CHARACTERIZATION OF A REGULAR FAMILY OF SEMIMARTINGALES BY LINE INTEGRALS
R. CHITASHVILI† AND M. MANIA
Abstract. A characterization of a regular family of semimartingales as a maximal family of processes with respect to which one can define a stochastic line integral with natural continuity properties is given.
According to the characterization of semimartingales by the theorem of Bichteler–Dellacherie–Mokobodzki (see [1],[2]) the class of semimartingales is the maximal class of processes with respect to which it is possible to define stochastic integrals of predictable processes with sufficiently natural properties, more exactly: an adapted cadlag processX = (Xt, t ≥0) is a semimartingale if and only if for every sequence of elementary predictable processes (Hn, n≥1), which converges uniformly to 0, the corresponding integral sums
Jt(Hn) = X i≤n−1
Hin(Xt∧ti+1−Xt∧ti)
converge to 0 in probability for eacht≥0.
The aim of this paper is to give an analogous characterization for a regular family of semimartingales XA = (Xa, a∈A) (see Definition 1 below) as a maximal family of processes relative to which one can define a stochastic line integral (along predictable curves u : Ω×[0,∞[→ A) with natural continuity properties.
Let on probability space (Ω,F, P) with filtrationF = (Ft, t≥0), satisfy-ing the usual conditions, a familyXA= ((X(t, a), t≥0), a∈A) of adapted cadlag processes be given, where (A,A) is a compact metric space with the metric r and Borel σ-algebra A. Denote by S,Mloc,M2loc,Aloc,V classes
of semimartingales, local martingales, locally square integrable martingales, processes of locally integrable variations and processes of finite variations (on every compact), respectively.
1991Mathematics Subject Classification. 60G44,60H05. Key words and phrases. Semimartingale, stochastic line integral.
525
For someλ >0 let ˜
Xta,λ=
X
s≤t
∆XsaI(|∆Xa s|>λ).
If Xa is a semimartingale then the process Xa −X˜a,λ is a special semi-martingale with the canonical decomposition
Xa−X˜a,λ=Ma,λ+Ba,λ,
where Ba,λ is a predictable process of finite variation and Ma,λ is a local martingale. Moreover,|∆Ba,λ| ≤λ,|∆Ma,λ| ≤2λand henceMa,λ∈ M2
loc
(see, e.g., [2]).
Definition 1. We say that a family XA = (Xa, a ∈ A) is a regular family of semimartingales if it satisfies the following conditions:
(A) the processXa = (X(t, a), t≥0) is a semimartingale for everya∈A; (B) there existsλ >0, a predictable, increasing processK= (Kt, t≥0) which dominates the characteristics of semimartingales (Xa−X˜a,λ,a∈A), i.e.,
hMa,λ, Ma,λi ≪K,Var(Ba,λ)≪K on every [0, t], for eacha∈A;
(C) the Radon–Nicod´ym derivative ϕ(a, b) =dhMa,λ−Mb,λi/dK (re-spectively,g(a) =dBa,λ/dK) is a continuous function of (a, b) (respectively, ofa) for almost every couple (ω, t) with respect to the measureµK, where
µK is the Doleans measure of K; a function ∆Xa
t is continuous in proba-bility uniformly on every compact;
(D) for eacht∈R+ a.s.
t
Z
0
sup a∈A
ψs(a, a)dKs+ t
Z
0
sup a∈A
|gs(a)|dKs+
+X s≤t
sup a∈A
|∆Xa|I(supa∈A|∆Xa|>λ)<∞.
Form∈ M2
loc denote byH(a, m),f(a, m) andψ(a, b) the derivatives
dhMa,λ, mi
dK ,
dhMa,λ, mi
dhm, mi ,
dhMa,λ, Mb,λi
dK ,
respectively. It was proved in [3] that condition (C) implies the existence of
µK-a.e.(µhmi-a.e., respectively) continuous inamodification of the function
Let us introduce the classes U and Ub of predictable processes taking values inAand inQ(A), respectively, whereQ(A) is the space of all prob-abilities on A. Assume that Q(A) is provided with the Levy–Prokhorov metric.
