1 seh ass eee @ Copyright by INCREsr, 1984
THE ITO-CLIFFORD INTEGRAL IV: A RADON-NIKODYM THEOREM AND BRACKET PROCESSES
C BARNETT, R F STREATER and I F WILDE
0 INTRODUCTION
The construction and various properties of the It6-Clifford stochastic integral have been discussed in [1, 2, 3] In particular, it was shown in [1] that any centred
t
L?-martingale is given as an [t6-Clifford stochastic integral; X, = \#6) dŸ,, 9
where ¥, = Y( 70 3) is the Fermi-field It was also shown in [1] that stochastic
integrals of the form \ faxX can be defined as elements of L3(2), the non-commu-
tative L?-space associated with the Clifford probability gage space We consider the
telationship between the stochastic integral with respect to ¥ and that with respect
to X Specifically, we prove a Radon-Nikodym theorem in the form: fax = \ ƒ⁄ZdV Using the Doob-Meyer decomposition of the submartingale X/X, given in {!], we define the pointed-bracket L1-process <X,, Y,> associated with L?-martin- gales (X,) and (Y,) The stochastic integral \ f4X is shown to be characterized, as a process, in terms of pointed-bracket processes These results parallel those of stan- dard (i.e commutative) probability theory (see, for example [8,9])
In Sections 1 and 2 we review and generalize some of the results from [1] The stochastic integral \ fdX is defined in Section 3 — this being a simplified version of that in [1] The Radon-Nikodym theorem is presented in Section 4, and in Section 5 an analogous result for stochastic integrals with respect to Wick martingales is
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Finally , in Section 7, we give a summary of the analogous results valid for “left?” rather than “‘right’’ integrals
A Doob-Meyer decomposition and stochastic integration with respect to martingales over an arbitrary probability gage space is considered in [4]
1 FOCK SPACE AND THE CLIFFORD ALGEBRA
We recall some notation and definitions from [1] Let A(L2(R*)) denote the
antisymmetric Fock space over L*(R†?), and, for øc /?(R+), let CŒÒ and A(u) ==
== C(u)* denote the creation and annihilation operators on A(L?(R*)) The fermion field is defined as ¥(u) = C(u) A(u), where u is the complex-conjugate of ø in £7(R+) The fermion fields satisfy the canonical anticommutation relations
(1.1) (1)ŸW(6) — W(c)Ÿ(u) = 2(, vw) 1
For each ¢ > 0, @, denotes the von Neumann algebra generated by the fields ¥(u) for ue L?(R*) with suppu ¢ (0, 7], and @ is the von Neumann algebra gene- rated by the increasing family {@, :f > 0} @ is the weakly closed Clifford algebra over L?(R*) [6, 11]
Let m denote the vector state m(x) = (Q,xQ), xe @, where Q is the Fock vacuum vector Then m is a faithful, central state on @&
For 1 <p < co, L?(Ø) ¡is the completion of @ with respect to the norm
|x|, = mCxi?)/?, xe @, and L®(@) is $ equipped with its C*-norm
The elements of L?(@) can be identified with closed (possibly unbounded)
operators on A(L?(R*)) [7,10] In fact, L?(@) consists of those closed operators Y on A(L?(R+)) affiliated to @ such that Q is in the domain of 'Y;?/? Similarly, one
defines L°(@,) for t = 0 and 1 < p < o, so that L*(%,) is a closed subspace of L?(@) Indeed, if @ is any von Neumann subalgebra of ¢, L?(@) is a closed subspace of L’(@) The conditional expectation given # is a contraction of L?(@) onto L’(#) for all 1 < p < oo, and is denoted m(-|@) We will write M, for m( @,),¢ 2 0
