GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 525-542
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 process X = (Xt , t ≥ 0) is a
semimartingale if and only if for every sequence of elementary predictable
processes (H n , n ≥ 1), which converges uniformly to 0, the corresponding
integral sums
X
Jt (H n ) =
Hin (Xt∧ti+1 − Xt∧ti )
i≤n−1
converge to 0 in probability for each t ≥ 0.
The aim of this paper is to give an analogous characterization for a regular
family of semimartingales X A = (X a , 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 filtration F = (Ft , t ≥ 0), satisfying the usual conditions, a family X A = ((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.
1991 Mathematics Subject Classification. 60G44,60H05.
Key words and phrases. Semimartingale, stochastic line integral.
525
c 1996 Plenum Publishing Corporation
1072-947X/96/1100-0525$09.50/0 
R. CHITASHVILI† AND M. MANIA
526
For some λ > 0 let
X̃ta,λ =
X
s≤t
∆Xsa I(|∆Xsa |>λ) .
If X a is a semimartingale then the process X a − X̃ a,λ is a special semimartingale with the canonical decomposition
X a − X̃ a,λ = M a,λ + B a,λ ,
where B a,λ is a predictable process of finite variation and M a,λ is a local
martingale. Moreover, |∆B a,λ | ≤ λ, |∆M a,λ | ≤ 2λ and hence M a,λ ∈ M2loc
(see, e.g., [2]).
Definition 1. We say that a family X A = (X a , a ∈ A) is a regular
family of semimartingales if it satisfies the following conditions:
(A) the process X a = (X(t, a), t ≥ 0) is a semimartingale for every a ∈ A;
(B) there exists λ > 0, a predictable, increasing process K = (Kt , t ≥ 0)
which dominates the characteristics of semimartingales (X a − X̃ a,λ , a ∈ A),
i.e.,
hM a,λ , M a,λ i œ K, Var(B a,λ ) œ K
on every [0, t], for each a ∈ A;
(C) the Radon–Nicodým derivative ϕ(a, b) = dhM a,λ − M b,λ i/dK (respectively, g(a) = dB a,λ /dK) is a continuous function of (a, b) (respectively,
of a) for almost every couple (ω, t) with respect to the measure µK , where
µK is the Doleans measure of K; a function ∆Xta is continuous in probability uniformly on every compact;
(D) for each t ∈ R+ a.s.
Zt
0
sup ψs (a, a)dKs +
a∈A
+
X
s≤t
Zt
0
sup |gs (a)|dKs +
a∈A
sup |∆X a |I(supa∈A |∆X a |>λ) < ∞.
a∈A
2
denote by H(a, m), f (a, m) and ψ(a, b) the derivatives
For m ∈ Mloc
dhM a,λ , mi
,
dK
dhM a,λ , mi
,
dhm, mi
dhM a,λ , M b,λ i
,
dK
respectively. It was proved in [3] that condition (C) implies the existence of
µK -a.e.(µhmi -a.e., respectively) continuous in a modification of the function
H(a, m) (respectively, f (a, m)) and such a version will be considered.
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 527
b of predictable processes taking
Let us introduce the classes U and U
values in A and in Q(A), respectively, where Q(A) is the space of all probabilities on A. Assume that Q(A) is provided with the Levy–Prokhorov
metric.
Denote by U 0 the class consisting of elements of U taking values in some
b 0 the class of elements of U
b taking values in
finite subset A0 of A and by U
the set of probability measures concentrated in some finite subset of A, i.e.,
for which there exists a finite set A0 ⊂ A such that ut (ω, A0 ) = 1 for each
(ω, t).
bd ) be the class of piecewise constant functions
Let Ud (respectively, U
b ) such that for some finite
consisting of elements u ∈ U (respectively, U
sequence of (deterministic) moments t1 < t2 < ... < tN , ut = uti for ti <
t ≤ ti+1 and uti is a Fti -measurable random variable taking values in A
(respectively, in Q(A)).
b0 = U
b0 ∩ U
bd .
Let Ud0 = U 0 ∩ Ud and U
d
b by the same symbols,
It is convenient to denote the elements of U and U
since every element u ∈ U can be considered as a degenerate element of
b . So we shall use the shortened notation: u = (ut (ω), t ≥ 0) for u ∈ U ,
U
b , while
u = u(C) = (ut (ω), t ≥ 0) = (ut (ω, C), t ≥ 0), C ∈ A for u ∈ U
the symbol H(u, m) for u ∈ U will be used instead of the exact notation
(Ht (ω, ut , m), t ≥ 0); the symbols f (u, m), ϕ(u, v), g(u) are understood in a
b , we sometimes use the expressions g(s, us ), Hs (us , m)
similar way.R For u ∈ U
R
instead of A gs (a)us (da), A Hs (a, m)us (da), respectively.
