LJILJANA PETROVI ´C
Received 20 February 2004 and in revised form 3 December 2004
We consider a problem (that follows directly from realization problem) on finding Mar-kovian representations for a given family of Hilbert spaces such that each of these two families provides exactly the same amount of information about some other family of Hilbert spaces.
1. Introduction
The study of Granger causality has been mainly preoccupied with time series. We will instead concentrate on continuous time processes. Many systems to which it is natural to apply tests of causality take place in continuous time. For example, this is generally the case within economy.
In the first part of this paper, we give a generalization of a causality relationship “Gis a cause ofEwithinH” which (in terms ofσ-algebras) was first given in [4] and which is based on Granger’s definition of causality [2].
In the second part, we relate concepts of causality to the stochastic realization problem. The approach adopted in this paper is that of [3]. However, since our results do not depend on probability distribution, we deal with arbitrary Hilbert spaces instead of those generated by Gaussian processes.
We suppose that it is known that a stochastic dynamic system (s.d.s.)S1with known outputsHcauses, in a certain sense, behaviour of some other s.d.s.S2, whose states (or some information about them)Eare given. The main problem, to be formulated more precisely below, is to determine Markovian representationsG(as a state space of an s.d.s.
S1) for a familyHsuch that familyGprovides exactly the same amount of information about the familyHas familyE(seeDefinition 2.10).
It is clear that all the results of this paper can be extended on theσ-algebras generated by finite-dimensional Gaussian random variables. But, in the case thatσ-algebras are arbitrary, the extensions of the proofs of this paper are nontrivial because one cannot take an orthogonal complement with respect to aσ-algebra as one can with respect to subspaces in a Hilbert space.
Copyright©2005 Hindawi Publishing Corporation
2. Preliminary notions and definitions
LetF=(Ft),t∈R, be a family of Hilbert spaces. We will think aboutFtas the information
available at timet, or as a current information. Total informationF<∞ carried byF is
defined byF<∞= ∨t∈RFt, while past and future information ofFattis defined asF≤t= ∨s≤tFsandF≥t= ∨s≥tFs, respectively. It is to be understood thatF<t= ∨s<tFsandF>t= ∨s>tFsdo not have to coincide withF≤t andF≥t, respectively;F<tandF>tare sometimes
referred to as the real past and real future ofFatt. Analogous notation will be used for familiesH=(Ht),G=(Gt),E=(Et), andJ=(Jt).
IfF1andF2are arbitrary subspaces of a Hilbert spaceᏴ, thenP(F1|F2) will denote the orthogonal projection ofF1ontoF2andF1F2will denote a Hilbert space generated by all elementsx−P(x|F2), wherex∈F1. IfF2⊆F1, thenF1F2coincides withF1∩F2⊥, whereF2⊥is the orthogonal complement ofF2inᏴ.
Possibly the weakest form of causality can be introduced in the following way.
Definition 2.1. It is said thatHis submitted toG(and written asH⊆G) ifH≤t⊆G≤tfor
eacht.
It will be said that familiesHandGare equivalent (and written asH=G) ifH⊆G andG⊆H.
Definition 2.2. It is said thatHis strictly submitted toG(and written asH≤G) ifHt⊆Gt
for eacht.
It is easy to see that strict submission implies submission and that the converse does not hold.
The notions of minimality and maximality of families of Hilbert spaces are specified in the following definition.
Definition 2.3. It will be said thatFis a minimal (resp., strictly minimal) family having a certain property if there is no familyF∗having the same property which is submitted (resp., strictly submitted) toF.
It will be said thatF is a maximal (resp., strictly maximal) family having a certain property if there is no familyF∗having the same property such that familyFis submitted (resp., strictly submitted) toF∗.
It should be understood that a minimal (resp., strictly minimal) and maximal (resp., strictly maximal) family having a certain property are not necessarily unique.
An important tool in definition of the Hilbert space s.d.s. is the concept of conditional orthogonality.
Definition 2.4(compare with [7,8] and conditional independence from [9]). IfF1and
F2are arbitrary Hilbert spaces, then it is said thatF is splitting forF1andF2 or thatF1 andF2are conditionally orthogonal givenF(and written asF1⊥F2|F) ifF1F⊥F2F. WhenFis trivial, that is,F= {0}, this reduces to the usual orthogonalityF1⊥F2.
The following result gives an alternative way of defining splitting.
