Statistical convergence of Markov experiments to diffusion limits

0.23 MB·English

by  Valentin Konakov

Statistical convergence of Markov experiments
to diffusion limits.
∗
2
Valentin Konakov†, Enno Mammen‡ and Jeannette Woerner§
1
0
2
January 30, 2012

n
a
Abstract
J

0
3 Assumethatoneobservesthek-th,2k-th,....,nk-thvalueofaMarkovchainX1,h,...,Xnk,h. That

means we assume that a high frequency Markov chain runs in the background on a very fine time
]
T
gridbutthatitisonlyobservedonacoarsergrid. Thisasymptoticsreflectsasetupoccurringinthe
S
. highfrequencystatistical analysisforfinancialdatawherediffusionapproximationsareusedonlyfor
h
at coarsertimescales. InthispaperweshowthatunderappropriateconditionstheL1-distancebetween
m
thejoint distribution of theMarkov chain and thedistribution of thediscretized diffusion limit con-
[
vergestozero. TheresultimpliesthattheLeCamdeficiencydistancebetweenthestatisticalMarkov

1
v experiment and its diffusion limit converges to zero. This result can be applied to Euler approxima-
7
tionsfor thejoint distribution ofdiffusions observed at points∆,2∆,,,,,n∆. Thejoint distribution
0
3 can be approximated by generating Euler approximations at the points ∆k−1,2∆k−1,,,,,n∆. Our
6
1. result implies that underour regularity conditions the Euler approximation is consistent for n→∞
0 if nk−2 →0.
2
1 Keywords: Markov chains, diffusion processes, deficiency distance, high frequency time series,
:
v
Euler approximations
i
X
Mathematics Subject Classifications: primary 62G07, secondary 60G60
r
a
Short title: Statistical convergenceof Markov experiments
∗Thisstudywascarriedoutwithin”TheNationalResearchUniversityHigherSchoolofEconomics”AcademicFundPro-
gramin2012-2013, researchgrantNo.11-01-0083. Supportbygrant436RUS113/467/81-2 fromtheDeutscheForschungs-
gemeinschaftisalsoacknowledged. TheresearchofEnnoMammenandJeannette Woerner wassupportedbytheGerman
Science Foundation (DFG) in the framework of the German-Swiss Research Group FOR 916 ”Statistical Regularization
andQualitativeConstraints”.
†HigherSchool ofEconomics,PokrovskiiBoulevard11,103012 Moscow,Russia. E-mail: [email protected]
‡Department of Economics, University of Mannheim, L7,3-5, 68229 Mannheim, Germany. E-mail:
[email protected]
§Technische Universit¨at Dortmund, Fakult¨at fu¨r Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany. E mail:
[email protected]
1 Introduction.
In this paper we consider approximations of the joint distribution of a partially observed Markov chain
by the law of a discretely observed diffusion. More precisely we consider a Markov chain X ,...,X
1,h nk,h
with values at nk time points. This time points are equalto h,2h,...,nkhwhere h is a time intervalthat
convergestozero. Weassumethatthisprocessisonlyobservedateachk-thpoint,i.e.atthetime points
kh,2kh,...,nkh. That means we assume that a high frequency Markov chain runs in the background on
a very fine time grid but that it is only observed on a coarser grid. This asymptotics reflects a set up
occurring in the high frequency statistical analysis for financial data where diffusion approximations are
usedforcoarsertimescales. Forthefinestscalediscretepatterninthepriceprocessesbecometransparent
that could not be modeled by diffusions. The joint distribution of the observed values of the Markov
chain is denoted by P . We assume that this joint distribution can be approximated by the distribution
h
of (Y∗,...,Y∗) where Y∗,...,Y∗ are the values of a diffusion Y on the equidistant grid kh,2kh,...,nkh,
1 n 1 n
i.e. Y(ikh)=Y∗. The joint distribution of (Y∗,...,Y∗) is denoted by Q .
i 1 n h
In this paper we show that
P Q 0
h h 1
k − k →
under some regularity conditions if
n
0.
k →
ThisresultcanbeappliedtotheasymptoticstudyofMarkovexperiments(P :θ Θ)whereΘisafinite
h,θ
∈
or infinite-dimensional parameter set. Suppose that for this family of Markov chains our assumptions
apply uniformly for θ Θ. Then one gets that sup P Q 0 where Q is the distribution
∈ θ∈Θk h,θ− h,θk1 → h,θ
of the discretized limiting diffusion. This implies that the Markov experiment (P : θ Θ) and the
h,θ
∈
diffusion experiment (Q : θ Θ) are asymptotically equivalent in the sense of Le Cam‘s statistical
h,θ
∈
theory of asymptotic equivalence of experiments. Asymptotic equivalence of nonparametric experiments
has been discussed in a series of papers starting with Brown and Low (1996) and Nussbaum (1996).
