On weak convergence of sequences of continuous local martingales

A NNALES DE L ’I. H. P., SECTION B

S HINTARO N AKAO

On weak convergence of sequences of continuous local martingales

Annales de l’I. H. P., section B, tome 22, n

^o

3 (1986), p. 371-380

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On weak convergence of sequences of continuous local martingales

Shintaro NAKAO

Department ^of Mathematics, ^Osaka University, Toyonaka, ^Osaka 560, Japan

Ann. Inst. Henri Poincaré,

Vol. 22, ^n° 3, 1986, p. 371-380. Probabilités et Statistiques

SUMMARY. -

Let {Mn(t) ~n-1,2,...

^{be a sequence} of Hilbert _space valued continuous local

martingales.

^We

give

^a necessary and sufficient condition for

which {

^is

tight

^{in C in terms}

of ( ( }. Using

^{this result we}

show the

preservation

^{of the local}

martingale property

^{under the weak} convergence in C

of {

Mots-clés : Continuous local martingale. Weak convergence.

RESUME. ^-

Soit { ^Mn(t) ~,~=1,2,...

^{une suite de}

martingales

locales conti-

nues a valeurs dans un espace de Hilbert. Nous donnons une condition necessaire et suffisante _pour

que {

^{soit tendue dans C a} l’aide d’une condition de tension

sur ~ ~ }.

A l’aide de ce

résultat,

^{nous montrons}

que la

propriete

^de

martingale

faible locale est conservee _par la convergence faible dans C.

1. INTRODUCTION

Let H be a real

separable

^Hilbert space with inner

product (,)

^and

be an orthonormal basis of H. Consider a

sequence { Mn(t) ~n-1,2,...

Classi,fication AMS : 60 G 44.

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

Vol. 22/86/03/371/ 10 ,’~ 3,00/~ Gauthier-Villars 371

372 S. NAKAO

of H-valued continuous local

martingales starting

^at

^0,

where the time interval I is

[0, T] ] (0

^{T +}

oo)

^or

[0,

⁺

oo).

^Denote

by dia ~ (t)

the vector of all

diagonal components

^of

( ~ (Mn, ek), (Mn, el) ~ where ( (Mn, ek), (M~‘, el) ~

^{is the}

quadratic

variational _process of the

martingales (Mn, ek)

^and

(M~‘, el) (cf. [3 ] [6 ] [9 ]).

^{We then}

regard dia ; Mn ~

as an

l1-valued

continuous _process in ^case that H is infinite-dimensional.

Rebolledo

[11 ] proved, using Lenglart’s inequality [8],

^{that in case}

of H = f~ the

tightness

^{in C}

of { ^Mn(t) ~n-1,~9.,_

^{and the}

tightness

^{in C of}

~ dia ~ M~‘ ~ (t) ~n-1 ~2,... { _ ~ M~‘ ~ {t) ~n= n2,...)

^are

equivalent

^{to each}

other,

^{where C is the} space of continuous

sample

functions. A main _purpose of this article is to show the above

equivalence

^{holds even if H} is infinite- dimensional

(Theorem

^A in Section

2).

By applying

^{Theorem A we} show that _any accumulation

point

^of

~ ~,~-

~,2,", under the weak convergence in C is also a continuous local

martingale (Theorem

^{B in Section}

3).

Moreover the

continuity

^of

the

mapping dia ( )>

under the weak convergence in C is stated in Theo-

rem C in Section 3.

In

Appendix

^we

give

^an

elementary proof

of Theorem A in case of H == [R which is best

adapted

^{to the situation} where all the processes considered

are continuous and which is based ^on

change

^{of time.}

2. TIGHTNESS

OF ~ M’~ ~ AND { dia ~ M~‘ ~ ~

For a real

separable

^Banach space

E,

^set

Co(I, E) _ ~

^{w: I ---~}

E,

^w is continuous and

v~(o)

⁼

0 }

and we define the usual metric on

Co(I, E) (see [5 ]);

^that

is, {

^converges

to w if and

only if {

^converges

uniformly

^{to w on each}

compact

^interval in I. Let

E))

^{be the}

topological 6-algebra

^of

Co(I, E).