The following results concerning splitting will be used later (for the proof, see the given reference).
Theorem2.6 (see [3]). The spaceF is a minimal one such thatF1⊥F2|Fif and only if
F=P(F1|S)for some spaceSsuch thatF2⊆S⊆(F2∨P(F2|F1))⊕(F1∨F2)⊥.
Corollary2.7 (see [3]). The spaceF⊆F1∨F2is a minimal one such thatF1⊥F2|F if and only ifF=P(F1|S)for some spaceSsuch thatF2⊆S⊆F2∨P(F2|F1).
In this paper, the following definition of Markovian property will be used.
Definition 2.8(compare with [7,8]). FamilyGwill be called Markovian ifP(G≥t|G≤t)= Gtfor eacht.
Now we give a definition of an s.d.s. in terms of Hilbert spaces using conditional or-thogonality relation. The characterizing property is the condition that past information of outputs and states and future information of outputs and states are conditionally or-thogonal given the current state.
Definition 2.9(see [3] and compare with [9]). An s.d.s. is a set of two families,H (out-puts) andG(states), that satisfy the condition
H<t∨G<t⊥H>t∨G>t|Gt. (2.1)
For given family of outputsH, any familyGsatisfying (2.1) is called a realization of an s.d.s. with those outputs.
It is clear that the realization of an s.d.s. is Markovian.
The next intuitively justifiable notion of causality has been proposed in [5].
Definition 2.10. It is said that Gis a cause ofEwithinH(and written asE|<G;H) if
E<∞⊆H≤∞,G⊆HandE<∞⊥H≤t|G≤tfor eacht.
Intuitively,E|<G;Hmeans that, for arbitraryt, information aboutE<∞provided by
H≤tis not “bigger” than that provided byG≤t. The meaning of this interpretation is
spec-ified in the next result.
Lemma 2.11 (see [1]). E|<G;Hif and only ifE<∞⊆H<∞,G⊆H, and P(E<∞|H≤t)=
P(E<∞|G≤t)for eacht.
A definition, analogous toDefinition 2.10, formulated in terms ofσ-algebras, was first given in [4]; however, a strict Hilbert space version of the definition in [4] contains also the conditionE⊆H(instead ofE<∞⊆H<∞) which does not have intuitive justification.
IfGandHare such thatG|<G;H, we will say thatGis its own cause withinH (com-pare with [4]). It should be mentioned that the notion of subordination (as introduced in [7,8]) is equivalent to the notion of being one’s own cause, as defined here.
IfGandHare such thatG|<G;G∨H(whereG∨His a family determined by (G∨ H)t=Gt∨Ht), we will say thatHdoes not causeG. It is clear that the interpretation
difficulty, it can be shown that this term and the term “Hdoes not anticipateG” (as introduced in [7,8]) are identical.
3. Causality and stochastic dynamic systems
Suppose that an s.d.s.S1causes, in a certain sense, behaviour of some other s.d.s.S2. It is natural to assume that outputsHof systemS1can be registered and that some informa-tionEabout the states (or perhaps states themselves) of systemS2 is given. Results that we will prove will tell us under which conditions concerning the relationships betweenH andEit is possible to find statesGof systemS1which are in a certain causality relation-ship in the sense ofDefinition 2.10withHandE. More precisely, the following two cases will be considered:
(i) states of an s.d.s.S1are a cause of outputs of the same system within available information about s.d.s.S2;
(ii) the available information aboutS2is a cause of outputs ofS1within states ofS1. This paper is a continuation of the papers [1,5,6].
We consider different kinds of causality between familiesG,H, andE, whileGandH are in the same relationship, that is,Gis a realization of an s.d.s. with outputsHin all cases.
The following results give the solutions of the problem (i).
Theorem 3.1. Let H and E be such that P(Et|H<∞)⊆ E≤t and P(E<t|H<∞)⊥
H>t|P(Et|H<∞)for eacht. If the familyEis Markovian, then the familyG, defined by
Gt=PEt|H<∞, t∈R, (3.1)
is a minimal realization (of an s.d.s. with outputsH) that causesHwithinE.
Proof. FromG≤t=P(E≤t|H<∞), it follows thatH<∞⊥E≤t|G≤t. Also, the definition ofG
and the assumptionP(Et|H<∞)⊆E≤timplyG≤t⊆E≤t, which together with the previous
orthogonality relation meansH|<G;E. The minimality ofGfollows fromTheorem 2.6 andCorollary 2.7.
