Additive Summable Processes and their Stochastic Integral

August 19, 2005

Abstract

We define and study a class of summable processes,called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D–B].

We relax the definition of a summable processes X : Ω×ℝ_{+} → E ⊂ L(F, G) by asking for the associated measure I_{X} to have just an additive extension to the predictable σ−algebra P, such that each of the measures (I_{X})_{z}, for z ∈ (L_{G}^{p})^{∗}, being σ−additive, rather than having a σ−additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processes X : Ω × ℝ_{+} → E ⊂ L(F, G) with integrable semivariation if c_{0} ∉ G.

Introduction

We study the stochastic integral in the case of Banach-valued processes, from a measure-theoretical point of view.

The classical stochastic integration (for real-valued processes) refers only to integrals with respect to semimartingale (Dellacherie and Meyer [DM78]). A similar technique has also been applied by Kunita [Kun70], for Hilbert valued processes, making use of the inner product. A number of technical difficulties emerge for Banach spaces, since the Banach space lacks an inner product.

Vector integration using different approaches were presented in several books by Dinculeanu [Din00], Diestel and Uhl [DU77], and Kussmaul [Kus77]. Brooks and Dinculeanu [BD76] were the first to present a version of integration with respect to a vector measure with finite semivariation. Later, the same authors [BD90] presented a stochastic integral with respect to so-called summable Banach-valued processes.

A Banach-valued process X is called summable if the Doleans-Dade measure I_{X} defined on the ring generated by the predictable rectangles can be extended to a σ-additive measure with finite semivariation on the corresponding σ-algebra P. The summable process X plays the role of the square integrable martingale in the classical theory: a stochastic integral H · X can be defined with respect to X as a cadlag modification of the process (∫_{[o,t]} H dI_{X})_{t≥0} of integrals with respect to I_{X} such that ∫_{[0,t]} HdI_{X} ∈ L_{G}^{p}

for every t ∈ ℝ_{+}.

In [Din00] Dinculeanu presents a detailed account of the integration theory with respect to these summable processes, from a measure-theoretical point of view.

Our attention turned to a further generalization of the stochastic integral. Besides the processes considered in the classical theory (Hilbert-valued square-integrable martingales and processes with integrable variation), the class of summable processes also includes processes with integrable semivariation, as long as the Banach space E satisfies some restrictions. To get rid of some of these restrictions, we redefine, in Section 2, the notion of summability: now we only require that I_{X} can be extended to an additive measure on P, but such that each of the measures (I_{X})_{z}, for z ∈ Z a norming space for L_{G}^{p}, is σ-additive. With this new notion of summability, called additive summability, the stochastic integral is then defined, in Section 5.1, as before. The rest of Chapter 5 is dedicated to proving the same type of properties of the stochastic integral as in Dinculeanu [Din00], namely measure theoretical properties.

In Section we will prove that there are more additive summable processes than summable processes by reducing the restrictions imposed on the space E.

1 Notations and definitions

Throughout this paper we consider S to be a set and R, D, Σ respectively a ring, a δ−ring, a σ−ring, and a σ−algebra of subsets of S, E, F, G Banach spaces with E ⊂ L(F, G) continuously, that is, |x(y)| ≤ |x||y| for x ∈ E and y ∈ F; for example, E = L(ℝ, E). If M is any Banach space, we denote by |x| the norm of an element x ∈ M, by M_{1} its unit ball of M and by M^{∗} the dual of M. A space Z ⊂ G^{∗} is called a norming space for G, if for every x ∈ G we have

|x| = sup_{z∈Z_{1}} |⟨x, z⟩|.

If m : R → E ⊂ L(F, G) is an additive measure for every set A ⊂ S the semivariation of m on A relative to the embedding E ⊂ L(F, G) (or relative to the pair (F, G)) is denoted by m̃_{F,G}(A) and defined by the equality

m̃_{F,G}(A) = sup | ∑_{i∈I} m(A_{i})x_{i}|,

where the supremum is taken for all finite families (A_{i})_{i∈I} of disjoint sets from R contained in A and all families (x_{i})_{i∈I} of elements from F_{1}.

2 Additive summable processes

The framework for this section is a cadlag, adapted process X : ℝ_{+} × Ω → E ⊂ L(F, G), such that X_{t} ∈ L_{E}^{p} for every t ≥ 0 and 1 ≤ p < ∞.

2.1 The Measures I_{X} and (I_{X})_{z}

Let S be the semiring of predictable rectangles and I_{X} : S → L_{E}^{p} the stochastic measure defined by

I_{X}({0} × A) = 1_{A}X_{0}, for A ∈ F_{0}

and

I_{X}((s, t] × A) = 1_{A}(X_{t} − X_{s}), for A ∈ F_{s}.

Note that I_{X} is finitely additive on S therefore it can be extended uniquely to a finitely additive measure on the ring R generated by S. We obtain a finitely additive measure I_{X} : R → L_{E}^{p} verifying the previous equalities.

Let Z ⊂ (L_{G}^{p})^{∗} be a norming space for L_{G}^{p}. For each z ∈ Z we define a measure (I_{X})_{z}, (I_{X})_{z} : R → F^{∗} by

⟨y, (I_{X})_{z}(A)⟩ = ⟨I_{X}(A)y, z⟩ = ∫ ⟨I_{X}(A)(ω)y, z(ω)⟩dP (ω), for A ∈ P and y ∈ F

where the bracket in the integral represents the duality between G and G^{∗}.

Since L_{E}^{p} ⊂ L(F, L_{G}^{p}), we can consider the semivariation of I_{X} relative to the pair (F, L_{G}^{p}). To simplify the notation, we shall write (Ĩ_{X})_{F,G} instead of (Ĩ_{X})_{F,L_{G}^{p}} and we shall call it the semivariation of I_{X} relative to (F, G):

2.2 Additive Summable Processes

Definition 1. We say that X is p-additive summable relative to the pair (F, G) if I_{X} has an additive extension I_{X} : P → L_{E}^{p} with finite semivariation relative to (F, G) and such that the measure (I_{X})_{z} is σ-additive for each z ∈ (L_{G}^{p})^{∗}.

If p = 1, we say, simply, that X is additive summable relative to (F, G).

Remark. 1) This definition is weaker that the definition of summable processes since here we don’t require the measure I_{X} to have a σ−additive extension to P.

2) The problems that might appear if (I_{X}) is not σ−additive are convergence problems (most of the convergence theorem are stated for σ−additive measures and extension problems (the uniqueness of extensions of measures usually requires σ−additivity).

3) Note that in the paper “The Riesz representation theorem and extension of vector valued additive measures” N. Dinculeanu and B. Bongiorno [BD01] (Theorem 3.7 II) proved that if each of the measures (I_{X})_{z} is σadditive and if I_{X} : R → L_{E}^{p} has finite semivariation relative to (F, G) then I_{X} has canonical additive extension I_{X} : P → (L_{E}^{p})^{∗∗} with finite semivariation relative to (F, (L_{E}^{p})^{∗∗}) such that for each z ∈ (L_{G}^{p})^{∗}, the measure (I_{X})_{z} is σ−additive on P and has finite variation |(I_{X})_{z}|.

Proposition 2. If X is p−additive summable relative to (ℝ, E) then X is p−summable relative to (ℝ, E).

Proof. If X is p−additive summable relative to (ℝ, E) then the measure I_{X} has an additive extension I_{X} : P → L_{E}^{p} with finite semivariation relative to (ℝ, E). Moreover for each z ∈ (L_{E}^{p})^{∗} the measure (I_{X})_{z} is σ-additive.

By Pettis Theorem, the measure I_{X} is σ−additive. Hence, the process X is p−summable.

2.3 The Integral ∫ HdI_{X}

Let X be a p-additive summable process relative to (F, G).