Recent work of nonparametric experiments that converge to diffusions include Milstein and Nussbaum
(1998), Genon-Catalot, Laredo and Nussbaum (2002), Brown, Wang and Zhao (2003), Wang (2002),
Dalalyan and Reiss (2006, 2007). Our result justifies approximating diffusion models for high frequency
financial processes that are observed on a coarser grid. We also outline that the Markov experiment
and its diffusion approximation differ in first order if n/k does not converge to zero. Then skewness
properties of the Markovchain do not vanishin firstorder. For a related paper see Duval andHoffmann
(2011). TheyconsiderestimationoftheintensityofadiscretelyobservedcompoundPoissonprocesswith
symmetric Bernoulli jumps. For this model they discuss limit experiments under different assumptions
on the limit of the difference between neighbored time points.
1
We only discuss Markov chains with continuous state space. The distribution of Markov chains with
discrete statespace cannotbe approximatedby the distribution ofcontinuousdiffusions. Forasymptotic
equivalenceofthe experiments(P :θ Θ)and(Q :θ Θ)one hastoshowthatthereexistMarkov
h,θ h,θ
∈ ∈
kernels K and L with sup K P Q 0 and sup P L Q 0. We expect
n n θ∈Θk n h,θ − h,θk1 → θ∈Θk h,θ − n h,θk1 →
that such results could be shown by using expansions for transition densities of Markov random walks.
The approach of this paper is based on expansions developed in Konakov and Mammen (2009). The
latter paper only considers Markov chains with continuous state space. To treat Markov random walks
their approachhas to be carried over to the case of discrete state spaces.
2 The main result.
We consider a Markov chain X in R that runs on very fine time grid and has the following form
l,h
X =X +m(X )h+√hξ , X =x R, l =0,...,nk 1. (1)
l+1,h l,h l,h l+1,h 0,h 0
∈ −
The innovation sequence (ξ ) is assumed to satisfy the Markov assumption: the conditional
l,h l=1,...,nk
distributionofξ giventhe pastX =x ,...,X =x depends onlyonthe lastvalueX =x and
l+1,h l,h l 0,h 0 l,h l
has a conditional density q(x , ). The conditional variance corresponding to this density is denoted by
l
·
σ2(x ) and the conditional ν-th order moment by µ (x ). The transition densities of (X ) given (X )
l ν l r,h l,h
are denoted by p (rh lh,x, ).
h l
− ·
In the following, C denotes a finite strictly positive constant whose meaning may vary from line to
line. We make the following assumptions.
(A1) It holds that yq(x,y)dy =0 for x R.
R ∈
R
(A2) There exist positive constants σ and σ⋆ such that the variance σ2(x)= y2q(x,y)dy satisfies
⋆ R
R
σ σ2(x) σ⋆
⋆
≤ ≤
for all x R.
∈
(A3) There exist a positive integer S′ > 1 and a real nonnegative function ψ(y), y R satisfying
∈
supy∈Rψ(y)<∞ and R|y|Sψ(y)dy <∞ with S =2S′+4 such that
R
Dνq(x,y) ψ(y), x,y R, 0 ν 4.
y ≤ ∈ ≤ ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
Moreover,for all x,y R, j 1
∈ ≥
Dνq(j)(x,y) Cj−1/2ψ j−1/2y ,0 ν 3
x ≤ ≤ ≤
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12)
2
for a constant C < . Here q(j)(x,y) denotes the usual j-fold convolution of q for fixed x as a
∞
function of y:
q(j)(x,y)= q(j−1)(x,u)q(x,y u)du,
−
Z
q(1)(x,y)=q(x,y).
Note that the last condition is very weak. It is motivated by (A2) and the classical local limit theorem.