^{We denote}

the space of all

probability

measures on

(Co(I, E), E))) by ~(Co(I, E))

and endow

~(Co{I, E))

with the weak convergence

topology.

Consider the

following subspace

^of

~(Co(I, E)) ;

is continuous local

martingale

^for every

/

^E

E’ j,

where E’ is the dual _space of E.

Annales de l’lnstitut Henri Poincaré - Probabilités ^et Statistiques

373

WEAK CONVERGENCE OF CONTINUOUS LOCAL MARTINGALES

For an orthonormal of

H,

define the

mapping ~):

Co(I, H) -~ Co(I, l2(J)) by

(2.1) ~(w)(t) _ ((w(t),

^{J for w E}

Co(I, H), t E I ,

where

for p

^{= 1. ?}

Then it is obvious that

is a

homeomorphism . According

^{to the}

mapping ~,

^we introduce the

mapping C:

by

We have then

by (2 . 2)

is a

homeomorphism

and

is a

homeomorphism.

We denote

by w(t) =

^the

sample path

^of

Co(I, l2(J)).

^{Since the}

0 0

process

(dia ( w ), P) = (( ( P)

^{is a}

ll(J)-valued

continuous _process

0

for P e

l2{J))),

^{we can define the}

mapping dia ( ):

in the

following

^{manner; for P E}

12(J)))

0

(2. 6) dia ( P )> ==

^the

probability

measure on

Co(I, ~1(J))

^induced

by

0

(dia ^w X P).

We

get

^the

following

^{theorem about the}

tightness

^{of a}

family

of H-valued continuous local

martingales.

THEOREM A. ^- Let

Pa E H)) (a

^E

A).

^{Then the}

following

^three

conditions are

equivalent

^to each other.

i )

^is

tight

ⁱⁿ

Co(I, H) .

ii) { ~(Px) ;

^{a E}

A ~

^is

tight

ⁱⁿ

Co(I, r2(J)) . iii ) { dia ~(Pa) ~ ;

^{a e}

A }

^is

tight

ⁱⁿ

Co(J, l l (J)) .

Vol. 22, ^n° 3-1986.

374 S. NAKAO

As stated in the introduction Rebolledo

[77] proved

the above theorem

using Lenglart’s inequality

^{in case of H = ~. So we will} prove the above theorem in case of dim H ⁼ + oo. For this purpose ^we estimate the tail behaviors

of { ~(Pa) ;

^{a E}

A ~ and { dia ( ~(Pa) ; uniformly

^{in oc e A.}

Let K be a subset of P

( p

⁼

1, 2).

It is well known

( [4 ])

^{that K is}

relatively compact

^{if and}

only

^if

Let I ⁼

[o, T ]

^and

Qa

^E

lp)) (a

^E

A). Noting (2 . 7),

it is easy ^{to see}

that { ^{Qa ;}

^{a E}

A }

^is

tight

ⁱⁿ

Co(I, lp)

^{if and}

only

^{if for any e > 0 and i}

== 1,2,...

there exist a

compact

^{set K ~ lp and}

03B4(i)

^> 0 such that

and

where w ⁼

(wl,

w2,

... )

^E

Co(I, lp).

From now on we denote

by

^{w =}

(wi, w2, ... )

^the

sample path ofCo(I,

We prepare ^a lemma for the

proof

^{of Theorem A.}

0

LEMMA 2.1. ^- Let I ⁼

[0, T] ]

^and

l2)) (a

^E

A).

^{Then the}

following

^two

properties

^are

equivalent

^to each other.

i )

For any 6 ^{> 0} there exists a

compact

^{set K}

c l2

such that

ii )

^For any £ ^> 0 there exists ^a

compact

^set

K c ll

such that

Proof

^{- We show that}

i ) implies ii).

