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α-Wiener Bridges: Singularity of Induced Measures andSample Path PropertiesMátyás Barczy a & Gyula Pap aa Faculty of Informatics , University of Debrecen , Debrecen, HungaryPublished online: 12 Apr 2010.

To cite this article: Mátyás Barczy & Gyula Pap (2010) α-Wiener Bridges: Singularity of Induced Measures and Sample PathProperties, Stochastic Analysis and Applications, 28:3, 447-466, DOI: 10.1080/07362991003704985

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Stochastic Analysis and Applications, 28: 447–466, 2010Copyright © Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362991003704985

�-Wiener Bridges: Singularity of InducedMeasuresand Sample Path Properties

MÁTYÁS BARCZY AND GYULA PAP

Faculty of Informatics, University of Debrecen, Debrecen, Hungary

Let us consider the process �X���t �t∈�0�T� given by the SDE dX���

t = − �T−t

X���t dt +

dBt, t ∈ �0� T�, where � ∈ �, T ∈ �0���, and �Bt�t≥0 is a standard Wiener process.In case of � > 0, the process X��� is known as an �-Wiener bridge, in case of� = 1 as the usual Wiener bridge. We prove that for all �� � ∈ �, � �= �, theprobability measures induced by the processes X��� and X��� are singular on(C�0� T����C�0� T��

). Further, we investigate regularity properties of X���

t as t ↑ T .

Keywords Induced measures; Sample path properties; Singularity; �-Wienerbridges.

Mathematics Subject Classification 60G30; 60G17; 60J60.

1. Introduction

There has been a lot of work concerning questions of absolute continuity andsingularity for various types of stochastic processes on finite and infinite timeintervals (see, e.g., [4, 8, 15]). However, most of the literature deal with timehomogeneous diffusion processes. In this article, we study absolute continuity andsingularity of �-Wiener bridges which are time inhomogeneous diffusion processes.We also present some results on the sample path behavior of these processes.

Let T ∈ �0��� be fixed. For all � ∈ �, we consider the process �X���t �t∈�0�T� given

by the stochastic differential equation (SDE)dX���t = − �

T − tX

���t dt + dBt� t ∈ �0� T��

X���0 = 0�

(1.1)

Received November 7, 2008; Accepted February 18, 2009The first author has been supported by the Hungarian Scientific Research Fund

under Grants No. OTKA–F046061/2004 and OTKA T-048544/2005. The second authorhas been supported by the Hungarian Scientific Research Fund under Grant No. OTKAT-048544/2005.

Address correspondence to Gyula Pap, Faculty of Informatics, University of Debrecen,Pf. 12, Debrecen H-4010, Hungary; E-mail: papgy@inf.unideb.hu

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448 Barczy and Pap

where �Bt�t≥0 is a one-dimensional standard Wiener process on a probability space�����P�. To our knowledge, these kind of processes have been first considered byBrennan and Schwartz [7] (see also Mansuy [14]). In Brennan and Schwartz [7],the SDE (1.1) is used to model the arbitrage profit associated with a given futurescontract in the absence of transaction costs. In case of � > 0 the process X��� isknown as an �-Wiener bridge, in case of � = 1 as the usual Wiener bridge. Byformula (5.6.6) in Karatzas and Shreve [9], the SDE (1.1) has a unique strongsolution, namely,

X���t =

∫ t

0

(T − t

T − s

)�

dBs� t ∈ �0� T�� (1.2)

defined on a filtered probability space(���� ��t�t∈�0�T��P

)constructed by the help of

the standard Wiener process B (see, e.g., [9, Section 5.2.A]). This filtered probabilityspace satisfies the so-called usual conditions, i.e., �����P� is complete, the filtration��t�t∈�0�T� is right-continuous, �0 contains all the P-null sets in � and � = �T−,where �T− �=

(⋃t∈�0�T� �t

).

In Section 2, we calculate the covariance of X���s and X

���t for all s� t ∈ �0� T� and

�, � ∈ �. Further, we recall a strong law of large numbers and a law of the iteratedlogarithm for continuous local martingales which will be used for proving regularityproperties of X���.

In Section 3, we prove that X���t → 0 almost surely as t ↑ T in case of � > 0

(see Lemma 3.1) which is why we can use the expression “�-Wiener bridge” for X���

in this case. Lemma 3.1 can be considered as a generalization of Lemma 5.6.9 inKaratzas and Shreve [9]. We will also examine what happens in case of � ≤ 0, seeRemark 3.5. Further, we investigate regularity properties of X���

t as t ↑ T . In case of� ≥ 1

2 , we have theorems of type of the law of the iterated logarithm for X���, seeTheorems 3.2 and 3.3.

In Section 4, we investigate the absolute continuity and singularity of theprobability measures induced by the processes X��� with different values of �.Namely, we show that for all �� � ∈ �, � �= �, the probability measures inducedby the processes X��� and X��� on

(C�0� T����C�0� T��

)are singular, where C�0� T�

is the space of continuous functions from �0� T� into �, and ��C�0� T�� denotesthe Borel -algebra on C�0� T�, see Theorem 4.1. We note that Prakasa Rao [15,Theorem 5] proved a similar statement for fractional Wiener processes. Namely,he showed that if �WHi

�t��t≥0, i = 1� 2, are two fractional Wiener processes withHurst indices H1� H2 ∈ �0� 1�, H1 �= H2, then the probability measures induced bythe processes WHi

, i = 1� 2, on(C�0������C�0����

)are singular, where ��C�0����

denotes the Borel -algebra on C�0���. We also note that our technique for theproof of Theorem 4.1 differs from the technique of Prakasa Rao [15, Theorem 5].Our proof is based on strong consistency of the maximum likelihood estimator of�, while the proof of Prakasa Rao [15, Theorem 5] is based on a Baxter type resultof Kurchenko [10] for second order quadratic variations (second order increments)for a fractional Wiener process. By giving a second proof of Theorem 4.1, wealso discuss the connections between strong consistency of the maximum likelihoodestimator of �, Hellinger processes (see, e.g., [8, Chapter IV]) and singularityof induced measures. Moreover, we study absolute continuity and singularity ofprobability measures induced by processes for which the diffusion coefficients in

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�-Wiener Bridges 449

the SDE (1.1) are not identically one. Giving two different proofs, we prove that aso-called dichotomy holds, see Theorem 4.5.