Denote byU0the class consisting of elements ofU taking values in some
finite subsetA0 ofAand byUb0the class of elements ofUb taking values in
the set of probability measures concentrated in some finite subset ofA, i.e., for which there exists a finite setA0⊂A such that ut(ω, A0) = 1 for each
(ω, t).
Let Ud (respectively, Ubd) be the class of piecewise constant functions consisting of elements u ∈ U (respectively, Ub) such that for some finite sequence of (deterministic) moments t1 < t2 < ... < tN, ut =uti forti < t ≤ ti+1 and uti is a Fti-measurable random variable taking values in A
(respectively, inQ(A)). LetU0
d =U0∩Ud andUbd0=Ub0∩Ubd.
It is convenient to denote the elements ofU andUb by the same symbols, since every element u ∈ U can be considered as a degenerate element of
b
U. So we shall use the shortened notation: u= (ut(ω), t ≥0) for u∈U,
u = u(C) = (ut(ω), t ≥ 0) = (ut(ω, C), t ≥ 0), C ∈ A for u ∈ Ub, while the symbol H(u, m) foru ∈ U will be used instead of the exact notation (Ht(ω, ut, m), t≥0); the symbolsf(u, m), ϕ(u, v), g(u) are understood in a similar way. Foru∈Ub, we sometimes use the expressionsg(s, us), Hs(us, m) instead ofRAgs(a)us(da),RAHs(a, m)us(da), respectively.
If the family of semimartingales XA satisfies conditions (A), (B), (C), (D) then for each u ∈ U (respectively, for each u ∈ Ub) one can define a stochastic line integral (respectively, a generalized stochastic line integral) with respect to the family of semimartingales (Xa, a∈A) along the curve
u= (ut,t≥0) as t
Z
0
X(ds, us) = t
Z
0
M(ds, us) + t
Z
0
gs(us)dKs+
X
s≤t
(X(s, u(s))− −X(s−, u(s))I(|X(s,u(s))−X(s−,u(s))|>λ), t≥0 (1)
resp., t
Z
0
X(ds, us) = t
Z
0
M(ds, us) + t
Z
0
Z
A
gs(a)us(da)dKs+
+X s≤t
Z
A
∆XsaI(|∆Xa
s|>λ)us(da), t≥0
, (2)
such that
DZ·
0
M(ds, us), mE
t= t
Z
0
Hs(us, m)dKs (3)
resp.,D
·
Z
0
M(ds, us), mE
t= t
Z
0
Z
A
Hs(a, m)us(da)dKs
(4)
for everym∈ L2(MA) (we recall thatHs(a, m) =dhMa,λ, mis/dKs). The stochastic line integral defined above does not depend on the choice ofλ, on the choice of a dominating processK (if for them conditions (A), (B), (C), (D) are fulfilled) and posseses the following properties:
(1) for eachu∈U0 (respectively, for eachu∈Ub0)
t
Z
0
X(ds, us) = X a∈A0
t
Z
0
I(us=a)X(ds, a)
resp., t
Z
0
X(ds, us) = X a∈A0
t
Z
0
us(a)X(ds, a)
and for every u∈Ud(respectively, for everyu∈Ubd)
t
Z
0
X(ds, us) =X i
[X(ti+1∧t, uti)−X(ti∧t, uti)],
resp., t
Z
0
X(ds, us) =X i
Z
A
[X(ti+1∧t, a)−X(ti∧t, a)]uti(da)
;
(2) forut=u1tI(t<τ)+u2tI(t≥τ)
t
Z
0
X(ds, us) = t∧τ
Z
0
X(ds, u1
s) + t
Z
t∧τ
X(ds, u2
(3) for eachu∈U (respectively, for eachu∈Ub)
∆ t
Z
0
(X(ds, u(s)) =X(t, u(t))−X(t−, u(t))
resp., ∆ t
Z
0
(X(ds, u(s)) =
Z
A
(X(t, a)−X(t−, a))ut(da)
;
(4) for eachu∈U (respectively, for eachu∈Ub)
D Z
M(ds, u(s)), Z
M(ds, u(s))E t=
t
Z
0
ψs(us, us)dKs
resp.,D Z M(ds, u(s)), Z
M(ds, u(s))E t=
t
Z
0
Z
A
ψs(a, a)us(da)dKs
;
(5) ifun→u µK-a.e. (in the sense of weak convergence of values in the space of probabilitiesQ(A) ifun∈Ub), then for eacht∈R
+
sup s≤t
t
Z
0
X(ds, uns)− t
Z
0
X(ds, us)
→0, n→ ∞
in probability.