Consider the additive measure I_{X} : P → L_{E}^{p} ⊂ L(F, L_{G}^{p}) with bounded semivariation Ĩ_{F,G} relative to (F, L_{G}^{p}) for which each measure (I_{X})_{z} is σadditive for every z ∈ Z.

Then we have

(Ĩ_{X})_{F,L_{G}^{p}} = sup{|m_{z}| : z ∈ Z, ∥z∥ ≤ 1, z ∈ (L_{F}^{p})^{∗}},

(See Corollary 23, Section 1.5 [?].)

If p is fixed, to simplify the notation, we can write Ĩ_{F,G} = Ĩ_{F,L_{G}^{p}}.

For any Banach space D we denote by F_{D}(Ĩ_{F,G}) or F_{D}(Ĩ_{F,L_{G}^{p}}) the space of predictable processes H : ℝ_{+} × Ω → D such that

Ĩ_{F,G}(H) = sup{∫ |H|d|(I_{X})_{z}| : ∥z∥_{q} ≤ 1} < ∞.

Definition 3. Let D = F. For any H ∈ F_{F} (Ĩ_{F,G}) We define the integral ∫ HdI_{X} to be the mapping z ↦ ∫ Hd(I_{X})_{z}.

Observe that if H ∈ F_{F,G}(X) := F_{F} (Ĩ_{F,G}) the integral ∫ Hd(I_{X})_{z} is defined and is a scalar for each z ∈ Z, hence the mapping z ↦ ∫ Hd(I_{X})_{z} is a continuous linear functional on (L_{G}^{p})^{∗} Therefore, ∫ HdI_{X} ∈ (L_{G}^{p})^{∗∗}

⟨∫ HdI_{X}, z⟩ = ∫ Hd(I_{X})_{z}, for z ∈ Z

and

| ∫ HdI_{X}| ≤ Ĩ_{F,G}(H).

Theorem 4. Let (H^{n})_{0≤n<∞} be a sequence of elements from F_{F,G}(X) such that |H^{n}| ≤ |H^{0}| for each n and H^{n} → H pointwise. Assume that

(i) ∫ H^{n}dI_{X} ∈ L_{G}^{p} for every n ≥ 1 and (ii) The sequence (∫ H^{n}dI_{X})_{n} converges pointwise on Ω, weakly in G. Then a) ∫ HdI_{X} ∈ L_{G}^{p} and

b) ∫ H^{n}dI_{X} → ∫ HdI_{X}, in the weak topology of L_{G}^{p}, as well as pointwise, weakly in G.

c) If (∫ H^{n}dI_{X})_{n} converges pointwise on Ω, strongly in G, then

∫ H^{n}dI_{X} → ∫ HdI_{X},

strongly in L_{G}^{1}.

Proof. This theorem was proved in [Din00] under the assumtion that I_{X} is σ−additive. But, in fact, only the σ-additivity of each of the measures (I_{X})_{z} was used. hence the same proof remains valid in our case.

2.4 The Stochastic Integral H · X

In this section we define the stochastic integral and we prove that the stochastic integral is a cadlag adapted process.

Let H ∈ F_{F,G}(X). Then, for every t ≥ 0 we have 1_{[0,t]}H ∈ F_{F,G}(X). We denote by ∫_{[0,t]} HdI_{X} the integral ∫ 1_{[0,t]}HdI_{X}. We define

∫_{[0,∞]} HdI_{X} := ∫_{[0,∞)} HdI_{X} = ∫ HdI_{X}.

Taking Z = (L_{G}^{p})^{∗}, for each H ∈ F_{F,G}(X) we obtain a family (∫_{[0,t]} HdI_{X})_{t∈ℝ_{+}} of elements of (L_{G}^{p})^{∗∗}.

We restrict ourselves to processes H for which ∫_{[0,t]} HdI_{X} ∈ L_{G}^{p} for each t ≥ 0. Since L_{G}^{p} is a set of equivalence classes, ∫_{[0,t]} HdI_{X} represents an equivalence class. We use the same notation for any random variable in its equivalence class. We are interested to see whether or not the process (∫_{[0,t]} HdI_{X})_{t≥0} is adapted and if it admits a cadlag modification.

Theorem 5. Let X : ℝ → E ⊂ L(F, G) be a cadlag, adapted, p-summable process relative to (F, G) and H ∈ F_{F,G}(X) such that ∫_{[0,t]} HdI_{X} ∈ L_{G}^{p} for every t ≥ 0.

Then the process (∫_{[0,t]} HdI_{X})_{t≥0} is adapted.

Proof. This is the equivalent of Theorem 10.4 in [Din00] and since in the proof was used the σ-additivity of the measures (I_{X})_{z} rather than σ-additivity of the measure I_{X} that proof will work for our case too.

It is not clear that there is a cadlag modification of the previously defined process (∫_{[0,t]} HdI_{X})_{t}. Therefore we use the following definition

Definition 6. We define L_{F,G}^{1}(X) to be the set of processes H ∈ F_{F,G}(I_{X}) that satisfy the following two conditions:

a) ∫_{[0,t]} HdI_{X} ∈ L_{G}^{p} for every t ∈ ℝ_{+};

b) The process (∫_{[0,t]} HdI_{X})_{t≥0} has a cadlag modification.

The processes H ∈ L_{F,G}^{1}(X) are said to be integrable with respect to X.

If H ∈ L_{F,G}^{1}(X), then any cadlag modification of the process (∫_{[0,t]} HdI_{X})_{t≥0} is called the stochastic integral of H with respect to X and is denoted by H · X or ∫ HdX:

(H · X)_{t}(ω) = (∫ HdX)_{t}(ω) = (∫_{[0,t]} HdI_{X})(ω), a.s.

Therefore the stochastic integral is defined up to an evanescent process. For t = ∞ we have

(H · X)_{∞} = ∫_{[0,∞]} HdI_{X} = ∫_{[0,∞)} HdI_{X} = ∫ HdI_{X}.

Note that if H : ℝ_{+} × Ω → F is an R-step process then we have

(H · X)_{t}(ω) = ∫_{[0,t]} H_{s}(ω)dX_{s}(ω),

that is, the stochastic integral can be computed pathwise.

The next theorem shows that the stochastic integral H · X is a cadlag process and it is cadlag in L_{G}^{p}.

Theorem 7. If X : ℝ_{+}×Ω → E ⊂ L(F, G) is a p-additive summable process relative to (F, G) and if H ∈ L_{F,G}^{1}(X), then: a) The stochastic integral H ·X is a cadlag, adapted process.

b) For every t ∈ [0, ∞) we have (H · X)_{t−} ∈ L_{G}^{p} and

(H · X)_{t−} = ∫_{[0,t)} HdI_{X}, a.s.

If (H · X)_{∞−}(ω) exists for each ω ∈ Ω, then

(H · X)_{∞−} = (H · X)_{∞} = ∫ HdI_{X}, a.s.

c) The mapping t ↦ (H · X)_{t} is cadlag in L_{G}^{1}.

Proof. a) Follows from the previous theorem and definition. b) and c) are proved as in theorem 10.7 in [Din00] since there was not used the σ-additivity of I_{X} but rather of (I_{X})_{z}.

2.5 The Stochastic Integral and Stopping Times

Let T be a stopping time. If A ∈ F_{T}, then the stopping time T_{A} is defined by T_{A}(ω) = T (ω) if ω ∈ A and T_{A}(ω) = ∞ if ω ∉ A. With this notation the predictable rectangles (s, t] × A with A ∈ F_{s} could be written as stochastic intervals (s_{A}, t_{A}]. Another notation we will use is I_{X}[0, T] for I_{X}([0, T] × Ω.

