On Local Linear Approximations to Diffusion Processes

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 906846,26pages

doi:10.1155/2011/906846

Research Article

On Local Linear Approximations to

Diffusion Processes

X. L. Duan,

1

_{Z. M. Qian,}

1

_{and W. A. Zheng}

2

1_{Department of Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles’,} Oxford OX1 3LB, UK

2_{Department of Statistics, SFS, ITCS, East China Normal University, Shanghai 200062, China}

Correspondence should be addressed to W. A. Zheng,[email protected]

Received 27 December 2010; Revised 29 April 2011; Accepted 27 June 2011

Academic Editor: Shyam Kalla

Copyrightq2011 X. L. Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Diffusion models have been used extensively in many applications. These models, such as those used in the financial engineering, usually contain unknown parameters which we wish to determine. One way is to use the maximum likelihood method with discrete samplings to devise statistics for unknown parameters. In general, the maximum likelihood functions for diffusion models are not available, hence it is difficult to derive the exact maximum likelihood estimator

MLE. There are many diﬀerent approaches proposed by various authors over the past years, see, for example, the excellent books and Kutoyants2004, Liptser and Shiryayev1977, Kushner and Dupuis2002, and Prakasa Rao1999, and also the recent works by A¨ıt-Sahalia1999,2004,

2002, and so forth. Shoji and Ozaki1998; see also Shoji and Ozaki1995and Shoji and Ozaki

1997proposed a simple local linear approximation. In this paper, among other things, we show that Shoji’s local linear Gaussian approximation indeed yields a good MLE.

1. Introduction

Diﬀusion processes are used as theoretical models in analyzing random phenomena evolved

in continuous time. These models may be described in terms of It ˆo’s type stochastic dif-ferential equations

dXtAXt, θdtσXt, θdWt, 1.1

whereWtt≥0is a Brownian motion, with some unknown parametersθto be determined in

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It is, however, diﬃculty to derive the maximum likelihood estimator for θ if the

diﬀusion coeﬃcienti.e., the volatilityσ is unknown. On the other hand, in practice, the

volatility is determined first by using the fact that

σ2T lim

n

j1

Xj−1T/n−XjT/n

2

in prob. 1.2

whenσ is a constant. Therefore we will limit ourselves on diﬀusion models with constant

volatility:

dXtAXt, θdtdWt. 1.3

Since there is no much diﬀerence at technical level, we will consider one-dimensional models

only. That is, we will assume throughout the paper thatW is a one-dimensional Brownian

motion, andXis real valued.

The distributionμX

T ofXtt≥0over a finite time interval0, T has a density with respect

to the Wiener measureμW

T the law of the Brownian motionW, given by the Cameron-Martin

formula:

dμX T dμW

T

exp

T

0

AXt, θdXt−

1 2

_T

0

AXt, θ2dt

, 1.4

which is in turn the likelihood function with continuous observation. In practice, only discrete valuesXt0, . . . , Xtn may be observed over the duration0, T , where 0t0 < t1 <· · ·< tn T

andti−ti−1 δ. The corresponding likelihood functionLθis the conditional expectation

under Wiener measure:

E Lθ|Xt0, . . . , Xtn

_n

j1pθ

δ, Xtj−1, Xtj

n j1G

δ, Xtj−1, Xtj

, 1.5

where pθt, x, y is the conditional probability density function of Xt given X0 x, and

Gt, x, yis the Gaussian density 1/√2πtexp{−|x−y|2_/₂_t}_see₁ _{. Since the denominator}

of1.5does not depend onθ, we may simply consider the numerator

LXt0, . . . , Xtn≡

n

j1

pθ

δ, Xtj−1, Xtj

, 1.6

as a likelihood function. Therefore, the MLE forθunder a discrete observation may be found

by solving either explicitly if possible or numerically the likelihood equation

∇LXt0, . . . , Xtn 0. 1.7

The diﬃculty with this approach is that, unless for a very special drift vector fieldA, an

explicit formula forpθt, x, yis not known. To overcome this diﬃculty, many approximation

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the diﬀusion model1.3by an approximation model for which an explicit formula for the likelihood function is available. One possible candidate is of course the Euler-Maruyama approximation

Xtj−Xtj−1 A

Xtj−1, θ

δξj

δ, 1.8

where {ξj} is an i.i.d. sequence with standard normal distribution N0,1 and X0 X0.

However, the likelihood functionL1X0, . . . , Xnfor this model is not, in general, close enough

to that of the diﬀusion model if measured in terms of the ratio of their corresponding

likelihood functions

LXt0, . . . , Xtn

L1Xt0, . . . , Xtn

. 1.9

The second approach is to discretize the likelihood function dμX_T/dμW_T for continuous

observations. In order to utilize this likelihood function, we need to handle the It ˆo integral

_T

0 AXt, θdXt which is defined only in probability sense. If A ∇f where f is a C1

-functionis a gradient field, then, according to It ˆo’s formula,

dμX_T

dμW_T exp

fXT, θ−

T

0

1 2

ΔfXT, θ 1

2|AXT, θ|

2

dt

, 1.10

here the right-hand side involves only the sampleX. This idea to get rid of It ˆo’s integral and

replace it by an ordinary one has far-reaching consequences, see the interesting paper2 for

some applications.