If the family of semimartingales X A satisfies conditions (A), (B), (C),
b ) one can define a
(D) then for each u ∈ U (respectively, for each u ∈ U
stochastic line integral (respectively, a generalized stochastic line integral)
with respect to the family of semimartingales (X a , a ∈ A) along the curve
u = (ut , t ≥ 0) as
Zt
X(ds, us ) =
’
resp.,
M (ds, us ) +
0
0
Zt
Zt
Zt
gs (us )dKs +
X
s≤t
0
(X(s, u(s)) −
− X(s−, u(s))I(|X(s,u(s))−X(s−,u(s))|>λ) , t ≥ 0
X(ds, us ) =
Zt
M (ds, us ) +
0
0
+
XZ
s≤t A
Zt
Z
(1)
gs (a)us (da)dKs +
0 A
∆Xsa I(|∆Xsa |>λ) us (da),
“
t≥0 ,
(2)
Rt
where ( 0 M (ds, us ), t ≥ 0) (see [3],[4]) is a unique element of the stable
space of martingales L2 (M A ) generated by the family M A = (M a,λ , a ∈ A)
R. CHITASHVILI† AND M. MANIA
528
such that
D Z·
M (ds, us ), m
0
’
resp.,
D Z·
E
M (ds, us ), m
0
E
t
=
Zt
=
Zt Z
Hs (us , m)dKs
(3)
0
t
Hs (a, m)us (da)dKs
0 A
“
(4)
for every m ∈ L2 (M A ) (we recall that Hs (a, m) = dhM a,λ , mis /dKs ).
The stochastic line integral defined above does not depend on the choice
of λ, on the choice of a dominating process K (if for them conditions (A),
(B), (C), (D) are fulfilled) and posseses the following properties:
b 0)
(1) for each u ∈ U 0 (respectively, for each u ∈ U
Zt
X(ds, us ) =
t
X Z
I(us =a) X(ds, a)
a∈A0 0
0
t
’
“
Zt
X Z
resp.,
X(ds, us ) =
us (a)X(ds, a)
a∈A0 0
0
bd )
and for every u ∈ Ud (respectively, for every u ∈ U
Zt
X(ds, us ) =
X
i
0
[X(ti+1 ∧ t, uti ) − X(ti ∧ t, uti )],
’
“
Zt
XZ
[X(ti+1 ∧ t, a) − X(ti ∧ t, a)]uti (da) ;
resp.,
X(ds, us ) =
i
0
A
(2) for ut = ut1 I(t<τ ) + u2t I(t≥τ )
Zt
0
X(ds, us ) =
Zt∧τ
0
X(ds, u1s )
+
Zt
t∧τ
X(ds, us2 );
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 529
b)
(3) for each u ∈ U (respectively, for each u ∈ U
∆
Zt
resp., ∆
Zt
0
’
(X(ds, u(s)) = X(t, u(t)) − X(t−, u(t))
(X(ds, u(s)) =
0
Z
“
(X(t, a) − X(t−, a))ut (da) ;
A
b)
(4) for each u ∈ U (respectively, for each u ∈ U
DZ
M (ds, u(s)),
Z
M (ds, u(s))
E
t
=
Zt
ψs (us , us )dKs
0
’
“
Z
Zt Z
DZ
E
resp.,
M (ds, u(s)), M (ds, u(s)) =
ψs (a, a)us (da)dKs ;
t
0 A
n
K
(5) if u → u µ -a.e. (in the sense of weak convergence of values in the
b ), then for each t ∈ R+
space of probabilities Q(A) if un ∈ U
ΠZt
Œ
Zt
Œ
Œ
n
Œ
Œ
sup Œ X(ds, us ) − X(ds, us )Œ → 0, n → ∞
s≤t
0
0
in probability.
The stochastic line integral was introduced by Gikhman and Skorokhod in
[5] for continuous process K and for the derivative dhM a −M b i/dK = ϕ(a, b)
which is continuous with respect to a, b uniformly in (ω, t).
Let H be a space of predictable processes H bounded by unity of the
form
X
H=
Hi I]ti ,ti+1 ] ,
i≤n−1
where t1 < t2 < · · · < tn and Hi ∈ Fti .