2. Preliminaries

First, we determine the covariance of X���s and X

���t for all s� t ∈ �0� T� and �, � ∈ �.

Lemma 2.1. Let T ∈ �0��� be fixed. For �, � ∈ �, let us consider the processes�X

���t �t∈�0�T� and �X

���t �t∈�0�T� given by the SDE (1.1). Then for all s� t ∈ �0� T�, the

covariance of X���s and X

���t is

Cov�X���s � X

���t � =

�T − s���T − t��

1− �− �

(T 1−�−� − �T − �s ∧ t��1−�−�

)if �+ � �= 1�

�T − s���T − t�� ln(

T

T − �s ∧ t�

)if �+ � = 1

(2.1)

Especially, for all t ∈ �0� T�, X���t is a normally distributed random variable with mean

EX���t = 0 and with variance

E�X���t �2 =

T

1− 2�

(T − t

T

)2�

− T − t

1− 2�if � �= 1

2�

�T − t��ln�T�− ln�T − t�� if � = 12

Proof. By Bauer [5, Lemma 48.2] and (1.2), �X���t �t∈�0�T� is a Gauss process

with mean EX���t = 0, t ∈ �0� T�, and with variance E�X���

t �2, t ∈ �0� T�. ByProposition 3.2.10 in Karatzas and Shreve [9], for all s� t ∈ �0� T�, we have

Cov(X���

s � X���t

) = E(X���

s X���t

) = E( ∫ s

0

(T − s

T − u

)�

dBu

∫ t

0

(T − t

T − v

)�

dBv

)= �T − s���T − t��

∫ s∧t

0

1�T − u��+�

du�

and hence we obtain (2.1). �

For proving regularity properties of X���, we recall a strong law of largenumbers and a law of the iterated logarithm for continuous local martingales.

Let T ∈ �0��� be fixed. In all what follows, if �Mt�t∈�0�T� is a continuouslocal martingale satisfying P�M0 = 0� = 1, then �Mt�t∈�0�T� denotes the quadraticvariation of M .

The following theorem is a modification of Theorem 3.4.6 in Karatzas andShreve [9] (due to Dambis, Dubins, and Schwartz, see also Theorem 1.6 inChapter V in [16]). In fact, our next Theorem 2.2 is Exercise 1.18 in Chapter V inRevuz and Yor [16].

Theorem 2.2. Let T ∈ �0��� be fixed and let(���� ��t�t∈�0�T��P

)be a filtered

probability space satisfying the usual conditions. Let �Mt�t∈�0�T� be a continuous local

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450 Barczy and Pap

martingale with respect to the filtration ��t�t∈�0�T� such that P�M0 = 0� = 1 andP�limt↑TMt = �� = 1. For each s ∈ �0���, define the stopping time

�s �= inf t ∈ �0� T� � Mt > s�

Then the time-changed process

�Bs �= M�s���s

�s≥0

is a standard Wiener process. In particular, the filtration ���s�s≥0 satisfies the usual

conditions and

P(Mt = BMt for all t ∈ �0� T�

) = 1

Now we formulate a strong law of large numbers for continuous localmartingales. Compare with Lépingle [11, Theorem 1] or with 3�� in Exercise 1.16in Chapter V in Revuz and Yor [16]. We note that the above-mentioned citationsare about continuous local martingales with time interval �0���, but they are alsovalid for continuous local martingales with time interval �0� T�, T ∈ �0���, withappropriate modifications in the conditions, see as follows.

Theorem 2.3. Let T ∈ �0��� be fixed and let(���� ��t�t∈�0�T��P

)be a filtered

probability space satisfying the usual conditions. Let �Mt�t∈�0�T� be a continuous localmartingale with respect to the filtration ��t�t∈�0�T� such that P�M0 = 0� = 1 andP�limt↑TMt = �� = 1. Let f � �1��� → �0��� be an increasing function such that∫ �

1

1f�x�2

dx < �

Then

P(limt↑T

Mt

f�Mt�= 0

)= 1

Now we present a law of the iterated logarithm for continuous localmartingales. Compare with Exercise 1.15 in Chapter V in Revuz and Yor [16]. Wenote that the above mentioned citation is about continuous local martingales withtime interval �0���, but the result is also valid for continuous local martingales withtime interval �0� T�, T ∈ �0���, with appropriate modifications in the conditions, seeas follows.

Theorem 2.4. Let T ∈ �0��� be fixed and let ����� ��t�t∈�0�T��P� be a filteredprobability space satisfying the usual conditions. Let �Mt�t∈�0�T� be a continuous localmartingale with respect to the filtration ��t�t∈�0�T� such that P�M0 = 0� = 1 andP�limt↑TMt = �� = 1. Then

P(lim sup

t↑T

Mt√2Mt ln�lnMt�

= 1)= P

(lim inf

t↑TMt√

2Mt ln�lnMt�= −1

)= 1

Theorem 2.4 simply follows from Theorem 2.2 and the law of the iteratedlogarithm for a standard Wiener process (see, e.g., [9, Theorem 2.9.23]).

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�-Wiener Bridges 451

3. Sample Paths Properties

The following Lemma 3.1 can be considered as a generalization of Lemma 5.6.9in Karatzas and Shreve [9]. Namely, our result with � = 1 gives back Lemma 5.6.9in Karatzas and Shreve [9]. Furthermore, Lemma 3.1 can be also considered as ageneralization of Corollary 4.4 in Becker-Kern [6] in the one-dimensional Browniancase. Namely, by (1.2), �X

���t �t∈�0�T� coincides with the process �Ut�t∈�0�T� defined

in (4.7) in Becker-Kern [6] in the one-dimensional Brownian case for all � > 0.In this case, Becker-Kern proved that Ut converges in probability to 0 as t ↑ T ,while we prove convergence with probability one. For historical fidelity, we remarkthat something similar to the statement of our Lemma 3.1 is stated on p. 1023 inMansuy [14] but without any proof.

Lemma 3.1. Let T ∈ �0��� and � > 0 be fixed, and let �Bt�t≥0 be a 1-dimensionalstandard Wiener process. The process �Y ���

t �t∈�0�T� defined by

Y���t �=

∫ t

0

(T − t

T − s

)�

dBs if t ∈ �0� T��

0 if t = T�

is a centered Gauss process with almost surely continuous paths.