Assume that

(2 .10)

holds. Set

Since it holds that

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

375

~

WEAK CONVERGENCE OF CONTINUOUS LOCAL MARTINGALES

we

get by Chebyshev’s inequality

On the other hand it is _easy to see that

00

~r=1 ~2~...

^{be a sequence such that}

E/2.

^Then

r = 1

in view of

(2.13)

^{for any r =}

1, 2,

^{... there}

exists nr

^{such that}

Putting

we

get

^from

(2.14)

r

B~

We then set

={

^c

^{= (cl,}

^c2,

^{... )}

^{E c}

^{~l1 ~} 203B2/~, 03A3 |cm| ~

^Er

for r ⁼

1, 2,

^...

}. Obviously ^K

^is compact and satisfies from

(2.12)

^and

(2 .15)

Consequently

^we

get

^from

(? .10)

We can prove the

implication ii)

^-~

i)

^{in the same} way ^as above. D We now return to the

proof

^{of Theorem A.}

Proofof

Theorem A. It is obvious that

i )

^and

ii )

^are

equivalent

^{to each}

other. We show the

equivalence

^between

ii )

^and

Hi).

We may ^assume that I ⁼

[0, T] ]

^{and dim H = + 00.} The Rebolledo’s result

[Cor. II,

^{3 .14 in}

Vol. 22, n° 3-1986.

376 _{S. NAKAO}

~l ~ J) implies

^{that the condition}

(2.8) with { ~(Pa) ~

^instead

of {

^is

equivalent

^{to the condition}

(2. 8) with { dia ~ ~(Pa) ~ ~

^instead

of {

On the other hand we

get by

Lemma 2.1 that the condition

(2.9)

^with

~ ~(Pa) ~

^{instead is}

equivalent

^to the condition

(2.9)

^with

{ dia ( ~(P~) ~ ~

^instead

of { Qa }. Consequently ii )

^and

iii )

^are

equivalent

to each other. The

proof

^is

completed.

^D

3. PRESERVATION

OF LOCAL MARTINGALE PROPERTY

First we show that any accumulation

point

^{of a} sequence of H-valued continuous local

martingales

^under the weak convergence is also ^a H-valued continuous local

martingale.

THEOREM B. ^-

H))

^{is closed in}

~(Co(I, H)).

0

Proof

²⁰¹⁴ We may ^assume H ⁼

l2. Let { P,~ ~

^converges

weakly

^to

00 0

P(Pn o o

^E

l2)),

^{P E}

-~(Co(I, l2))).

^Denote

by Rn

^the

probability

^measure

0

on

Co(I, l2)

^x

Co(I, ll)

^induced

by ((w, dia ~ w ~ ),

^{Theorem A}

implies {

^is

^tight.

Therefore there exists a

subsequence { n’ ~

^such

that { R~. ~

converges

weakly

^{to R. For N =}

1, 2,

^{... set}

For each i ⁼

1, 2, ...,

s2, ... , sp, ti , t2, ... ,

t~

^{_ s t} and bounded continuous function

f

^on

(12)P

^{x we have}

Since aN is R-a. s. continuous on

and

the

equality (3.1)

holds with R instead of

Rn"

^{Therefore we have}

Next we state a

continuity property

^{of the}

mapping dia ~ )>

^defined

by (2 : 6).

THEOREM C. ^-

dia ~ ~ :

¹

l2))

^--~

[1))

^is continuous.

Proof

Within the situation of the above

proof,

^{we can show that}

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

377

WEAK CONVERGENCE OF CONTINUOUS LOCAL MARTINGALES

(Wf - wi, R)

^{is a} continuous

martingale

for any ^{i =}

1, 2,

^{.... Therefore R}

~ , ~ ~

~ ~ 0

is the

probability

measure on

Co(I, f)

^x

Co(I, h)

^induced

by ((w, dia ( w ~ ), P).

, ~ ~ ~

0

This

implies that { Rn ~

₀ ^converges

weakly

^{to R.}

Hence { dia ~ Pn ~ }

^converges

weakly

^to

dia ( P ~

^{and the}

proof

^is

completed.

