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 = (X_{t}, 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

J_{t}(H^{n}) = ∑_{i≤n−1} H_{i}^{n}(X_{t∧t_{i+1}} − X_{t∧t_{i}})

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 = (F_{t}, 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, M_{loc}, M_{loc}^{2}, A_{loc}, V classes of semimartingales, local martingales, locally square integrable martingales, processes of locally integrable variations and processes of finite variations (on every compact), respectively.

526 R. CHITASHVILI^{†} AND M. MANIA

525 For some λ > 0 let

1991 Mathematics Subject Classification. 60G44,60H05.

Key words and phrases. Semimartingale, stochastic line integral.

1072-947X/96/1100-0525$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

X̃_{t}^{a,λ} = ∑_{s≤t} ΔX_{s}^{a}I_{(|ΔX_{s}^{a}|>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, |ΔB^{a,λ}| ≤ λ, |ΔM^{a,λ}| ≤ 2λ and hence M^{a,λ} ∈ M_{loc}^{2} (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 = (K_{t}, t ≥ 0)

which dominates the characteristics of semimartingales (X^{a} −X̃^{a,λ}, a ∈ A), i.e.,

⟨M^{a,λ}, M^{a,λ}⟩ ≪ K, Var(B^{a,λ}) ≪ K

on every [0, t], for each a ∈ A;

(C) the Radon–Nicodým derivative φ(a, b) = d⟨M^{a,λ} − M^{b,λ}⟩/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 μ^{K}, where μ^{K} is the Doleans measure of K; a function ΔX_{t}^{a} is continuous in probability uniformly on every compact;

(D) for each t ∈ R_{+} a.s.

∫_{0}^{t} sup_{a∈A} ψ_{s}(a, a)dK_{s} + ∫_{0}^{t} sup_{a∈A} |g_{s}(a)|dK_{s} +

+ ∑_{s≤t} sup_{a∈A} |ΔX^{a}|I_{(sup_{a∈A} |ΔX^{a}|>λ)} < ∞.

For m ∈ M_{loc}^{2} denote by H(a, m), f(a, m) and ψ(a, b) the derivatives

\frac{d⟨M^{a,λ}, m⟩}{dK}, \frac{d⟨M^{a,λ}, m⟩}{d⟨m, m⟩}, \frac{d⟨M^{a,λ}, M^{b,λ}⟩}{dK},

respectively. It was proved in [3] that condition (C) implies the existence of μ^{K}-a.e.(μ^{⟨m⟩}-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

Let us introduce the classes U and Û of predictable processes taking 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 finite subset A_{0} of A and by Û^{0} the class of elements of Û taking values in the set of probability measures concentrated in some finite subset of A, i.e., for which there exists a finite set A_{0} ⊂ A such that u_{t}(ω, A_{0}) = 1 for each (ω, t).

Let U_{d} (respectively, Û_{d}) be the class of piecewise constant functions consisting of elements u ∈ U (respectively, Û) such that for some finite sequence of (deterministic) moments t_{1} < t_{2} < ... < t_{N}, u_{t} = u_{t_{i}} for t_{i} < t ≤ t_{i+1} and u_{t_{i}} is a F_{t_{i}}-measurable random variable taking values in A (respectively, in Q(A)).

Let U_{d}^{0} = U^{0} ∩ U_{d} and Û_{d}^{0} = Û^{0} ∩ Û_{d}.

It is convenient to denote the elements of U and Û by the same symbols, since every element u ∈ U can be considered as a degenerate element of Û. So we shall use the shortened notation: u = (u_{t}(ω), t ≥ 0) for u ∈ U, u = u(C) = (u_{t}(ω), t ≥ 0) = (u_{t}(ω, C), t ≥ 0), C ∈ A for u ∈ Û, while the symbol H(u, m) for u ∈ U will be used instead of the exact notation (H_{t}(ω, u_{t}, m), t ≥ 0); the symbols f(u, m), φ(u, v), g(u) are understood in a similar way. For u ∈ Û, we sometimes use the expressions g(s, u_{s}), H_{s}(u_{s}, m) instead of ∫_{A} g_{s}(a)u_{s}(da), ∫_{A} H_{s}(a, m)u_{s}(da), respectively.