Proof. By Bauer [5, Lemma 48.2], �Y ���t �t∈�0�T� is a centered Gauss process. To prove

almost surely continuity, we follow the method of the proof of Lemma 5.6.9 inKaratzas and Shreve [9]. For all t ∈ �0� T� and � ∈ �, let

M���t �=

∫ t

0

1�T − s��

dBs

Then �M���t �t∈�0�T� is a continuous, square-integrable martingale with respect to the

filtration induced by B and with quadratic variation

M���t �=∫ t

0

1�T − s�2�

ds =

T 1−2�

1− 2�

(1−

(1− t

T

)1−2�)if � �= 1

2�

− ln(1− t

T

)if � = 1

2�

t ∈ �0� T�

(3.1)

Then

limt↑T

M���t =

� if � ≥ 1

2�

T 1−2�

1− 2�if � <

12

(3.2)

Hence, in case of � ≥ 12 , Theorem 2.2 implies that there exists a standard one-

dimensional Wiener process �Wt�t≥0 on �����P� such that

P(M

���t = WM���t for all t ∈ �0� T�

) = 1

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452 Barczy and Pap

First, we consider the case of � > 12 . Let us define the function f� � �1��� → �0���

by f��x� �= x�/�2�−1�, x ≥ 1. Then f� is strictly monotone increasing and∫ �

1

1f��x�

2dx =

∫ �

1x−2�/�2�−1�dx = 2�− 1 < ��

hence, we may apply Theorem 2.3 and then we obtain

P(limt↑T

M���t

f��M���t�= 0

)= 1� � >

12

We have

Y���t = �T − t��M

���t = �T − t��f��M���t�

M���t

f��M���t��

where t ∈ �0� T� is such that M���t ≥ 1. Here

�T − t��f��M���t� ≤ �T − t��f�

(1

2�− 11

�T − t�2�−1

)= �2�− 1�−�/�2�−1��

where t ∈ �0� T� is such that M���t ≥ 1. Hence we conclude P(limt↑T Y

���t = 0

) = 1.Now we consider the case of � = 1

2 . Let us define the function f1/2�x� �= ex/2,x ∈ �1���. Then f1/2 is strictly monotone increasing and∫ �

1

1f1/2�x�

2dx =

∫ �

1e−xdx = e−1 < ��

hence, we may apply Theorem 2.3 and then we obtain

P(limt↑T

M�1/2�t

f1/2�M�1/2�t�= 0

)= 1

We have

Y�1/2�t = �T − t�1/2M

�1/2�t = �T − t�1/2f1/2�M�1/2�t�

M�1/2�t

f1/2�M�1/2�t��

where t ∈ �0� T� is such that M�1/2�t ≥ 1. Here

�T − t�1/2f1/2�M�1/2�t� = �T − t�1/2 exp{12ln(

T

T − t

)}= T 1/2�

where t ∈ �0� T� is such that M�1/2�t ≥ 1. Hence, we conclude P(limt↑T Y

�1/2�t = 0

)= 1.Finally, we consider the case of 0 < � < 1

2 . Using (3.2) we have Proposition 1.26in Chapter IV and Proposition 1.8 in Chapter V in Revuz and Yor [16] imply thatthe limit M���

T �= limt↑T M���t exists almost surely. Since

Y���t = �T − t��M

���t � t ∈ �0� T��

we get limt↑T Y���t = 0 almost surely. �

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�-Wiener Bridges 453

By Lemma 3.1, we can say that in case of � > 0, the process X��� has an almostsurely continuous extension. Later, in Remark 3.5, we examine the possibility ofsuch an almost surely continuous extension of �X���

t �t∈�0�T� in case of � ≤ 0.Now we prove some results about the asymptotic behavior of X

���t as t ↑ T .

Theorem 2.4 has the following consequences on X���.

Theorem 3.2. If � > 12 , then

P

(lim sup

t↑T

X���t√

2�T−t�

2�−1 ln(ln 1

T−t

) = 1

)= P

(lim inf

t↑TX

���t√

2�T−t�

2�−1 ln(ln 1

T−t

) = −1

)= 1

(3.3)

Especially,

P(lim sup

t↑T

X���t

�T − t��= �

)= P

(lim inf

t↑TX

���t

�T − t��= −�

)= 1 (3.4)

Proof. With the notation introduced in the proof of Lemma 3.1, we have X���t

�T−t��=

M���t , t ∈ �0� T�, and the quadratic variation M���t, t ∈ �0� T�, of the continuous

martingale X���t

�T−t��, t ∈ �0� T�, is given in (3.1). Using (3.2) we have limt↑TM���t = �.

Then, by Theorem 2.4, in case of � > 12 we get

P(lim sup

t↑T

X���t√

D��T �t�= 1

)= P

(lim inf

t↑TX

���t√

D��T �t�= −1

)= 1�

where for all t ∈ �0� T�,

D��T �t� �= 2�T − t�2�T 1−2�

1− 2�

(1−

(1− t

T

)1−2�)ln

(ln(T 1−2�

1− 2�

(1−

(1− t

T

)1−2�)))

Hence, to prove (3.3) it is enough to check that

limt↑T

2�T − t�2� T 1−2�

1−2�

(1− (

1− tT

)1−2�)ln(ln(

T 1−2�

1−2�

(1− (

1− tT

)1−2�)))

2�T−t�

2�−1 ln(ln 1

T−t

) = 1

This is satisfied, since, by using L’Hospital’s rule twice, we get

limt↑T

ln(ln(

T 1−2�

1−2�

(1− (

1− tT

)1−2�)))

ln(ln 1

T−t

) = limt↑T

�T − t�1−2� − T 1−2�

�T − t�1−2�= 1

Using (3.3) and the decomposition

X���t

�T − t��= X

���t√

2�T−t�

2�−1 ln(ln 1

T−t

)√2�T − t�1−2�

2�− 1ln(ln

1T − t

)� t ∈ �0� T��

we have (3.4). �

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454 Barczy and Pap

The next theorem is about the limit behavior of X�1/2�t as t ↑ T .