One can also use approximations to the probability density functionpθt, x, yand

construct functions which are close to the maximum likelihood function. There are a great

number of articles devoted to this approach, such as 3–5 , for example. The diﬃculty,

however, is that evenft, x, yis a uniform approximation ofpθt, x, y, there is no guarantee

that the approximate likelihood function _j ft, xj−1, xj would tend to

j pθt, xj−1, xj

whenn → ∞.

In this paper we consider the linear diﬀusion approximation proposed by Shoji and

Ozaki 6 to the diﬀusion model1.3, which leads to the following approximation of the

likelihood functionLXt0, . . . , Xtn:

L2Xt0, . . . , Xtn

n

j1

hj

δ, Xtj−1, Xtj

, 1.11

where tj jT/nso that Xtj is a sample with fixed duration δ tj −tj−1 over0, T , and

hjt, x, yis the probability transition density of the following linear diﬀusion model

dXt

AXtj−1, θ

Xt−Xtj−1

dtdWt, 1.12

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The approximation1.12is called the local linearization of the diﬀusion model1.3,

which has been studied in Shoji and Ozaki6 . Shoji has showed numerically that the local

linearizations do yield better estimates. Shoji’s approximation was revisited in Prakasa Rao

7 , without a definite conclusion.

The main goal of the paper is to proveTheorem 3.1which implies that the local linear

approximations1.12is eﬃcient for the propose of deriving MLE with discrete samples.

The paper is organized as follows. InSection 2, we present the MLE for linear models

such as1.12. InSection 3, we state our main result for Shoji’s local linear approximation,

and give some comments about the conditions on the sampling data. Our main theorem

provides a deterministic convergence rate for the likelihood functions. InSection 4, we prove

that the likelihood function for the local linear approximation converges to the

Cameron-Martin density but only in probability sense. Sections5,6, and7are devoted to the proof of

our main result. InSection 5, we state the main tool, a representation formula for diﬀusions,

established by Qian and Zheng8 . InSection 6, we develop the main technical estimates

in order to proveTheorem 3.1, whose proof is completed inSection 7.Section 8contains a

discussion about the Euler-Maruyama approximation which concludes the paper.

2. Linear Diffusions

Let us begin with the MLE of parametersa,b, andσ >0 for the linear diﬀusion modelMishra

and Bishwal9 discussed a similar model:

dXt b−aXtdtσdWt, 2.1

whose finite-dimensional distributions are Gaussian, determined through the probability

transition functionht, x, y. Fortunately we have an explicit formula forh. Indeed the linear

equation2.1may be solved explicitly and its solution is given by the formula

Xte−atX0b

a

1−e−atσ

t

0

e−at−sdWs, 2.2

formula6.8of Karatzas and Shreve10 , page 354, and therefore

ht, x, y √1

πσ

a

1−e−2atexp

− a

1−e−2at

_y₋_e−at_x₋_b/a₁₋_e−at2

σ2

. 2.3

Suppose we have a discrete sample observed over the equal time scale during the

period0, T ,XiT/n,i0, . . . , n. According to the Markov property, their joint distribution, or

the maximum likelihood function

La, b, σ;x0, . . . , xn μx0

n

j1

hδ, xj−1, xj

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where δ T/n, and μx is the probability density function of the initial distribution. Therefore the logarithmic of the maximum likelihood function

la, b, σ;{xi} logμx0

n

j1

logh

1

n, xj−1, xj

logμx0−n

2logπ−nlogσ

n

2loga−

n

2log

1−e−2aT/n

− na

1−e−2aT/n 1

σ2_T

n

j1

xj− b

a−

xj−1− b

a

e−aT/n

2

.

2.5

The maximum likelihood estimates for a, b, and σ are the stationary points of l, that is

solutions to the equation∇l0. Setρe−aT/n_{. Then}_a₋_n/T_log_ρ_and_β_b/a_.

Proposition 2.1. The maximum likelihood estimates for the linear diﬀusion model2.1with discrete

observations are given by

a n

T log

_n

j1x2j−1−1/n _n

j1xj−1 2 n

j1xj−1xj−1/nnj1xj−1nj1xj,

β 1

n

j1

xj

1

n

ρ

1−ρxn−x0,

σ2 1

T

2a

1−ρ2

n

j1

xj−ρx j−1−

1−ρβ2.

2.6

As an interesting consequence we have the following.

Corollary 2.2. The maximum likelihood estimatorsa, b,σto the linear diﬀusion model2.1are

not suﬃcient statistics whilea, b,σ, X 0, Xnare suﬃcient.