If the family of semimartingales X^{A} satisfies conditions (A), (B), (C), (D) then for each u ∈ U (respectively, for each u ∈ Û) one can define a stochastic line integral (respectively, a generalized stochastic line integral) with respect to the family of semimartingales (X^{a}, a ∈ A) along the curve u = (u_{t}, t ≥ 0) as

∫_{0}^{t} X(ds, u_{s}) = ∫_{0}^{t} M(ds, u_{s}) + ∫_{0}^{t} g_{s}(u_{s})dK_{s} + ∑_{s≤t}(X(s, u(s)) −
− X(s−, u(s))I_{(|X(s,u(s))−X(s−,u(s))|>λ)}, t ≥ 0 (1)
(resp., ∫_{0}^{t} X(ds, u_{s}) = ∫_{0}^{t} M(ds, u_{s}) + ∫_{0}^{t} ∫_{A} g_{s}(a)u_{s}(da)dK_{s} +

+ ∑_{s≤t} ∫_{A} ΔX_{s}^{a}I_{(|ΔX_{s}^{a}|>λ)}u_{s}(da), t ≥ 0),  (2)

where (∫_{0}^{t} M(ds, u_{s}), t ≥ 0) (see [3],[4]) is a unique element of the stable space of martingales L^{2}(M^{A}) generated by the family M^{A} = (M^{a,λ}, a ∈ A) such that

⟨ ∫_{0}^{·} M(ds, u_{s}), m⟩_{t} = ∫_{0}^{t} H_{s}(u_{s}, m)dK_{s}  (3)

(resp., ⟨ ∫_{0}^{·} M(ds, u_{s}), m⟩_{t} = ∫_{0}^{t} ∫_{A} H_{s}(a, m)u_{s}(da)dK_{s})  (4)

for every m ∈ L^{2}(M^{A}) (we recall that H_{s}(a, m) = d⟨M^{a,λ}, m⟩_{s}/dK_{s}).

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:

(1) for each u ∈ U^{0} (respectively, for each u ∈ Û^{0})

∫_{0}^{t} X(ds, u_{s}) = ∑_{a∈A^{0}} ∫_{0}^{t} I_{(u_{s}=a)}X(ds, a)

(resp., ∫_{0}^{t} X(ds, u_{s}) = ∑_{a∈A^{0}} ∫_{0}^{t} u_{s}(a)X(ds, a))

and for every u ∈ U_{d} (respectively, for every u ∈ Û_{d})

∫_{0}^{t} X(ds, u_{s}) = ∑_{i}[X(t_{i+1} ∧ t, u_{t_{i}}) − X(t_{i} ∧ t, u_{t_{i}})],

(resp., ∫_{0}^{t} X(ds, u_{s}) = ∑_{i} ∫_{A} [X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)]u_{t_{i}}(da));

(2) for u_{t} = u_{t}^{1}I_{(t<τ)} + u_{t}^{2}I_{(t≥τ)}

∫_{0}^{t} X(ds, u_{s}) = ∫_{0}^{t∧τ}X(ds, u_{s}^{1}) + ∫_{t∧τ}^{t} X(ds, u_{s}^{2});

(3) for each u ∈ U (respectively, for each u ∈ Û)

Δ ∫_{0}^{t}(X(ds, u(s)) = X(t, u(t)) − X(t−, u(t))

(resp., Δ ∫_{0}^{t}(X(ds, u(s)) = ∫_{A} (X(t, a) − X(t−, a))u_{t}(da));

(4) for each u ∈ U (respectively, for each u ∈ Û)

⟨ ∫ M(ds, u(s)), ∫ M(ds, u(s))⟩_{t} = ∫_{0}^{t} ψ_{s}(u_{s}, u_{s})dK_{s}

(resp., ⟨ ∫ M(ds, u(s)), ∫ M(ds, u(s))⟩_{t} = ∫_{0}^{t} ∫_{A} ψ_{s}(a, a)u_{s}(da)dK_{s});

(5) if u^{n} → u μ^{K}-a.e. (in the sense of weak convergence of values in the space of probabilities Q(A) if u^{n} ∈ Û), then for each t ∈ R_{+}

sup_{s≤t} | ∫_{0}^{t} X(ds, u_{s}^{n}) − ∫_{0}^{t} X(ds, u_{s})| → 0, n → ∞

in probability.

The stochastic line integral was introduced by Gikhman and Skorokhod in [5] for continuous process K and for the derivative d⟨M^{a}−M^{b}⟩/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

H = ∑_{i≤n−1} H_{i}I_{]t_{i},t_{i+1}]},

where t_{1} < t_{2} < ⋯ < t_{n} and H_{i} ∈ F_{t_{i}}.