Theorem 3.3. We have

P

(lim sup

t↑T

X�1/2�t√

2�T − t�(ln 1

T−t

)(ln ln ln 1

T−t

) = 1

)

= P

(lim inf

t↑TX

�1/2�t√

2�T − t�(ln 1

T−t

)(ln ln ln 1

T−t

) = −1

)= 1 (3.5)

Especially,

P(lim sup

t↑T

X�1/2�t√T − t

= �)= P

(lim inf

t↑TX

�1/2�t√T − t

= −�)= 1

Proof. With the notation introduced in the proof of Lemma 3.1, we have X�1/2�t√T−t

=M

�1/2�t , t ∈ �0� T�, and the quadratic variation M�1/2�t, t ∈ �0� T�, of the continuous

martingale X�1/2�t√T−t

, t ∈ �0� T�, is given in (3.1). Using (3.2) we have limt↑TM�1/2�t = �.Then, by Theorem 2.4, we get

P

(lim sup

t↑T

X�1/2�t√

2�T − t�(ln T

T−t

)(ln ln ln T

T−t

) = 1

)

= P

(lim inf

t↑TX

�1/2�t√

2�T − t�(ln T

T−t

)(ln ln ln T

T−t

) = −1

)= 1�

which yields (3.5). �

The next theorem is about the limit behavior of X���t as t ↑ T in case of � < 1

2 .

Theorem 3.4. If � < 12 , then

P(limt↑T

X���t

�T − t��= M

���T

)= 1� (3.6)

where M���T is a normally distributed random variable with mean 0 and with variance

T 1−2�

1−2� . Consequently,

P(limt↑T

X���t

�T − t��= 0

)= 1 for all � < �� (3.7)

P(limt↑T

X���t

�T − t��= −�

)= P

(limt↑T

X���t

�T − t��= �

)= 1

2for all � > � (3.8)

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�-Wiener Bridges 455

Proof. By (1.2), using the notations introduced in the proof of Lemma 3.1, we get

M���t = X

���t

�T − t��=∫ t

0

1�T − s��

dBs� t ∈ �0� T�

By (3.2), since � < 12 ,

limt↑T

M���t =T 1−2�

1− 2�< ��

and, hence, Proposition 1.26 in Chapter IV and Proposition 1.8 in Chapter V inRevuz and Yor [16] imply that the limit M

���T �= limt↑T M

���t exists almost surely.

Using that M���t is normally distributed with mean 0 and with variance M���t

for all t ∈ �0� T�, we have the random variable M���T is also normally distributed

with mean 0 and with variance T 1−2�

1−2� . Indeed, normally distributed random variablescan converge in distribution only to a normally distributed random variable, bycontinuity theorem (see, e.g., Shiryaev [17, p. 304]). This implies (3.6). Hence, for all�� � ∈ �, we get

X���t

�T − t��= �T − t��−�M

���t � t ∈ �0� T� (3.9)

If � < �, then using (3.9) and that P�limt↑T M���t = M

���T � = 1, we get (3.7). If � > �,

using that P�M���T = 0� = 0, we have (3.9) implies that

P(limt↑T

X���t

�T − t��∈ −����

)= 1

Since P�M���T > 0� = P�M���

T < 0� = 12 , we get (3.8). �

Remark 3.5. In case of � = 0, the process �X�0�t �t∈�0�T� is a standard Wiener process

and, hence, it can be extended to an almost surely continuous process �Y�0�t �t∈�0�T�

with the definition Y�0�T �= BT . In case of � < 0, there does not exist any almost surely

continuous process �Y ���t �t∈�0�T� such that P�X���

t = Y���t � = 1 for all t ∈ �0� T�. Indeed,

by (3.8), we get

P(limt↑T

X���t = −�

)= P

(limt↑T

X���t = �

)= 1

2for all � < 0

4. Singularity of Induced Measures

For probability measures P1 and P2 on a measurable space �����, equivalence andsingularity of them will be denoted by P1 ∼ P2 and P1 ⊥ P2, respectively.

Using that for all � ∈ �, the process X��� has continuous paths (by the definitionof strong solution, see, for example, [8, Definition 2.24, Chapter III]), we have

P( ∫ t

0

�X���u �2

�T − u�2du < �

)= 1� ∀� ∈ �� t ∈ �0� T� (4.1)

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456 Barczy and Pap

For all � ∈ � and t ∈ �0� T�, let PX����t denote the law of the process �X���s �s∈�0�t� on(

C�0� t����C�0� t��), where ��C�0� t�� denotes the Borel -algebra on C�0� t�. Using

Theorem 7.20 in Liptser and Shiryaev [12] and (4.1), we get PX����t ∼ PX�0��t and

dPX����t

dPX�0��t

�X��� ��0�t�� = exp{−�

∫ t

0

X���u

T − udX���

u − �2

2

∫ t

0

�X���u �2

�T − u�2du} (4.2)

Here, PX�0��t is nothing else but the Wiener measure on �C�0� t����C�0� t���.

We recall that for all t ∈ �0� T�, the maximum likelihood estimator (MLE) �̂�X����

t

of the parameter � based on the observation �X���s �s∈�0�t� is defined by

�̂�X����t �= argmax

�∈�ln(dPX����t

dPX�0��t

(X���

∣∣�0�t�

))

By (4.1) and (4.2), for all t ∈ �0� T�, there exists a unique MLE �̂�X����t of the

parameter � based on the observation �X���s �s∈�0�t� given by

�̂�X����t = −

∫ t

0X���s

T−sdX���

s∫ t

0�X

���s �2

�T−s�2ds

� t ∈ �0� T�

To be more precise, by (4.1), for all t ∈ �0� T�, the MLE �̂�X����t exists P-almost surely.

As a special case of Theorem 3.12 in Barczy and Pap [2], the MLE of � is stronglyconsistent, that is,

P(limt↑T

�̂�X����t = �

)= 1� � ∈ � (4.3)

For all � ∈ �, let PTX����T

be the law of the process �X���t �t∈�0�T� given by the

SDE (1.1) on �C�0� T����C�0� T���.

Theorem 4.1. For all �� � ∈ �, � �= �, we have PTX����T

⊥ PTX����T

. In other words, the

laws of the processes �X���t �t∈�0�T� and �X

���t �t∈�0�T� on �C�0� T����C�0� T��� are singular

for all �� � ∈ �, � �= �.