DEFINITION 1.1 An L’(@)-martingale is a family {X,:t > 0} with Y,¢L7(¢)
for all ¢20 and such that 4,X, = X, for all O<t < 5s It follows that Y,¢€L°(@,),
t 20
It is convenient to define here the parity operator B and establish some of its properties Let Q° denote the linear space of even polynomials in the fields Y(v), ue L?(R*), and let Q denote the von Neumann subalgebra of @ generated by Q? DEFINITION 1.2 The parity operator B is the map f: L°(@) > L°(%),
Trang 3THE IT6-CLIEFORD INTEGRAL IV 257
Evidently, B: L°(¢@) — L°(@) is continuous, and B(f)* = B(f*) for fe L(G) Moreover, using the fact that m(m(-!Q)|Q) = m(-'Q), we see that B(f)=f for fe L?(@) Furthermore, since m(-|Q) defines the orthogonal projection of L*(@)
onto L7(Q), it follows that B : L?(@) > £7(@) is self-adjoint and unitary
Proposition 1.3 (i) B(f) =f for all fe L?(Q), 1 < p < ow
(ii) B(x) = — x for any odd polynomial x in the fields ¥(u), ue L?(R*) (iii) 8: 9 > @ is o-weakly continuous
(iv) B(hg) = BA) B(g) for he L°(@), gE LG), with \/p + 1/q=1
(v) B: L°(@) ¬ L?(@) is isometric for 1 S p < 00
(vi) If fe D(@) and f > 0, then B(f) 2 0
(vil) B: L°(@,) = L°(@,) for allt 20, le p< om
Proof (i) Trivial using m(f| Q) = f for fe L?(Q)
(it) If x is an odd polynomial in the fields and ye Q°, one sees that yQ and
xQ are orthogonal in A(Z?(R+)) Hence m(y*x) = 0 By continuity it follows that
m(y*x) = 0 for all ye OQ, and so m(x|Q) =0 Thus B(x) = —~x
(iii) By continuity and self-adjointness of B on L?(@), we have m(gB(/h)) = => m(B(g)h) for he L°(@), ge L(G) with I/p + 1/g = 1 In particular, this holds
for g == 1, p = oo But L(@) is the predual of ¢ [10] under the pairing Z1(@)x x L°(@) > C, (g, h) > m(gh), and so we deduce that B: L°(@) > L°(@) is o-weakly continuous
(iv) Any polynomial in the fields can be written as a linear combination of an odd and an even polynomial Now if x’, x’ are odd and y’, y’”’ are even polynomials
in the fields, we have that x'x”, y'y“ are even and x'y“ and xyˆ are odd Hence,
using (i) and (ii),
BU’ -+ y’) (x“ + y’)) —= 8Œ” + xy” + yx’ +4 yy”) =
— xx” — xy yx” + yy” —
=Œ'+zy')8Œ“+zy)
By continuity, it follows that B(xy) = B(x)8() for any x, ye @ and, again by con- tinuity, the result follows
(v) From (iv), and the fact that B(x) = 0 implies that Ø?(x) = x = 0, we see that f is an automorphism of ¢ Hence B: @ — @ is isometric By duality it fol- lows that B: L1(@) > L1(@) is a contraction, and so by interpolation [7] B: L’(@) > — L£°(@) is a contraction for | < p < oo Now, forfe L’(¢),1 <p < ~,
Flo = NPA lle < WBC < Wile
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(vi) lf fe L(@) with f > 0, then, there is geL*(@) such that f.:: g*g
Indeed, we can take g = f/2 Then, by (iv), B(f) = B(g*g) = (g)*f(g) > 0 (vii) This follows from (i), (ii), (iii) and (v) Q.E.D
REMARK 1.4 Properties (i) and (ii) imply that the definition of 8 given here agrees with that in [1] Indeed, the above results are readily obtained using the definition of B given in [1], but the proofs given here seem to be interesting in their own right
2 THE ITO-CLIFFORD INTEGRAL