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

( ∫_{0}^{t} H_{s}dX_{s}^{a}, H ∈ H, a ∈ A)

is stochastically bounded.

We shall use the following statements proved by Mémin and Slomiński

Proposition 1 ([7]). A family of semimartingales ((X_{t}^{a}, 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 (⟨M^{a,λ}, M^{a,λ}⟩_{t}, a ∈ A) (see [8]).

Proposition 2 ([7]). Let ((X_{t}^{n}, 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_{s≤t} |M_{s}^{n,λ} − M_{s}^{λ}| → 0, n → ∞,  (5)
sup_{s≤t} |B_{s}^{n,λ} − B_{s}^{λ}|, n → ∞  (6)

in probability, where M^{λ} (respectively, B^{λ}) is a martingale part (respectively, the part of finite variation) of the semimartingale X − ∑ ΔXI_{(ΔX>λ)}.

Let us consider now a family X^{A} = (X^{a}, a ∈ A) of cadlag adapted processes and for each piecewise constant function u from Û_{d}^{0} , associated with a subdivision 0 = t_{0} < t_{1} < ⋯ < t_{n} < ∞, define a new stochastic process

J_{t}(u, X^{A}) = ∑_{i} ∫_{A} [X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)]u_{t_{i}}(da)  (7)

which takes the form

J_{t}(u, X^{A}) = ∑_{i}[X(t_{i+1} ∧ t, u_{t_{i}}) − X(t_{i} ∧ t, u_{t_{i}})]  (8)

for functions u from the class U_{d}.

It is easy to see that if X^{A} is a family of semimartingales, the process (J_{t}(u, X^{A}), t ≥ 0) defined by expressions (8),(7) coincides with the stochastic line integral and with the generalized stochastic line integral, respectively.

Theorem 1. Let X^{A} = (X^{a}, a ∈ A) be a family of adapted cadlag processes. If for each sequence (u^{n}, n ≥ 1) ∈ Û_{d}^{0} converging uniformly to some u ∈ Û (in the sense of weak convergence of values in the space of probabilities Q(A)) the corresponding integral sums J_{t}(u^{n}, 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.

Proof. Let, for each sequence (u^{n}, n ≥ 1) ∈ Û_{d}^{0} converging uniformly to some u ∈ Û,

|J_{t}(u^{n}, X^{A}) − J_{t}(u^{m}, 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 L^{0}(P). Then there exists a measure Q equivalent to P, with bounded density, such that

sup_{γ∈G} E^{Q}γ ≤ C < ∞,

where L^{0}(P) is the space (of classes of equivalence) of finite random variables provided with the topology of convergence in probability.

Let us take as G the image of the set Û_{d}^{0} under the mapping J_{t}. Evidently, G is a convex (as an image of a convex set) and bounded subset of L^{0}(P). Therefore, applying Lemma 1 for G = J_{t}(Û_{d}^{0}) and using expression (8) for J_{t}(u, X^{A}), we obtain

E^{Q} ∑_{i}[X(t_{i+1} ∧ t, u_{t_{i}}) − X(t_{i} ∧ t, u_{t_{i}})] ≤ C  (9)

for each u ∈ U_{d}^{0}.

Let ũ be a function from U^{0}, associated with the subdivision 0 = t_{0} < t_{1} < ⋯ < t_{n} < ∞, such that

ũ_{i} = \begin{cases}a & if E^{Q}[X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)/F_{t_{i}}] ≥ \\ & ≥ E^{Q}[X(t_{i+1} ∧ t, b) − X(t_{i} ∧ t, b)/F_{t_{i}}] \\ b & otherwise \end{cases} .

From (9) we have

E^{Q}J_{t}(ũ, X^{A}) = E^{Q} ∑_{i} E^{Q}[X(t_{i+1} ∧ t, ũ_{i}) − X(t_{i} ∧ t, ũ_{i})/F_{t_{i}}] =

= E^{Q} ∑_{i}[I_{(E^{Q}[X(t_{i+1},a)−X(t_{i},a)/F_{t_{i}}]≥E^{Q}[X(t_{i+1},b)−X(t_{i},b)/F_{t_{i}}])}E^{Q} ×

×[X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)/F_{t_{i}}] +

+I_{(E^{Q}[X(t_{i+1},a)−X(t_{i},a)/F_{t_{i}}]<E^{Q}[X(t_{i+1},b)−X(t_{i},b)/F_{t_{i}}])}E^{Q} ×