Proof. First we check that for all � ∈ � and t ∈ �0� T�,

∫ t

0

X���s

T − sdX���

s = 12

(�X

���t �2

T − t−∫ t

0

�X���s �2

�T − s�2ds − ln

(T

T − t

)) (4.4)

By Itô’s rule (see, e.g., [12, Theorem 4.4]), we get

d

(X

���t

T − t

)= X

���t

�T − t�2dt + 1

T − tdX���

t

= X���t

�T − t�2dt − �

X���t

�T − t�2dt + 1

T − tdBt� t ∈ �0� T� (4.5)

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�-Wiener Bridges 457

Now we verify that �X���t �t∈�0�T� and

(X���t

T−t

)t∈�0�T� are continuous semimartingales

adapted to the filtration induced by B. Consider the decomposition

X���t = �T − t��

∫ t

0

1�T − s��

dBs� t ∈ �0� T�

Here, the deterministic function �T − t��, t ∈ �0� T�, is monotone and hence has afinite variation over each finite interval of �0� T�, and then, by Jacod and Shiryaev[8, Proposition 4.28, Chapter I], it is a semimartingale. Since

∫ t

0

1�T − s��

dBs� t ∈ �0� T��

is a martingale with respect to the filtration induced by B, using Theorem 4.57 inChapter I in Jacod and Shiryaev [8] with the function f�x� y� �= xy, x� y ∈ �, wehave �X���

t �t∈�0�T� is a continuous semimartingale adapted to the filtration induced byB. Similarly as above, using that 1

T−t, t ∈ �0� T�, is continously differentiable, and

hence has a finite variation over each finite interval of �0� T�, one can get(

X���t

T−t

)t∈�0�T�

is a continuous semimartingale adapted to the filtration induced by B. Moreover, by(4.5), the cross-variation process of the continuous martingale parts of the processes�X

���t �t∈�0�T� and

(X���t

T−t

)t∈�0�T�

equals

∫ t

0

1T − s

ds = ln(

T

T − t

)� t ∈ �0� T�

Hence, by integration by parts formula (see, e.g., [9, p. 155]), we have for allt ∈ �0� T�,

∫ t

0

X���s

T − sdX���

s = X���t

T − tX

���t −

∫ t

0X���

s d(

X���s

T − s

)− ln

(T

T − t

)

= �X���t �2

T − t−∫ t

0

�X���s �2

�T − s�2ds −

∫ t

0

X���s

T − sdX���

s − ln(

T

T − t

)�

which yields (4.4). Hence, �̂�X����t = At�X

���� for all t ∈ �0� T�, where At �

C�0� T�\ 0� → �, defined by

At�x� �=− x�t�2

T−t+ ∫ t

0x�s�2

�T−s�2ds + ln

(T

T−t

)2∫ t

0x�s�2

�T−s�2ds

� x ∈ C�0� T�\ 0�� t ∈ �0� T�

For all � ∈ �, let us introduce the following subset of C�0� T�,

S� �={x ∈ C�0� T�\ 0� � lim

t↑TAt�x� = �

}

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458 Barczy and Pap

We check that S� ∈ ��C�0� T��. By Problem 2.4.1 in Karatzas and Shreve [9], underthe metric

��x� y� �=�∑n=1

12n

supu∈�0�n�

��x���u��− y���u�� � ∧ 1�� x� y ∈ C�0� T��

the set C�0� T� is a complete, separable metric space, where � � �0��� �→ �0� T�,��u� �= 2T

�arctan�u�, u ≥ 0. For all t ∈ �0� T�, let Lt � C�0� T� → �,

Lt�x� �=∫ t

0

x�s�2

�T − s�2ds� x ∈ C�0� T�

Let x ∈ C�0� T� be fixed. We show that for all t ∈ �0� T�, Lt is continuous at the pointx ∈ C�0� T�. Indeed, for all y ∈ C�0� T�, we have

�Lt�x�− Lt�y�� ≤ sups∈�0�t�

(�x�s�+ y�s���x�s�− y�s��) ∫ t

0

1�T − s�2

ds

≤ sups∈�0�t�

(�2�x�s�� + �y�s�− x�s����x�s�− y�s��) ∫ t

0

1�T − s�2

ds

If y ∈ C�0� T� is such that � �= sups∈�0�t� �y�s�− x�s�� < 1 and n0 ∈ � is such thatn0 > �−1�t�, then

��x� y� ≥ 12n0

supu∈�0�n0�

��y���u��− x���u��� ∧ 1�

≥ 12n0

supu∈�0��−1�t��

�y���u��− x���u��� = �

2n0�

and, hence,

�Lt�x�− Lt�y�� ≤ �

(1+ 2 sup

s∈�0�t��x�s��

) ∫ t

0

1�T − s�2

ds ≤ K�t���x� y��

where K�t� �= 2n0(1+ 2 sups∈�0�t� �x�s��

) ∫ t

01

�T−s�2ds, which yields the continuity of Lt

at x. Consequently, At is continuous for all t ∈ �0� T�. Consider the decomposition

S� =⋂�>0

⋃t∈�0�T�

⋂s∈�t�T�

{x ∈ C�0� T�\ 0� � �As�x�− �� ≤ �

}=

�⋂n=1

�⋃m=1

⋂s∈�T− 1

m �T�∩�+

{x ∈ C�0� T�\ 0� � �As�x�− �� ≤ 1

n

}�

where �+ denotes the set of positive rational numbers. Since As is continuous forall s ∈ �0� T�, we have{

x ∈ C�0� T�\ 0� � �As�x�− �� ≤ 1n

}∈ ��C�0� T��� s ∈ �0� T�� n ∈ ��

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�-Wiener Bridges 459

and, hence, S� ∈ ��C�0� T��. For all �� � ∈ �, � �= �, we have S� ∩ S� = ∅ and,by (4.3),

PTX����T

�S�� = P(limt↑T

�̂�X����t = �

)= 1� PT

X����T�S�� = P

(limt↑T

�̂�X����t = �

)= 1�

PTX����T

�S�� = PTX����T

�S�� = 0�

which implies the assertion by definition of singularity. �

In what follows, we will study the connections between the technique of theproof of our Theorem 4.1 and the very general results on singularity and absolutecontinuity due to Jacod and Shiryaev [8, Chapter IV]. In fact, we also present asecond proof of Theorem 4.1.