Let X : ℝ_{+} × Ω → E ⊂ L(F, G) be an additive summable process

Proposition 8. For any stopping time T we have X_{T} ∈ L_{E}^{p} and I_{X}[0, T] = X_{T} for T simple. For any decreasing sequence (T_{n}) of simple stopping times such that T_{n} ↓ T, and for every z ∈ (L_{G}^{p})^{∗} we have

⟨I_{X}([0, T])y, z⟩ = lim_{n}⟨X_{T_{n}}y, z⟩,  (1)

where the bracket represents the duality between L_{G}^{p} and (L_{G}^{p})^{∗}.

Proof. Assume first that T is a simple stopping time of the form

T = ∑_{1≤i≤n} 1_{A_{i}}t_{i},

with 0 < t_{i} ≤ ∞, t_{i} ≠ t_{j} for i ≠ j, A_{i} ∈ F_{t_{i}} are mutually disjoint and ⋃_{1≤i≤n} A_{i} = Ω. Then [0, T] = ⋃_{1≤i≤n}[0, t_{i}] × A_{i} is a disjoint union. Hence I_{X}([0, T]) = ∑_{i} I_{X}([0, t_{i}] × A_{i}) = ∑_{i} 1_{A_{i}}X_{t_{i}} = X_{T}. Since I_{X} : P → L_{E}^{p}, we conclude that X_{T} ∈ L_{E}^{p}.

Next, assume that (T_{n}) is a sequence of simple stopping times such that T_{n} ↓ T. Then [0, T_{n}] ↓ [0, T]. Since (I_{X})_{z} is σ-additive in F^{∗}, for any y ∈ F and z ∈ (L_{G}^{p})^{∗}, we have

⟨I_{X}([0, T])y, z⟩ = ⟨(I_{X})_{z}([0, T]), y⟩ = lim_{n→∞}⟨(I_{X})_{z}([0, T_{n}]), y⟩

= lim_{n→∞}⟨I_{X}([0, T_{n}]y, z⟩ = lim_{n→∞}⟨X_{T_{n}}y, z⟩.

and the relation (4.1) is proven. It remains to prove that X_{T} ∈ L_{E}^{p}. Since X_{T_{n}}(ω) → X_{T} (ω) it follows that X_{T} is F−measurable. We will prove that |X_{T_{n}}| ∈ L^{p} to deduce that X_{T_{n}} ∈ L_{G}^{p}.

We saw before that for y ∈ F and z ∈ (L_{G}^{p})^{∗} the sequence ⟨(I_{X})([0, T_{n}])y, z⟩ is convergent hence bounded, i.e.

sup_{n} |⟨(I_{X})([0, T_{n}])y, z⟩| < ∞, for y ∈ F, z ∈ (L_{G}^{p})^{∗}.

By the Banach-Steinhauss Theorem, we have

sup_{n} ∥I_{X}([0, T_{n}]y∥_{L_{G}^{p}} < ∞, for y ∈ F

hence

sup_{n} ∥I_{X}([0, T_{n}]∥_{L_{E}^{p}} < ∞.

or sup_{n} ∥X_{T_{n}}]∥_{L_{E}^{p}} < ∞, which is equivalent to sup_{n} ∫ |X_{T_{n}}|^{p}dP < ∞. Now |X_{T} |^{p} = lim |X_{T_{n}}|^{p} = lim inf |X_{T_{n}}|^{p}. If we apply Fatou Lemma we get:

∫ |X_{T} |^{p}dP = ∫ lim inf |X_{T_{n}}|^{p} ≤ lim inf ∫ |X_{T_{n}}|^{p}dP ≤ sup ∫ |X_{T_{n}}|^{p}dP < ∞.

therefore X_{T} ∈ L_{G}^{p}.

Proposition 9. Let S ≤ T be stopping times and h : Ω → F be an F_{S}- measurable, simple random variable. Then for any pair (T^{n})_{n}, (S^{n})_{n} of sequences of simple stopping times, with T^{n} ↓ T, S^{n} ↓ S, such that S^{n} ≤ T^{n} for each n, we have

⟨∫ h1_{(S,T]} dI_{X}, z⟩ = lim_{n}⟨h(X_{T^{n}} − X_{S^{n}}), z⟩, for z ∈ (L_{G}^{p})^{∗},  (2)

where the bracket represents the duality between L_{G}^{p} and (L_{G}^{p})^{∗}.

Proof. First we prove that there are two sequences (T^{n}) and (S^{n}) of simple stopping times such that T^{n} ↓ T, S^{n} ↓ S and S^{n} ≤ T^{n}. In fact, there are two sequences of simple stopping times T^{n} and P^{n} such that P^{n} ↓ S and T^{n} ↓ T. Consider, now, S^{n} = P^{n} ∧ T^{n}. Since P^{n} and T^{n} are stopping times, S^{n} is a stopping time and S^{n} ≤ T^{n}. On the other hand, observe that S ≤ S^{n} ≤ P^{n} and lim P^{n} = S. Therefore lim_{n→∞}S^{n} = S too. So we have S^{n} ↓ S and S^{n} ≤ T^{n}.

Now we want to prove (4.2). Assume first h = 1_{A}y with A ∈ F_{S} and y ∈ F. Then

∫ h1_{(S,T]} dI_{X} = ∫ 1_{A}y1_{(S,T]} dI_{X} = ∫ 1_{(S_{A},T_{A}]}y dI_{X} = I_{X}((S_{A}, T_{A}])y.

For any sequence of simple stopping times (T^{n}) and (S^{n}) with T^{n} ↓ T, S^{n} ↓ S and S^{n} ≤ T^{n}, we have T_{A}^{n} ↓ T_{A} and S_{A}^{n} ↓ S_{A}. Therefore, applying Proposition 8 for every z ∈ (L_{G}^{p})^{∗}, we get

⟨∫ h1_{(S,T]} dI_{X}, z⟩ = ⟨I_{X}((S_{A}, T_{A}])y, z⟩ = ⟨[I_{X}([0, T_{A}]) − I_{X}([0, S_{A}])]y, z⟩

= lim_{n→∞}⟨X_{T_{A}^{n}}y, z⟩ − lim_{n→∞}⟨X_{S_{A}^{n}}y, z⟩ = lim_{n→∞}⟨(X_{T_{A}^{n}} − X_{S_{A}^{n}})y, z⟩

= lim_{n→∞}⟨1_{A}(X_{T^{n}} − S_{X^{n}})y, z⟩ = lim_{n→∞}⟨h(X_{T^{n}} − X_{S^{n}}), z⟩

Then the equality holds for any F_{S}-step function h.

Proposition 10. Let S ≤ T be stopping times and assume that either

(i) h : Ω → ℝ is a simple, F_{S}-measurable function and H ∈ L_{F,G}^{1}(X), or (ii) The measure I_{X} is σ−additive, h : Ω → F is a simple, F_{S}-measurable function and H ∈ L_{ℝ,E}^{1}(X).

If ∫ 1_{(S,T]}H dI_{X} ∈ L_{G}^{p} in case (i) and ∫ 1_{(S,T]}H dI_{X} ∈ L_{E}^{p} in case (ii) then

∫ h1_{(S,T]}H dI_{X} = h ∫ 1_{(S,T]}H dI_{X}.