(A4) Thefunctionsm(x)andσ(x)andtheirderivativesuptotheordersixarecontinuousandbounded.
Furthermore, D6σ(x) is Ho¨lder continuous of order 1.
x
(A5) There exists κ < 1 and a constant C >0 such that
5
C−1k−κ <hk<C.
The Markov chain X , see (1), is an approximation to the following stochastic differential equation
l,h
in R:
dY =m(Y )ds+σ(Y )dW , Y =x R, s [0,T], (2)
s s s s 0 0
∈ ∈
where(W ) isthe standardWiener process. The conditionaldensityofY , givenY =xis denotedby
s s≥0 t s
p(t s,x, ). WealsowriteY(s)forY . ThejointdistributionofY ontheequidistantgridkh,2kh,...,nkh
s
− ·
is denoted by Q .
h
Our main result is stated in the following theorem.
Theorem 1. Assume (A1)–(A5) and nk−1 0. Then it holds that P Q 0.
h h 1
→ || − || →
Remark 1. Theorem1canbe generalizedtohigherdimensionsandtothenonhomogenouscase. We
only treat the univariate homogenous case for simplicity. In our proof we make use of the representation
(4) from Dacunha-Castelle and Florens-Zmirou (1986) that is only available for the univariate case. For
multivariate reducible diffusions one can apply the Hermite expansion given in A¨ıt-Sahalia (2008).
Remark 2. TheassumptionsofTheorem1allowtoapplysecondorderexpansionsforthe transition
densities of Markov chains that have been developed in Konakov and Mammen (2009). In the proof of
Theorem1wemakeonlyuseoffirstorderexpansions. Forthisreasontheassumptionscouldbeweakened.
E.g. we expect that one needs only four derivatives in (A4) instead of six. We do not pursue this here
because we will need the second order expansions for getting the results in the following theorem.
Theorem 2. Assume (A1)–(A5), nh1+δ 0 and nk−2 0, where δ >0 is chosen such that the state-
→ →
ment of Theorem 4 holds for this choice. Suppose that the third conditional moment µ (x) of innovations
3
of the Markov chain fulfills µ (x) 0. Then it holds that P Q 0.
3 h h 1
≡ || − || →
3
Remark 3. This result can be applied to Euler approximations of diffusions and to Markov chains
with symmetric innovations. For Euler schemes that approximate the joint density of a diffusion at
points ∆,2∆,...,n∆ it means that one has to generate Euler approximations of the diffusions at points
∆k−1,2∆k−1,...,n∆ where k is chosen such that nk−2 0 and n(∆/k)−(1+δ) 0 . The joint
→ ∞ → →
distribution of the Euler values at the points ∆,2∆,...,n∆ is then the approximation of the joint dis-
tribution of the diffusion at these points. Under the regularity assumptions of Theorem 2 the Euler
approximation is consistent. A more detailed discussion of the necessity of the above assumptions on k
will be given elsewhere.
We now show that our assumption on the growth of k in Theorem 1 is sharp. For this purpose we
consider a simple model ofMarkovchains that convergeto a Gaussianprocessand we show that for this
case P Q does not converge to zero if the condition on the growth of k in Theorem 1 is not met.
h h 1
|| − ||
Theorem 3. Assume (A1)–(A5) for Markov chains with m(x) 1 and innovation density q(x, )=q()
≡ · ·
not depending on x. We assume that nk−1 c for a constant c = 0. Furthermore, suppose, that
→ 6
µ (x)=µ =0 and that kh 0. Then P Q does not converge to zero.
3 3 h h 1
6 → || − ||
3 Proofs.
TheproofofTheorem1willbedividedintoseverallemmas. Fortheproofwewillmakeuseoftheresults
inKonakovandMammen(2009)where Edgeworthtype expansionsofp weregivenfor nonhomogenous
h
MarkovchainsinRd ford 1. Wenowrestatetheirmainresultforone-dimensionalhomogenousMarkov
≥
chains. To formulate their result we need some additional notation.
We will use the following differential operators L and L :
Lf(t,x,y)= 1σ2(x)∂2f(t,x,y)e+m(x)∂f(t,x,y),
2 (∂x)2 ∂x
1 ∂2f(t,x,y) ∂f(t,x,y)
L˜f(t,x,y)= σ2(y) +m(y) . (3)
2 (∂x)2 ∂x
We also need the following convolution type binary operation :
⊗
t
f g(t,x,y)= du f(u,x,z)g(t u,z,y)dz.