DEFINITION 2.1 An L°(@)-valued process on [O, f] 1s a map f: [0,7] > L?/) such that f(s)¢ 2°(@,) for 0 < s < ¢ Such a process is said to be elementary if it is of theform f = gy, ,) for some 0 <r <1 < tandgeL(@) Note that f's)& e€ L’(@,) implies that ge L7(@,) An L?-process on [0,?] is said to be simple if, on [0, £), it is a finite sum of elementary L’-processes Denote by ((0, tJ, (⁄)) the linear space of simple L?(%)-valued processes on {0, /}
For wé Li,,(R*), set ¥ (uv) = (17a, 4) Then it was shown in [1] thất (0) :
t > 0} is an L®-martingale adapted to the filtration {@,: 1 2 O}
DEFINITION 2.2 If ƒ=: 8y; Sr <1 <<, is an elementary L°-valued process on [0, /], the /6-Cljfford stochastic integral of f with respect to 'Pj(u) is defined to be f(s) d¥ (6) =: g(W,(0) — (0) So am ? The integral Me) for ƒe ⁄(0,}, L**) ¡is defined by linearity 6
Let § (0, f], ‘u(s);? ds) denote the subspace of processes in L7((0, r], u(s) "dy: £°(@)) Then as in [1], one sees that #({0,t], L°) is dense in $([0, 1], u(s; ds), and that the Ité-Clifford stochastic integral is well-defined, by continuity, as an element of L°(@) for fe (0, 2], 'u(s)!?ds) and satisfies the isometry property:
(2.1)
Ce fis) a,c) = ( /0)1#IwG)84e
9
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t
in [1] that for each fe §,,,(Rt, |u(s)|? ds), JUes# 130} is a centred L?- -martingale
If u(s) = 1 for se R*, write W, for W,(u) Using Theorem 4.1 of [1], we have
the following representation theorem
THEOREM 2.3 Let (X,) be an L?-martingale (adapted to the family {@,: t > 0}) Then there is a unique element X of $1o.(R*, ds) such that
t
X,=Xs+ 0) d¥, fort >0
0
Proof Since (X, — X,) is centred, the existence of X follows from Theorem 4.1
of [I] The uniqueness follows immediately from the isometry property equation (2.1)
Q.E.D Our first aim is to generalize this theorem
Lemma 2.4 For ue Li,,(R*) and he §((0, ¢], |u(s)|?ds), we have
t
(2.2) Vi đW (u)= eo) d¥
0
Proof We first note that Au € §((0, zt], ds) and so the right-hand side is well-de-
fined If h is elementary and u = y;5,,s.) for some 0 < 5, < sy < ¢, the result is clear
Now, for any ve L?(R*), ||¥(v)|l.o = ||o||, so ø > W(e) is a linear continuous map from L?(R+) into L°(@) Hence, by linearity and continuity, the result holds for A elementary and ue Lj,,(R*+) But then again by linearity and continuity the result holds for arbitrary ue £2,(R*) and Ae (0, ft], !u(s)|*? ds) Q.E.D
REMARK This is a Radon-Nikodym theorem
THEOREM 2.5 Let we Lj.(R*) and let (X,) be an L?-martingale There is £ gE H,,.(R*, lus)? ds) such that X,= Xp) + \ 0) d¥(u), 20, if and only if 0 {s: Ấ(s) # 0} © {s:u(s) #0} up to a set of (Lebesgue) measure zero If such g exists, it is unique
Proof Let E= {seR*: u(s) £0}, and set v=u+y,c Then v #0
Trang 6260 C BARNETT, R F STREATER and ï F WILDE {s: X(s) #0} <{s: u(s)#0} up to a set of Lebesgue measure zero, we have Ý=z„Ý=
ypuh == ygu (Lebesgue) almost everywhere Putting g=y,h, we see that £eH,,,.(R*, :u(s)? ds), and by Lemma 2.4, for t > 0,
t t t
X,—-X= \ X(s)d¥, = \relsu(s (s) d¥, = \z@04#,00,
0 Ũ
The uniqueness of g follows from the isometry property, equation 2.1
Conversely, suppose that there is g¢ %,,.(Rt, ,¿(s),? de) such that X, =: Xe + t a \s(av.uo, t > 0 Then, by Lemma 2.4, 0 t X,—*⁄;,= G66, t20 â
Hence, since gu  §,,.(R*, ds), it follows from Theorem 2.3 that X = gu, and the
proof is complete Q.E.D
Remark As is customary, for convenience (and brevity) we have treated ele- ments of L3,.(R*) and § as if they were maps rather than equivalence classes of
maps