3. Diffusion Models

We consider the diﬀusion model 1.3. Our approach and our conclusions are applicable

to multidimensional cases as long as the diﬀusion coeﬃcients are constant. For simplicity,

we only consider one-dimensional case. The question is to estimate θ under a discrete

observation{x0, . . . , xn}over the time scaleδin the time interval0, T . Then, up to a constant

factor, its maximum likelihood function

Lx0, . . . , xn

n

j1

pδ, xj−1, xj

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wherept, z, yis the transition probability density ofXt we have dropped the subscript

θfor simplicity. The approximation maximum likelihood function, proposed in6 , is given

by

L2x0, . . . , xn

n

j1

hj

δ, xj−1, xj

, 3.2

wherehjt, x, yis the transition density function to the linear diﬀusion model

dXt

Axj−1, θ

Xt−xj−1

dtdWt, 3.3

which is the first-order approximation to1.3.

In what follows we assume thatAhas bounded first and second derivatives and

_A_{x, θ}_, _A_{x, θ}_≤₂_C

0, 3.4

for some constantC0>0 independent of parametersθ.

The main result of the paper is follows.

Theorem 3.1. Assume thatA·, θand A·, θare bounded uniformly inθ. LetT > 0 be a fixed

time andC >0 be a constant. Suppose{x_jn}_j_≤_n(n1,2, . . .) is a family of discrete samples such that

xn_j −x_jn₋₁2≤Cδn, xn_j≤C, 3.5

for all pairj, nsuch thatj ≤n,n1,2, . . ., whereδn T/n. Then

lim n→ ∞

Lxn

0, . . . , xnn

L2

xn

0, . . . , xnn

1, 3.6

whereLandL2are defined in3.1and3.2withδδn T/n.

The convergence in3.6happens in a deterministic sense, and therefore conditions

such as|xn_j−xn_j₋₁|2_≤_Cδ_n_and_|xn

j| ≤Care reasonable. The first condition, that is|xjn−xnj−1|2≤

Cδn, just says the “variance” of the sample cannot be too big. Since

j

_X_jT/n₋_X_j₋₁_T/n2_−→

T in probability, 3.7

so that on average we should have|xn

j −xnj−1|2 ≤ Cδn. SinceXthas continuous sample

paths, so that{Xωt:t∈0, T }for a fixed sample pointωis bounded. Sincexnj are sampled

from the fixed duration 0, T , thus we can assume that {xn

j}is bounded, though here we

have a countable many samples. It is possible to relax this constraint, for example, we may

impose that|xn

j| ≤Cnαwithα <1/2, but for simplicity we only consider the bounded case.

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From the asymptotic of the transition density functionpt, x, y, it is easy to see that

lim δ→0

pδ, xj−1, xj

hj

δ, xj−1, xj

1, 3.8

for eachj, while, as our observation{x0, . . . , xn}happens over a fixed time interval 0, T ,

the ratio 3.6 as n → ∞ is really an infinite product, its behavior thus depends on the

global behavior ofpt, x, y. Although there are many results about bounds ofpt, x, y in

the literaturesee2,11 e.g., the best we could find are those which yield3.8uniformly

inxj, none of them yields the precise limit3.6. In fact, the proof of3.6depends on careful

estimates onpt, x, ythrough a representation formula established in8 .

4. Linear Diffusion Approximations

Without losing generality, we may assume thatT1. LetXj/nbe a discrete observation of the

diﬀusion model1.3attj j/nj0, . . . , n. For simplicity, writeXj/nasXjif no confusion

may arise. Consider the family of linear diﬀusions

dXj_t AXj−1, θ

AXj−1, θ

Xj_t−Xj−1

dtdWt, 4.1

withXj_j₋₁_/nXj−1. Let

bjA

Xj−1, θ

−AXj−1, θ

Xj−1,

ajA

Xj−1, θ

.

4.2

Then

X_tjeajt−tj−1_X

j−1

eajt−tj−1−₁

aj bj

_t

tj−1

eajt−s_d_W

s, 4.3

so that

hj

δ, Xj−1, Xj

PXtjjXj|X

j

tj−1Xj−1

2aj

e2ajδ−₁

1

√

2πe

−aj/e2aj δ−1Xj−Xj−1−eaj δ−1/ajbjajXj−12_,

4.4

whereδ1/n. The approximating likelihood function is

L2δ

n

j1

hj

δ, Xj−1, Xj

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We need to compare this function to the likelihood function with continuous observation— the Cameron-Martin density, which, however, should be discounted with respect to the

Wiener measure. Thus we have to renormalizeL2δagainst the discrete version of Brownian

motion, which is given by

L2δ

n

j1

hj

δ, Xj−1, Xj

Gδ, Xj−1, Xj

, 4.6

where

Gδ, x, y √1

2πδexp

−x−y

2

2δ

. 4.7

Hence its logarithmic

l2δ

1 2

j

log 2δaj

e2ajδ−₁

1 2δ

j

_X_j₋_X_j₋₁2

− 1

2

j 2aj e2ajδ−₁

Xj−Xj−1−

eajδ−₁

aj

bjajXj−1

2

.

4.8

Proposition 4.1. One has

lim δ↓0

l2δ l 4.9

uniformly inθ, in probability with respect to the Wiener measure, wherelis the log of the

Cameron-Martin density1.4.

Proof. LetDjXj−Xj−1. Then

l2δ 1

2

j

log 2δaj

e2ajδ−₁

1 2δ

j

1− 2δaj

e2ajδ−₁

D2_j

−

j 1 eajδ₁

bjajXj−1

2eajδ−1

aj

2 j

1 eajδ₁

bjajXj−1

Dj.