FromG≤t⊆E≤t, the fact thatP(G≥t|G≤t)=P(E≥t|G≤t) which follows fromG<∞=
H<∞, (3.1) and the assumption thatEis Markovian, we get
PG≥t|G≤t
=PE≥t|G≤t
=PPE≥t|E≤t
|G≤t
=PEt|G≤t
. (3.2)
The relationH|<G;E(becauseH<∞=G<∞) particularly impliesG<∞G≤t⊥Et, so
that (3.2) becomes
PG≥t|G≤t
=PEt|G<∞=PEt|H<∞=Gt, (3.3)
which means that G is Markovian. This fact with the assumptionG<t⊥H>t|Gt gives
G≤t⊥H>t∨G≥t|Gt. However, since H<t⊆G≤t (which is an obvious consequence of
Corollary 3.2. LetH be its own cause withinEandP(E<t|H≤t)⊥H>t|P(Et|H≤t)for
eacht. If Eis Markovian, then the familyG, defined by
Gt=PEtH≤t, t∈R, (3.4)
is a minimal realization (of an s.d.s. with outputsH) that causesHwithinE.
The following result does not requireEto be Markovian but provides a realization whose present information attis equal to its total information accumulated up tot. Theorem3.3. LetHandEbe such thatH<∞⊆E<∞andP(Et|S)⊆E≤tfor eachtwhereS
is some space such thatH<∞⊆S⊆H<∞∨P(H<∞|E≤t). FamilyGis a minimal realization
(of an s.d.s. with outputsH) that causesHwithinEif and only if it is defined by
Gt=P
E≤t|S
, t∈R. (3.5)
Proof. Let (3.5) hold. FromTheorem 2.6andCorollary 2.7, it follows thatGtis a minimal space such thatH<∞⊥E≤t|Gtif and only if it is defined by (3.5). SinceGt=G≤t, it follows
thatH<∞⊥E≤t|G≤t, which becauseG⊆E(it follows from the assumptionP(Et|S)⊆E<t
and (3.5)) is equivalent toH|<G;E.
FromH⊆G(it is clear from (3.5)) andGt=G≤t, it follows immediately thatGis a
realization of an s.d.s. with outputsH.
The converse is trivial.
The next example illustrates the above result.
Example 3.4. LetZ(t)=2
n=1 t
0gn(t,u)dZn(u),t∈[0, 1], be a proper canonical represen-tation of the process{Z(t), t∈[0, 1]}(see [7,8]) and let the process{X(t), t∈[0, 1]} be defined byX(t)=0th(u)dZ1(u), t∈[0, 1]. ThenF≤Xt⊆F≤Zt for eacht. Further, for
Y(t)=P(Z(t)|FX
<∞)=
1
0g1(t,u)dZ1(u),t∈[0, 1], we have thatF≤Yt⊆F≤Zt. According to
Theorem 3.3(forH=FX=(FX
t ),t∈[0, 1],FtX= {c·X(t),c∈R},E=FZandGt=FY),
it follows that familyFY
t =P(F≤Zt|F<X∞),t∈[0, 1], is a minimal realization (of an s.d.s.
with outputsF≤Xt) that causesF≤XtwithinZ(t).
In the remaining part of the paper, we consider the problem (ii) formulated above. In part (i), we considered realizationsGsuch thatG⊆E, that is, the given familyEis a natural “framework” in which we find realizationsGof an s.d.s.S1. However, in the case (ii), whereE⊆G, the familyEis submitted to unknown familyG, so that we will assume that all considered families of Hilbert spaces are submitted to some given “framework” familyFof Hilbert spaces.
The following theorem considers the problem of determining the possible statesG(of an s.d.s. with outputsH) such that the familyEis a cause of outputsHwithinG. Theorem3.5. (1)LetHandEbe such thatH⊆E. Each Markovian familyGsuch that E|<E;GandG<t⊥H>t|Gtfor eachtis a realization (of an s.d.s. with outputsH) andEis a
(2)LetH,E, andJbe such thatH⊆E, as well asE|<E;JandP(J<t|E≤t)⊥H>t|P(Jt|E≤t)
for eacht. If Jis Markovian, then the familyG, defined by
Gt=PJt|E≤t, t∈R, (3.6)
is minimal among the realizations (of an s.d.s. with outputsH) such thatEis a cause ofH withinG.