3 EXTENSION OF THE ITO-CLIFFORD INTEGRAL
We shall construct stochastic integrals with respect to an arbitrary L?-martin-
gale and, in the next section, relate them to the Ité-Clifford integral by means of a Radon-Nikodym theorem The construction here is a simplified version of that given in § 7 of [1] DEFINITION 3.! For an L?-martingale (Y,), let zy denote the Borel measure on t R* given by x([0, /]) =| (\X(s){3ds, where X is given by Theorem 2.3 0
We shall sometimes use 4 to denote Lebesgue measure on Rt
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We shall extend the definition of the stochastic integral to more general inte-
grands using a contraction property First we recall the following result from [I] t
THEOREM 3.3 If (X,) is an L?-martingale, then Z, == X”X, — \ |8(Ÿ))J*ds is 0
an L}-martingale
Proof Use Theorem 2.3 and [l, Theorem 3.18) Q.E.D
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COROLLARY 3.6 For any fe #((0, t], ux), let (g,) be a sequence in (|0, r], L®) t such that g, 2 f in X((0, t}, wy) Then there exists Lyin £,(8)dX, This limit is 9 t independent of the particular sequence (g,) converging to ƒ; and is denoted \ f(s) dX, a Furthermore t t ' 9 I\v ax! < [Faux 0 ie 0
Trang 9THE ITÔ-CLIFFORD INTEGRAL IV 263
Proof By linearity, we may suppose that f is elementary: say, f= 8Xtr, x)» O0<r<t<t, geL(G%,) Then t \/ dX, = g(X, — X= al Rs) 4%, = leroy, = 0 (since left-multiplication by an element of L® is continuous from L2(@) —¬ L2 (Ø)) f = ƒ)#() dW, Q.E.D 9
THEOREM 4.2 (Radon-Nikodym theorem for stochastic integrals) Let (X,)
be an L?-martingale and let ƒ (0, t], uy) Then ƒŠ e 6(0, t], ds) and t
f(s) dX, = \ f(s)X(s) aƠ,
9
.ô
Proof Let (g,) be a sequence in S((0, t],L°(@)) such that g„>/ƒ in H((0, t], ux) By passing to a subsequence if necessary we may suppose that
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COROLLARY 4.3 Let (X,) be an L?-martingale and let ge X((0, t], ny) for t each t > 0 Then if Y, = \ g(s)dX,, 1 > 0, we have 0 t t e9 dY,= | fet dự, 0 0 for any fe # (0, 4), wy) rc
Proof By the theorem, \vay = \/et But Y = \ dX= \ sav, again by
the theorem Hence Y = gX and the result follows Q.E.D
5 STOCHASTIC INTEGRALS WITH RESPECT TO WICK MONOMIAL MARTINGALES
For given real zy , ,u,, € Li,,(R*), the Wick monomial martingale W(u,, ,4,55),
s 20 is defÑned as W0, ., tạ; S) = : nfte, vị) - (0,#Xra 3): Whe€T€ : :
denotes Wick ordering It was shown in [I] that W, = W(u,, , H„; s), $20 is t
an L®-martingale, and that stochastic integrals \ S(s)dW, can be constructed as 0
elements of L7(%) for fe §((0, t], dv), the set of processes in L((0, t], dv; £7(@)), where
v is the measure on R* given by v({s, t]) = 4, — a, and a;eR is a, =Jf? W,=: == WW t Writing W, = ( Ws) dW,, we have, by the isometry property 0 t a, = m(WeW,) = init liữ@)¡3d» Ũ
Hence v is the measure jy
The closure of S([0, t], L°(@)) in L*({0, t], duy; 2(@)) is equal to
(0, /], duy) and so Theorem 4.3 has a sharper analogue here
THEOREM 5.1 Let fe 6([0, /], duy) Then ƒW e 9(0, r], ds) and
t t
\/@ dW, = V6 W(s)a¥
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Proof This is analogous to that of Theorem 4.2 Q.E.D REMARK 5.2 The isometry property for fe §({0, 4], diy) can be written as r Vv 0 =| I F(S)IIE | WC) [Bds This follows because dv = djty = || W(s)||2 ds We see from the theorem that t rev) -\ I f(s) Wésy|B ds and so t t \ LAs) Ws) [3ds -\ I/@)18II#@) l ds