The stochastic line integral was introduced by Gikhman and Skorokhod in [5] for continuous processKand for the derivativedhMa−Mbi/dK=ϕ(a, b) which is continuous with respect toa, buniformly in (ω, t).
Let H be a space of predictable processes H bounded by unity of the form
H = X
i≤n−1
HiI]ti,ti+1],
wheret1< t2<· · ·< tn andHi∈ Fti.
Definition 2 ([6]). A family of processes ((Xa, t≥0), a∈A) satisfies the U.T. (uniform tightness) condition if for eacht >0 the set
Zt
0
HsdXsa, H∈ H, a∈A
is stochastically bounded.
Proposition 1 ([7]). A family of semimartingales((Xa
t, t≥0), a∈A)
satisfies the condition U.T. if and only if there exists λ > 0 such that for each t∈R+
1. the familyVar( ˜Xa,λ)t, a∈A)is stochastically bounded; 2. the family([Ma,λ, Ma,λ]t, a∈A)is stochastically bounded; 3. the familyVar(Ba,λ)t, a∈A)is stochastically bounded.
Remark . This proposition was proved in [7] forA ={1,2, . . .}. For an arbitrary setA this statement is proved without any changess. Note that sinceMa,λ is a local martingale with bounded jumps, the Lenglart inequal-ity implies that condition 2 is equivalent to the stochastic boundedness of the family (hMa,λ, Ma,λit, a∈A) (see [8]).
Proposition 2 ([7]). Let ((Xn
t, t ≥ 0), n ≥ 1) be a sequence of
semi-martingales satisfying the condition U.T. and letXn converge to some
pro-cessX in probability uniformly on every compact. Then the limiting process X is also a semimartingale and, moreover, for eacht >0
sup s≤t|M
n,λ
s −Msλ| →0, n→ ∞, (5) sup
s≤t
|Bsn,λ−Bλs|, n→ ∞ (6)
in probability, whereMλ(respectively,Bλ)is a martingale part(respectively,
the part of finite variation)of the semimartingale X−P∆XI(∆X>λ).
Let us consider now a family XA = (Xa, a ∈ A) of cadlag adapted processes and for each piecewise constant function ufrom Ub0
d , associated with a subdivision 0 = t0 < t1 < · · · < tn < ∞, define a new stochastic process
Jt(u, XA) =X i
Z
A
[X(ti+1∧t, a)−X(ti∧t, a)]uti(da) (7)
which takes the form
Jt(u, XA) =X i
[X(ti+1∧t, uti)−X(ti∧t, uti)] (8)
for functionsufrom the classUd.
Theorem 1. Let XA= (Xa, a∈A) be a family of adapted cadlag
pro-cesses. If for each sequence(un, n≥1)∈Ub0
d converging uniformly to some
u∈Ub (in the sense of weak convergence of values in the space of probabili-tiesQ(A)) the corresponding integral sums Jt(un, XA)are fundamental in
probability for everyt, then the difference Xa−Xb is a semimartingale for
each paira∈A, b∈A.
Proof. Let, for each sequence (un, n ≥ 1) ∈ Ub0
d converging uniformly to someu∈Ub,
|Jt(un, XA)−Jt(um, XA)| →0, n→ ∞, m→ ∞,
in probability (for eacht∈R+). We must show thatXa−Xb∈S for each
a, b∈A. To prove this, we use the same idea as in the proof of the theorem of Bichteler–Dellacherie–Mokobodzki which consists in the following: the continuity of the functionalJand the semimartingale property are invariant under an equivalent change of measure and it is sufficient to construct a law
Q equivalent to P such that the process Xa−Xb be a quasimartingale relative to the measureQ.