×[X(t_{i+1} ∧ t, b) − X(t_{i} ∧ t, b)/F_{t_{i}}] =

= E^{Q} ∑_{i} E^{Q}[X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)/F_{t_{i}}] ∨ E^{Q} ×

×[X(t_{i+1} ∧ t, b) − X(t_{i} ∧ t, b)/F_{t_{i}}] ≤ 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,

E^{Q} ∑_{i}[−X(t_{i+1} ∧ t, a) + X(t_{i} ∧ t, a)] =

= −E^{Q} ∑_{i} E^{Q}[X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a)/F_{t_{i}}] ≤ C,  (11)

and

−E^{Q} ∑_{i} E^{Q}[X(t_{i+1} ∧ t, b) − X(t_{i} ∧ t, b)/F_{t_{i}}] ≤ C.  (12)

Therefore it follows from (10), (11), (12) and from the equality |x − y| = 2 max(x, y) − x − y that

E^{Q} ∑_{i} |E^{Q}[X(t_{i+1}∧t, a)−X(t_{i}∧t, a)−(X(t_{i+1}∧t, b)−X(t_{i}∧t, b))/F_{t_{i}}]| ≤ C.

Thus

sup_{t_{1}<···<t_{n}} E^{Q} ∑_{i} |E^{Q}[X(t_{i+1} ∧ t, a) − X(t_{i} ∧ t, a) −

−(X(t_{i+1} ∧ t, b) − X(t_{i} ∧ t, b))/F_{t_{i}}]| ≤ 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:

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.

Evidently, for every u ∈ Û_{d}^{0} and b ∈ A the process (J_{t}(u, X^{A}) − X_{t}^{b}, t ≥ 0) is a semimartingale. One can show that the family of semimartingales (J(u, X^{A}) − X^{b}, u ∈ Û_{d}^{0}) satisfies the condition U.T. (for each fixed b ∈ A), i.e., the set

( ∫_{0}^{t} H_{s}d(J_{s}(u, X^{A}) − X_{s}^{b}), u ∈ Û_{d}^{0}, H ∈ H)  (13)

is stochastically bounded for every t ∈ R_{+}.

It follows from the continuity property of the functional J that the set (J_{t}(u, X^{A}), u ∈ Û_{d}^{0}) and hence the set (J_{t}(u, X^{A}) − J_{t}(v, X^{A}), u, v ∈ Û_{d}^{0}), is stochastically bounded for each t ∈ R_{+}. Therefore it is sufficient to show that for each H ∈ H and u ∈ Û_{d}^{0} there exists a pair u^{1}, u^{2} ∈ Û_{d}^{0} such that

∫_{0}^{t} H_{s}d(J_{s}(u, X^{A}) − X_{s}^{b}) = J_{t}(u^{1}, X^{A}) − J_{t}(u^{2}, X^{A}).  (14)

Without loss of generality we can assume that the function H ∈ H has the same intervals of constancy as the function u.

For H = ∑_{i≤n−1} H_{i}I_{]t_{i},t_{i+1}]} and u(ω, t, da) = ∑_{i≤n−1} u_{i}(ω, da)I_{]t_{i},t_{i+1}]} let us define

u^{1}(ω, t, da) = ∑_{i≤n−1}(H_{i}^{+}(ω)u_{i}(ω, da) + (1 − H_{i}^{+}(ω))ε_{ω,t}^{b}(da))I_{]t_{i},t_{i+1}]},
u^{2}(ω, t, da) = ∑_{i≤n−1}(H_{i}^{−}(ω)u_{i}(ω, da) + (1 − H_{i}^{−}(ω))ε_{ω,t}^{b}(da))I_{]t_{i},t_{i+1}]},

where H^{+} = max(H, 0), H^{−} = − min(H, 0) and ε^{b} is the measure concentrated at the point b ∈ A for each ω, t.