We can formulate an analogue of Theorem 3.18 of [I]
THEOREM 5.3 Let fe §([0, t], duy) Then Z=|\ = (8/6909)
9 is a centred L1-martingale (on [0, t})
Proof Use Theorem 5.1 together with [l, Theorem 3.18] Q.E.D f
As a corollary, we note that Âu ”) has a Doob-Meyer type decomposition for any fe §,,,.(R*, dyy)
6 THE POINTED-BRACKET PROCESS
We shall define the so-called pointed-bracket process corresponding to a pair of L?-martingales, and give a characterization of the process given by the ltô- -Clifford integral
Trang 12266 C BARNETT, R F STREATER and [ F WILDE
Notice that (X,, X,> is the increasing L!-process given in the Doob-Meyer decom-
position of X;'X, by Theorem 3.3
Clearly, ¢-,-> is a sesquilinear map from tx Mt into the set of processes in Lioce(R*, di; L1(@)), where Mt denotes the set of L°-martingales
For any bounded Borel set E in R*, set (X, Y>(£) = \ B(Y(s))*B(X(s)) ds
E
Then ¢X, Y> defines an L1(%)-valued countably additive Borel vector measure on any bounded interval [0, ] in R*
Lemma 6.2 For X¥, Ye M, (X, YD has bounded variation on any interval {0, t] in R+, and is continuous with respect to Lebesgue measure on the Borel sets in {0, ?} Proof By polarization, it is enough to show that (X, X> has bounded variation
on {0, t] Now, the variation of <X, X> on [0, ¢] is defined to be the set function Ev> |\(X, X>| (E) = sup YX, XE ia
where E is any Borel set in [0, t] , and the supremum is taken over all partitions of
E into a finite number of disjoint Borel sets
But for any such partition {£;} of a Borel set E in [0, t], we have SIE X9Œ)h = XIŒfG)2##G9)06:] = +, i =Y "(| 0(@)34:)= ‡ (since \ BCX(s))*B(X(s)) ds is a non-negative element of L1(¢ ) E + = m(5 ÂI#(#@)#4s] —|ÂIB(#f@Pas] = vor DE! \ 1 : £ i
Hence, for any ¢ > 0, |X, X>| ((0, ¢]) =: Ji<X, X3 ([0, z)ii¡, and so <X, X3
has finite variation on [0, t] The same is then true of <X, Y}
If Eis a Borel set in (0, t] with Lebesgue measure /(£) = 0, then
(X, YE) = \UŒG)/8f@) ds = 0 Q.E.D
E
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Lemma 6.3 Let t 2 0, and suppose that f:[0, t] + L®(@) is the limit 2 a.e on (0, t] of a uniformly bounded sequence (g,) in F([0, t], L°(G)) Then, for any X,YeEM, f is integrable with respect to <X, Y> and L1{Ø)-lim g,d(X, Y) = \ fd¢X, YY exists, where [dCX, Y> is the Bartle integral £ Proof This follows immediately from Bartle’s bounded convergence theorem [5] Q.E.D,
DEFINITION 6.4 Let Z[0, £] denote the set of maps /: [0, /] > L®(2) satisfying the hypothesis of Lemma 6.3
Lemma 6.5 For fe P[0, t] and X eM, we have
() 8Œ) Z0 ¡];
Œ 8Œ)X < 5(0, ¿], ds; 12(2));
t t
(iii) \%09 dv, = (\ 8(#@)194#,
Proof (i) If (g,) is a uniformly bounded sequence in Y((0, t], L°(@)) such that |le„(s) — /(s)ll¿ —> 0 Ã a.e on [0, ¢], then, using Proposition 1.3, it follows that (B(g,)) is also a uniformly bounded sequence in F([0, rj, L°(¢)) and B(g,) > B(f) in
L™(@) A a.e on [0, ¢] In other words, B(f) € PO, #]
(ii) There is a sequence (h,) in (0, t], L°(¢)) such that h, > Xin §((0, ¢], ds; L?(@)) Then with the notation of part (i) above, and passing to a subsequence if
necessary, we see that f(g,)h, > B(f)X in L2(Ø) 2 a.e on [0,r] Hence B(f)X is a
A measurable L®-process on [0, f]
It is easy to see that B(g,)h, is a Cauchy sequence in §((0, f], ds) and so the result follows