⊗ −
Z0 ZR
We now introduce the following differential operators
µ (x)
[f](t,x,y) = 3 D3f(t,x,y),
F1 6 x
µ (x) 3σ4(x)
[f](t,x,y) = 4 − D4f(t,x,y).
F2 24 x
4
The Gaussian transition densities p(t,x,y) are defined as
1
p(t,x,y) = (2π)−e1/2σ(y)−1t−1/2exp (y x tm(y))2σ(y)−2 .
−2t − −
(cid:18) (cid:19)
We are now inethe position to state the Edgeworth type expansion for Markov chain transition densities
from Konakov and Mammen (2009).
Theorem 4. (Konakov and Mammen, 2009).Assume (A1)–(A5). Then there exists a constant δ > 0
such that the following expansion holds, uniformly for 0 i n 1:
≤ ≤ −
S′
y x
sup (kh)1/2 1+ − p (kh,x,y) p(kh,x,y)
h
x,y∈R h1/2π (kh,(cid:12)(cid:12)(cid:12)(cid:12)x√,yk)h (cid:12)(cid:12)(cid:12)(cid:12)hπ!(×kh(cid:12)(cid:12)(cid:12),x,y) =O(−h1+δ),
1 2
− −
(cid:12)
where S′is defined in Assumption (A3) and where (cid:12)(cid:12)
π (t s,x,y) = (p [p])(t s,x,y),
1 1
− ⊗F −
π (t s,x,y) = (p [p])(t s,x,y)+p [p [p]](t s,x,y)
2 2 1 1
− ⊗F − ⊗F ⊗F −
1
+ p (L2 L2)p(t s,x,y).
2 ⊗ ⋆− −
Here the operator L is defined as L, but with the coefficients “frozen” at the point x.
⋆
e
We denote nowthe signedmeasureonRn definedby the products ofp+h1/2π asQ1 andthe signed
1 h
measure defined by the products of p+h1/2π +hπ as Q2.
1 2 h
Proof of Theorem 1. Theorem 1 immediately follows from the following two lemmas.
In all lemmas of this section we make the assumptions of Theorem 1.
Lemma 1. It holds that:
Q1 Q =o(1) for n .
|| h− h||1 →∞
Lemma 2. It holds that:
P Q1 =o(1) for n .
|| h− h||1 →∞
The hardpartofthese two lemmas is the proofofLemma 1. For the proofofthe twolemmas we will
use a series of lemmas that are stated and proved now. We will come back to the proofs of Lemmas 1
and 2 afterwards.
5
In our proofs we make use of the following representation of transition densities. For the transition
densityp(t s,x,ξ)ofthediffusion(2)thefollowingformulaholds,seeformula(3.2)inDacunha-Castelle
−
and Florens-Zmirou (1986)
p(t s,x,y) = p(t s,x,y) (4)
− −
1
Eexp (t s) g[z (S(x),S(y))+ (t s)B ]dδ ,
×b − δ − δ
(cid:20) Z0 (cid:21)
p
where for 0 δ 1 B is a Brownianbridge. Furthermore,for u 0 we put g(u)= 1 C2(u)+C′(u)
≤ ≤ δ ≥ −2
and z (x,y)=(1 δ)x+δy with (cid:0) (cid:1)
δ
−
1 (S(y) S(x))2
p(t s,x,y) = exp − +H(y) H(x) , (5)
− 2π(t s)σ(y) − 2(t s) −
− (cid:20) − (cid:21)
x du
b S(x) = p ,
σ(u)
Z0
S(x) m(u) 1
H(x) = C(u)du with C(u)= σ′(u) (6)
σ(u) − 2
Z0
for x,y,s,t R.
∈
Note that under our assumptions g is bounded, g(x) M, and, hence, for t s kh
| |≤ − ≤
1
Eexp (t s) g[z (S(x),S(ξ))+ (t s)B ]dδ exp[Mkh] C∗ (7)
δ δ
− − ≤ ≤
(cid:20) Z0 (cid:21)
p
for some constant C∗ > 0 because of (A5). For the proof of Lemma 1 we make use of the following
lemmas.