D

Applying

^{the above}

theorems,

^{we can}

easily get

^the

following

^remark

about the central limit theorem for ^a sequence of Hilbert space valued continuous local

martingales.

Remark. Let V ⁼ _{I ,2,...} be a

nonnegative definite, symmetric

real matrix such

that

_{i= 1} ^{vii + oo and}

^Pw

^E

^l2))

^{be a Wiener}

measure such that the mean vector is zero and the covariance function

o o

is

((t

^{A Consider a}

sequence {Pn} ^{(Pn E} l2))).

^{Then the}

following

^two conditions are

equivalent

^to each other.

o o

i )

converges

weakly

^to

Pw

ⁱⁿ

Co(I, f2).

o

P,~ ~ ~

^converges

weakly

^{to the} distribution of v22t,

... )

in

Co(I, [1)

^{and for} any

I, j

⁼

1, 2,

^{... and t >}

0 {

^the distribution of

~ ~ 0

(( (t), P n) on [~ }

^converges

weakly

^{to the} 6-distribution at

Vijt

^{on ~.}

Finally

^{we note that such} infinite-dimensional central limit theorem for

current valued stochastic _processes are

investigated by

^Ochi

[1 D ] and

^Ikeda

andOchi

[ ~ ].

Vol. 22, ^{n° 3-1986.}

378 S. NAKAO

APPENDIX

As stated in the introduction, in the case of H ⁼ (~, ^{Theorem A is} proved by ^Rebolledo

as a consequence of a general ^{Aldous theorem for} right continuous processes and using Lenglart’s inequality (cf. [7] ] [Il ]). ^The considered processes Mn may be discontinuous and

only the limits

of { and ( ( M" ~ ~

^{are assumed} continuous. The proof ^is quite heavy.

We give ^{here a direct} simple proof adapted ^{to the} special ^{case of} continuous processes.

LEMMA A.I. ^- Let M(t) (t E [0, T]) ^{be an R-valued} continuous martingale ^defined

on an usual filtered probability space (S~, ~, P, ~t) starting ^{at 0 with E}

[M(T)4

] ^{+ ~o.}

For each finite

partition 0394 = {0 =

so si ^{... =}

T}

^of [0, T], ^set

Then there exists a universal positive constant C such that

where

and

Proof ^{- It} is easy ^{to see} that the left hand side of (A. 2)

where [s ^s Sj+ ¹ ( j ⁼ 0, 1, ... , l - 1 ). Applying ^Schwarz’s inequality ^{to the}

second part ^{of the} right hand side of the ahc»t inequality, ^we get (A..2). D LEMMA A. 2. ^- Let M"(t ) (n ⁼ 1, 2, ... ) ^{be an} R-valued continuous martingale ^defined

on a filtered probability space (Q, j~, P, ^with M"(0) ^{= 0.} Suppose that there exists a

stochastic process M ^on (S2, J~, P) such that with probability

one

converges ^to M

uniformly ^{on each} compact ^interval. Further suppose that there exists a constant f3 ^{> 0}

such that sup _n E [

~4+~]

^cx~ (t ^> 0). ^{Then M is a} continuous martingale ^with E [ ] :r (t ^> 0) and satisfies

Proof ^- It is obvious that M is a continuous martingale ^{with E} [ ~ M(t) ~4 ] ^oo (t ^> 0).

Ito formula implies

Annales de l’Institut Henri Poincaré - Probabilités ^et Statistiques

WEAK CONVERGENCE OF CONTINUOUS LOCAL MARTINGALES 379

We then get ^{from the} assumption

For any finite partition A

= ~ 0

⁼ so sl ^... Sz ⁼

T ~

^of [0, T we ^have

where t) ^and IA(M, t) ^{are defined} in the same way ^as (A .1).

