̃x , we obtain the subexponential growth ondition on kx , that is
kx ≤ k 1 + ek|x| (x ∈ R)
As a onsequene
ks dSi (s) ≤ k + k e dSi (s) ≤ k + k ek|s| dZxi∆0 (s)
Again, the last integral is E ek|Yi∆ | | Y0 = x0 . As ondition (E1) holds, Lemma 2
implies that
h
i
sup E ek|Yi∆ | | Y0 = x0 < ∞
k|s|
i=1,··· ,n−1
Finally
x0 ,n
0 Y0,T − Qx0,T
≤kx0 ∆3/2−ε + ∆3/2−ε
≤d∆3/2−ε n =
n−2
X
i=0
k 1+
sup
i=1,··· ,n−1
d 1/2−ε
∆
T
h
i
E ek|Yi∆ | | Y0 = x0
4.
Conlusion
In this paper we proved two onvergene results about the Quasi-EA simulation
sheme.
We shown the onvergene of the law of the BBM Z to the law of the diusion proess X when both are started at the same value and the time interval
[0, ∆] shrinks to zero. The onvergene is obtained with respet to the total variation distane
and an upper bound for the rate of the onvergene is shown to be
Ox ∆3/2−ε ∀ε > 0. The notation underlines that this speed of onvergene is not
neessarily uniform in x.
We also extend this onvergene to the global ase of a xed time interval [0, T ].
In this ase [0, T ] is uniformly partitioned in n intervals and the onvergene is
obtained as n ↑ ∞. The main diulty that the starting point for X and Y on
eah single interval is not the same anymore is overome using the oupling method.
CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism
britishTHE STRONG WEAK CONVERGENCE OF THE QUASI-EA
13
We are thus able to onstrut a suessful myopi oupling of the Quasi-EA and
the diusion. Consequently we obtain the onvergene with respet to the total
variation distane and an upper bound for the rate of onvergene O ∆1/2−ε ∀ε >
0.
Very eient algorithms to sample from the parametri family of densities {hx,T }x∈X
are introdued in Peluhetti [2007℄. However, while the quasi-EA is more eient
than the Euler sheme, a brief simulation study suggests that Preditor-Corretor
shemes result in a more aurate simulation.
As already noted, the hypothesis used in the derivation of these results inlude
the onditions that permits the simulation of X using EA3. It is possible to weaken
the ondition used in our work and prove the onvergene of the Quasi-EA even in
models where EA3 an not be applied, and this will be the fous of future researh.
This ould be worked out in future researh. The main ontribute in this paper is
to obtain an insight into the role of the BBM Z in the ontext of EA.
Referenes
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A. Beskos, O. Papaspiliopoulos, and G.O. Roberts. A new fatorisation of diusion
measure and nite sample path onstrution. Methodology and Computing in
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Valentine Genon-Catalot. Personal omuniation, 2007.
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Rennes & Eole Normale Supérieure de Cahan, antenne de Bretagne, 2005.
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thesis, IMQ - Universitá Commeriale Luigi Booni, 2007.
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simulating sdes under the total variation norm. 2006.
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CRiSM Paper No. 08-18, www.warwick.ac.uk/go/crism