Lemma 3. For all c > 0 there exists a constant C > 0 such that the following estimates hold for
0 t s c
≤ − ≤
∂ p(t s,x,y) y x
p(t s,x,y) C − (√t s+ | − |), (8)
∂x − ≤ √t s − √t s
(cid:12) (cid:12) − −
(cid:12) ∂ (cid:12) p(t s,x,y) y x
(cid:12)(cid:12) p(t s,x,y)(cid:12)(cid:12) C − (√t s+ | − |), (9)
∂y − ≤ √t s − √t s
(cid:12) (cid:12) − −
(cid:12)(cid:12)(cid:12)∂2 p(t s,x,y)(cid:12)(cid:12)(cid:12) Cp(t−s,x,y)(1+√t s+ |y−x|)2, (10)
∂y2 − ≤ t s − √t s
(cid:12) (cid:12) − −
(cid:12)(cid:12)(cid:12) ∂2 p(t s,x,y)(cid:12)(cid:12)(cid:12) Cp(t−s,x,y)(1+√t s+ |y−x|)2, (11)
∂x2 − ≤ t s − √t s
(cid:12) (cid:12) − −
(cid:12)(cid:12)(cid:12) ∂3 p(t s,x,y)(cid:12)(cid:12)(cid:12) Cp(t−s,x,y)(1+√t s+ |y−x|)3, (12)
∂x3 − ≤ (t s)3/2 − √t s
(cid:12) (cid:12) − −
∂(cid:12)(cid:12)(cid:12)4 p(t s,x,y)(cid:12)(cid:12)(cid:12) Cp(t−s,x,y)(1+√t s+ |y−x|)4, (13)
∂x2∂y2 − ≤ (t s)2 − √t s
(cid:12) (cid:12) − −
∂2 (cid:12)(cid:12) ∂2 (cid:12)(cid:12) p(t s,x,y) y x y x 2 y x 3
(cid:12) p(t s,x,y)(cid:12) C − 1+ | − | + − + − . (14)
(cid:12)(cid:18)∂x∂y − ∂x2(cid:19) − (cid:12) ≤ √t−s √t−s (cid:12)√t−s(cid:12) (cid:12)√t−s(cid:12) !
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Proof(cid:12)of Lemma 3. Weprovethe(cid:12)second,thethirdandthelastinequal(cid:12)ity. The(cid:12)rem(cid:12)ainingin(cid:12)equalities
6
can be proved exactly in the same way. From (5) we obtain
∂ σ′(y) (S(y) S(x))2
p(t s,x,y) = exp − +H(y) H(x)
∂y − − 2π(t s)σ2(y) − 2(t s) −
− (cid:20) − (cid:21)
1 (S(y) S(x))2
b p
+ exp − +H(y) H(x)
2π(t s)σ(y) − 2(t s) −
− (cid:20) − (cid:21)
(S(y) S(x))
p H′(y) −
× − (t s)σ(y)
(cid:18) − (cid:19)
σ′(y) S(y) S(x)
= p(t s,x,y) +H′(y) − , (15)
− −σ(y) − (t s)σ(y)
(cid:20) − (cid:21)
∂2 ∂ b σ′(y) S(y) S(x)
p(t s,x,y) = p(t s,x,y) +H′(y) −
∂y2 − ∂y − −σ(y) − (t s)σ(y)
(cid:20) − (cid:21)
b +pb(t s,x,y) (σ′(y))2−σ(y)σ′′(y) +H′′(y) 1−σ′(y)(S(y)−S(x))
− " σ2(y) − (t s)σ2(y) #
−
b σ′(y) S(y) S(x) 2
= p(t s,x,y) +H′(y) −
− −σ(y) − (t s)σ(y)
(cid:20) − (cid:21)
b +p(t s,x,y) (σ′(y))2−σ(y)σ′′(y) +H′′(y)
− " σ2(y)
b 1 (S(y) S(x)) σ′(y)
+ − (16)
−(t s)σ2(y) (t s) σ2(y)
− − (cid:21)
It follows from (15) and (16) and our assumptions that
∂ p(t s,x,y) S(y) S(x)
p(t s,x,y) C − √t s+ | − | , (17)