Proof. Assume first hypothesis (i). Let (T^{n}) and (S^{n}) be two sequences of simple stopping times such that T^{n} ↓ T, S^{n} ↓ S and S^{n} ≤ T^{n}. Assume H = 1_{(s,t]×A}y with A ∈ F_{s} and y ∈ F. Then T^{n} ∧ t ↓ T ∧ t, S^{n} ∧ s ↓ S ∧ s. Let z ∈ (L_{G}^{p})^{∗}. Then

⟨∫ h1_{(S,T]}H dI_{X}, z⟩ = ⟨∫ h1_{A}y1_{(S∨s,T ∧t]} dI_{X}, z⟩,

where the bracket represents the duality between L_{G}^{p} and (L_{G}^{p})^{∗}. By (4.2), for the simple F_{S∨s}-measurable function h1_{A}y and the stopping times (S ∨ s) ≤ (T ∧ t) we have

⟨h ∫ 1_{(S,T]}H dI_{X}, z⟩ = ⟨∫ 1_{(S,T]}HdI_{X}, hz⟩ = ⟨∫ 1_{(S∨s,T ∧t]}1_{A}ydI_{X}, hz⟩

= lim⟨1_{A}y(X_{T^{n}∧t} − X_{S^{n}∨s}), hz⟩

= lim⟨h1_{A}y(X_{T^{n}∧t} − X_{S^{n}∨s}), z⟩ = ⟨∫ h1_{A}y1_{(S∨s,T ∧t]}dI_{X}, z⟩

= ⟨∫ h1_{A}y1_{(s,t]}1_{(S,T]}dI_{X}, z⟩ = ⟨∫ hH1_{(S,T]}dI_{X}, z⟩

If H = 1_{{0}×A}y with A ∈ F_{0} and y ∈ F, since 1_{(S,T]}1_{{0}×A} = 0 we have

⟨h ∫ 1_{(S,T]}H dI_{X}, z⟩ = 0 = ⟨∫ hH1_{(S,T]}dI_{X}, z⟩.

It follows that for any B ∈ R we have

⟨∫ h1_{(S,T]}1_{B}y dI_{X}, z⟩ = ⟨h ∫ 1_{(S,T]}1_{B}y dI_{X}, z⟩.  (*)

The class M of sets B ∈ P for which the above equality holds for all z ∈ (L_{G}^{p})^{∗} is a monotone class: in fact, let B_{n} be a monotone sequence of sets from M and let B = lim B_{n}. For each n we have

∫ h1_{(S,T]}1_{B_{n}}yd(I_{X})_{z} = ⟨h ∫ 1_{(S,T]}1_{B_{n}}ydI_{X}, z⟩.

Since h1_{(S,T]}1_{B_{n}}y is a sequence of bounded functions converging to h1_{(S,T]}1_{B}y (h is a step-function) with |h1_{(S,T]}1_{B_{n}}y| ≤ |h||y|, we can apply Lebesgue Theorem and conclude that ∫ h1_{(S,T]}1_{B_{n}}yd(I_{X})_{z} → ∫ h1_{(S,T]}1_{B}yd(I_{X})_{z}. Using the same reasoning we can conclude that ∫ 1_{(S,T]}1_{B_{n}}yd(I_{X})_{hz} → ∫ 1_{(S,T]}1_{B}yd(I_{X})_{hz}. hence we have

⟨∫ h1_{(S,T]}1_{B}y dI_{X}, z⟩ = lim_{n}⟨∫ h1_{(S,T]}1_{B_{n}}y dI_{X}, z⟩ = lim_{n}⟨h ∫ 1_{(S,T]}1_{B_{n}}y dI_{X}, z⟩

= ⟨h lim_{n} ∫ 1_{(S,T]}1_{B_{n}}y dI_{X}, z⟩ = ⟨h ∫ 1_{(S,T]}1_{B}y dI_{X}, z⟩

Since the class M of sets satisfying equality (*) is a monotone class containing R we conclude that the equality (*) is satisfied by all B ∈ P.

It follows that for any predictable, simple process H and for each z ∈ (L_{G}^{p})^{∗} we have

⟨∫ h1_{(S,T]}H dI_{X}, z⟩ = ⟨h ∫ 1_{(S,T]}H dI_{X}, z⟩  (**)

Consider now the general case. If H ∈ L_{F,G}^{1}(X), then there is a sequence (H^{n}) of simple, predictable processes such that H^{n} → H and |H^{n}| ≤ |H|. We apply Lebesgue’s Theorem and deduce that

∫ h1_{(S,T]}H^{n} d(I_{X})_{z} → ∫ h1_{(S,T]}H d(I_{X})_{z},  (1)

and

∫ 1_{(S,T]}H^{n} d(I_{X})_{hz} → ∫ 1_{(S,T]}H d(I_{X})_{hz}.  (2)

By equality (**) for each n we have

∫ h1_{(S,T]}H^{n} d(I_{X})_{z} = ⟨∫ h1_{(S,T]}H^{n} dI_{X}, z⟩ = ⟨h ∫ 1_{(S,T]}H^{n} dI_{X}, z⟩
=⟨∫ 1_{(S,T]}H^{n} dI_{X}, hz⟩ = ∫ 1_{(S,T]}H^{n} d(I_{X})_{hz}

By (1) and (2) we deduce that

∫ h1_{(S,T]}H d(I_{X})_{z} = ∫ 1_{(S,T]}H d(I_{X})_{hz},

which is equivalent to

⟨∫ h1_{(S,T]}H dI_{X}, z⟩ = ⟨∫ 1_{(S,T]}H dI_{X}, hz⟩.

We conclude that

∫ h1_{(S,T]}H dI_{X} = h ∫ 1_{(X,T]}H dI_{X}, a.e.

Assume now hypothesis (ii). Since the measure I_{X} is σ−additive the process X is summable. Then observe that the assumptions of (ii) are the same as the assumptions in Proposition 11.5 (ii) of [Din00]. Hence

∫ h1_{(S,T]}H dI_{X} = h ∫ 1_{(X,T]}H dI_{X},

which concludes our proof.

Proposition 11. Let X : ℝ × Ω → E ⊂ L(F, G) be a p-additive summable process relative to (F, G) and T a stopping time.

a) For every z ∈ (L_{G}^{p})^{∗} and every B ∈ P we have:

(I_{X^{T}})_{z}(B) = (I_{X})_{z}(B ∩ [0, T]).

b) The measure I_{X^{T}} : R → L_{E}^{p} has finite semivariation relative to (F, L_{G}^{p})

c) If T is a simple stopping time then the process X^{T} is summable.

Proof. a) First we prove that if T and S are simple stopping times then we have I_{X}((S, T]) = X_{T} − X_{S}.

Assume that T is a simple stopping time of the form

T = ∑_{1≤i≤n} 1_{A_{i}}t_{i},

with 0 < t_{i} ≤ ∞, t_{i} ≠ t_{j} for i ≠ j, A_{i} ∈ F_{t_{i}} are mutually disjoint and ⋃_{1≤i≤n} A_{i} = Ω. Then [0, T] = ⋃_{1≤i≤n}[0, t_{i}] × A_{i} is a disjoint union. Hence I_{X}([0, T]) = ∑_{i} I_{X}([0, t_{i}] × A_{i}) = ∑_{i} 1_{A_{i}}X_{t_{i}} = X_{T}. Since (S, T] = [0, T] − [0, S] and I_{X} is an additive measure, we have I_{X}((S, T]) = I_{X}([0, T]) − I_{X}([0, S]) = X_{T} − X_{S}.

Next observe that if T is a simple stopping time then T ∧t is also a simple stopping time. In fact, if T = ∑_{1≤i≤n} 1_{A_{i}}t_{i} then T ∧ t = ∑_{i:t_{i}<t} 1_{A_{i}}t_{i} + ∑_{i:t_{i}≥t} 1_{A_{i}}t which is a simple stopping time.

Now we prove that for B ∈ R we have

I_{X^{T}} (B) = I_{X}([0, T] ∩ B).

In fact, for A ∈ F_{0} we have

I_{X^{T}} ({0} × A) = 1_{A}X_{0} = I_{X}({0} × A) = I_{X}([0, T] ∩ ({0} × A)).

For s < t and A ∈ F_{s} we have,

I_{X^{T}} ((s, t] × A) = 1_{A}(X_{t}^{T} − X_{s}^{T}) = 1_{A}(X_{T ∧t} − X_{T ∧s})
=1_{A}(I_{X}((T ∧ s, T ∧ t]) = 1_{A} ∫ 1_{(s,t]}1_{[0,T]}dI_{X}
= ∫ 1_{A}1_{(s,t]}1_{[0,T]}dI_{X} = I_{X}([0, T] ∩ ((s, t] × A)).  (*)

We used the above Proposition 10 with h = 1_{A}, (S, T] = (s, t] and H = 1_{[0,T]}.