It is easy to check that for u^{1}, u^{2} defined above equality (14) is true and hence the family of semimartingales (J(u, X^{A}) − X^{b}, u ∈ Û_{d}^{0}) satisfies the 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 b (e.g., we write Y^{a} instead of Y^{a,b} and M^{a,λ} instead of M^{a,b,λ} below). Let

Ỹ_{t}^{a,λ} = ∑_{s≤t} Δ(Y_{s}^{a})I_{[|Δ(Y_{s}^{a})|>λ]},
X̃_{t}^{u,λ} = ∑_{s≤t} Δ(J_{s}(u, Y^{A}))I_{[|Δ(J_{s}(u,Y^{A}))|>λ]},

and let

Y_{t}^{a} − Ỹ_{t}^{a,λ} = M^{λ}(t, a) + B^{λ}(t, a), M^{λ}(·, a) ∈ M_{loc}, B^{λ}(·, a) ∈ A_{loc},
J_{t}(u, Y^{A}) − Ỹ_{t}^{u,λ} = M^{u,λ}(t) + B^{u,λ}(t), M^{u,λ} ∈ M_{loc}, B^{u,λ} ∈ A_{loc},

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

J_{t}(u, Y^{A}) = ∑_{a∈A^{0}} ∫_{0}^{t} I_{(u_{s}=a)}d(Y_{s}^{a})

for each u ∈ U_{d}^{0} and it is easy to see that for each u ∈ U^{0} (the more so for each u ∈ U_{d}^{0}, but not for each Û_{d}^{0})

Ỹ_{t}^{u,λ} = ∑_{a∈A_{0}} nt_{0}^{t}I_{(u_{s}=a)}dỸ_{s}^{a,λ}.

Therefore the uniqueness of the canonical decomposition of the special semimartingales implies that for every u ∈ U_{d}^{0}

M^{u,λ} = J(u, M^{A}), B^{u,λ} = J(u, B^{A}),  (15)

where

J_{t}(u, M^{A}) = ∑_{i}(M^{λ}(t_{i+1} ∧ t, u_{t_{i}}) − M^{λ}(t_{i} ∧ t, u_{t_{i}})) =

= ∑_{a∈A_{0}} ∫_{0}^{t} I_{[u_{s}=a]}dM_{s}^{λ,a}, ∈ M_{loc}^{2},

J_{t}(u, B^{A}) = ∑_{i}(B^{λ}(t_{i+1} ∧ t, u_{t_{i}}) − B^{λ}(t_{i} ∧ t, u_{t_{i}})) =

∑_{a∈A_{0}} ∫_{0}^{t} I_{[u_{s}=a]}dB_{s}^{λ,a}, ∈ A_{loc},

and an easy calculation shows that the square characteristic of the martingale J(u, M^{A}) is equal to

⟨J(u, M^{A})⟩_{t} = ∑_{a∈A_{0}} ∫_{0}^{t} I_{[u_{s}=a]}d⟨M^{λ,a}⟩_{s}.

Since the family of semimartingales (J(u, Y^{A}), u ∈ Û_{d}^{0}) satisfies the condition U.T., it follows from Proposition 1 that the family of random variables (⟨M^{u,λ}⟩_{t}, Var(B^{u,λ})_{t}, Var(Ỹ^{u,λ})_{t}, u ∈ Û_{d}^{0}) is stochastically bounded for each t ≥ 0. Therefore from equality (15) we have

lim_{N} sup_{u∈U_{d}^{0}} P (⟨J(u, M^{A})⟩_{t} ≥ N) = 0,  (16)
lim_{N} sup_{u∈U_{d}^{0}} P (Var(J(u, B^{A}))_{t} ≥ N) = 0,  (17)

and

lim_{N} sup_{u∈U_{d}^{0}} P (Var(Ỹ^{u,λ})_{t} ≥ N) = 0.  (18)

Let for each t ∈ R_{+}

K_{t}^{M} = ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{t},
K_{t}^{B} = ess sup_{u∈U_{d}^{0}} |J_{t}(u, B^{A})|.

Evidently, K_{s}^{M} ≤ K_{t}^{M}, K_{s}^{B} ≤ K_{t}^{B} a.s. for each s ≤ t. Let us prove that P (K_{t}^{M} < ∞) = P (K_{t}^{B} < ∞) = 1 for every t ∈ R_{+}.

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_{u∈U_{d}^{0}} E^{Q}⟨J(u, M^{A})⟩_{t} < ∞.  (19)

Suppose that

Q(ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{t} = ∞) > α > 0.  (20)

Then there exists some 0 ≤ s ≤ t for which (a.s.)

E^{Q}(ess sup_{u∈U_{d}^{0}}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s})/F_{s}) = ∞.

For each fixed s, t ∈ R_{+}, s ≤ t, consider the family of random variables

G_{s,t}^{Q}(u) = E^{Q}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s}/F_{s}), u ∈ U^{0}.

This family satisfies the ε-lattice property (with ε = 0). Indeed, if u^{1}, u^{2} ∈ U_{d}^{0} then we define u^{3} ∈ U_{d}^{0} by

u^{3} = I_{Ω×[0,s]}v + I_{D×]s,t]}u^{1} + I_{D^{c}×]s,t]}u^{2},

where D = {ω : G_{s,t}(u^{1}) ≥ G_{s,t}(u^{2})} and v is an arbitrary element of U_{d}^{0}, for which we have

G_{s,t}(u^{3}) ≥ max(G_{s,t}(u^{1}), G_{s,t}(u^{2})).

Therefore according to Lemma 16.11 of [9]

E^{Q}(ess sup_{u∈U_{d}^{0}}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s})/F_{s}) =
= ess sup_{u∈U_{d}^{0}} E^{Q}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s}/F_{s}).  (21)

Thus from (20) and (21) we have

ess sup_{u∈U_{d}^{0}} E^{Q}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s}/F_{s}) = ∞ a.s.

which contradicts (19), since, according to the definition of ess sup (keeping the lattice property for G^{Q} in mind) there exists a sequence (u^{n}, n ≥ 1) ∈ U_{d}^{0} such that

E^{Q}(⟨J(u^{n}, M^{A})⟩_{t} − ⟨J(u^{n}, M^{A})⟩_{s}/F_{s}) → ∞ a.s.

The equality P (K_{t}^{B} < ∞) = 1 is proved in a similar way.

Let us prove now that

⟨M^{a}, M^{a}⟩_{t} ≪ K_{t}^{M}, t ∈ R_{+},
(Var B^{a})_{t} ≪ K_{t}^{B}, t ∈ R_{+},

for every a ∈ A.

Without loss of generality we can assume that the random variables K_{t}^{M} and K_{t}^{B} are integrable for each t ∈ R_{+} (otherwise, one can use a localizing sequence of stopping times (τ_{n}, n ≥ 1) with EK_{t∧τ_{n}}^{M} < ∞ for each n ≥ 1. Such a sequence exists, since P (K_{t}^{M} + B_{t}^{M} < ∞) = 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

G_{s,t}(u) = E(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s}/F_{s}), u ∈ U_{d}^{0}

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.

Therefore using (21) we obtain

E(K_{t}^{M} − K_{s}^{M}/F_{s}) =

= E(ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{t} − ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{s}/F_{s}) ≥

≥ ess sup_{u∈U_{d}^{0}} E(⟨J(u, M^{A})⟩_{t}/F_{s}) − ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{s} =

= ess sup_{u∈U_{d}^{0}} E(⟨J(u, M^{A})⟩_{t} − ess sup_{u∈U_{d}^{0}}⟨J(u, M^{A})⟩_{s}/F_{s}) ≥

≥ ess sup_{u∈U_{d}^{0}} E(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s}/F_{s}) =
= E(ess sup_{u∈U_{d}^{0}}(⟨J(u, M^{A})⟩_{t} − ⟨J(u, M^{A})⟩_{s})/F_{s}) ≥
≥ E(⟨M^{a}⟩_{t} − ⟨M^{a}⟩_{s}/F_{s}),

and hence the process (K_{t}^{M} − ⟨M^{a}⟩_{t}, t ≥ 0) is a submartingale. Since every predictable local martingale of finite variation is constant (see, e.g., [10]), we conclude that the process (K_{t}^{M} − ⟨M^{a}⟩_{t}, t ≥ 0) is increasing for each a ∈ A. Evidently, this implies that the measure d⟨M^{a}⟩_{t}(ω) is absolutely continuous with respect to the measure dK_{t}^{M}(ω) for almost all ω ∈ Ω, i.e., ⟨M^{a}⟩ ≪ K^{M} for each a ∈ A.

Indeed, suppose H is some bounded positive predictable process such that E ∫_{0}^{t} H_{s}dK_{s}^{M} = 0. Then

E ∫_{0}^{t} H_{s}d⟨M^{a}⟩_{s} + E ∫_{0}^{t} H_{s}d(K_{s}^{M} − ⟨M^{a}⟩_{s}) = 0

implies that E ∫_{0}^{t} H_{s}d⟨M^{a}⟩_{s} = 0, since K^{M} −⟨M^{a}⟩ is an increasing process. In a similar way one can prove that

(Var B^{a})_{t} ≪ K_{t}^{B}, t ∈ R_{+},

for each a ∈ A. Thus the increasing process K_{t} = K_{t}^{M} + K_{t}^{B} 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 u ∈ Û^{0}

J̄_{t}(u, X^{A}) = ∑_{a∈A^{0}} ∫_{0}^{t} u_{s}(a)dX_{s}^{a}.

Evidently,

J̄_{t}(u, X^{A}) = ∑_{a∈A^{0}} ∫_{0}^{t} I_{(u_{s}=a)}dX_{s}^{a}

if u ∈ U^{0}, and

J̄(u, X^{A}) = J(u, X^{A})

for every u ∈ Û_{d}^{0}.

Theorem 3. X^{A} is a regular family of semimartingales if and only if for each sequence (u^{n}, n ≥ 1) ∈ Û^{0} uniformly converging to some u ∈ Û, the corresponding integral sums Ĵ_{t}(u^{n}, X^{A}) are fundamental in probability uniformly on every compact.

Proof. The necessity part of the theorem is proved in [3].

Sufficiency. Let for each sequence (u^{n}, n ≥ 1) ∈ Û^{0} converging uniformly to some u ∈ Û

sup_{s≤t} |Ĵ_{t}(u^{n}, X^{A}) − Ĵ(u^{m}, X^{A})| → 0  (22)

as n → ∞, m → ∞, for every t ≥ 0. We must show that the family of semimartingales X^{A} is regular. Since J̄(u, X^{A}) = J(u, X^{A}) for every u ∈ Û_{d}^{0}, it follows from Theorem 2 that X^{A} 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 μ^{K}-a.e. continuous modification of the Radon–Nicodým derivative

φ(a, a′) = d⟨M^{a} − M^{a′}⟩/dK.

Let A_{0}^{m} = (a_{1}^{m}, a_{2}^{m}, …, a_{i(m)}^{m}) be a \frac{1}{m}-net of the compact A. Suppose that the function φ(a, a′) is separable (or we can choose such a version).

The function

φ^{∗}(a, a′) = lim_{m} sup_{i:r(a_{i}^{m},a)≤ \frac{1}{m}} sup_{j:r(a_{j}^{m},a′)≤ \frac{1}{m}} φ(a_{i}^{m}, a_{j}^{m})

is upper continuous a.e. with respect to the measure μ^{K} and it is easy to see that the μ^{K}-a.e. continuous modification of the derivative of φ(a, a′) exists if and only if

φ_{t}^{∗}(u_{t}, u_{t}) = 0, 0 ≤ t < ∞,

μ^{K}-a.e. for all u ∈ U. Let there exist u ∈ U such that

μ^{K}[(ω, t) : φ^{∗}(u, u) > 0] > 0.

We can construct two sequences (u^{m}, m ≥ 1), (v^{m}, m ≥ 1) ∈ U^{0} (i.e., u^{m} and v^{m} are taking a finite number of values for each m), such that u^{m} → u, v^{m} → u uniformly and

φ(u^{m}, v^{m}) → φ^{∗}(u, u), μ^{K} − a.e,  (23)

as m → ∞.

Put u_{t}^{m}(ω) = a_{i_{m}(ω,t)}^{m}, v_{t}^{m}(ω) = a_{j_{m}(ω,t)}^{m}, where the indices a_{i_{m}}^{m} and a_{j_{m}}^{m} are selected so that

r(a_{i_{m}(ω,t)}^{m}, u_{t}(ω)) ≤ \frac{1}{m}, r(a_{j_{m}(ω,t)}^{m}, u_{t}(ω)) ≤ \frac{1}{m}

and

φ(a_{i_{m}(ω,t)}^{m}, a_{j_{m}(ω,t)}^{m}) ≥ sup_{i:r(a_{i}^{m},u_{t}(ω))≤ \frac{1}{m}} sup_{j:r(a_{j}^{m},u_{t}(ω))≤ \frac{1}{m}} φ(a_{i}^{m}, a_{j}^{m}) − \frac{1}{m}.