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SincebjAXj−1, θ−ajXj−1andajAXj−1, θ, so that

l2δ 1

2

j

log 2δaj

e2ajδ−₁

1 2δ

j

1− 2δaj

e2ajδ−₁

D2_j

−

j 1 eajδ₁A

Xj−1, θ

2eajδ−1

aj

2 j

1 eajδ₁A

Xj−1, θ

Dj.

4.11

However,

lim δ↓02

j 1 eajδ₁A

Xj−1, θ

Dj

1

0

AXt, θdXt,

lim δ↓0

j 1 eajδ₁A

Xj−1, θ

2eajδ−1

aj

1 2

1

0

|A|2

Xt, θdt,

lim δ↓0

1 2

j

log 2δaj

e2ajδ−₁ −

1 2

₁

0

AXt, θdt,

lim δ↓0

1 2δ

j

1− 2δaj

e2ajδ−₁

D_j2 1

2

₁

0

AXt, θdt

4.12

in probability. The claim thus follows immediately.

5. A Representation Formula

From this section, we develop necessary estimates in order to prove Theorem 3.1. In this

section, we recall the main tool in our proof, a representation formula proved by Qian and

Zheng8 . Based on this formula, we prove the main estimate6.65, which has independent

interest, in the next section. We conclude the proof ofTheorem 3.1inSection 7.

Letx∈R. Consider the linear diﬀusion

dXt

Ax, θ Ax, θXt−x

dtdWt, 5.1

whose probability transition function is also denoted byht, z, y. Recall thatpt, z, yis the

probability transition function of the diﬀusion defined by1.3. The strong solution of5.1

is given by

XtX0σa, t2aX0b

_t

0

eat−s_d_W

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so that

ht, z, y √ 1

2πσ2a, texp

− 1

2σ2a, t2

y−z−σa, t2baz2

, 5.3

wherebAx, θ−xAx, θ,aAx, θandσa, t eat₋₁_/a_.

Observe that for anya∈R,t → σa, tis increasing, and

lim t↓0

σa, t

√

t 1 for anya. 5.4

We will also use the fact that

σ2a, t

e2at₋₁

2a

eat₁

2 σa, t. 5.5

Lemma 5.1. Forx∈R, andCz Az, θ−Ax, θ−Ax, θz−x. Then

|Cz| ≤C0min

2|z−x|,1 2|z−x|

2

. 5.6

Our main tool is a representation formula5.7discovered in8 . LetXt,Pxbe the

solution to the linear stochastic diﬀerential equation5.1.

Proposition 5.2. Forx, y∈RandT >0 one has

pT, x, y

hT, x, y 1

T

0

Px

UtCXt

∇hT−t, Xt, y

hT, x, y

dt, 5.7

where

Utexp

t

0

CXsdWs−1

2

t

0

|C|2

Xsds

, 5.8

which is a martingale under the probabilityPx_.

To prove3.6, we need to estimate the double integral appearing on the right-hand

side of5.7, which requires a precise estimate for

ItPx

UtCXt

∇hT−t, Xt, y

hT, x, y

, 5.9

which can be achieved since we know the precise formhT, x, y. Of course, if we knew the

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Our arguments are based on the fact thatUtis a martingale underPx, together with some delicate estimates for the functional integral

Px

∇hT−t, Xt, y

hT, x, y

p

, 5.10

which will be done in the next section.

6. Main Estimates

We use the notations established in the previous section. LetT >0,x,y∈Randd y−x.

Then

∇zh

T−t, z, y

hT−t, z, y

2aeaT−t

e2aT−t₋₁

y−eaT−tz−σa, T−t2b, 6.1

and therefore

∇zh

T−t, Xt, y

hT, x, y

2aeaT−t

e2aT−t₋₁

hT−t, Xt, y

hT, x, y

×y−eaT−tXt−σa, T−t2b

σ2a, TeaT−t

σ2a, T−t3 e

1/2σ2a,T2ST2

×e−1/2σ2a,T−t2|y−eaT−tXt−σa,T−t2b|

2

,

6.2

where

ST y−x−σa, T2axb. 6.3

Fort∈0, Tandp >1 we set

βpt

a

pe2aT₋₁₋_p₋₁_e2aT−t₋₁σa, T

2_b_ax₋_d_,

αpt

_e2aT−t₋₁

pe2aT₋₁₋_p₋₁_e2aT−t₋₁,

6.4

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Lemma 6.1. For anyp >1 one has

1

p

σa, T2bax−d≤

e2aT₋₁

a βpt≤

σa, T2bax−d,

1

p

e2aT−t₋₁

e2aT−₁ ≤αpt≤

e2aT−t₋₁

e2aT−₁ ,

6.5

for allt∈0, T .

Proof. The two inequalities follow from the fact that

e2aT₋₁

pe2aT₋₁₋_p₋₁_e2aT−t₋₁ 6.6

assumes its maximum 1 and minimum1/p.