First, we recall that the proof of Theorem 4.1 is based on the strong consistencyof the MLE of �, see Barczy and Pap [2, Theorem 3.12]. A short outline of the proofof Theorem 3.12 in Barczy and Pap [2] (specifying for �-Wiener bridges) sounds asfollows. Using the explicit form of the Laplace transform of

∫ t

0�X

���u �2

�T−u�2du, t ∈ �0� T�,

due to Barczy and Pap [2, Theorem 4.1], one can check that

limt↑T

E exp{−∫ t

0

�X���u �2

�T − u�2du}= 0� ∀� ∈ �

Hence,

P(limt↑T

∫ t

0

�X���u �2

�T − u�2du = �

)= 1� ∀� ∈ �� (4.6)

which easily implies strong consistency of the MLE of �. It will turn out that ifwe apply Theorem 4.23 in Jacod and Shiryaev [8, Chapter IV] for proving PT

X����T⊥

PTX����T

with �� � ∈ �, � �= �, then we have to check condition (4.6). We also notethat the fact that condition (4.6) has to be checked is in accordance with part (i) ofTheorem 1 in Ben-Ari and Pinsky [4]. But we emphasize that Ben-Ari and Pinsky’sresult is valid for time-homogeneous diffusions and hence we can not use it for�-Wiener bridges. By giving a second proof of Theorem 4.1, we shed more light onthe role of condition (4.6).

Second Proof of Theorem 4.1. Let �� � ∈ �, � �= � be fixed. Let us introduce theprocess �X̃���

t �t≥0 given by

X̃���t �= X

���

��t�� t ≥ 0�

where � � �0��� �→ �0� T�, ��t� = 2T�arctan�t�, t ≥ 0. Then, by the SDE (1.1) and a

change of variable, we get for all t ≥ 0,

X̃���t = −�

∫ ��t�

0

X���s

T − sds + B��t� = −�

∫ t

0

X���

��u�

T −��u���u�du+ B��t��

where ��t� �= ddt��t�, t ≥ 0, and �B��t��t≥0 is a Wiener process with variance function

��t�, t ≥ 0, see Definitions 4.9 in Chapter I in Jacod and Shiryaev [8].

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Let us consider the filtered space �C�0������ ��t�t≥0�, where � is the Borel-algebra ��C�0���� on C�0��� and �t, t ≥ 0, defined as follows. For all t ≥ 0, let

�t �=⋂�>0

�−1t+�����

where �t � C�0��� → C�0��� defined by ��tx��s� �= x�t ∧ s� for s ≥ 0, x ∈ C�0���.Then the filtration ��t�t≥0 is right-continuous, since for all t ≥ 0,

�t =⋂�>0

�−1t+���� = ⋂

�>0

⋂�>0

�−1t+�+���� = ⋂

�>0

�t+�

Moreover, since �−1t ��� ⊂ �t for all t ≥ 0, by Problem 2.4.2 in Karatzas and

Shreve [9], we get

� =

(⋃t≥0

�t

)

Let PX̃��� and PX̃��� denote the law of the processes �X̃���t �t≥0 and �X̃

���t �t≥0 on

�C�0������, respectively. We check that

PTX����T

⊥ PTX����T

⇐⇒ PX̃��� ⊥ PX̃���

Indeed, by definition, PTX����T

⊥ PTX����T

means that there exist so called distinguishingsets S� and S� in ��C�0� T�� such that S� ∩ S� = ∅ and

P( � ∈ � � �X

���t ����t∈�0�T� ∈ S��

) = P( � ∈ � � �X

���t ����t∈�0�T� ∈ S��

) = 1

Similarly, PX̃��� ⊥ PX̃��� means that there exist S̃� and S̃� in � such that S̃� ∩ S̃� = ∅and

P({� ∈ � �

(X̃

���t ���

)t≥0

∈ S̃�}) = P

({� ∈ � �

(X̃

���t ���

)t≥0

∈ S̃�}) = 1

Using that

� ∈ � � �X̃���t ����t≥0 ∈ S̃�� =

{� ∈ � �

(X

���

��t����)t≥0

∈ S̃�}

= {� ∈ � �

(X

���t ���

)t∈�0�T� ∈ �−1�̃S��

}�

where �−1�̃S�� �= f ��−1 � f ∈ S̃��, singularity of PX̃��� and PX̃��� withdistinguishing sets S̃�� S̃� ∈ � implies singularity of PT

X����Tand PT

X����Twith

distinguishing sets �−1�̃S����−1�̃S�� ∈ ��C�0� T��. The converse statement can be

thought over similarly.Hence, by Corollary 2.8 in Chapter IV in Jacod and Shiryaev [8], to prove the

assertion it is enough to check that the measures PX̃��� and PX̃��� are locally equivalentwith respect to each other (where the restrictions of the measures refers to thegiven filtration ��t�t≥0) and that PX̃��� �limt→� h

�1/2�t < �� = 0, where �h�1/2�

t �t>0 is the

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�-Wiener Bridges 461

Hellinger process of order 1/2 between PX̃��� and PX̃��� . Using that the continuity ofthe process X��� implies that the process

∫ t

0

(X

���

��u�

)2�T −��u��2

��u�2du� t ≥ 0�

does not jump to infinity (for the definition of jumping to infinity, see, for example,[8, Definitions 5.8(ii) in Chapter III]), by (4.1) and a generalization of part (b) and(c) of Theorem 4.23 in Chapter IV in Jacod and Shiryaev [8], we have the measuresPX̃��� and PX̃��� are locally equivalent with respect to each other and the process

��− ��2

8

∫ t

0

x���u��2

�T −��u��2��u�du� x ∈ C�0���� t > 0� (4.7)

is a version of the Hellinger process(h�1/2�t

)t>0

. Indeed, using the notations ofSections 3a and 4b in Chapter IV in Jacod and Shiryaev [8], we have C�t� = ��t� =∫ t