Next we consider the general case, with T a stopping time.

For A ∈ F_{0} we have

I_{X^{T}} ({0} × A) = 1_{A}X_{0} = I_{X}({0} × A) = I_{X}([0, T] ∩ ({0} × A)).

Let y ∈ F and z ∈ (L_{G}^{p})^{∗}. We have

⟨(I_{X^{T}})_{z}({0} × A), y⟩ = ⟨I_{X^{T}} ({0} × A)y, z⟩
=⟨I_{X}([0, T] ∩ ({0} × A))y, z⟩ = ⟨(I_{X})_{z}([0, T] ∩ ({0} × A)), y⟩  (1)

For s < t and A ∈ F_{s} we have,

I_{X^{T}} ((s, t] × A) = 1_{A}(X_{t}^{T} − X_{s}^{T}) = 1_{A}(X_{T ∧t} − X_{T ∧s})  (2)

Let T_{n} be a sequence of simple stopping times such that T_{n} ↓ T. Let y ∈ F and z ∈ (L_{G}^{p})^{∗}. We have by (2):

⟨(I_{X^{T}})_{z}((s, t] × A), y⟩ = ⟨I_{X^{T}} ((s, t] × A)y, z⟩ = ⟨1_{A}(X_{T ∧t} − X_{T ∧s})y, z⟩

= lim_{n→∞}⟨1_{A}(X_{T_{n}∧t} − X_{T_{n}∧s})y, z⟩.

By (*) we have:

lim_{n→∞}⟨1_{A}(X_{T_{n}∧t} − X_{T_{n}∧s})y, z⟩ = lim_{n→∞}⟨I_{X}([0, T_{n}] ∩ ((s, t] × A))y, z⟩

= lim_{n→∞}⟨(I_{X})_{z}([0, T_{n}] ∩ ((s, t] × A)), y⟩ = ⟨(I_{X})_{z}([0, T] ∩ ((s, t] × A)), y⟩ (3)

since (I_{X})_{z} is σ-additive. By (1) and (3) and the fact that (I_{X^{T}})_{z} is σ-additive we deduce that

(I_{X^{T}})_{z}(B) = (I_{X})_{z}(B ∩ [0, T]), for all B ∈ R  (4)

Since (I_{X})_{z} is σ-additive we deduce that (I_{X^{T}})_{z} is σ-additive, hence it can be extended to a σ-additive measure on P. Since (I_{X^{T}})_{z}(B) = (I_{X})_{z}(B ∩ [0, T]) for all B ∈ R we deduce that

(I_{X^{T}})_{z}(B) = (I_{X})_{z}(B ∩ [0, T]), for all B ∈ P,

b) Let A be a set in R. By Proposition 4.15 in [Din00] we have svar_{F,L_{G}^{p}}I_{X^{T}} (A) < ∞ if and only if var(I_{X^{T}})_{z}(A) < ∞ for each z ∈ (L_{G}^{p})^{∗}. But

sup_{z∈((L_{G}^{p})^{∗})_{1}} var(I_{X^{T}})_{z}(A) = sup_{z∈((L_{G}^{p})^{∗})_{1}} var(I_{X})_{z}(A ∩ [0, T])

= svar_{F,L_{G}^{p}}I_{X}(A ∩ [0, T]) < ∞,

and Assertion b) is proved.

c) Assume T is a simple stopping time. By the equality (*) we have

I_{X^{T}} (B) = I_{X}([0, T] ∩ B), for B ∈ R.

Since X is p-additive summable relative to (F, G), I_{X} has a canonical additive extension I_{X} : P → L_{G}^{p}. The equality

I_{X^{T}} (A) = I_{X}([0, T] ∩ A), for A ∈ P,

defines an additive extension of I_{X^{T}} to P. Since the measure I_{X} has finite semivariation relative to (F, L_{G}^{p}) (X is additive summable), the measure I_{X^{T}} has finite semivariation relative to (F, L_{G}^{p}) also. Moreover, for each z ∈ (L_{G}^{p})^{∗}, by Assertion a), the measure (I_{X^{T}})_{z} defined on P is σ−additive. Therefore X^{T} is additive summable. We have |(I_{X^{T}})_{z}|(A) = |(I_{X})_{z}|([0, T] ∩ A) for A ∈ P since |(I_{X})_{z}| is the canonical extension of its restriction on R. Then |(I_{X^{T}})_{z}| is the canonical extension of its restriction to R. it follows that I_{X^{T}} is the canonical extension of its restriction to R.

The next theorem gives the relationship between the stopped stochastic integral and the integral of the process 1_{[0,T]}H. The same type of relation was proved in Theorem 11.6 in [Din00].

Theorem 12. Let H ∈ L_{F,G}^{1}(X) and let T be a stopping time. Then 1_{[0,T]}H ∈ L_{F,G}^{1}(X) and

(1_{[0,T]}H) · X = (H · X)^{T} .

Proof. Suppose first that T is a simple stopping time of the form

T = ∑_{1≤i≤n} 1_{A_{i}}t_{i}

with 0 ≤ t_{1} ≤ t_{2} ≤ … t_{n} ≤ +∞, A_{i} ∈ F_{t_{i}} mutually disjoint and with union Ω. Then for t ≥ 0 we have

(H · X)_{t}^{T} (ω) = ∑_{1≤i≤n}(H · X)_{t_{i}∧t}(ω)1_{A_{i}}(ω).

In fact, for ω ∈ Ω there is 1 ≤ i ≤ n such that ω ∈ A_{i}. Then T (ω) = t_{i}, hence

(H · X)_{t}^{T} (ω) = (H · X)_{t_{i}∧t}(ω).

On the other hand

(1_{[0,T]}H) · X)_{t}(ω) = ∑_{1≤i≤n}(H · X)_{t_{i}∧t}(ω)1_{A_{i}}(ω).

In fact,

(∫_{[0,t]} 1_{[0,T]}HdI_{X})(ω) = (∫_{[0,t]} ∑_{1≤i≤n} 1_{[0,t_{i}]}1_{A_{i}}HdI_{X})(ω) = ∑_{1≤i≤n}(∫_{[0,t_{i}∧t]} 1_{A_{i}}HdI_{X})(ω)

= ∑_{1≤i≤n}(∫_{[0,∞]} H1_{A_{i}}dI_{X})(ω) − ∑_{1≤i≤n}(∫_{(t_{i}∧t,∞]} 1_{A_{i}}HdI_{X})(ω)

= (∫_{[0,∞]} HdI_{X})(ω) − ∑_{1≤i≤n} 1_{A_{i}}(ω)(∫_{(t_{i},∞]} HdI_{X})(ω)

= ∑_{1≤i≤n} 1_{A_{i}}(ω)(∫_{[0,∞]} HdI_{X})(ω) − ∑_{1≤i≤n} 1_{A_{i}}(ω)(∫_{(t_{i},∞]} HdI_{X})(ω)

= ∑_{1≤i≤n} 1_{A_{i}}(ω)(∫_{[0,t_{i}∧t]} HdI_{X})(ω) = ∑_{1≤i≤n}(H · X)_{t_{i}∧t}(ω)1_{A_{i}}(ω),

where the 4th equality is obtained by applying Proposition 10, with h = 1_{A_{i}}. Hence, for T simple, we have1_{[0,T]}H ∈ L_{F,G}^{1}(X) and

(1_{[0,T]}H) · X = (H · X)^{T} .

Now choose T arbitrary. Then there is a decreasing sequence (T_{n}) of simple stopping times, such that T_{n} ↓ T.