The possibility of constructing such a measureQis based on the following lemma whose proof one can find in [1].
Lemma 1. Let G be a bounded convex set of L0(P). Then there exists
a measure Qequivalent toP, with bounded density, such that
sup γ∈GE
Qγ≤C <∞,
where L0(P) is the space(of classes of equivalence) of finite random
vari-ables provided with the topology of convergence in probability.
Let us take asGthe image of the setUb0
d under the mappingJt. Evidently,
Gis a convex (as an image of a convex set) and bounded subset of L0(P).
Therefore, applying Lemma 1 forG=Jt(Ubd0) and using expression (8) for
Jt(u, XA), we obtain
EQX
i
[X(ti+1∧t, uti)−X(ti∧t, uti)]≤C (9)
for eachu∈U0
d.
Letue be a function fromU0, associated with the subdivision 0 =t 0 <
t1<· · ·< tn<∞, such that
e ui=
a ifEQ[X(t
i+1∧t, a)−X(ti∧t, a)/Fti]≥
≥EQ[X(t
i+1∧t, b)−X(ti∧t, b)/Fti] b otherwise
From (9) we have
EQJt(
e
u, XA) =EQX i
EQ[X(t
i+1∧t,eui)−X(ti∧t,eui)/Fti] =
=EQX
i
[I(EQ[X(t
i+1,a)−X(ti,a)/Fti]≥EQ[X(ti+1,b)−X(ti,b)/Fti])E
Q×
×[X(ti+1∧t, a)−X(ti∧t, a)/Fti] +
+I(EQ[X(ti
+1,a)−X(ti,a)/Fti]<EQ[X(ti+1,b)−X(ti,b)/Fti])E
Q× ×[X(ti+1∧t, b)−X(ti∧t, b)/Fti] =
=EQX
i
EQ[X(ti+1∧t, a)−X(ti∧t, a)/Fti]∨E
Q×
×[X(ti+1∧t, b)−X(ti∧t, b)/Fti]≤C. (10)
On the other hand, since the familyYA= (−Xa,a∈A) satisfies the same continuity property as the familyXA, using for the familyYAan inequality analogous to (9), we have, for each fixedaandb∈A,
EQX
i
[−X(ti+1∧t, a) +X(ti∧t, a)] = =−EQX
i
EQ[X(ti+1∧t, a)−X(ti∧t, a)/Fti]≤C, (11)
and
−EQX
i
EQ[X(ti+1∧t, b)−X(ti∧t, b)/Fti]≤C. (12)
Therefore it follows from (10), (11), (12) and from the equality |x−y|= 2 max(x, y)−x−y that
EQX
i
|EQ[X(ti+1∧t, a)−X(ti∧t, a)−(X(ti+1∧t, b)−X(ti∧t, b))/Fti]| ≤C.
Thus
sup t1<···<tn
EQX
i
|EQ[X(ti+1∧t, a)−X(ti∧t, a)− −(X(ti+1∧t, b)−X(ti∧t, b))/Fti]| ≤C
for each t ∈ R+, hence Xa−Xb is a quasimartingale with respect to the
measureQ, and according to Girsanov’s theorem the processXa−Xb will be a semimartingale relative to the measureP.
Theorem 2. Let the conditions of Theorem1be satisfied. Then for each b∈Athe family(Xa−Xb, a∈A)is a dominated family of semimartingales,
i.e., for each fixedb∈Athe family(Xa−Xb, a∈A)satisfies condition (B)
of Definition1.
Proof. According to Theorem 1 the process Xa−Xb is a semimartingale for each a, b ∈ A. Let us prove that for every fixed b ∈ A the family of semimartingales (Xa−Xb, a∈A) is dominated.