Definition 2 ([6]). A family of processes ((X a , t ≥ 0), a ∈ A) satisfies
the U.T. (uniform tightness) condition if for each t > 0 the set
’ Zt
0
Hs dXsa , H ∈ H, a ∈ A
“
is stochastically bounded.
We shall use the following statements proved by Mémin and Slomiński
[7].
R. CHITASHVILI† AND M. MANIA
530
Proposition 1 ([7]). A family of semimartingales ((Xta , 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 family Var(X̃ a,λ )t , a ∈ A) is stochastically bounded;
2. the family ([M a,λ , M a,λ ]t , a ∈ A) is stochastically bounded;
3. the family Var(B a,λ )t , a ∈ A) is stochastically bounded.
Remark . This proposition was proved in [7] for A = {1, 2, . . . }. For an
arbitrary set A this statement is proved without any changess. Note that
since M a,λ is a local martingale with bounded jumps, the Lenglart inequality implies that condition 2 is equivalent to the stochastic boundedness of
the family (hM a,λ , M a,λ it , a ∈ A) (see [8]).
Proposition 2 ([7]). Let ((Xtn , t ≥ 0), n ≥ 1) be a sequence of semimartingales satisfying the condition U.T. and let X n converge to some process X in probability uniformly on every compact. Then the limiting process
X is also a semimartingale and, moreover, for each t > 0
sup |Msn,λ − Msλ | → 0, n → ∞,
(5)
sup |Bsn,λ − Bsλ |, n → ∞
(6)
s≤t
s≤t
in probability, where M λ (respectively, B λ ) is a martingale
Ppart (respectively,
the part of finite variation) of the semimartingale X − ∆XI(∆X>λ) .
Let us consider now a family X A = (X a , a ∈ A) of cadlag adapted
b 0 , associated
processes and for each piecewise constant function u from U
d
with a subdivision 0 = t0 < t1 < · · · < tn < ∞, define a new stochastic
process
XZ
Jt (u, X A ) =
[X(ti+1 ∧ t, a) − X(ti ∧ t, a)]uti (da)
(7)
i
A
which takes the form
Jt (u, X A ) =
X
i
[X(ti+1 ∧ t, uti ) − X(ti ∧ t, uti )]
(8)
for functions u from the class Ud .
It is easy to see that if X A is a family of semimartingales, the process
(Jt (u, X A ), t ≥ 0) defined by expressions (8),(7) coincides with the stochastic line integral and with the generalized stochastic line integral, respectively.
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 531
Theorem 1. Let X A = (X a , a ∈ A) be a family of adapted cadlag prob 0 converging uniformly to some
cesses. If for each sequence (un , n ≥ 1) ∈ U
d
b (in the sense of weak convergence of values in the space of probabiliu∈U
ties Q(A)) the corresponding integral sums Jt (un , X A ) are fundamental in
probability for every t, then the difference X a − X b is a semimartingale for
each pair a ∈ A, b ∈ A.
b 0 converging uniformly to
Proof. Let, for each sequence (un , n ≥ 1) ∈ U
d
b,
some u ∈ U
|Jt (un , X A ) − Jt (um , X A )| → 0, n → ∞, m → ∞,
in probability (for each t ∈ R+ ). We must show that X a − X b ∈ 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 functional J and 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 X a − X b be a quasimartingale
relative to the measure Q.
The possibility of constructing such a measure Q is 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 Q equivalent to P , with bounded density, such that
sup E Q γ ≤ C < ∞,
γ∈G
where L0 (P ) is the space (of classes of equivalence) of finite random variables provided with the topology of convergence in probability.
b 0 under the mapping Jt . Evidently,
Let us take as G the image of the set U
d
G is a convex (as an image of a convex set) and bounded subset of L0 (P ).
b 0 ) and using expression (8) for
Therefore, applying Lemma 1 for G = Jt (U
d
A
Jt (u, X ), we obtain
X
EQ
(9)
[X(ti+1 ∧ t, uti ) − X(ti ∧ t, uti )] ≤ C
i
Ud0 .
for each u ∈
Let u
e be a function from U 0 , associated with the subdivision 0 = t0 <
t1 < · · · < tn < ∞, such that

Q

a if E [X(ti+1 ∧ t, a) − X(ti ∧ t, a)/Fti ] ≥
u
ei =
≥ E Q [X(ti+1 ∧ t, b) − X(ti ∧ t, b)/Fti ] .