Note first that since (H · X) is cadlag we have

(H · X)^{T_{n}} → (H · X)^{T} .  (1)

Moreover for t ≥ 0 we have 1_{[0,T_{n}∧t]}H ↓ 1_{[0,T_{n}∧t]}H pointwise. Since 1_{[0,T_{n}∧t]}H ∈ L_{F,G}^{1}(X), for each (z ∈ L_{G}^{p})^{∗} we have 1_{[0,T_{n}∧t]}H ∈ L_{F}^{1} (|(I_{X})_{z}|), hence

⟨∫ 1_{[0,T_{n}∧t]}HdI_{X}, z⟩ = ∫ 1_{[0,T_{n}∧t]}Hd(I_{X})_{z} → ∫ 1_{[0,T ∧t]}Hd(I_{X})_{z} = ⟨∫ 1_{[0,T ∧t]}HdI_{X}, z⟩.

By Theorem 4 we conclude that ∫ 1_{[0,T ∧t]}HdI_{X} = ∫_{[0,t]} 1_{[0,T]}HdI_{X} ∈ L_{G}^{p} and

∫ 1_{[0,T_{n}∧t]}HdI_{X} → ∫ 1_{[0,T ∧t]}HdI_{X},

or

∫_{[0,t]} 1_{[0,T_{n}]}HdI_{X} → ∫_{[0,t]} 1_{[0,T]}HdI_{X}.

Since for each n we have (1_{[0,T_{n}]}H · X)_{t} = (H · X)_{t}^{T_{n}}, by (1) we deduce that ∫_{[0,t]} 1_{[0,T]}HdI_{X} = (H · X)_{t}^{T} . Hence the mapping t ↦ ∫_{[0,t]} 1_{[0,T]}HdI_{X} is cadlag, from which we conclude that 1_{[0,T]}H ∈ L_{F,G}^{1}(X). Moreover

(1_{[0,T]}H · X)_{t} = (H · X)_{T ∧t} = (H · X)_{t}^{T} .

The next corollary is a useful particular case of the previous theorem:

Corollary 13. For every stopping time T we have

1_{[0,T]} · X = X^{T} .

Proof. Taking H = 1 ∈ L_{F,G}^{1}(X) and applying Theorem 12 we conclude that 1_{[0,T]} · X = X^{T}.

The following theorem gives the same type of results as Theorem 11.8 in[Din00].

Theorem 14. Let S ≤ T be stopping times and assume that either

(i) h : Ω → ℝ is a simple, F_{S}-measurable function and H ∈ L_{F,G}^{1}(X), or (ii) The measure I_{X} is σ−additive, h : Ω → F is a simple, F_{S}-measurable function and H ∈ L_{ℝ,E}^{1}(X). Then 1_{(S,T]}H and h1_{(S,T]}H are integrable with respect to X and

(h1_{(S,T]}H) · X = h[(1_{(S,T]}H) · X].

Proof. Note that

1_{(S,T]}H = 1_{[0,T]}H − 1_{[0,S]}H

Assume first the case (i). Applying Theorem 12 for 1_{[0,T]}H and 1_{[0,S]}H we conclude that 1_{(S,T]}H ∈ L_{F,G}^{1}(X).

If for each t ≥ 0 we apply Proposition 10, we obtain

∫_{[0,t]} h1_{(S,T]}HdI_{X} = h ∫_{[0,t]} 1_{(S,T]}HdI_{X}.

Since 1_{(S,T]}H ∈ L_{F,G}^{1}(X) we deduce that h1_{(S,T]}H ∈ L_{F,G}^{1}(X) and

((h1_{(S,T]}H) · X)_{t} = h((1_{(S,T]}H) · X)_{t},

which concludeds the proof of case (i). Case (ii) is treated similarly.

2.6 The Integral ∫ HdI_{X^{T}}

In this section we define the set of processes integrable with respect to the measure I_{X^{T}} with finite semivariation relative to the pair (F, L_{G}^{p}).

Let X : ℝ_{+} ×Ω → E ⊂ L(F, G) be a cadlag, adapted process and assume X is p-additive summable relative to (F, G).

Consider the additive measure I_{X} : P → L_{E}^{p} ⊂ L(F, L_{G}^{p}) with bounded semivariation Ĩ_{F,G} relative to (F, L_{G}^{p}), such that each of the measures (I_{X})_{z} with z ∈ (L_{G}^{p})^{∗} is σ-additive.

To simplify the notations denote m = I_{X^{T}}. We proved in the previous proposition that the measure m : R → L_{E}^{p} ⊂ L(F, L_{G}^{p}) has bounded semivariation relative to (F, L_{G}^{p}), on R, and for each z ∈ (L_{G}^{p})^{∗} the measures m_{z}, is σ-additive. In order for the process X^{T} to be additive summable we need the measure m : R → L_{E}^{p} to have an extension m : P → L_{E}^{p} with finite semivariation and such that each of the measures m_{z} with z ∈ (L_{G}^{p})^{∗} is σ−additive. Applying Theorem 7 from Bongiorno–Dinculeanu, citeBD2001, the measure m has a unique canonical extension m : P → (L_{E}^{p})^{∗∗}, with bounded semivariation such that for each z ∈ (L_{G}^{p})^{∗} the measure m_{z}, is σ-additive and has bounded variation |m_{z}|, therefore X^{T} is summable.

Then we have

m̃_{F,L_{G}^{p}} = sup{|m_{z}| : z ∈ (L_{G}^{p})^{∗}, ∥z∥_{q} ≤ 1}.

We denote by F_{F,G}(X^{T}) the space of predictable processes H : ℝ_{+} ×Ω → F such that

m̃_{F,G}(H) = m̃_{F,L_{G}^{p}}(H) = sup{∫ |H|d|m_{z}| : ∥z∥ ≤ 1} < ∞.

Let H ∈ F_{F,G}(X^{T}); then H ∈ L_{F}^{1} (|m_{z}|) for every z ∈ (L_{G}^{p})^{∗}, hence the integral ∫ Hdm_{z} is defined and is a scalar. The mapping z ↦ ∫ Hdm_{z} is a linear continuous functional on (L_{G}^{p})^{∗}, denoted ∫ Hdm. Therefore, ∫ Hdm ∈ (L_{G}^{p})^{∗∗},

⟨∫ Hdm, z⟩ = ∫ Hdm_{z}, for z ∈ (L_{G}^{p})^{∗}.

We denote by L_{F,G}^{1}(X^{T}) the set of processes H ∈ F_{F,G}(I_{X}^{T}) satisfying the following two conditions:

a) ∫_{[0,t]} Hdm ∈ L_{G}^{p} for every t ∈ ℝ_{+};

b) The process (∫_{[0,t]} Hdm)_{t≥0} has a cadlag modification.

Theorem 15. Let X : ℝ → E ⊂ L(F, G) be a p-additive summable process relative to (F, G) and T a stopping time.

a) We have H ∈ F_{F,G}(X^{T}) iff 1_{[0,T]}H ∈ F_{F,G}(X) and in this case we have:

∫ HdI_{X^{T}} = ∫ 1_{[0,T]}HdI_{X}.

b) We have H ∈ L_{F,G}^{1}(X^{T}) iff 1_{[0,T]}H ∈ L_{F,G}^{1}(X) and in this case we have:

H · X^{T} = (1_{[0,T]}H) · X.

If H ∈ L_{F,G}^{1}(X), then H ∈ L_{F,G}^{1}(X^{T}), 1_{[0,T]}H ∈ L_{F,G}^{1}(X) and

(H · X)^{T} = H · X^{T} = (1_{[0,T]}H) · X.