(iii) Using the canonical anticommutation relations, equation (1.1), and Pro-
position 1.3, one sees [l, Lemma 3.15] that for any 0 <r <s and ge L™(G,), (g¥ (xe, s1))* = ¥ (xc, 38" = ÿB(g)*v Œợ, si):
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From the definition of the stochastic integral and the continuity of the adjoint
t +
*: £6) > L*(%), ¡t follows that (\ nso.) = \0G)*sy, for any />0
‘ 0
and fe 9((0, t], ds) Q.E.D
THEOREM 6.6 Let (X,), (Y,) be L?(@)-martingales Then for any t 2 0 and e #0, /], ny)n Z0, 0Ì; ny) me haye (i (ser = (sex, Và: 0 9 t t Proof For fe (0, r], nyx)n “((0, t], wy), we have \ướy= \ vay and 0 9 t t Ự dX = \ f*Xd¥, by Theorem 4.2 Hence, by definition 0 9 (x \ve 2) = \ #8G)Ÿ@))*8(#0)) ds = 9 a t = Ya Fs)" f(s)*X(s)) ds = (\ ƒ*dX, vy: Q.E.D
THEOREM 6.7 Let (X,), (Y;) be L*°(@)-martingales and let fe A(0, t} Then
(i) \racx, Y= (i, (\roaxsy,
ặ õ
and
t
(i) (race, Yỳ* = ‹Ñ fs) ax?) ry
Trang 15THE ITÔ-CLIFFORD INTEGRAL IV 269
Now, by Lemma 6.5 (iii),
f t
Yi = Ye+ (( Fav.) =Y#+ dd,
But, for he L°(@), ||B(A)* ll, = |All, and so py = py Moreover, P[0, 7] S t t c#Z(0 /], uy) and thus van: = G06)" d¥, Thus, again by Lemma 6.5 9 (iii), we have t r (\rave)’=\ {8(ŸG))/(s)*}d#, = \ ŸG)8ŒG))* d#, Hence t "ˆ t (i, (\rare) ) -| B(ŸG)8(f(s))*)*B LỄ (s)) đã = ƒG)B(Ÿ@))*8(Ÿ0))ds
This proves (i)
The proof of (ii) follows from (i) using (X,, ¥,>* = <Y,,X, Q.E.D
This theorem has as a corollary a characterization of the stochastic integral in
terms of pointed-bracket processes
THEOREM 6.8 Let(X,) be an L*(@)-martingale, and let f: R*+ + L°(Ø) be t such that the restriction of f to (0, t} belongs to Y{0, t) for allt = 0 Then (\ fax ) is the unique centred L?(@)-martingale, Z,, say, such that
f
Me ¥*) =(Y,,Z,)
ọ for any L?(¢)-martingale (Y,)
Proof By Theorem 6.7, (Z,) satisfies ft
le, X*) = €Œ,, Z2
0
lf (Z/) is a centred 7/2-martingale also satisfying this equation, then
Trang 16270 C BARNETT, R F STREATER and I F WILDE
t
which implies that \ iB(Z(s) — 2’(s));2ds=0 for t > 0 Thus Z = Z’ in §,,.(R*, ds)
0
Since (Z,) and (Z;) are both centred, it follows that they are equal Q.E.D
7 THE LEFT STOCHASTIC INTEGRAL
Results analogous to those of the preceding sections can be obtained for the left stochastic integral \ dXf which is defined similarly to the right stochastic integral Indeed, if (X,) is an L?-martingale, then, in terms of the left stochastic integral, Theorem 2.3 becomes t t X,=X)+ Roar, =Xe+ \t, f(Ẽ@)), +>0 The analogues of Theorem 3.4 and Corollary 3.6 yield the contraction property: t lexso, <lusoideno ‹ 0 for fe Z0, t], Hx)
The analogue of Theorem 4.2 is:
THEOREM 7.1 Let (X,) be an L*-martingale and let fe #((0, t], ux) Then
B(X) fe §((0, 1], ds) and
\ax./@ = ler pov
For left integrals, Theorem 6.6 becomes as follows
THEOREM 7.2 Let (X,), (Y,) be L°(@)-martingales Then for t > 0 and
Trang 17THE ITÔ-CLIFFORD INTEGRAL IV 271
Proof As for that of Theorem 6.7
THEOREM 7.4 Let (X,) be an L?(@)-martingale, and suppose that f ¢ P{0, t] for t all t > 0 Then ( \ax f is the unique centred L?(€)-martingale, N,, say, such that é \ ex, Y>ƒ= (N,, YỌ 0 Proof Just as for that of Theorem 6.8 REFERENCES 1 Barnett, C.; STREATER, R F.; WILDE, I F., The Ité-Clifford integral, J Functional Ana- lysis, 48(1982), 172—212
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