(3)If H⊆Eand if given “framework” familyFsuch thatE|<E;F, then the familyG, defined by
Gt=F≤t, t∈R, (3.7)
is strictly maximal among the realizations (of an s.d.s. with outputsH) such thatEis a cause of HwithinG.
Proof. (1) According to Lemma 2.5, the assumption G<t⊥H>t|Gt is equivalent to
P(H>t|G≤t)⊆Gt. From that and the assumption thatGis Markovian family, we get
Gt=P
G≥t|G≤t
=PG≥t∨H>t|G≤t
(3.8)
which is equivalent toG≤t⊥G≥t∨H>t|Gt. However, sinceH<t⊆G≤t(which is an
obvi-ous consequence ofH⊆EandE|<E;G), the last relation means thatGis a realization of an s.d.s. with outputsH. FromE|<E;GandH⊆E, it follows thatH|<E;Gholds.
(2) From (3.6), it follows thatG≤t=E≤t and immediately we getH|<E;G. According
toDefinition 2.10, it is clear that the familyG, defined by (3.6), is a minimal family such thatH|<E;G. From the assumptions thatE|<E;Jand the fact thatJis Markovian, we get P(G≥t|G≤t)=P(J≥t|E≤t)=P(P(J≥t|J≤t)|E≤t)=P(Jt|E≤t)=Gt which means thatG
is Markovian. Now, according to part (1) of this theorem, it follows that the familyG, defined by (3.6), is a realization (of an s.d.s. with outputsH) such thatH|<E;G.
(3) SinceG≤t=F≤t, the assumptionE|<E;Fis equivalent toE|<E;G, so that because
ofH⊆E, it follows thatH|<E;G. FromGt=G≤tandH⊆G, it immediately follows that
Gis a realization of an s.d.s. with outputsH. From the fact thatFis a “framework” family (i.e.,G⊆F), it is clear thatGis a strictly maximal realization with given properties.
It is easy to see that for given outputsHof an s.d.s.S1and informationEabout an s.d.s.
S2, the familyG, defined by (3.6), is not a unique minimal realization (of an s.d.s.S1) such thatH|<E;G. For each familyJ∗⊆F which satisfies conditions fromTheorem 3.5(2), withG∗t =P(Jt∗|E≤t),t∈R, is defined a minimal realization (of an s.d.s.S1) such that H|<E;G∗. All these minimal realizations have the past information equivalent toE≤t, t∈R, but their present information attis different.
Example 3.6. Let AandB be arbitrary Hilbert spaces and letH=(Ht), E=(Et), and
J=(Jt),t∈ {1, 2, 3}, be defined by
H1=A, H2=B, H3=A∨B,
E1=A, E2=B, E3=A,
J1=A, J2=A∨B, J3=B.
(3.9)
It is easy to see thatJis Markovian,E|<E;JandP(J<t|E≤t)⊥H>t|P(Jt|E≤t) for eacht.
If the familyGis defined by (3.6), then
G1=A, G2=A∨B, G3=B. (3.10)
According toTheorem 3.5(2),Gis a minimal realization (of an s.d.s. with outputsH) and H|<E;G. However, the familyG∗=(G∗t),t∈ {1, 2, 3}, defined by
G∗1 =A, G∗2 =A∨B, G∗3 = {0}, (3.11)
is another realization of the same s.d.s. andH|<E;G∗. Obviously,G∗≤G.
The problem of determining a strictly minimal realizationG(of an s.d.s. with outputs H) such thatH|<E;Gis still open. If it would be possible to find a strictly minimal family Jmbetween familiesJ∗⊆Fthat satisfy conditions fromTheorem 3.5(2), this strictly
min-imal familyJmwould define a strictly minimal familyGm(with (3.6)) among all families
GofTheorem 3.5(2). It is clear that if there exists such strictly minimal family, it cannot be necessarily unique, so that a strictly minimal realization with given properties is not necessarily unique.
Remark 3.7. It is of interest to find conditions for the existence of realizations with certain properties less restrictive than those obtained in this paper.
Acknowledgment
This research was supported by the Science Fund of Serbia, through the Mathematical Institute, Belgrade.
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Ljiljana Petrovi´c: Faculty of Economics, Belgrade University, 11000 Belgrade, Serbia and Montenegro