Proof. a) Define m : R → E by m(B) = I_{X^{T}} (B) for B ∈ R. We proved in Theorem 11 (a) that for every z ∈ (L_{G}^{p})^{∗} we have

m_{z}(B) = (I_{X})_{z}(B ∩ [0, T]), for all B ∈ R.  (*)

Since (I_{X})_{z}((·) ∩ [0, T]) is a σ−additive measure, with bounded variation on P satisfying (*) and since P is the σ−algebra generated by R, by the uniqueness theorem 7.4 in [Din00] we conclude that

m_{z}(B) = (I_{X})_{z}(B ∩ [0, T]), for all B ∈ P.

Let H ∈ F_{F,G}(X^{T}) = ⋂_{∥z∥_{q}≤1, z∈(L_{G}^{p})^{∗}} L_{F}^{1} (m_{z}). From the previous equality we deduce that

∫ Hdm_{z} = ∫ 1_{[0,T]}Hd(I_{X})_{z},

therefore

∫ HdI_{X^{T}} = ∫ 1_{[0,T]}HdI_{X},

and this is the equality in Assertion a).

b) To prove Assertion b) we replace H with 1_{[0,t]}H in the previous assertion and deduce that 1_{[0,t]}H ∈ F_{F,G}(X^{T}) iff 1_{[0,t]}1_{[0,T]}H ∈ F_{F,G}(X) and in this case we have

∫_{[0,t]} HdI_{X^{T}} = ∫_{[0,t]} 1_{[0,T]}HdI_{X}.

It follows that H ∈ L_{F,G}^{1}(X^{T}) iff 1_{[0,T]}H ∈ L_{F,G}^{1}(X) and in this case we have

(H · X^{T})_{t} = ((1_{[0,T]}H) · X)_{t}.

If now H ∈ L_{F,G}^{1}(X), then, from Theorem 12 we deduce that 1_{[0,T]}H ∈ L_{F,G}^{1}(X) and

(1_{[0,T]}H) · X = (H · X)^{T} .

2.7 Convergence Theorems

Assume X is p-additive summable relative to (F, G). In this section we shall present several convergence theorems.

Lemma 16. Let (H^{n}) be a sequence in L_{F,G}^{1}(X) and assume that H^{n} → H in F_{F,G}(X).Then there is a subsequence (r_{n}) such that

(H^{r_{n}} · X)_{t} → (H · X)_{t} = ∫_{[0,t]} HdI_{X}, a.s., as n → ∞,

uniformly on every bounded time interval.

Proof. Since H^{n} is a convergent sequence in F_{F,G}(X) there is a subsequence H^{r_{n}} of (H^{n}) such that

Ĩ_{F,G}(H^{r_{n}} − H^{r_{n+1}}) ≤ 4^{−n}, for each n.

Let t_{0} > 0. Define the stopping time

u_{n} = inf{t : |(H^{r_{n}} · X)_{t} − (H^{r_{n+1}} · X)_{t}| > 2^{−n}} ∧ t_{0}.

By Theorem 12 applied to the stopping time u_{n}, we obtain

(H^{r_{n}} · X)_{u_{n}} = (H^{r_{n}} · X)_{∞}^{u_{n}} = ((1_{[0,u_{n}]}H^{r_{n}}) · X)_{∞} = ∫_{[0,u_{n}]} H^{r_{n}}dI_{X},

hence

E(|(H^{r_{n}} · X)_{u_{n}} − (H^{r_{n+1}} · X)_{u_{n}}|) = E(| ∫_{[0,u_{n}]} H^{r_{n}}dI_{X} − ∫_{[0,u_{n}]} H^{r_{n+1}}dI_{X}|)
= E(| ∫_{[0,u_{n}]}((H^{r_{n}} − H^{r_{n+1}})dI_{X}|) = (∥ ∫_{[0,u_{n}]}(H^{r_{n}} − H^{r_{n+1}})dI_{X}∥_{L_{G}^{1}}
≤ ∥ ∫_{[0,u_{n}]}(H^{r_{n}} − H^{r_{n+1}})dI_{X}∥_{L_{G}^{p}} ≤ Ĩ_{F,G}(H^{r_{n}} − H^{r_{n+1}}) ≤ 4^{−n}.  (*)

Using inequality (*) and following the same techniques as in Theorem 12.1 a) in [Din00] one could show first that the sequence (H^{r_{n}} · X)_{t} is a Cauchy sequence in L_{G}^{p} uniformly for t < t_{0} and then conclude that

(H^{r_{n}} · X)_{t} → ∫_{[0,t]} HdI_{X},

uniformly on every bounded time interval.

Theorem 17. Let (H^{n}) be a sequence from L_{F,G}^{1}(X) and assume that H^{n} → H in F_{F,G}(X). Then:

a) H ∈ L_{F,G}^{1}(X).

b) (H^{n} · X)_{t} → (H · X)_{t}, in L_{G}^{p}, for t ∈ [0, ∞].

c) There is a subsequence (r_{n}) such that

(H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s., as n → ∞,

uniformly on every bounded time interval.

Proof. For every t ≥ 0 we have 1_{[0,t]}H^{n} → 1_{[0,t]}H in F_{F,G}(X). Since the integral is continuous, we deduce that

(H^{n} · X)_{t} = ∫_{[0,t]} H^{n}dI_{X} → ∫_{[0,t]} HdI_{X}, in (L_{G^{∗}}^{p})^{∗}.

Since H^{n} ∈ L_{F,G}^{1}(X) we have ∫_{[0,t]} H^{n}dI_{X} ∈ L_{G}^{p} and

(H^{n} · X)_{t} → ∫_{[0,t]} HdI_{X}, in L_{G}^{p}.

From the previous lemma we deduce that there is a subsequence (H^{r_{n}}) such that

(H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s., as n → ∞,

uniformly on every bounded time interval. Since (H^{r_{n}} · X) are cadlag it follows that the limit is also cadlag, hence H ∈ L_{F,G}^{1}(X) which is Assertion a). Hence

(H · X)_{t} = ∫_{[0,t]} HdI_{X}, a.s.

and therefore (H^{n}·X)_{t} → (H ·X)_{t}, in L_{G}^{p}, which is Assertion b). Also observe that for the above susequence (H^{r_{n}}) we have

(H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s., as n → ∞,

uniformly on every bounded time interval.

We can restate Theorem 17 as:

Corollary 18. L_{F,G}^{1}(X) is complete.

Next we state an uniform convergence theorem. Uniform convergence implies convergence in L_{F,G}^{1}(X).

Theorem 19. Let (H^{n}) be a sequence from F_{F,G}(X). If H^{n} → H pointwise uniformly then H ∈ F_{F,G}(X) and H^{n} → H in F_{F,G}(X).

If, in addition, for each n, H^{n} is integrable, i.e. H^{n} ∈ L_{F,G}^{1}(X) then a) H ∈ L_{F,G}^{1}(X) and H^{n} → H in L_{F,G}^{1}(X);

b) For every t ∈ [0, ∞] we have (H^{n} · X)_{t} → (H · X)_{t}, in L_{G}^{p}.

c) There is a subsequence (r_{n}) such that (H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s. as n → ∞, uniformly on any bounded interval.

Proof. Assertion a) is immediate. Assertions b), c) and d) follow from Theorem 17.

Now we shall state Vitali and Lebesgue-type Convergence Theorems. They are direct consequences of the convergence Theorem 17 and of the uniform convergence Theorem 19.

Theorem 20. (Vitali). Let (H^{n}) be a sequence from F_{F,G}(X) and let H be an F -valued, predictable process. Assume that (i) Ĩ_{F,G}(H^{n}1_{A}) → 0 as Ĩ_{F,G}(A) → 0, uniformly in n and that any one of the conditions (ii) or (iii) below is true: (ii) H^{n} → H in Ĩ_{F,G}-measure; (iii) H^{n} → H pointwise and I_{F,(L_{G}^{p})^{∗}} is uniformly σ-additive (this is the case if H^{n} are real-valued, i.e., F = ℝ).