Evidently, for every u ∈ Ub0
d and b ∈ A the process (Jt(u, XA)−Xtb,
t≥0) is a semimartingale. One can show that the family of semimartingales (J(u, XA)−Xb, u∈Ub0
d) satisfies the condition U.T. (for each fixedb∈A), i.e., the set
Zt
0
Hsd(Js(u, XA)−Xsb), u∈Ubd0, H ∈ H
(13)
is stochastically bounded for everyt∈R+.
It follows from the continuity property of the functional J that the set (Jt(u, XA), u ∈Ub0
d) and hence the set (Jt(u, XA)−Jt(v, XA), u, v ∈Ubd0), is stochastically bounded for eacht∈R+. Therefore it is sufficient to show
that for eachH ∈ Hand u∈Ub0
d there exists a pairu1, u2∈Ubd0 such that t
Z
0
Hsd(Js(u, XA)−Xsb) =Jt(u1, XA)−Jt(u2, XA). (14)
Without loss of generality we can assume that the functionH ∈ Hhas the same intervals of constancy as the functionu.
For H =Pi≤n−1HiI]ti,ti+1] and u(ω, t, da) = P
i≤n−1ui(ω, da)I]ti,ti+1]
let us define
u1(ω, t, da) = X i≤n−1
(Hi+(ω)ui(ω, da) + (1−Hi+(ω))εbω,t(da))I]ti,ti+1],
u2(ω, t, da) = X i≤n−1
(Hi−(ω)ui(ω, da) + (1−Hi−(ω))εbω,t(da))I]ti,ti+1],
where H+ = max(H,0), H− =−min(H,0) and εb is the measure concen-trated at the pointb∈A for eachω, t.
It is easy to check that foru1, u2 defined above equality (14) is true and
hence the family of semimartingales (J(u, XA)−Xb, u∈Ub0
d) satisfies the condition U.T.
b(e.g., we writeYa instead ofYa,bandMa,λinstead ofMa,b,λbelow). Let ˜
Yta,λ=
X
s≤t ∆(Ya
s)I[|∆(Ya s)|>λ],
˜
Xtu,λ=
X
s≤t
∆(Js(u, YA))I
[|∆(Js(u,YA))|>λ],
and let
Yta−Y˜ a,λ
t =Mλ(t, a) +Bλ(t, a), Mλ(·, a)∈ Mloc, Bλ(·, a)∈ Aloc,
Jt(u, YA)−Y˜tu,λ=Mu,λ(t) +Bu,λ(t), Mu,λ∈ Mloc, Bu,λ∈ Aloc,
be the canonical decomposition of the special semimartingales Ya−Y˜a,λ andJ(u, YA)−Y˜u,λ, respectively.
SinceYa∈S (for anya∈A), one can write the processJ(u, Ya) in the form
Jt(u, YA) = X a∈A0
t
Z
0
I(us=a)d(Y
a s) for eachu∈U0
d and it is easy to see that for eachu∈U0 (the more so for eachu∈U0
d, but not for eachUbd0) ˜
Ytu,λ=
X
a∈A0
ntt0I(us=a)dY˜
a,λ s .
Therefore the uniqueness of the canonical decomposition of the special semi-martingales implies that for everyu∈U0
d
Mu,λ=J(u, MA), Bu,λ=J(u, BA), (15) where
Jt(u, MA) =X i
(Mλ(t
i+1∧t, uti)−M
λ(t
i∧t, uti)) =
= X a∈A0
t
Z
0
I[us=a]dM
λ,a
s ,∈ M2loc,
Jt(u, BA) =X i
(Bλ(ti+1∧t, uti)−B
λ(t
i∧t, uti)) = X
a∈A0
t
Z
0
I[us=a]dB
λ,a
and an easy calculation shows that the square characteristic of the martin-galeJ(u, MA) is equal to
hJ(u, MA)it= X a∈A0
t
Z
0
I[us=a]dhM
λ,ais.