b otherwise
R. CHITASHVILI† AND M. MANIA
532
From (9) we have
u, X A ) = E Q
E Q Jt (e
=E
Q
X
i
X
i
E Q [X(ti+1 ∧ t, u
ei ) − X(ti ∧ t, u
ei )/Fti ] =
[I(E Q [X(ti+1 ,a)−X(ti ,a)/Fti ]≥E Q [X(ti+1 ,b)−X(ti ,b)/Fti ]) E Q ×
×[X(ti+1 ∧ t, a) − X(ti ∧ t, a)/Fti ] +
+I(E Q [X(ti+1 ,a)−X(ti ,a)/Fti ]<E Q [X(ti+1 ,b)−X(ti ,b)/Fti ]) E Q ×
×[X(ti+1 ∧ t, b) − X(ti ∧ t, b)/Fti ] =
X
= EQ
E Q [X(ti+1 ∧ t, a) − X(ti ∧ t, a)/Fti ] ∨ E Q ×
i
×[X(ti+1 ∧ t, b) − X(ti ∧ t, b)/Fti ] ≤ C.
(10)
On the other hand, since the family Y A = (−X a , a ∈ A) satisfies the same
continuity property as the family X A , using for the family Y A an inequality
analogous to (9), we have, for each fixed a and b ∈ A,
X
EQ
[−X(ti+1 ∧ t, a) + X(ti ∧ t, a)] =
= −E
Q
X
i
i
E Q [X(ti+1 ∧ t, a) − X(ti ∧ t, a)/Fti ] ≤ C,
(11)
and
−E Q
X
i
E Q [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
X
EQ
|E Q [X(ti+1 ∧t, a)−X(ti ∧t, a)−(X(ti+1 ∧t, b)−X(ti ∧t, b))/Fti ]| ≤ C.
i
Thus
sup
t1 <···<tn
EQ
X
i
|E Q [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 X a − X b is a quasimartingale with respect to the
measure Q, and according to Girsanov’s theorem the process X a − X b will
be a semimartingale relative to the measure P .
The following theorem gives a characterization of a dominated family of
semimartingales:
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 533
Theorem 2. Let the conditions of Theorem 1 be satisfied. Then for each
b ∈ A the family (X a −X b , a ∈ A) is a dominated family of semimartingales,
i.e., for each fixed b ∈ A the family (X a − X b , a ∈ A) satisfies condition (B)
of Definition 1.
Proof. According to Theorem 1 the process X a − X b is a semimartingale
for each a, b ∈ A. Let us prove that for every fixed b ∈ A the family of
semimartingales (X a − X b , a ∈ A) is dominated.
b 0 and b ∈ A the process (Jt (u, X A ) − Xtb ,
Evidently, for every u ∈ U
d
t ≥ 0) is a semimartingale. One can show that the family of semimartingales
b 0 ) satisfies the condition U.T. (for each fixed b ∈ A),
(J(u, X A ) − X b , u ∈ U
d
i.e., the set
’ Zt
0
A
Hs d(Js (u, X ) −
Xsb ), u
b 0, H ∈ H
∈U
d
“
(13)
is stochastically bounded for every t ∈ R+ .
It follows from the continuity property of the functional J that the set
b 0 ),
b 0 ) and hence the set (Jt (u, X A ) − Jt (v, X A ), u, v ∈ U
(Jt (u, X A ), u ∈ U
d
d
is stochastically bounded for each t ∈ R+ . Therefore it is sufficient to show
b 0 there exists a pair u1 , u2 ∈ U
b 0 such that
that for each H ∈ H and u ∈ U
d
d
Zt
0
Hs d(Js (u, X A ) − Xsb ) = Jt (u1 , X A ) − Jt (u2 , X A ).
(14)
Without loss of generality we can assume that the function H ∈ H has the
same intervals
Pof constancy as the function u.
P
For H = i≤n−1 Hi I]ti ,ti+1 ] and u(ω, t, da) = i≤n−1 ui (ω, da)I]ti ,ti+1 ]
let us define
X
(Hi+ (ω)ui (ω, da) + (1 − Hi+ (ω))εbω,t (da))I]ti ,ti+1 ] ,
u1 (ω, t, da) =
i≤n−1
2
u (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 concentrated at the point b ∈ A for each ω, t.
It is easy to check that for u1 , u2 defined above equality (14) is true and
b 0 ) satisfies the
hence the family of semimartingales (J(u, X A ) − X b , u ∈ U
d
condition U.T.