Since

βpt

αpt

a

e2aT−t₋₁σa, T 2

bax−d,

e2aT−t₋₁

e2aT₋₁

β2pt2

α2pt2

a

e2aT₋₁

σa, T2bax−d2,

6.7

so that

∇zh

T−t, Xt, y

hT, x, y

σ2a, T

σ2a, T−t3e

e2aT−t₋₁_/_e2aT₋₁_β

2pt2/α2pt2aT−t

×e−1/2σ2a,T−t2|y−eaT−tXt−σa,T−t2b|

2 ,

6.8

which yield, together with5.7, the following.

Lemma 6.2. One has

pT, x, y

hT, x, y 1

_T

0

2aeaT−t

e2aT−t₋₁

e2aT₋₁

e2aT−t₋₁ee

2aT−t₋₁_/_eaT₋₁_β

2pt2/α2pt2

×Px_{U

tCXtKt}dt,

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where

Kt

y−eaT−tXt−

eaT−t₋₁

a b

×e−a/e2aT−t−1|y−eaT−tXt−eaT−t−1/ab|2_.

6.10

Let

Jp_t p

!

Px_|K_t_|p

Dp_t 2p

Px_e2p/2p−10t|C|2Xsds|C_X

t|2p

.

6.11

Lemma 6.3. Chooseζ >0 such that

Pexp

ζsup

0,1

|Wt|2

c <∞. 6.12

Then for anyT >0 andλ >0, such that

4λC0

eaT

e2aT₋₁

a ≤ζ, 6.13

one has

Px_eλ₀t|C|2_X

sds

≤c ∀t≤T. 6.14

Proof. Let

YtXt−x−σa, t2axb

t

0

eat−sdWs.

6.15

Then

|C|2_X

s≤4C0|Xs−x|2

≤4C0Ysσa, s2axb

2

≤4C0σa, s4|axb|24C0C2|Ys|2,

6.16

and therefore

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On the other hand

Yseas

s

0

e−ardWr, 6.18

so thatMs≡e−asYsis a martingale with

M_s

_s

0

e−2ardr e

2as₋₁

2a . 6.19

ThusMsis a time change of a standard Brownian motion, and

MsBe2as₋₁_/₂_a, 6.20

for some standard Brownian motionBtt≥0. Since

2λC0

1 eat

e2at₋₁

2a ≤ζ, 6.21

so we have

Px _e4λC0

t

0eas|Ms|2ds

Pexp

4λC0 t

0

easBe2as₋₁_/₂_a2ds

Pexp

4λC0

_e2at₋₁_/₂_a

0

s √

2as1|Bs|

2_d_s

≤Pexp

⎛ ⎝₄_λC₀ 1

eat

e2at₋₁ 2a

2

sup

0,e2at₋₁_/₂_a

|Bs|2

⎞ ⎠

Pexp

4λC0

1 eat

e2at₋₁ 2a sup₀,1 |Bs|

2

≤c.

6.22

Corollary 6.4. Forp >1 andT >0 to be such that

16pC0

2p−1 1

eaT

e2aT₋₁

2a ≤ζ, 6.23

one has

(15)

In what follows, we always assume thatT >0 is chosen such that the condition6.23

is satisfied. Next we estimateDp_t, which is provided in the following.

Lemma 6.5. Letp >1. Then

Dp_t≤C1

σ2a, t2σa, t2axb2, 6.25

where the positive constantC1depends only onp,ζ, andC0.

Proof. Let

Ft Pxe2p/2p−10t|C|

2_X

sds|C_X

t|2p

. 6.26

Then, by the H ¨older inequality

Ft

Px_e4p/2p−1t

0|C|

2_X

sds

Px_|C_X

t|4p

≤C0

Px_|C_X_t_|4p_.

6.27

Next we estimate the expectationPx_|C_X

t|4p. Since

|CXt| ≤ C0

2 |Xt−x|

2

≤C0|Yt|2C0σa, T2axb2,

6.28

so that

|CXt|4p≤24p−1C40p|Yt|8p24p−1C40pσa, T8paxb8p. 6.29

On the other hand

Px_|Y

t|8p 1

√

2πσ2a, t

|z|8p_exp

− |z|2

σ2a, t2

dz

σ2a, t8p,

6.30

so that

4p

!

Px_|C_X_t_|4p_≤₂4p−1/4p_C

(16)

Lemma 6.6. LetT >0 satisfy condition6.23,x, y ∈R, andp >1 andq > 1 such that1/p

1/q 1. Then

pT, x, y

hT, x, y ≤1exp

e2aT−t₋₁

eaT₋₁

β2pt2

α2pt2

×

T

0

2aeaT−t

e2aT−t₋₁

e2aT₋₁

e2aT−t₋₁D

p_tJ2p_tdt.

6.32

Proof. Since

Ute−q−1/2

t

0|C|2Xsdsq_exp

t

0

qCXsdWs−

1 2q

2 _t

0

|C|2_X

sds

6.33

is a martingale underPx_{, so that}

Px_U

te−q−1/2

t

0|C|2Xsds

q

1. 6.34

By the H ¨older inequality we deduce that

|Px_U

tCXtKt| ≤D

p_tJ2p_t. 6.35

Equation6.32, follows from the representation6.9.

Lemma 6.7. Letp >1. Then

Jp_t≤p 2p−1_α

pt11/pe−β

2 p

×

⎧ ⎨ ⎩pεp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

a βpt

⎫ ⎬ ⎭.

6.36

Proof. We have

Px,_|K_t_|p

-≤Px

y−eaT−tXt−e

aT−t₋₁

a b

p

×e−pa/e2aT−t−1|y−eaT−tXt−eaT−t−1/ab|2

.

(17)

Under the probabilityPx_,

Zt≡eaT−t

Xt−eatx−e

at₋₁

a b

_t

0

eaT−sdWs

6.38

is a central normal distribution with variance

_t

0

e2aT−sds e

2aT₋_e2aT−t

2a . 6.39

In terms ofZtanddy−x

Px,_|K_t_|p-_≤_Px

Zte aT₋₁

a bax−d

p

×e−pa/e

2aT−t₋₁_|_Z

teaT−1/abax−d|2

⎫ ⎪ ⎬ ⎪ ⎭.

6.40

Making change of variable

N

2a

e2aT₋_e2aT−tZt. 6.41

Then, underPx_,_N_{has the standard normal distribution}_N₀_,₁_{, so that}

Px_|K_t_|p_≤_P

⎧ ⎨ ⎩

e2aT₋_e2aT−t

2a N

eaT₋₁

a bax−d

p

×e−pa/e2aT−t−1|

√

e2aT₋_e2aT−t_/₂_a_N_eaT₋₁_/a_b_ax₋_d_|2

⎫ ⎬ ⎭

√1

2π

R ⎧ ⎨ ⎩

e2aT₋_e2aT−t

2a z

eaT₋₁

a bax−d

p

×e−pa/e2aT−t−1|√e2aT₋_e2aT−t_/₂_a_z_eaT₋₁_/a_b_ax₋_d_|2₋_z2_/₂

⎫ ⎬ ⎭dz.

(18)

Let us simplify the last integral. Indeed, set

η

pe2aT₋_p₋₁_e2aT−t₋₁

e2aT−t−₁ z,

Et2p

_e2aT₋_e2aT−t

2e2aT−t₋₁

eaT₋₁

a bax−d

×

a

pe2aT₋_p₋₁_e2aT−t₋₁.

6.43

Then we rewrite the term appearing in the exponential in the last line of6.42

− pa

e2aT−t₋₁

e2aT₋_e2aT−t

2a z

eaT₋₁

a bax−d

2

−1

2

η2₂_E

tη

− pa

e2aT−t−₁

eaT₋₁

a bax−d

2

−1

2

ηEt

21

2E

2

t −

pa

e2aT−t₋₁

eaT₋₁

a bax−d

2

,

6.44

together with

1

2E

2

t−

pa

e2aT−t₋₁

eaT₋₁

a bax−d

2

− pa

pe2aT−_p−₁_e2aT−t−₁

eaT₋₁

a bax−d

2 6.45

the inequality6.42may be rewritten as follows:

Px_|K_t_|p_≤ _√1

2πe

−pa/pe2aT₋_p₋_1e2aT−t₋₁_eaT₋₁_/a_b_ax₋_d2

×

R

e2aT₋_e2aT−t

2a z

eaT₋₁

a bax−d

p

×e−1/2ηEt2_d_z.

(19)

Making change of variable in the last integral

vηEt

e2aT−t₋₁ zEt, 6.47

so that

dz

e2aT−t₋₁

pe2aT₋_p₋₁_e2aT−t₋₁dv,

e2aT₋_e2aT−t

2a z

eaT₋₁

a bax−d

_e2aT−t₋₁_e2aT₋_e2aT−t

2ape2aT₋_p₋₁_e2aT−t₋₁v

e2aT−t−1

eaT₋₁

a bax−d

.

6.48

Thus6.46yields that

Px_|K_t_|p_≤

⎛ ⎝

e2aT−t₋₁ 2a

⎞ ⎠

p1

2a

×e−pa/pe2aT−p−1e2aT−t−1eaT−1/abax−d2Qt,

6.49

where

Qt

R

_e2aT₋_e2aT−t

pe2aT₋_p₋₁_e2aT−t₋₁z

!

2ae2aT−t₋₁_eaT₋₁_/a_b_ax₋_d

p

e−1/2z2

√

2π dz

≤2p−1_ε

p

_e2aT₋_e2aT−t

p

2p−1

!

2ae2aT−t₋₁_eaT₋₁_/a_b_ax₋_d

pe2aT−_p−₁_e2aT−t−₁

p

εp

1

√

2π

R|z|

p_e−1/2z2

dz.

(20)

Therefore

Jp_t≤

⎛ ⎝

e2aT−t₋₁

2a

⎞ ⎠

11/p

2a

1/2p

×e−a/pe2aT−p−1e2aT−t−1eaT−1/abax−d2

×

⎧ ⎨ ⎩p

!