Since the family of semimartingales (J(u, YA), u∈Ub0
d) satisfies the condi-tion U.T., it follows from Proposicondi-tion 1 that the family of random variables (hMu,λit,Var(Bu,λ)t,Var( ˜Yu,λ)t, u∈Ubd0) is stochastically bounded for each
t≥0. Therefore from equality (15) we have lim
N usup∈U0
d
P(hJ(u, MA)it≥N) = 0, (16) lim
N usup∈U0
d
P(Var(J(u, BA))t≥N) = 0, (17)
and
lim N usup∈U0
d
P(Var( ˜Yu,λ)t≥N) = 0. (18)
Let for eacht∈R+
KM
t = ess supu∈U0
dhJ(u, M
A)it,
KB
t = ess supu∈U0
d|Jt(u, B
A)|. Evidently, KM
s ≤ KtM, KsB ≤ KtB a.s. for each s ≤t. Let us prove that
P(KM
t <∞) =P(KtB <∞) = 1 for everyt∈R+.
It follows from (16) and Lemma 1 that there exists a measureQequivalent toP, with bounded density, such that (for eacht∈R+)
sup u∈U0
d
EQhJ(u, MA)it<∞. (19)
Suppose that
Q(ess supu∈U0
dhJ(u, M
A)it=∞)> α >0. (20) Then there exists some 0≤s≤t for which (a.s.)
EQ(ess supu∈U0
d(hJ(u, M
A)it− hJ(u, MA)is)/Fs) =∞. For each fixeds, t∈R+, s≤t,consider the family of random variables
This family satisfies theε-lattice property (withε= 0). Indeed, ifu1, u2∈
U0
d then we defineu3∈Ud0 by
u3=I
Ω×[0,s]v+ID×]s,t]u1+IDc×]s,t]u2,
where D={ω :Gs,t(u1)≥Gs,t(u2)} andv is an arbitrary element of U0
d, for which we have
Gs,t(u3)≥max(Gs,t(u1), Gs,t(u2)).
Therefore according to Lemma 16.11 of [9]
EQ(ess supu∈U0
d(hJ(u, M
A)it− hJ(u, MA)is)/Fs) = = ess supu∈U0
dE
Q(hJ(u, MA)it− hJ(u, MA)is/Fs). (21) Thus from (20) and (21) we have
ess supu∈U0
dE
Q(hJ(u, MA)it− hJ(u, MA)is/Fs) =∞ a.s.
which contradicts (19), since, according to the definition of ess sup (keeping the lattice property forGQin mind) there exists a sequence (un, n≥1)∈U0
d such that
EQ(hJ(un, MA)it− hJ(un, MA)is/Fs)→ ∞ a.s. The equalityP(KB
t <∞) = 1 is proved in a similar way. Let us prove now that
hMa, Mait≪KtM, t∈R+,
(VarBa)t≪KtB, t∈R+,
for everya∈A.
Without loss of generality we can assume that the random variablesKM t andKB
t are integrable for eacht∈R+ (otherwise, one can use a localizing
sequence of stopping times (τn, n≥1) with EKtM∧τn <∞for each n≥1.
Such a sequence exists, sinceP(KM
t +BtM <∞) = 1 for eacht∈R+, and
the processesKM, KB are predictable).
Similarly to the above, we can show that for each fixeds, t∈R+, s≤t,
the family of random variables
Gs,t(u) =E(hJ(u, MA)it− hJ(u, MA)is/Fs), u∈U0
d
Therefore using (21) we obtain
E(KM
t −KsM/Fs) = =E(ess supu∈U0
dhJ(u, M
A)it−ess sup u∈U0
dhJ(u, M
A)is/Fs)≥ ≥ess supu∈U0
dE(hJ(u, M
A)it/Fs)−ess sup u∈U0
dhJ(u, M
A)is= = ess supu∈U0
dE(hJ(u, M
A)it−ess sup u∈U0
dhJ(u, M
A)is/Fs)≥ ≥ess supu∈U0
dE(hJ(u, M
A)it− hJ(u, MA)is/Fs) = =E(ess supu∈U0
d(hJ(u, M
A)it− hJ(u, MA)is)/Fs)≥ ≥E(hMait− hMais/Fs),
and hence the process (KM
t − hMait, t≥0) is a submartingale. Since every predictable local martingale of finite variation is constant (see, e.g., [10]), we conclude that the process (KM
t − hMait, t ≥ 0) is increasing for each
a ∈ A. Evidently, this implies that the measure dhMait(ω) is absolutely continuous with respect to the measuredKM
t (ω) for almost allω∈Ω, i.e., hMai ≪KM for eacha∈A.