∂y − ≤ √t s − √t s
(cid:12) (cid:12) − (cid:18) − (cid:19)
(cid:12)(cid:12)(cid:12)∂2 pb(t s,x,y)(cid:12)(cid:12)(cid:12) Cpb(t−s,x,y) 1+√t s+ |S(y)−S(x)| 2. (18)
∂y2 − ≤ t s − √t s
(cid:12) (cid:12) − (cid:18) − (cid:19)
(cid:12) (cid:12) b
It is easy to see(cid:12)thatb (cid:12)
(cid:12) (cid:12)
∂ 1
Eexp (t s) g[z (S(x),S(y))+ (t s)B ]dδ
δ δ
∂y − −
(cid:12) (cid:20) Z0 (cid:21)(cid:12)
(cid:12) 1 p (cid:12)
(cid:12) (cid:12)
(cid:12) C(t s)Eexp (t s) g[zδ(S(x),S(y))+ (t(cid:12) s)Bδ]dδ , (19)
≤ − − −
(cid:20) Z0 (cid:21)
∂2 1 p
Eexp (t s) g[z (S(x),S(y))+ (t s)B ]dδ
∂y2 − δ − δ
(cid:12) (cid:20) Z0 (cid:21)(cid:12)
(cid:12) 1 p (cid:12)
(cid:12)(cid:12) C(t s)2Eexp (t s) g[zδ(S(x),S(y))+ (t(cid:12)(cid:12) s)Bδ]dδ . (20)
≤ − − −
(cid:20) Z0 (cid:21)
p
The secondandthe thirdinequalityofthe statementofthe lemma nowfollowfromourassumptionsand
from (4), (7), (17)–(20).
It remains to show (14). For a proof of this claim note that
∂2 ∂2 S(y) S(x)
pˆ(t s,x,y) = pˆ(t s,x,y) − H′(x)
∂x∂y − ∂x2 − − (t s)σ(x) −
(cid:18) (cid:19) (cid:20)(cid:18) − (cid:19)
σ′(y) (S(y) S(x))(σ−1(x) σ−1(y))
+H′(y) H′(x)+ − −
× −σ(y) − t s
(cid:18) − (cid:19)
σ′(x) (S(y) S(x)) 1 σ−1(x) σ−1(y)
+ H′′(x) − − .
− − σ2(x) t s − σ(x) t s
(cid:18) − − (cid:19)(cid:21)
7
Claim (14) follows from our assumptions and (4).
Put
π (kh,x,y)
δ (x,y) = √h 1 .
1
p (kh,x,y)
We will also make use of the following bound:
Lemma 4. There exists a constant C such that for x,y R
∈
C y x
δ (x,y) (1+ | − |)3.
1
| |≤ √k √kh
Proof of Lemma 4. Note that by definition of π :
1
kh ∂3
6h−1/2δ (x,y)p(kh,x,y) = du p(u,x,ξ)µ (ξ) p(kh u,ξ,y)dξ
1 3 ∂ξ3 −
Z0 Z
kh/2 kh
= du... + du...
Z0 Zkh/2
+ + . (21)
1 2
ℑ ℑ
We now apply the estimates of Lemma 3 to obtain the upper bounds for and in (21). For
1 2
ℑ ℑ
u [kh,kh] we apply two times integrations by parts. From our assumptions on µ (ξ) and from (7), (9)
∈ 2 3
and (10) we obtain that
kh ∂2 ∂
= du [p(u,x,ξ)µ (ξ)] p(kh u,ξ,y)dξ
|ℑ2| (cid:12)(cid:12)Zkh/2 Z ∂ξ2 3 ∂ξ − (cid:12)(cid:12)
(cid:12)(cid:12) kh ∂2 ∂ (cid:12)(cid:12)
≤ (cid:12) du ∂ξ2[p(u,x,ξ)µ3(ξ)] ∂ξp(kh−u,ξ,y) (cid:12)dξ
Zkh/2 Z (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
kh (cid:12) p(u,x,ξ) (cid:12)(cid:12) S(ξ) S(x) 2(cid:12)
C du (cid:12) 1+√u(cid:12)+(cid:12) | − | (cid:12)
≤ u √u
Zkh/2 Z (cid:18) (cid:19)
p(kh u,ξ,y) S(y) S(ξ)
− √kh u+ | − | dξ
× √kh u − √kh u
− (cid:18) − (cid:19)
C kh du
exp[2Mkh] p(u,x,ξ)p(kh u,ξ,y)
≤ kh √kh u −
Zkh/2 − Z
2
1+√u+ |S(ξ)−S(x)| √bkh u+b|S(y)−S(ξ)| dξ. (22)
× √u − √kh u
(cid:18) (cid:19) (cid:18) − (cid:19)
For u [0,kh] we get from (7), (9) and (10) again by applying integration by parts:
∈ 2
kh/2 ∂ ∂2
= du [p(u,x,ξ)µ (ξ)] p(kh u,ξ,y) dξ
|ℑ1| ∂ξ 3 ∂ξ2 −
Z0 Z (cid:12) (cid:12)(cid:12) (cid:12)
kh/2 (cid:12)(cid:12) p(u,x,ξ) (cid:12)(cid:12)S(cid:12)(cid:12)(ξ) S(x) (cid:12)(cid:12)
C du (cid:12) √u+(cid:12)|(cid:12) − | (cid:12)
≤ √u √u
Z0 Z (cid:18) (cid:19)
2
p(kh u,ξ,y) S(y) S(ξ)
− 1+√kh u+ | − | dξ
× kh u − √kh u
− (cid:18) − (cid:19)
C kh/2 du
exp[2Mkh] p(u,x,ξ)p(kh u,ξ,y)
≤ kh √u −
Z0 Z
2
S(ξ) S(x) S(y) S(ξ)
√u+ | − | b 1+√kbh u+ | − | dξ. (23)
× √u − √kh u
(cid:18) (cid:19)(cid:18) − (cid:19)
8
We now use the following substitution:
u′ = kh u,
−
1/2
kh (S(ξ) S(y))
z(ξ) = −
kh u′ √u′
(cid:18) − (cid:19)
u′ 1/2 (S(y) S(x))
+ − .
kh u′ √kh
(cid:18) − (cid:19)
Note that
dξ = (kh u′)1/2(u′)1/2(kh)−1/2σ(ξ)dz, (24)
−
(S(y) S(x))2 kh (S(ξ) S(y))2
z2+ − = −
kh (kh u′) (u′)
−
(u′) (S(y) S(x))2
+ −
(kh u′) kh
−
(S(ξ) S(y))(S(y) S(x)) (S(y) S(x))2
+2 − − + −
kh u′ kh
−
(S(ξ) S(y))2 (S(ξ) S(y))2 (S(ξ) S(y))(S(y) S(x))
= − + − +2 − −
u′ kh u′ kh u′
− −
(S(y) S(x))2
+ −
kh u′
−
(S(ξ) S(y))2 (S(ξ) S(x))2
= − + − . (25)
u′ kh u′
−
From (24) and (25) we get that
C kh du′
exp[2Mkh]exp[H(y) H(x)]
1
|ℑ | ≤ kh − √kh u′
Zkh/2 −
1 1
× 2π(kh u′)σ(ξ) 2π(u′)σ(y)
Z −
(S(ξ) S(x))2 (S(y) S(ξ))2 S(ξ) S(x)
expp − p − √kh u′+ | − |
× − 2(kh u′) − 2(u′) × − √kh u′
(cid:20) − (cid:21) (cid:18) − (cid:19)
S(y) S(ξ) 2
1+√u′+ | − | dξ
× √u′
(cid:18) (cid:19)
C exp[H(y) H(x)] (S(y) S(x))2
exp[2Mkh] − exp −
≤ kh √2πkhσ(y) − 2kh
(cid:18) (cid:19)
kh du′ dz z2
exp
× √kh u′ √2π − 2
Zkh/2 − Z (cid:18) (cid:19)
2
u′ S(y) S(x) kh u′
1+√u′+ | − | + z −
× rkh √kh | |r kh !
kh u′ S(y) S(x) u′
√kh u′+ − | − | + z
× − r kh √kh | |rkh!
C kh du′ dz z2
exp[2Mkh]p(kh,x,y) exp
≤ kh √kh u′ √2π − 2
Zkh/2 − Z (cid:18) (cid:19)
2
1+√kbh+ |S(y)−S(x)| + z √kh+ |S(y)−S(x)| + z
× √kh | | √kh | |
(cid:18) (cid:19) (cid:18) (cid:19)
9