0 ��s�ds, t ≥ 0, and

�s�x� = −�x���s��

T −��s���s�� x ∈ C�0���� s ≥ 0�

�′s�x� = −�

x���s��

T −��s���s�� x ∈ C�0���� s ≥ 0�

�̃s�s� =�s�x�− �′

s�x�

��s�= −��− ��

x���s��

T −��s�� x ∈ C�0���� s ≥ 0

Hence, using the very same arguments given in the proof of Theorem 4.23 in Jacodand Shiryaev [8, Chapter IV], we get (4.7). By a change of variable, we have

∫ t

0

x���u��2

�T −��u��2��u�du =

∫ ��t�

0

x�s�2

�T − s�2ds� x ∈ C�0���� t ≥ 0�

and, hence, PX̃���

(limt→� h

�1/2�t < �) = 0, � ∈ �, is equivalent with (4.6), i.e., to

prove the assertion it is enough to verify (4.6). As it was mentioned earlier, as aspecial case of the proof of Theorem 3.12 in Barczy and Pap [2] we get (4.6). �

Remark 4.2. If �� � ∈ �, � �= �, by Theorem 4.1, we have PTX����T

⊥ PTX����T

.Moreover, in the proof of the theorem, we also constructed disjoint sets S�and S� in ��C�0� T�� that distinguish between the measures in the sense thatPTX����T

�S�� = 1 and PTX����T

�S�� = 1. We note that for some special time-homogeneous(one-dimensional) diffusions Ben-Ari and Pinsky [4, Propositions 1–3] also gave“illuminating” distinguishing sets.

Remark 4.3. In case of � ≥ 0, by Lemma 3.1, one can define a probability measurePTY ����T

on �C�0� T����C�0� T���, as the law of the process �Y���t �t∈�0�T� given in

Lemma 3.1. Then, by Theorem 4.1, for all �� � ≥ 0, � �= �, we get PTY ����T

⊥ PTY ����T

.Indeed, since PT

X����T⊥ PT

X����T, there exist sets S� and S� in ��C�0� T�� such that S� ∩

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462 Barczy and Pap

S� = ∅ and PTX����T

�S�� = PTX����T

�S�� = 1. For all B ∈ ��C�0� T��, let us introduce thenotation

B̃ �={x ∈ C�0� T� � x ��0�T� ∈ B and ∃ lim

t↑Tx�t� ∈ �

}

Then we have S̃�� S̃� ∈ ��C�0� T��, S̃� ∩ S̃� = ∅ and PTY ����T

�̃S�� = PTY ����T

�̃S�� = 1. As aspecial case, we also have for all � > 0, the probability measure PT

Y ����Tand the

standard Wiener measure PY �0��T on �C�0� T����C�0� T��� are singular.

Remark 4.4. We note that Theorem 4.1 is not an astonishing result. One can easilyformulate conditions on a general time-inhomogeneous diffusion process underwhich the same kind of singularity holds. Namely, let us consider a process

(X

���t

)t≥0

given by the SDE {dX���

t = �a(t� X

���t

)dt + dBt� t ≥ 0�

X���0 = 0�

(4.8)

where a � �0���×� → � is a known Borel-measurable function, �Bt�t≥0 is astandard Wiener process, and � ∈ � is an unknown parameter. Let us suppose thatthe SDE (4.8) has a unique strong solution

(X

���t

)t≥0

for all � ∈ �. For all � ∈ �, let

us denote by P� the law of(X

���t

)t≥0

on �C�0������C�0�����. Let us suppose thatfor all t > 0 and all � ∈ �,

P( ∫ t

0a(s� X���

s

)2ds < �

)= 1

As it is explained in details in the second proof of Theorem 4.1, using Theorem 4.23in Chapter IV in Jacod and Shiryaev [8], we get for all �1, �2 ∈ �, �1 �= �2,

P�1⊥ P�2

⇐⇒ P(limt→�

∫ t

0a(u�X��i�

u

)2du = �

)= 1 for i = 1 or i = 2

Concerning singularity of P�1and P�2

, the point is that whether the imposedconditions can be checked for a given diffusion process. And in this respect, time-inhomogeneous diffusions in general represent a hard task.

Concerning the SDE (1.1) one can ask why the diffusion coefficient in the SDE(1.1) is identically 1. The point is only that it is supposed to be a known andpositive constant. Remember that in many cases the measures induced by processeswith different diffusion coefficients are singular and continuous-time statisticalinference for this type of model is often trivial. In what follows we consider thisphenomenon in details. For all T ∈ �0���, � ∈ � and > 0, let us introduce thetime-inhomogeneous diffusion process

(X

����t

)t∈�0�T� given by the SDEdX

����t = − �

T − tX

����t dt + dBt� t ∈ �0� T��

X����0 = 0�

(4.9)

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�-Wiener Bridges 463

where �Bt�t≥0 is a 1-dimensional standard Wiener process. By formula (5.6.6) inKaratzas and Shreve [9], the SDE (4.9) has a unique strong solution, namely,

X����t =

∫ t

0

(T − t

T − s

)�

dBs� t ∈ �0� T�

For all t ∈ �0� T�, let PX�����t be the law of the process(X����

s

)s∈�0�t� given by the SDE

(4.9) on(C�0� t����C�0� t��

).

Theorem 4.5. For all �1� �2 ∈ �, 1 > 0, 2 > 0 and t ∈ �0� T�, the followingdichotomy holds:

PX��1�1��t ∼ PX��2�2��t if 1 = 2�

PX��1�1��t ⊥ PX��2�2��t if 1 �= 2

First Proof. In case of 1 = 2, the equivalence of PX��1�1��t and PX��2�2��t followsfrom Theorem 7.20 or Theorem 7.19 in Liptser and Shiryaev [13] and from (4.1).