Indeed, suppose H is some bounded positive predictable process such thatER0tHsdKsM = 0. Then
E
t
Z
0
HsdhMais+E t
Z
0
Hsd(KsM − hMais) = 0
implies thatER0tHsdhMais= 0, sinceKM− hMaiis an increasing process. In a similar way one can prove that
(VarBa)t≪KtB, t∈R+,
for eacha∈A. Thus the increasing processKt=KtM+KtB dominates the characteristics of semimartingales (Ya, a∈A).
LetXA = (Xa, a∈A) be a family of semimartingales and let for each
u∈Ub0
¯
Jt(u, XA) = X a∈A0
t
Z
0
us(a)dXsa. Evidently,
¯
Jt(u, XA) = X a∈A0
t
Z
0
I(us=a)dX
a s
ifu∈U0, and
¯
for everyu∈Ub0
d.
Theorem 3. XA is a regular family of semimartingales if and only if
for each sequence (un, n ≥ 1)∈Ub0 uniformly converging to some u∈ Ub,
the corresponding integral sums Jbt(un, XA) are fundamental in probability
uniformly on every compact.
Proof. The necessity part of the theorem is proved in [3].
Sufficiency. Let for each sequence (un, n≥1)∈Ub0converging uniformly
to someu∈Ub
sup s≤t
|Jbt(un, XA)−Jb(um, XA)| →0 (22) asn→ ∞, m→ ∞,for everyt≥0. We must show that the family of semi-martingalesXA is regular. Since ¯J(u, XA) =J(u, XA) for everyu∈Ub0
d, it follows from Theorem 2 thatXAis a dominated family of semimartingales, i.e., condition (B) of Definition 1 is satisfied.
Let us show that the familyXA satisfies conditions (C), (D).
First we prove that there exists aµK-a.e. continuous modification of the Radon–Nicod´ym derivative
ϕ(a, a′) =dhMa−Ma′i/dK.
LetAm
0 = (am1 , am2, . . . , aim(m)) be a m1-net of the compactA. Suppose that the functionϕ(a, a′) is separable (or we can choose such a version).
The function
ϕ∗(a, a′) = lim m i:r(asupm
i ,a)≤
1
m
sup j:r(am
j ,a ′)≤1
m
ϕ(ami , amj )
is upper continuous a.e. with respect to the measureµK and it is easy to see that the µK-a.e. continuous modification of the derivative of ϕ(a, a′)
exists if and only if
ϕ∗t(ut, ut) = 0, 0≤t <∞,
µK-a.e. for allu∈U. Let there existu∈U such that
µK[(ω, t) :ϕ∗(u, u)>0]>0.
We can construct two sequences (um, m≥ 1),(vm, m ≥1) ∈ U0 (i.e., um and vm are taking a finite number of values for each m), such that um→
u, vm→uuniformly and
Put um
t (ω) =amim(ω,t), v
m
t (ω) =amjm(ω,t), where the indices a
m im anda
m jm
are selected so that
r(amim(ω,t), ut(ω))≤
1
m, r(a
m
jm(ω,t), ut(ω))≤
1
m
and
ϕ(amim(ω,t), amjm(ω,t))≥ sup i:r(am
i ,ut(ω))≤m1
sup j:r(am
j ,ut(ω))≤m1
ϕ(ami , amj )− 1
m.