2p−1_ε

p

_e2aT₋_e2aT−t

p

2p−1

2e2aT−t₋₁

_a_eaT₋₁_/a_b_ax₋_d2

⎫ ⎪ ⎬ ⎪ ⎭, 6.51

which is equivalent to the required inequality.

Lemma 6.8. Let

Hp_t≡exp

e2aT−t₋₁

eaT₋₁

β2pt2

α2pt2

J2p_t. 6.52

Then

pT, x, y

hT, x, y ≤1

T

0

2aeaT−t

e2aT−t₋₁

e2aT−1

e2aT−t₋₁D

p_tHp_tdt, 6.53

Hp_t≤ 2p22p−1_α

2pt11/2pe−2p−1e

2aT₋_e2aT−t2

/e2aT−t−1e2aT₋₁_β

2pt2

×

⎧ ⎨ ⎩p εp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

a β2pt

⎫ ⎬ ⎭.

6.54

Proof. Indeed, byLemma 6.7we have

Hp_t≤ 2p22p−1_α

2pt11/2pe−β

2

2pe2aT−t−1/e2aT−1β2pt2/α2pt2

× ⎧ ⎨ ⎩p εp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

a βpt

⎫ ⎬ ⎭.

6.55

On the other hand

−β2

2p

e2aT−t₋₁

e2aT₋₁

β2pt2

α2pt2

p−1β2pt2

e2αT₋_e2αT−t

(21)

thus

Hp_t≤ 2p22p−1_α

2pt11/2pep−1β2pt

2_e2αT₋_e2αT−t_/_e2aT₋₁

×

⎧ ⎨ ⎩p

εp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

a β2pt

⎫ ⎬ ⎭.

6.57

Let

Bp_t≡ep−1β2pt2e2αT−e2αT−t/e2aT−1

×

⎧ ⎨ ⎩p

εp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

a β2pt

⎫ ⎬ ⎭.

6.58

Then6.53and6.54imply that

pT, x, y

hT, x, y ≤1

2p √

22p−1

e2aT₋₁1/4p

_T

0

2aeaT−t

e2aT−t₋₁1−1/4pD

p_tBp_tdt. 6.59

Lemma 6.9. One has

Bp_t≤ep−1β2pt2e2αT−e2αT−t/e2aT−1

×

⎧ ⎨ ⎩p εp

e2aT₋_e2aT−t

2a

e2aT−t₋₁

e2aT₋₁ ST

⎫ ⎬ ⎭.

6.60

In particular

Bp_t≤p ε_p

e2aT₋₁

2a ST,

pT, x, y

hT, x, y ≤1

2p √

22p−1

e2aT₋₁1/4p

⎧ ⎨ ⎩p εp

e2aT₋₁

2a ST

⎫ ⎬ ⎭

×

_T

0

2aeaT−t_ep−1β2pt2e2αT−e2αT−t/e2aT−1

e2aT−t₋₁1−1/4p D

p_tdt.

6.61

Proof. Let

Gp_t 2ae

aT−t

e2aT−t₋₁

e2aT₋₁

e2aT−t₋₁

2p

22p−1_α

(22)

Then

pT, x, y

hT, x, y ≤1

T

0

Gp_tDp_tBp_tdt. 6.63

Since

Gp_t 2ae

aT−t

e2aT−t₋₁1−1/4p

e2aT₋₁1/4p

2p

22p−1

×

e2aT₋₁

2pe2aT₋₁₋₂_p₋₁_e2aT−t₋₁

11/2p

≤ 2aeaT−t

2p √

22p−1

e2aT−t₋₁1−1/4p

e2aT₋₁1/4p,

6.64

which implies the required estimate.

By collecting all estimates we have established, we may obtain the following.

Proposition 6.10. There is a constantC2>0 depending only onζandCdsuch that

pT, x, y

hT, x, y ≤1C2e

|a|T_ep−1a/e2aT₋₁_S_T2

eaT₁

2 axb

2

× eaT−1

a

⎛ ⎝p _ε

p

e2aT₋₁

2a ST

⎞ ⎠_,

6.65

where

ST e

aT₋₁

a bax−

y−x. 6.66

Proof. Indeed

pT, x, y

hT, x, y ≤1C1e

p−1a/e2aT₋₁_S_T2

e2aT₋₁

a

₋1/4p⎛

⎝p _ε_p

e2aT₋₁

2a ST

⎞ ⎠

×

T

0

eaT−t

e2aT−t₋₁

a

1/4p−1

(23)

≤1Cep−1a/e2aT−1ST2

e2aT₋₁

a

−1/4p⎛

⎝p _ε_p

e2aT₋₁

2a ST

⎞ ⎠

×

eaT₁

2 axb

2

×eaT−1

a

_T

0

eaT−t

e2aT−t₋₁

a

1/4p−1

dt.

6.67

While,

T

0

eaT−t

e2aT−t₋₁_/₂_a1−1/4pdt

_e2aT₋₁_/₂_a

0

1

s1−1/4p

1

√

12asds

≤4pe|a|T

e2aT₋₁ 2a

1/4p

,

6.68

and it thus yields our key estimate6.65.