Since for each u, v ∈ U^{0}

⟨J̄(u, M^{A}) − J̄(v, M^{A})⟩_{t} = ∫_{0}^{t} φ_{s}(u_{s}, v_{s})dK_{s}

we have

lim inf_{m} E⟨J̄(u^{m}, M^{A}) − J̄(v^{m}, M^{A})⟩_{t} = lim inf_{m} E ∫_{0}^{t} φ_{s}(u_{s}^{m}, v_{s}^{m})dK_{s} ≥

≥ E ∫_{0}^{t} φ_{s}^{∗}(u_{s}, u_{s})dK_{s} > 0  (24)

for some t ≥ 0. Since the family of semimartingales J̄(u, X^{A}), uÛ^{0}) satisfies the condition U.T. (the proof is similar to the corresponding assertion of Theorem 2) and

sup_{s≤t} |J̄_{s}(u^{n}, X^{A}) − J̄_{s}(u^{m}, X^{A})| → 0, m, n → ∞,  (25)

it follows from Proposition 2 that

sup_{s≤t} |J̄_{s}(u^{n}, M^{A}) − J̄_{s}(u^{m}, M^{A})| → 0, m, n → ∞,  (26)
sup_{s≤t} |J̄_{s}(u^{n}, B^{A}) − J̄_{s}(u^{m}, B^{A})| → 0, m, n → ∞,  (27)

where

J̄(u, M^{A}) = ∑_{a∈A^{0}} ∫_{0}^{t} I_{(u_{s}=a)}dM_{s}^{a,λ}, J̄(u, B^{A}) = ∑_{a∈A^{0}} ∫_{0}^{t} I_{(u_{s}=a)}dB_{s}^{a,λ}

coincide with canonical decomposition terms of the special semimartingale J̄(u, X^{A}) − ∑ ΔJ̄(u, X^{A})I_{(|J̄(u,X^{A})|>λ)} for each u ∈ U^{0}.

Obviously, (24) implies that the continuity condition (26) for J(u, M^{A}) with respect to u does not hold and therefore φ(a, a′) is continuous in (a, a′) with respect to the measure μ^{K}.

The existence of a μ^{K}-a.e. continuous modification of the derivative g_{t}(a) = dB(t, a)/dK_{t} is proved analogously.

Let us prove now that the family X^{A} satisfies condition (D). Evidently, for each u ∈ U^{0}

⟨J̄(u, M^{A})⟩_{t} = ∫_{0}^{t} ψ_{s}(u_{s}, u_{s})dK_{s}

and it is easy to see that the μ^{K}-a.e. continuity of φ(a, a′) implies the μ^{K}- a.e. continuity of the function ψ(a, b) along the diagonal (i.e., there exists a μ^{K}-a.e. continuous modification of the function ψ(a, a)). Therefore

∫_{0}^{t} sup_{a∈A} ψ_{s}(a, a)dK_{s} = ess sup_{u∈U^{0}}⟨J(u, M^{A})⟩_{t} < ∞ a.s.

for each t ∈ R_{+}. The inequality

∫_{0}^{t} sup_{a∈A} g_{s}(a)dK_{s} < ∞

is proved in a similar way.

Since

ΔJ_{t}(u, X^{A}) = ∑_{a∈A^{0}} I_{(u_{s}=a)}(X(t, a) − X(t−, a)) = X(t, u(t)) − X(t−, u(t)),

relation (18) can be rewritten as

lim_{n→∞} sup_{u∈U^{0}} P {∑_{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 sup_{u∈U^{0}} ∑_{s≤t} |X(t, u(t)) − X(t−, u(t))|I_{(|X(t,u(t))−X(t−,u(t))|>λ)} < ∞

a.s. for each t ∈ R_{+}.

Since the continuity property (22) of the functional J̄ implies that a function ΔX_{t}^{a} is continuous with respect to a in probability uniformly on each [0, t] and A is a compact subset of some metric space we obtain that

∑_{s≤t} sup_{a∈A} Δ|X(s, a)|I_{(sup_{a∈A} Δ|X(s,a)|>λ)} =

= ess sup_{u∈U} ∑_{s≤t} |X(t, u(t)) − X(t−, u(t))|I_{(|X(t,u(t))−X(t−,u(t))|>λ)} =

= ess sup_{u∈U^{0}} ∑_{s≤t} |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 is a locally bounded predictable process then the usual stochastic integral ∫_{0}^{t} u_{s}dX_{s} satisfies relations (3), (4), and hence the stochastic integral u · M 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 = (X_{t}, t ≥ 0) be an adapted continuous process and consider the family X^{A} = (aX, a ∈ R). For an elementary predictable process u ∈ H

J_{t}(u, X^{A}) = ∑_{i} u_{i}[X(t_{i+1} ∧ t) − X(t_{i} ∧ t)]

and it follows from Theorem 1 that the continuity of the functional J(u) with respect to the class {u ∈ U_{d}^{0} : |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 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