Since for eachu, v∈U0
hJ¯(u, MA)−J¯(v, MA)it= t
Z
0
ϕs(us, vs)dKs
we have
lim inf m Eh
¯
J(um, MA)−J¯(vm, MA)it= lim inf m E
t
Z
0
ϕs(ums, vms )dKs≥
≥E
t
Z
0
ϕ∗s(us, us)dKs>0 (24)
for somet≥0. Since the family of semimartingales ¯J(u, XA), uUb0) satisfies
the condition U.T. (the proof is similar to the corresponding assertion of Theorem 2) and
sup s≤t
|J¯s(un, XA)−J¯s(um, XA)| →0, m, n→ ∞, (25) it follows from Proposition 2 that
sup s≤t
|J¯s(un, MA)−J¯s(um, MA)| →0, m, n→ ∞, (26) sup
s≤t
|J¯s(un, BA)−J¯s(um, BA)| →0, m, n→ ∞, (27) where
¯
J(u, MA) = X a∈A0
t
Z
0
I(us=a)dM
a,λ
s ,J¯(u, BA) =
X
a∈A0
t
Z
0
I(us=a)dB
a,λ s
coincide with canonical decomposition terms of the special semimartingale ¯
J(u, XA)−P∆ ¯J(u, XA)I
Obviously, (24) implies that the continuity condition (26) for J(u, MA) with respect toudoes not hold and thereforeϕ(a, a′) is continuous in (a, a′)
with respect to the measureµK.
The existence of a µK-a.e. continuous modification of the derivative
gt(a) =dB(t, a)/dKtis proved analogously.
Let us prove now that the familyXA satisfies condition (D). Evidently, for eachu∈U0
hJ¯(u, MA)it= t
Z
0
ψs(us, us)dKs
and it is easy to see that theµK-a.e. continuity ofϕ(a, a′) implies theµK -a.e. continuity of the functionψ(a, b) along the diagonal (i.e., there exists aµK-a.e. continuous modification of the functionψ(a, a)). Therefore
t
Z
0
sup a∈A
ψs(a, a)dKs= ess supu∈U0hJ(u, MA)it<∞ a.s.
for eacht∈R+. The inequality
t
Z
0
sup a∈A
gs(a)dKs<∞
is proved in a similar way. Since
∆Jt(u, XA) = X a∈A0
I(us=a)(X(t, a)−X(t−, a)) =X(t, u(t))−X(t−, u(t)),
relation (18) can be rewritten as lim
n→∞usup∈U0 P{X
s≤t
|X(t, u(t))−X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ)> n}= 0,
which implies that ess supu∈U0
X
s≤t
|X(t, u(t))−X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ)<∞
a.s. for eacht∈R+.
Since the continuity property (22) of the functional ¯J implies that a function ∆Xa
each [0, t] andA is a compact subset of some metric space we obtain that
X
s≤t sup
a∈A∆|X(s, a)|I(supa∈A∆|X(s,a)|>λ)= = ess supu∈U
X
s≤t
|X(t, u(t))−X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ)=
= ess supu∈U0 X
s≤t
|X(t, u(t))−X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ)<∞
a.s. for eacht≥0, and hence condition (D) is fulfilled.
Remark . A usual stochastic integral, evidently, corresponds to the case
X(t, a) =aX(t), a∈R, whereXis a semimartingale. It is easy to see that if
uis a locally bounded predictable process then the usual stochastic integral
Rt
0usdXssatisfies relations (3), (4), and hence the stochastic integralu·M
coincides (up to an evanescent set) with the stochastic line integral with respect to the family of martingales (aM, a∈R) along the curveu.
Conversely, let X = (Xt, t ≥ 0) be an adapted continuous process and consider the family XA = (aX, a ∈ R). For an elementary predictable processu∈ H
Jt(u, XA) =X i
ui[X(ti+1∧t)−X(ti∧t)]
and it follows from Theorem 1 that the continuity of the functionalJ(u) with respect to the class{u∈U0
d : |u| ≤1}implies that the processX=X1−X0 is a semimartingale (since the processX0= 0 is a semimartingale).
Acknowledgement
Careful reading of the manuscript by the referee and by Prof. T. Sher-vashidze is highly appreciated.
This research was supported by International Science Foundation Grant No. MXI000.
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(Received 15.03.1995) Author’s address:
