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Stability of Stochastic Integrals Under Change of Filtration

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Stability of Stochastic Integrals Under Change of Filtration View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Stochastic Processes and their Application\ SO ( 1994) 22 t-233 221 North-Holland Stability of stochastic integrals under change of filtration Eric V. Slud Received 10 February 1992 Revised 11 May 1993 Let (0, 7, P) be a probability space equipped with two filtrations (.7,) and (5,) satisfying the usual conditions. Assume that X is a semimartingale and that h is locally bounded and predictable for each of the two filtrations 1.7,) and (# ,), New examples of such processes are given. Utilizing and extending partial results of Zheng ( 1982). this paper extends the available results on the relationship between the stochastic integral processes J’h, dX, taken respectively in the sense of (5,) and of (,G’,), In particular, it is shown that these stochastic integrals differ at most by a continuous process with quadratic variation defined and equal to 0. If both stochastic integrals are (2, n E,) semimartingales, then it is proved that the stochastic integral j’h, dX, taken in (~7,) sense is indistinguishable from that taken in (S’,) sense. AMS Subject Class~ficcrtim: 6OG07, 6OG44 filtratiomp4 predictable process;p4 predictable projectiomp4 purely discontinuous;p4 quadratic variation;p4 semimartingale;p4 special semimartingale;p4 usual conditions Introduction In areas of application for stochastic integrals, the mathematical modeler often has some leeway in choosing a family of a-fields for which the stochastic integrands h will be predictable and the relevant stochastic integrators X will be semimartingales. It is therefore of great interest to know whether the random variable defined by the resulting stochastic integral jh dX depends on the choice of underlying filtration. A general proof that the stochastic integral is always the same has not previously been available, and is given in this paper for bounded predictable integrands and semimartingale integrators in the important case (Theorem 3.3) where the stochastic integrals jh dX taken in the sense of two different filtrations (7,) and ( W,) are both semimartingales with respect to the filtration {F, n .F,}. More generally. a result of Zheng ( 1982) is extended in Theorem 3.2 to prove that stochastic Corur.\]>mderre to: Prof. Eric V. Slud, Mathematics Department. University of Maryland, College Park, MD 20742, USA. Supported by Office of Naval Research contracts NO00 14-89-J. IO5 1 and NO00 14.92.COOl Y. 03044149/94/57.00 G 1994-Elsevier Science B.V. All rights reservfed SSDl0304-J140(93)E0042~D integrals with respect to different filtrations (.%,I and (25,) can differ as stochastic processes only by a continuous process with quadratic variation 0. In Section 2 of the paper, we give new examples ( 1) of an indicator process h which is predictable with respect to each of two filtrations (F,) and (Z,) satisfying the usual conditions but not with respect to (.F-,n?.F,],and (2) of a continuous semimartingale X (with martingale part a Wiener process), simultaneously with respect to filtrations {F,) and (,Y’,), which is not also a semimartingale with respect to { 9, V Yr). 1. Decomposition of stochastic integrals Let (0, l~T,)taO, P) be a filtered probability space satisfying the usual conditions from the general theory of processes, i.e., ( Fr] is a right-continuous nondecreasing family of com- pleted a-fields. Throughout the present paper, (F:,) will denote another filtration on (0, P) satisfying these conditions; h, denotes a locally bounded predictable process with respect to each of { FrJ and ( Ft}; and X, denotes a semimartingale (i.e., local martingale plus bounded-variation process) with respect to each of (.F,) and { ,Yr] which is pathwise almost surely right-continuous with limits from the left. Our general references for semimartingales and stochastic integrals are the books of Dellacherie and Meyer ( 1982) and Jacod and Shiryaev ( 1987). The stochastic integral rh, dx,Y= j/~~1,,~G ,, dX, has a well-defined sense with respect to each of the filtrations (.F,) and {F,). Define a new filtration [ j%vt) by A?, = .Frn .‘%,:it too satisfies the usual conditions. Despite the assumption that h, is predictable with respect to each of {F,} and (27,}, it will not always be predictable for A?,. (See Section 2, paragraph ( 1)) for an example.) In addition, local boundedness of h, with respect to each of { Ft) and (S’,) does not imply local boundedness with respect to (A?,), since in general h, evaluated at a stopping-time inf( s > 0: 1h,, 1 > K} may not be bounded, so that there need not exist (X,] stopping-times up to which h is bounded). However, a theorem of Stricker (Dellach- erie and Meyer, 1982, VII.60) implies that a (Fr) semimartingale X, adapted to (27,) c (9,) is also a (dy;I) semimartingale. If h, does happen to be locally bounded and predictable with respect to {A?,}, then the stochastic integral I%,, dX, has a well-defined sense as a (Z,) stochastic integral, and since A?