Denote by Y a (respectively, by J(u, Y A )) the difference X a −X b (respectively, J(u, X A ) − X b ). Since b is fixed, we omit (for convenience) the index
R. CHITASHVILI† AND M. MANIA
534
b (e.g., we write Y a instead of Y a,b and M a,λ instead of M a,b,λ below). Let
X
Ỹta,λ =
∆(Ysa )I[|∆(Ysa )|>λ] ,
s≤t
X̃tu,λ
=
X
∆(Js (u, Y A ))I[|∆(Js (u,Y A ))|>λ] ,
s≤t
and let
Yta − Ỹta,λ = M λ (t, a) + B λ (t, a), M λ (·, a) ∈ Mloc , B λ (·, a) ∈ Aloc ,
Jt (u, Y A ) − Ỹtu,λ = M u,λ (t) + B u,λ (t), M u,λ ∈ Mloc , B u,λ ∈ Aloc ,
be the canonical decomposition of the special semimartingales Y a − Ỹ a,λ
and J(u, Y A ) − Ỹ u,λ , respectively.
Since Y a ∈ S (for any a ∈ A), one can write the process J(u, Y a ) in the
form
t
X Z
A
Jt (u, Y ) =
I(us =a) d(Ysa )
a∈A0 0
for each u ∈ Ud0 and it is easy to see that for each u ∈ U 0 (the more so for
b 0)
each u ∈ Ud0 , but not for each U
d
X
Ỹtu,λ =
ntt0 I(us =a) dỸsa,λ .
a∈A0
Therefore the uniqueness of the canonical decomposition of the special semimartingales implies that for every u ∈ Ud0
M u,λ = J(u, M A ), B u,λ = J(u, B A ),
where
Jt (u, M A ) =
X
i
=
(M λ (ti+1 ∧ t, uti ) − M λ (ti ∧ t, uti )) =
t
X Z
a∈A0 0
A
Jt (u, B ) =
X
i
(B λ (ti+1 ∧ t, uti ) − B λ (ti ∧ t, uti )) =
t
X Z
a∈A0 0
I[us =a] dMsλ,a , ∈ M2loc ,
I[us =a] dBsλ,a , ∈ Aloc ,
(15)
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 535
and an easy calculation shows that the square characteristic of the martingale J(u, M A ) is equal to
A
hJ(u, M )it =
t
X Z
a∈A0 0
I[us =a] dhM λ,a is .
b 0 ) satisfies the condiSince the family of semimartingales (J(u, Y A ), u ∈ U
d
tion U.T., it follows from Proposition 1 that the family of random variables
b 0 ) is stochastically bounded for each
(hM u,λ it , Var(B u,λ )t , Var(Ỹ u,λ )t , u ∈ U
d
t ≥ 0. Therefore from equality (15) we have
lim sup P (hJ(u, M A )it ≥ N ) = 0,
(16)
lim sup P (Var(J(u, B A ))t ≥ N ) = 0,
(17)
lim sup P (Var(Ỹ u,λ )t ≥ N ) = 0.
(18)
N u∈U 0
d
N u∈U 0
d
and
N u∈U 0
d
Let for each t ∈ R+
KtM = ess supu∈U 0 hJ(u, M A )it ,
d
KtB = ess supu∈U 0 |Jt (u, B A )|.
d
KsM
≤ KtM , KsB ≤ KtB a.s. for each s
P (KtB < ∞) = 1 for every t ∈ R+ .
Evidently,
≤ t. Let us prove that
P (KtM < ∞) =
It follows from (16) and Lemma 1 that there exists a measure Q equivalent
to P , with bounded density, such that (for each t ∈ R+ )
sup E Q hJ(u, M A )it < ∞.
(19)
Q(ess supu∈U 0 hJ(u, M A )it = ∞) > α > 0.
(20)
u∈Ud0
Suppose that
d
Then there exists some 0 ≤ s ≤ t for which (a.s.)
E Q (ess supu∈U 0 (hJ(u, M A )it − hJ(u, M A )is )/Fs ) = ∞.
d
For each fixed s, t ∈ R+ , s ≤ t, consider the family of random variables
Q
A
A
0
GQ
s,t (u) = E (hJ(u, M )it − hJ(u, M )is /Fs ), u ∈ U .
R. CHITASHVILI† AND M. MANIA
536
This family satisfies the ε-lattice property (with ε = 0). Indeed, if u1 , u2 ∈
Ud0 then we define u3 ∈ Ud0 by
u3 = IΩ×[0,s] v + ID×]s,t] u1 + IDc ×]s,t] u2 ,
where D = {ω : Gs,t (u1 ) ≥ Gs,t (u2 )} and v is an arbitrary element of Ud0 ,
for which we have
Gs,t (u3 ) ≥ max(Gs,t (u1 ), Gs,t (u2 )).
Therefore according to Lemma 16.11 of [9]
E Q (ess supu∈U 0 (hJ(u, M A )it − hJ(u, M A )is )/Fs ) =
d
= ess supu∈U 0 E Q (hJ(u, M A )it − hJ(u, M A )is /Fs ).
d
(21)
Thus from (20) and (21) we have
ess supu∈U 0 E Q (hJ(u, M A )it − hJ(u, M A )is /Fs ) = ∞ a.s.
d
which contradicts (19), since, according to the definition of ess sup (keeping
the lattice property for GQ in mind) there exists a sequence (un , n ≥ 1) ∈ Ud0
such that
E Q (hJ(un , M A )it − hJ(un , M A )is /Fs ) → ∞ a.s.
The equality P (KtB < ∞) = 1 is proved in a similar way.
Let us prove now that
hM a , M a it œ KtM , t ∈ R+ ,
(Var B a )t œ KtB , t ∈ R+ ,
for every a ∈ A.
Without loss of generality we can assume that the random variables KtM
and KtB are integrable for each t ∈ R+ (otherwise, one can use a localizing
M
sequence of stopping times (τn , n ≥ 1) with EKt∧τ
< ∞ for each n ≥ 1.
n
M
M
Such a sequence exists, since P (Kt + Bt < ∞) = 1 for each t ∈ R+ , and
the processes K M , K B are predictable).
Similarly to the above, we can show that for each fixed s, t ∈ R+ , s ≤ t,
the family of random variables
Gs,t (u) = E(hJ(u, M A )it − hJ(u, M A )is /Fs ), u ∈ Ud0
also satisfies the ε-lattice property (with ε = 0) and equality (21) is also
valid, if we replace E Q by the mathematical expectation E with respect to
the basic measure P .
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 537
Therefore using (21) we obtain
E(KtM − KsM /Fs ) =
= E(ess supu∈U 0 hJ(u, M A )it − ess supu∈U 0 hJ(u, M A )is /Fs ) ≥
d
d
≥ ess supu∈U 0 E(hJ(u, M A )it /Fs ) − ess supu∈U 0 hJ(u, M A )is =
d
d
A
= ess supu∈U 0 E(hJ(u, M )it − ess supu∈U 0 hJ(u, M A )is /Fs ) ≥
d
d
≥ ess supu∈U 0 E(hJ(u, M A )it − hJ(u, M A )is /Fs ) =
d
= E(ess supu∈U 0 (hJ(u, M A )it − hJ(u, M A )is )/Fs ) ≥
d
≥ E(hM a it − hM a is /Fs ),
and hence the process (KtM − hM a it , t ≥ 0) is a submartingale. Since every
predictable local martingale of finite variation is constant (see, e.g., [10]),
we conclude that the process (KtM − hM a it , t ≥ 0) is increasing for each
a ∈ A. Evidently, this implies that the measure dhM a it (ω) is absolutely
continuous with respect to the measure dKtM (ω) for almost all ω ∈ Ω, i.e.,
hM a i œ K M for each a ∈ A.
Indeed,
R t suppose H is some bounded positive predictable process such
that E 0 Hs dKsM = 0. Then
E
Zt
0
Rt
a
Hs dhM is + E
Zt
0
Hs d(KsM − hM a is ) = 0
implies that E 0 Hs dhM a is = 0, since K M − hM a i is an increasing process.
In a similar way one can prove that
(Var B a )t œ KtB , t ∈ R+ ,
for each a ∈ A. Thus the increasing process Kt = KtM + KtB dominates the
characteristics of semimartingales (Y a , a ∈ A).
Let X A = (X a , a ∈ A) be a family of semimartingales and let for each
b0
u∈U
t
X Z
A
J¯t (u, X ) =
us (a)dXsa .
a∈A0 0
Evidently,
J¯t (u, X ) =
A
t
X Z
I(us =a) dXsa
a∈A0 0
0
if u ∈ U , and
¯ X A ) = J(u, X A )
J(u,
R. CHITASHVILI† AND M. MANIA
538
b 0.
for every u ∈ U
d
Theorem 3. X A is a regular family of semimartingales if and only if
b 0 uniformly converging to some u ∈ U
b,
for each sequence (un , n ≥ 1) ∈ U
n
A
the corresponding integral sums Jbt (u , X ) are fundamental in probability
uniformly on every compact.
Proof. The necessity part of the theorem is proved in [3].
b 0 converging uniformly
Sufficiency. Let for each sequence (un , n ≥ 1) ∈ U
b
to some u ∈ U
b m , X A )| → 0
sup |Jbt (un , X A ) − J(u
(22)
s≤t
as n → ∞, m → ∞, for every t ≥ 0. We must show that the family of semib 0 , it
¯ X A ) = J(u, X A ) for every u ∈ U
martingales X A is regular. Since J(u,
d
A
follows from Theorem 2 that X is a dominated family of semimartingales,
i.e., condition (B) of Definition 1 is satisfied.
Let us show that the family X A satisfies conditions (C), (D).
First we prove that there exists a µK -a.e. continuous modification of the
Radon–Nicodým derivative
0
ϕ(a, a0 ) = dhM a − M a i/dK.
1
m
m m
Let Am
0 = (a1 , a2 , . . . , ai(m) ) be a m -net of the compact A. Suppose that
the function ϕ(a, a0 ) is separable (or we can choose such a version).
The function
ϕ∗ (a, a0 ) = lim
m
sup
sup
1
1
i:r(aim ,a)≤ m
j:r(ajm ,a0 )≤ m
ϕ(aim , am
j )
is upper continuous a.e. with respect to the measure µK and it is easy to
see that the µK -a.e. continuous modification of the derivative of ϕ(a, a0 )
exists if and only if
ϕ∗t (ut , ut ) = 0, 0 ≤ t < ∞,
µK -a.e. for all u ∈ U . Let there exist u ∈ U such that
µK [(ω, t) : ϕ∗ (u, u) > 0] > 0.
We can construct two sequences (um , m ≥ 1), (v m , m ≥ 1) ∈ U 0 (i.e., um
and v m are taking a finite number of values for each m), such that um →
u, v m → u uniformly and
ϕ(um , v m ) → ϕ∗ (u, u), µK − a.e,
as m → ∞.
(23)
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 539
m
m
m
m
m
Put um
t (ω) = aim (ω,t) , vt (ω) = ajm (ω,t) , where the indices aim and ajm
are selected so that
r(am
im (ω,t) , ut (ω)) ≤
1
1
, r(am
jm (ω,t) , ut (ω)) ≤
m
m
and
m
ϕ(am
im (ω,t) , ajm (ω,t) ) ≥
sup
sup
1
i:r(aim ,ut (ω))≤ m
1
,ut (ω))≤ m
j:r(am
j
m
ϕ(am
i , aj ) −
1
.
m
Since for each u, v ∈ U 0
¯ M A )it =
¯ M A ) − J(v,
hJ(u,
Zt
ϕs (us , vs )dKs
0
we have
¯ m , M A )it = lim inf E
¯ m , M A ) − J(v
lim inf EhJ(u
m
m
≥E
Zt
Zt
0
m
ϕs (um
s , vs )dKs ≥
ϕ∗s (us , us )dKs > 0
(24)
0
b 0 ) satisfies
¯ X A ), uU
for some t ≥ 0. Since the family of semimartingales J(u,
the condition U.T. (the proof is similar to the corresponding assertion of
Theorem 2) and
sup |J¯s (un , X A ) − J¯s (um , X A )| → 0, m, n → ∞,
(25)
s≤t
it follows from Proposition 2 that
sup |J¯s (un , M A ) − J¯s (um , M A )| → 0, m, n → ∞,
(26)
sup |J¯s (un , B A ) − J¯s (um , B A )| → 0, m, n → ∞,
(27)
s≤t
s≤t
where
¯ M A) =
J(u,
t
X Z
a∈A0 0
¯ BA) =
I(us =a) dMsa,λ , J(u,
t
X Z
I(us =a) dBsa,λ
a∈A0 0
coincide with
canonical decomposition terms of the special semimartingale
0
¯ X A ) − P ∆J(u,
¯ X A )I(|J(u,X
J(u,
A )|>λ) for each u ∈ U .
¯
R. CHITASHVILI† AND M. MANIA
540
Obviously, (24) implies that the continuity condition (26) for J(u, M A )
with respect to u does not hold and therefore ϕ(a, a0 ) is continuous in (a, a0 )
with respect to the measure µK .
The existence of a µK -a.e. continuous modification of the derivative
gt (a) = dB(t, a)/dKt is proved analogously.
