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6.1102F 230411I. TITLE ticgau4a Secunty, CLawf~cation, aWeak- convergence of the variations, iterated integrals, n Doleans-Dide e.ponent aso

* 12. PERSONAL AUTHORIS) sequences of semimartingales"Avram - F,

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* IELD GROUP SUB seGx. -Keywords:

It. ABSTRACT 'cont"InU O1 '"ePse I f neceDIWy and identify by blocil number,

If X is a sequence of semimartingales, converging to a senunmirtingale(n) (n)

X, and such that [X, X converges to [X,XI, then all higher order variat ions(n)

and all the iterated integrals of X converge jointly to the respective

functionals of X.

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CENTER FOR STOCHASTIC PROCESSES

Department of StatisticsUniversity of North CarolinaChapel Hill, North Carolina

WEAK CONVERGENCE OF THE VARIATIONS, 1ITERATED INTEGRALS,

AND DOLEANS-DADE EXPONENTIALS OF S1.QUNCES OF SEMIMARTINGALES

by

Florin Avram

Technical Report No. 135

March 1980

t r6 0249

- -o m h 1i

WEAK CONVERGENCE OF THE VARIATIONS, ITERATED INTEGRALS,

AND DOLEANS-DADE EXPONENTIALS OF SEQUENCES OF SEMIMARTINGALES

by

Florin Avram

University of North Carolina at Chapel Hill

Abstract

(n)If X is a sequence of semimartingales, converging to a semimartingale

(n)(n)X, and such that [ X, X ] converges to [X,X], then all higher order variations

(n)and all the iterated integrals of X converge jointly to the respective

functionals of X.wI.,

AT?' Y. .m'1

. o . +: J:- 2.

AMS 1980 Subject Classifications: Primary, 60F17; Secondary, 60105.

Keywords and Phrases: Semimartingales, weak J1 Skoronod topology, variations,multiple integrals, Dolians-Dade exponential.

This research supported by the Air Force Office of Scientific Research

Contract No. F49620 85C 0144.

I,-,

S

. . ,.:.,','- ' .,. ' +-,.%'%L,',-+.+., -,'-.+, ' +.+.,.. '"'" .,.- .""'''"."-". -" . .,', ,'" - +," .' .- -.- ''''. . . . "."",'."." ".."-"- .- ".

.. . .. . . . . ,. , I Ii. + , +L + " + - ' + - + -! ' + + ' +

° t ! ' ' '

1. Introduction

(n)A. Let X t be a sequence of semimartingales, with tE [0,1], such that

(n)(I.1) x x,

where X is a semimartingale, and WO denotes weak convergence on D[OI]

with respect to the Jj-Skorohod topology.

-We investigate thelconvergence of the variations, iterated integrals

(n)and Dolans Dade exponentials.of X , which are defined as follows: For Y

a semimartingale, /

r Y for k= 1t

(1.2) V k(Y) = [YY]t = t+ 2 (AY s) for k= 2k t tt( ~ (AY) for k >t3

sst

Y for k-t

(1.3) k (Y)t trI (Y)s dY for k 2

(4E ) Ik-l s(1.4) E(XY) exp[XY - RY t:(XAYs

t t T ts_

- - . .W - .-. w.- - P - ._ .-... X.,- - .- RJ

2

(n)

Ik ( t 1 ,nn tIk( )t

I.&

r.

Corollary: Ifl>(n) w(JJ)

(1. 9) x >x

and the condition of Jacod (1983) holds:

(1.10) lim sup P{Var(B hn)1 >b} = 0b-o n-o

h~n(where h is a truncation function and (Bh'n) is the previsible projection

tn

of the truncated semimartingale X), then (1.5), (1.6), (1.7) and (1.8) hold.

Proof: cf. Jacod (1983), Theorem 5.1.1, (1.9) and (1.10) imply (1.5).

2. Proofs

Introduce the following notation: For any real number x,

'" >a,'-'"x := x'l{x 1>a}

x X.1

We establish now the following:(n)

Lemma 1: a) Suppose X are semimartingales such that

(n)(n)

(2.1) lim lim PQ X, X ]>b} 0,b-~ n-o

2and let f(x) be any real function such that f(x)=o(x2), as x-0. Then,

for all c,

(2.2) lim lim P{ f jf(AX-a) c = 0.a- 0 n- s

4

(n) (n) 2 o

Proof: a) Note first that Estf(A Xs)jX t (X) " ts Xs-

Since the Do1e'ans-Dade exponential

EXt=exp{x t - 2 [X X]I + f[/X T(X) ts tt

it remains only to note that the functional:

X a:D 2)[0,1] -~ D ()[0,1]

Is continuous a.s., if both spaces are endowed with the respective J1 topo-

*logies. Letting then a -0, as in the proof of Lemma 1, one gets:

* ((n), (n)(n) f(n)< ~ £( (n)>, X X t 9 x( X-), R 'eAX>))

s(J) [1x'x f(AX R II(AX >)t 5 5s~t s!- t

2 2*since enl+ x) -x + =o(x ), and since (1.5) implies (2.1). Finally,

* applying the continuous functional

(4)

P D [0,1] D[0,1]'

O(Z19 Z ,Z , exp[Z -Z + Z Izl2394 1 2 2 3 4

* we get that(n) ()

E(X X ) -4')E(XX).

*Since (1.6) is equivalent to (1.7) (by the use of the polynomial mapping),

* and (1.6) trivially implies (1.5), Theorem I is proved.

References

I.Avram, .,Taqqu, M.S.: Symetric Polynomials of Random VariablesAttracted to an Infinitely Divisible Law. To appear in Z. fUr Wahr.

Z. Billingsley, ?.: Convergence of probability -measures. Wiley: New York,(1968).

. Dnk-'n. 7.3. . Mandelbaun, A.: Symmetr~c szatistilcs, Poisson jDoint pro-cesses and miltioie Wiener integ-rals. Ann. of Statistics 11, 739-745,

-. Denker, M. , Grillenberger, Chr. , Keller, 0.: Anote on invarianceorincfioles for von Mises' statistics. Metrika (to appear 1985).

* ~ . cinilv -2 .. :. : Soecta' unctions, orobabilizy semigrouvs and 1-amiltonSlos. oo. iot-es in Mlath. , 696. Springer Verlag: New York (197S).

licoDc 2. hecr~rneST L~ic;e ?our Les Prccessus. E-cole d'Eta de?rcoao:.. de esc .aint-F-lour Y=--l9SS. Laot. Noteas --n Mt

-. >ln acc,,,Ms : nvariance 7rfnciola --:r svnetriz sta:is-

Mjev.er...... n cours sur !os intoo4rales st-ochastfocues. Sen. Probl X,c:. Notzes '-.a'1-. ill. 2_53-00. :_-rincer -erlac.- N ew Yrk ~l7 6,

iuL .....ae .. : Avooi distribution a:sv~erc statist-.cs.__________5 17:-0 (19 30) .

-71.

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