‘,c .F, and A?, C LiFr,the stochastic integral processes I%, dX, taken in the sense of { Ff} or (F,} agree with that in the sense of (F,) (Dellacherie and Meyer, 1982, Theorem VIII. 13). Therefore the difference (1.1) is indistinguishable from (i.e., almost surely pathwise identical to) 0. Exactly the same argument, with 9, Cfr and F/r C f ,, can be made if X is a semimartingale for the enlarged filtration { &,) = (7, V F,), with respect to which h is necessarily predictable. Thus we have: Lemma 1.1. The d@erence (1.1) between (.F,) and (F,] sense stochastic integrals is indistinguishable from 0 if either (a) h is { .Frfl .F’,) predictable or (b) X is a {F, V Fr) semimartingale. 0 In the general case where h is not necessarily predictable and locally bounded for (Z,], it is not known whether ( 1.1) is indistinguishable from 0. Protter ( 1989, p. 146) cites an example of Jeulin ( 1980) to the effect that one but not the other of the integrals in ( 1.1) may be well-defined for integrands h which are not locally bounded. The problem addressed here has not directly affected work on applications of stochastic integrals because, for the large class of locally bounded integrands h and semimartingale integrators covered by Lemma 1.1, the (F,} and (3 ,} integrals are known to be equal. Since we are concerned in this paper only with almost-sure properties of ( 1 . 1 ), we begin by reducing without loss of generality to the case where X has ‘special’ semimartingale decomposition X, = p, + B, with respect to (F,] and X, = v, + C, with respect to ( Fr}, where p, and v, are respectively (.F,) and ( W,) martingales such that p() = vc’o= 0 and E( I_L,)’and E( v,)’ are finite for each finite t, and where B and C are predictable bounded-variation processes. Indeed, by Snicker’s change-of-law theorem (Dellacherie and Meyer, 1982, VI1.58), there exists a probability measure Q on 0 equivalent to P such that the assertion about X as a ( ( Tr}, Q) semimartingale holds, and a simple modification of the proof given by Dellacherie and Meyer allows Q to be chosen so that the assertions about X as a ( cS’,) semimartingale also hold. Since Q and P are equivalent, the almost-sure properties of ( 1.1) are the same for Q as for P. 2. Examples of predictable processes and semimartingales for two non-nested filtrations In this Section, we develop examples of processes h and X which do not satisfy the hypotheses of Lemma 1.1. Here ( Fr ) and (ii,) will be defined as non-nested filtrations satisfying the usual conditions. First h is constructed to be predictable with respect to each of ( Yr) and (~~,) but not with respect to (9, n F’,}.Then X, is defined as a Wiener process plus a bounded-variation process which is a semimartingale for (Y,} and ( 2Fy}but not for (7, V .Y,). The author has not been able to find explicit examples of this kind in the literature. Since one might suspect that continuous semimartingales X with some special properties (such as Gaussian diffusions, or processes with Wiener distribution) could not arise with more than one semimartingale representation, we produce also an example of two Gaussian diffusions which are distributional Wiener processes but differ from each other by a non- trivial Gaussian bounded-variation process. ( 1) Let ( T, ( U,, V,, i > 1) ) be a system of jointly independent random variables defined on the Bore1 product-probability space ( 0, 3, P), such that T has unit exponential distri- butionandallU~andV,areUniform[O,l].Forn~1andn~l1(1-t)<n+1,define where N denotes the null sets, and put OF, = a( -Y, [ Ui < T] , i >, 1) and .!9, = a( ,V, [ V, < T] , i> 1) fort> 1. Since maxi,,, .!Ji and max,,,, V, both converge to 1 almost surely as n + m, evidently.~,_=.~, andy,_=??,,and [T~l]~.~,_n7,_.However A?“,=9,r)5r=~(~V) forO<t<l. (2.1) For instance, F’on570=~([UI<T], _N)na([V,<T], N)=v(N), a fact which together with the independence of the pairs ( U,, Vi) supplies the easy proof of (2.1) Define the random variable r= 1 + f,Ti, ,. Clearly ris a predictable stopping-time for each of (F,) and (.F,], since it is the pointwise increasing limit respectively of the (9,) stopping-times 1- lln+Z tmaxrs,zU,BT1and ofthe ]grl stopping-times 1 - l/n +ILmax,s,,v,pT1.However, T cannotbe{~“,}predictablesince~,_=a(~”S)by(2.1)andO<P(~=l}<l.
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