Similarly we have a lower bound

pT, x, y

hT, x, y ≥1−C3e

|a|T_ep−1a/e2aT₋₁_S_T2

eaT₁

2 axb

2

× eaT−1

a

⎛ ⎝p _ε_p

e2aT₋₁

2a ST

⎞ ⎠_,

6.69

whereC3depends only onζandC0.

7. Proof of

Theorem 3.1

We are now in a position to proveTheorem 3.1. We may assume thatT 1, so thatδn 1/n.

Letxn_j j 0,1, . . . , nbe discrete samplings with time scaleδ δn 1/non0,1 . By our

assumptions,|xn_j −x_jn₋₁|2≤Cδn, and|xn_j| ≤Cfor all pairj, nsuch that 0≤j ≤nandn≥1.

For simplicity we writexjforxjnif no confusion may arise.

In the proof below, we will useCito denote nonnegative constants which may depend

on C, T 1 and the bounds of A and A appearing in our diﬀusion model 1.3, but

independent ofn.

Recall thathjt, x, yis the probability transition density function of the diﬀusion3.3,

that is,

dXt

bjajXt

dtdWt, 7.1

where

bj A

xj−1, θ

−xj−1A

xj−1, θ

, ajA

xj−1, θ

(24)

According to6.65we have

pδ, xj−1, xj

hj

δ, xj−1, xj

≤1C8e|aj|δep−1aj/e

2aj δ₋₁_S2

j

eajδ₁

2

ajxj−1bj

2

×eaj−1δ−1

aj−1

⎛ ⎝p _ε_p

e2aj−1δ₋₁ 2aj−1 Sj

⎞ ⎠_,

7.3

where

Sj

eajδ−₁

aj

bjajxj−1

−xj−xj−1

. 7.4

Sinceajandxjare bounded,

_a_j_x_j₋₁_b_j_A

xj−1, θ≤C4

1xj−1

≤C9,

7.5

so that

Sj≤ e

ajδ−₁

ajδ C9δC

δ≤C10

δ. 7.6

Thus

p−1aj

e2ajδ−₁S

2

j

p−1ajδ

e2ajδ−₁

S2

j

δ ≤C11,

eajδ₁

2

ajxj−1bj

2_≤

C12.

7.7

Therefore

pδ, xj−1, xj

hj

δ, xj−1, xj

≤1C13

eaj−1δ₋₁

aj−1δ

⎛ ⎝p _ε

p

e2aj−1δ₋₁ 2aj−1δ

δSj

⎞ ⎠_δ

≤1C14

δδ1C14

1

n3/2,

7.8

where we have used7.6. It follows that

lim δ→0

n

j1

pδ, xj−1, xj

hj

δ, xj−1, xj

≤ lim n→ ∞

1C14 1

n3/2

n

1.

(25)

Similarly we have

lim n→ ∞

n

j1

pδ, xj−1, xj

hj

δ, xj−1, xj

≥1. 7.10

Therefore

lim n→ ∞

Lxn₀, . . . , xn

n

L2

xn

0, . . . , xnn

1, 7.11

and the proof ofTheorem 3.1is complete.

8. The Euler-Maruyama Approximation

Recall that the Euler-Maruyama approximation to1.3is a Markov chain given by

XjXj−1A

θ, Xj−1

δξj

δ, 8.1

where {ξj} is an i.i.d. random sequence, with standard normal N0,1. The conditional

distribution ofXj givenXj−1 xj−1 is Gaussian with meanxj−1Aθ, xj−1δand variance

δso that the likelihood function is given as

L1x0, . . . , xn 1

2πδn/2

n

j1

exp

−xj−xj−1−A

θ, xj−1

δ2

2πδ

. 8.2

Applying the representation formula5.7we have the following.

Proposition 8.1. It holds that

Lx0, . . . , xn

L1x0, . . . , xn

j

1−δeα2j/2δ

δ

0

P

Ujscj

Xsj, θ

_W_s_α_j

δ−s3/2e

−Wsαj2/2δ−s

ds

,

8.3

whereWtt≥0is the standard Brownian motion,X

j

sxj−1λj−1sWs,

Ujt exp

t

0

cj

Xsj, θ

dWs−

1 2

_t

0 _c_j2

Xsj, θ

ds

, 8.4

λj Axj−1, θ, αjλj−1δ−xjxj−1, and

cjz, θ Az, θ−A

xj−1, θ

. 8.5

(26)

Proposition 8.2. IfAx, θis bounded and Lipschitz continuous, uniformly inθ, then the maximum likelihood function with discrete sampling is stable, in the sense that

j

1−C01δeα

2

j/2δ

≤ Lx0, . . . , xn

L1x0, . . . , xn ≤

j

1C02δeα

2

j/2δ

, 8.6

for some constantC01andC02, whereαj Axj−1, θδ−xj−xj−1.

However, this estimate does not lead to the same result as for the local linear

approximation. It is not knownto our best knowledgewhether

lim n→ ∞

Lxn₀, . . . , xn

n

L1

xn

0, . . . , xnn

1 8.7

holds or not under similar conditions inTheorem 3.1.

Acknowledgment

The authors thank Dr. Zeng for providing a list of errors contained in the first version of the paper.

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