Let us prove now that the family X A satisfies condition (D). Evidently,
for each u ∈ U 0
Zt
A
¯
hJ(u, M )it = ψs (us , us )dKs
0
and it is easy to see that the µK -a.e. continuity of ϕ(a, a0 ) implies the µK a.e. continuity of the function ψ(a, b) along the diagonal (i.e., there exists
a µK -a.e. continuous modification of the function ψ(a, a)). Therefore
Zt
0
sup ψs (a, a)dKs = ess supu∈U 0 hJ(u, M A )it < ∞ a.s.
a∈A
for each t ∈ R+ . The inequality
Zt
0
sup gs (a)dKs < ∞
a∈A
is proved in a similar way.
Since
X
I(us =a) (X(t, a) − X(t−, a)) = X(t, u(t)) − X(t−, u(t)),
∆Jt (u, X A ) =
a∈A0
relation (18) can be rewritten as
X
lim sup P {
n→∞ u∈U 0
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
X
ess supu∈U 0
|X(t, u(t)) − X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ) < ∞
s≤t
a.s. for each t ∈ R+ .
Since the continuity property (22) of the functional J¯ implies that a
function ∆Xta is continuous with respect to a in probability uniformly on
CHARACTERIZATION OF SEMIMARTINGALES BY LINE INTEGRALS 541
each [0, t] and A is a compact subset of some metric space we obtain that
X
sup ∆|X(s, a)|I(supa∈A ∆|X(s,a)|>λ) =
s≤t
= ess supu∈U
X
s≤t
= ess supu∈U 0
X
s≤t
a∈A
|X(t, u(t)) − X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ) =
|X(t, u(t)) − X(t−, u(t))|I(|X(t,u(t))−X(t−,u(t))|>λ) < ∞
a.s. for each t ≥ 0, and hence condition (D) is fulfilled.
Remark . A usual stochastic integral, evidently, corresponds to the case
X(t, a) = aX(t), a ∈ R, where X is a semimartingale. It is easy to see that if
u
R tis a locally bounded predictable process then the usual stochastic integral
u dXs satisfies relations (3), (4), and hence the stochastic integral u · M
0 s
coincides (up to an evanescent set) with the stochastic line integral with
respect to the family of martingales (aM, a ∈ R) along the curve u.
Conversely, let X = (Xt , t ≥ 0) be an adapted continuous process and
consider the family X A = (aX, a ∈ R). For an elementary predictable
process u ∈ H
X
ui [X(ti+1 ∧ t) − X(ti ∧ t)]
Jt (u, X A ) =
i
and it follows from Theorem 1 that the continuity of the functional J(u) with
respect to the class {u ∈ Ud0 : |u| ≤ 1} implies that the process X = X 1 −X 0
is a semimartingale (since the process X 0 = 0 is a semimartingale).
Acknowledgement
Careful reading of the manuscript by the referee and by Prof. T. Shervashidze is highly appreciated.
This research was supported by International Science Foundation Grant
No. MXI000.
References
1. C. Dellacherie and P. A. Meyer, Probabilités et potentiel II. Hermann,
Paris, 1980.
2. J. Jacod, Calcul Stochastique et problèmes des martingales. Lecture
Notes in Math. 714, Springer, Berlin etc., 1979.
3. R. J. Chitashvili and M. G. Mania, On optimal controls in the problem
of locally absolutely continuous change of measure. Probabability theory
542
R. CHITASHVILI† AND M. MANIA
and mathematical statistics. Proceedings of the 4th international Vilnius
conference, vol. 1, 331–356, VNU Science Press, 1986.
4. R. J. Chitashvili, Martingale ideology in the theory of controlled
stochastic processes. Probability theory and mathematical statistics, 4th
USSR–Japan symposium, Proceedings, 73–92, Lecture Notes in Math. 1021,
Springer, Berlin, 1983.
5. I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations.
(Russian) Naukova Dumka, Kiev, 1982.
6. C. Stricker, Lois de sémimartingales et critères de compacité. Lecture
Notes in Math. 1123, Springer, Berlin etc., 1985.
7. J. Mémin and L. Slomiński, Condition UT et stabilité en loi des solutions d’équations differéntielles stochastiques. Séminaire de Probabilités,
XXV, 162–177, Lecture Notes in Math. 1485, Springer, Berlin, 1991.
8. R. Sh. Liptzer and A. N. Shiryaev, Martingale theory. (Russian)
Nauka, Moscow, 1986.
9. R. J. Elliott, Stochastic calculus and applications. Springer, Berlin,
1982.
10. C. Dellacherie, Capacités et processus stochastiques. Ergebnisse der
Math. 67, Springer, Berlin, 1972.
(Received 15.03.1995)
Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Alexidze St., Tbilisi 380093
Republic of Georgia