Let us suppose now that 1 �= 2. For all � ∈ � and > 0, by giving a directproof, we show the following Baxter type result

P(limn→�

n∑j=1

(X

����jt/n − X

����

�j−1�t/n

)2 = t2

)= 1 (4.10)

By the SDE (4.9), we have

n∑j=1

(X

����jt/n − X

����

�j−1�t/n

)2= �2

n∑j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)2

− 2�n∑

j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)�Bjt/n − B�j−1�t/n�

+ 2n∑

j=1

�Bjt/n − B�j−1�t/n�2 (4.11)

It is known that

P(limn→�

n∑j=1

�Bjt/n − B�j−1�t/n�2 = t

)= 1� (4.12)

(see, e.g., [12, Lemma 4.3]). Moreover, by Lagrange’s mean value theorem, one canthink it over that for all t ∈ �0� T� and for all continuous functions f � �0� t� → �,we have

limn→�

n∑j=1

( ∫ jt/n

0f�x�dx −

∫ �j−1�t/n

0f�x�dx

)2

= 0

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464 Barczy and Pap

Since for all t ∈ �0� T�, the process(X����u

T−u

)u∈�0�t� is continuous, we have

P

(limn→�

n∑j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)2

= 0

)= 1 (4.13)

Now we check that

P

(limn→�

n∑j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)�Bjt/n − B�j−1�t/n� = 0

)= 1

(4.14)

By Cauchy–Schwarz’s inequality, we have

n∑j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)�Bjt/n − B�j−1�t/n�

≤√√√√ n∑

j=1

( ∫ jt/n

0

X����u

T − udu−

∫ �j−1�t/n

0

X����u

T − udu)2√√√ n∑

j=1

�Bjt/n − B�j−1�t/n�2

with probability one, and then (4.12) and (4.13) implies (4.14). By (4.12), (4.13),and (4.14), using (4.11) we have (4.10). Then, using the definition of singularity ofmeasures, (4.10) implies that PX���1��t ⊥ PX���2��t for all � ∈ � and 1 �= 2. In caseof �1 �= �2 and 1 �= 2 we have PX��1�1��t ∼ PX��2�1��t and PX��2�1��t ⊥ PX��2�2��t, whichimply that PX��1�1��t ⊥ PX��2�2��t.

Second Proof. Using Baxter’s theorem due to Baxter [3, Theorem 1], we show thatfor all � ∈ � and > 0,

P(limn→�

2n∑k=1

(X

����kt/2n − X

����

�k−1�t/2n)2 = t2

)= 1� (4.15)

which is also enough (like (4.10)) to ensure that PX���1��t ⊥ PX���2��t for all � ∈ � and1 �= 2. For all t ∈ �0� T�, �X����

s �s∈�0�t� is a Gauss process with identically 0 meanfunction, and to have right to apply Baxter’s theorem, we need to check that thecovariance function of �X����

s �s∈�0�t� is continuous on �0� t�× �0� t� and has uniformlybounded second derivatives on �0� t�× �0� t�\ �s� s� � s ∈ �0� t��.

In case of � �= 12 , by (2.1), we get for all u� v ∈ �0� t�,

r�v� u� �= Cov�X����v � X����

u � = 2 �T − v���T − u��

1− 2�

(T 1−2� − �T − �v ∧ u��1−2�

)

Then r�v� u�, v� u ∈ �0� t�, is continuous, and clearly, if 0 ≤ v < u ≤ t, then

�r

�v�v� u� = 2 T

1−2�

1− 2��T − u����T − v��−1�−1�− 2

1− 2��T − u���1− ���T − v�−��−1��

�r

�u�v� u� = 2 T 1−2�

1− 2��T − v����T − u��−1�−1�− 2

1− 2��T − v�1−���T − u��−1�−1�

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�-Wiener Bridges 465

If 0 ≤ u < v ≤ t, then

�r

�v�v� u� = 2 T 1−2�

1− 2��T − u����T − v��−1�−1�− 2

1− 2��T − u�1−���T − v��−1�−1��

�r

�u�v� u� = 2 T

1−2�

1− 2��T − v����T − u��−1�−1�− 2

1− 2��T − v���1− ���T − u�−��−1�

This implies that �X����s �s∈�0�t� has uniformly bounded first derivatives on �0� t�×

�0� t�\ �s� s� � s ∈ �0� t��, and similarly one can check that the second derivatives alsoadmit this property. Moreover, for all u ∈ �0� t�,

D+�u� � = limv↓u

r�u� u�− r�v� u�

u− v= lim

v↓ur�v� u�− r�u� u�

v− u= lim

v↓u�r

�v�v� u�

= − 2�

1− 2�T 1−2��T − u�2�−1 + 2�

1− 2�

Similarly, for all u ∈ �0� t�,

D−�u� � = limv↑u

r�u� u�− r�v� u�

u− v= lim

v↑ur�v� u�− r�u� u�

v− u= lim

v↑u�r

�v�v� u�

= − 2�

1− 2�T 1−2��T − u�2�−1 + 2�1− ��

1− 2�

Hence, for all t ∈ �0� T�,

∫ t

0�D−�u�−D+�u��du = 2

∫ t

0

(1− �

1− 2�− �

1− 2�

)du = 2

∫ t

01 du = 2t�

and then Baxter’s theorem due to Baxter [3, Theorem 1] yields (4.15).In case of � = 1

2 , similarly to the case � �= 12 , one can check that the covariance

function of �X�1/2��s �s∈�0�t� is continuous on �0� t�× �0� t�, and has uniformly bounded

first and second derivatives on �0� t�× �0� t�\ �s� s� � s ∈ �0� t��. Moreover, by (2.1),for all u ∈ �0� t�,

D+�u� = limv↓u

2√�T − v��T − u� ln

(T

T−u

)− 2�T − u� ln(

TT−u

)v− u

= 2 ln(

T

T − u

)√T − u

ddu

√T − u = −2

2ln(

T

T − u

)

Similarly, by (2.1), for all u ∈ �0� t�,

D−�u� = limv↑u

2√�T − v��T − u� ln

(T

T−v

)− 2�T − u� ln(

TT−u

)v− u

= 2√T − u

ddu

(√T − u ln

(T

T − u

))= −2

2ln(

T

T − u

)+ 2

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466 Barczy and Pap

Hence, for all t ∈ �0� T�,∫ t

0�D−�u�−D+�u��du = 2

∫ t

01 du = 2t�

and then Baxter’s theorem due to Baxter [3, Theorem 1] yields (4.15). �

We note that the same dichotomy that we have in Theorem 4.5 holds forOrnstein–Uhlenbeck processes (see, e.g., [1, p. 226]).

Remark 4.6. For all � ∈ � and > 0, let PX���� denote the law of the process�X����

s �s∈�0�T� on �C�0� T����C�0� T���. By the proof of Theorem 4.1, we get PX��1�1� ⊥PX��2�2� for all �1� �2 ∈ � and 1 > 0, 2 > 0.

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