Since ( ( (T)2 ~

is uniformly integrable, ^{Lemma A. 1} implies ^{that for} any 8 ^> 0 there exists a positive ^constant

~ such that

On the other hand it is easy ^{to see} that lim an(~, 2) ^{= 0 for} any d. Therefore we have

n -~ o0

Combining (A. 4) ^with (A. 5), ^{we then} get (A. 3). ⁰

Proof of ^{T heorem A in case} of ^{H = -} First ^we prove that iii ) implies ii). ^Let (n = 1, 2, ...) ^{be an} R-valued continuous local martingale ^with M"(0) ^{= 0 defined on a} filtered probability space (Om .~n, Pn, ^{We assume}

that ( ( M" ~ ~ is

tight ⁱⁿ Co([0, ⁺ oo), (~).

Then for _{any 11} ⁼ 1, 2, ^... there exist an extension

(SZ;"

^Pn,

~r~)

^of

(Q,

^~n, Pn,

~t)

^and

a one-dimensional Brownian motion Bn(t) ^on

(S~n,

~n, P~" ^with Bn(0) = 0 ^{such that} Mn(t ) ⁼ B"( ~ (t )). ^The tightness

M" ~ ) ~

implies ^the tightness

of (

^Mn

~.

Next we prove that ii ) implies iii ). ^Let Mn(t) (n ⁼ 1, 2, ... ) ^{be an} [?-valued continuous local martingale ^{defined on a filtered} probability space

(Q,

^{~, P, with} Mn(0) ^{= 0. We}

assume that with probability

one { M"(t ) ~

^{converges uniformly on each} compact interval.

It is sufficient to prove

that { ( Mn ~ ~

^is tight ⁱⁿ Co( [0, T ], R) (T ^> 0). ^{For N =} 1, 2, ^...

and n ⁼ 1, 2, ^... put

It is _easy to see that for _{any e} > 0 there exists No ^{such that}

(A . 6) P(Sn(N 0) = T) ^{> 1 - E for n =} 1, 2, ^....

Since {

Mn(t ^A

S,~(N)) ~

^is tight ⁱⁿ Co( [0, T ], ^{for N = 1, 2, ..., we have} by ^{Lemma A. 2}

that { (

(t ^{A is}

tight

ⁱⁿ Co( [0, ^{for N =} 1, 2, ^.... Therefore the tightness

in Co( [0, T ],

of ( ( Mn ~ ~

^{follows from} (A. 6). ⁰ REFERENCES

[1] ^D. ALDOUS, Stopping ^{times and} tightness, ^Ann. Probability, ^t. 6, 1978, p. 335-340.

[2] ^P. BILLINGSLEY, Convergence of probability ^measures, Wiley ^and Sons, 1968.

Vol. 22, ^{n° 3-1986.}

380 S. NAKAO

[3] ^C. DELLACHERIE, ^{P. A.} MEYER, Probabilities and potential B, North-Holland, 1982.

[4] N. DUNFORD and J. T. SCHWARTZ, ^Liner operators, ^Part I, Interscience, ^1958.

[5] ^N. IKEDA, ^Y. OCHI, Central limit theorems and random currents, ^to appear in the

Proceedings of the 3rd Bad Honnef Conference ^on Stochastic Differential Systems.

[6] ^N. IKEDA, ^S. WATANABE, Stochastic differential equations ^and diffusion processes,

North-Holland, Kodansha, ^1981.

[7] ^J. JACOD, ^J. MEMIN, ^M. MÉTIVIER, ^On tightness ^and stopping times, Stoch. Processes

Appl., ^t. 14, 1983, p. 109-146.

[8] ^E. LENGLART, Relation de domination entre deux processus, Ann. Inst. Henri Poincaré,

t. 13, 1977, p. 171-179.

[9] ^M. MÉTIVIER, Semimartingales, ^{Walter de} Gruyter, ^1982.

[10] ^Y. OCHI, Limit theorems for a class of diffusion processes, ^to appear in Stochastic Processes and their Applications.

[11] ^R. REBOLLEDO, ^La méthode des martingales appliquée à l’étude de la convergence

en loi de processus, Bull. Soc. Math. France, ^t. 62, p. 1-125.

(Manuscrit rcyrr lo ?9 novembre 1985)

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Annales de l’Institut Henri Poincaré - Probabilités et Statistiques