Then:

a) H ∈ F_{F,G}(X) and H^{n} → H in F_{F,G}(X).

Conversely, if H^{n}, H ∈ F_{F,G}(B, X) and H^{n} → H in F_{F,G}(X), then conditions (i) and (ii) are satisfied.

Under the hypotheses (i) and (ii) or (iii), assume, in addition, that H^{n} ∈ L_{F,G}^{1}(X) for each n. Then

b) H ∈ L_{F,G}^{1}(X) and H^{n} → H in L_{F,G}^{1}(X);

c) For every t ∈ [0, ∞] we have (H^{n} · X)_{t} → (H · X)_{t}, in L_{G}^{p};

d) There is a subsequence (r_{n}) such that (H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s., as n → ∞, uniformly on any bounded interval.

Theorem 21. (Lebesgue). Let (H^{n}) be a sequence from F_{F,G}(X) and let H be an F -valued predictable process. Assume that

(i) There is a process ϕ ∈ F_{ℝ}(B, I_{F,G}) such that

|H^{n}| ≤ ϕ for each n;

and that any one of the conditions (ii) or (iii) below is true: (ii) H^{n} → H in Ĩ_{F,G}-measure; (iii) H^{n} → H pointwise and I_{F,L_{G^{∗}}^{q}} is uniformly σ-additive (this is the case if H^{n} are real valued, i.e., F = ℝ).

Then:

a) H ∈ F_{F,G}(B, X) and H^{n} → H in F_{F,G}(X).

Assume, in addition that H^{n} ∈ L_{F,G}^{1}(X) for each n. Then

b) H ∈ L_{F,G}^{1}(X) and H^{n} → H in L_{F,G}^{1}(X);

c) For every t ∈ [0, ∞] we have (H^{n} · X)_{t} → (H · X)_{t}, in L_{G}^{p};

d) There is a subsequence (r_{n}) such that (H^{r_{n}} · X)_{t} → (H · X)_{t}, a.s., as n → ∞, uniformly on any bounded interval.

2.8 Summability of the Stochastic Integral

Assume X is p-additive summable relative to (F, G). In this section we are studying the additive summability of the stochastic integral H · X for F −valued processes H.

If H is a real valued processes then in order for the stochastic integral H · X to be defined we need each of the measure (I_{X})_{z}, for z ∈ (L_{E}^{P})^{∗}, to be σ−additve, hence the measure I_{X} would be σ−additive. Therefore the process X would be summable. In this case the summability of the stochastic integral is proved in Theorem 13.1 of [Din00].

The next theorem shows that if H is F −valued then the measure I_{H·X} is σ−additive even if I_{X} is just additive.

Theorem 22. Let H ∈ L_{F,G}^{1}(X) be such that ∫_{A} HdI_{X} ∈ L_{G}^{p} for A ∈ P. Then the measure I_{H·X} : R → L_{G}^{p} has a σ−additive extension I_{H·X} : P → L_{G}^{p} to P.

Proof. We first note that H ·X : ℝ_{+} ×Ω → G = L(ℝ, G) is a cadlag adapted process with (H · X)_{t} ∈ L_{G}^{p} for t ≥ 0 ( by the definition of H · X).

Since ∫_{A} HdI_{X} ∈ L_{G}^{p} for every A ∈ P, by Proposition ??, with m = I_{X} and g = H, we deduce that HI_{X} is σ−additive on P.

Next we prove that for any predictable rectangle A ∈ R we have

I_{H·X}(A) = ∫_{A} HdI_{X}.  (1)

In fact, consider first A = {0} × B with B ∈ F_{0}. Using Proposition 10 for h = 1_{B} we have

I_{H·X}({0} × B) = 1_{B}((H · X)_{0}) = 1_{B} ∫_{{0}} HdI_{X}
= ∫_{{0}} 1_{B}HdI_{X} = ∫_{{0}×B} HdI_{X};

Let now A = (s, t] × B with B ∈ F_{s}. Using Proposition 10 for h = 1_{B} and (S, T] = (s, t] we have

I_{H·X}((s, t] × B) = 1_{B}((H · X)_{t} − (H · X)_{s})
=1_{B}(∫_{[0,t]} HdI_{X} − ∫_{[0,s]} HdI_{X}) = 1_{B} ∫_{(s,t]} HdI_{X}
= ∫_{(s,t]} 1_{B}HdI_{X} = ∫_{(s,t]×B} HdI_{X};

and the desired equality is proved.

Since the measure A ↦ ∫_{A} HdI_{X} is σ−additive for A ∈ P it will follow that I_{H·X} can be extended to a σ-additive measure on P by the same equality

I_{H·X}(A) = ∫_{A} HdI_{X}, for A ∈ P.  (2)

The next theorem states the summability of the stochastic integral.

Theorem 23. Let H ∈ L_{F,G}^{1}(X) be such that ∫_{A} HdI_{X} ∈ L_{G}^{p} for A ∈ P. Then:

a) H · X is p-summable, hence p-additive summable relative to (ℝ, G) and

dI_{H·X} = d(HI_{X}).

b) For any predictable process K ≥ 0 we have

(Ĩ_{H·X})_{ℝ,G}(K) ≤ (Ĩ_{X})_{F,G}(KH).

c) If K is a real-valued predictable process and if KH ∈ L_{F,G}^{1}(X), then K ∈ L_{ℝ,G}^{1}(H · X) and we have

K · (H · X) = (KH) · X.

Proof. By Theorem 22 we know that the measure I_{H·X} is σ−additive. Therefore To prove (a) we only need to show that the extension of I_{H·X} to P has finite semivariation relative to (ℝ, L_{G}^{p}).

Let z ∈ (L_{G}^{p})^{∗}. From the equality (2) in Theorem 22 we deduce that for every A ∈ P, and we have

(I_{H·X})_{z}(A)⟩ = ⟨I_{H·X}(A), z⟩ = ⟨∫_{A} HdI_{X}, z⟩ = ∫_{A} Hd(I_{X})_{z}.

From this we deduce the inequality

|(I_{H·X})_{z}|(A) ≤ ∫_{A} |H|d|(I_{X})_{z}|, for A ∈ P.  (*)

Taking the supremum for z ∈ (L_{G}^{P})_{1}^{∗} we obtain

sup{|(I_{H·X})_{z}|(A), z ∈ (L_{G}^{P})_{1}^{∗}} ≤ sup{∫_{A} |H|d|(I_{X})_{z}|, z ∈ (L_{G}^{P})_{1}^{∗}}

≤ sup{∫ |1_{A}H|d|(I_{X})_{z}|, z ∈ (L_{G}^{P})_{1}^{∗}}, for A ∈ P.

Therefore

(Ĩ_{H·X})_{ℝ,G}(A) ≤ (Ĩ_{X})_{F,G}(1_{A}H) < ∞, for A ∈ P.

It follows that H ·X is p-summable, hence p−additive summable, relative to (ℝ, G) and this proves Assertion a).

Since the extension to ¶ of the measure I_{X·H} is σ−additive and has finite semivariation b) and c) follow from Theorem 13.1 of [Din00].

2.9 Summability Criterion

Let Z ⊂ L_{E^{∗}}^{q} be any closed subspace norming for L_{E}^{p}. The next theorem differs from the summability criterion in [Din00] by the fact that the restrictive condition c_{0} ∉ E was not imposed. Also note that this theorem does not give us necessary and sufficient conditions for the sumability of the precess.

Theorem 24. Let X : ℝ_{+} × Ω → E be an adapted, cadlag process such that X_{t} ∈ L_{E}^{p} for every t ≥ 0. Then the Assertions a)–d) below are equivalent.

a) I_{X} : R → L_{E}^{p} has an additive extension I_{X} : P → Z^{∗} such that for each g ∈ Z, the real valued measure ⟨I_{X}, g⟩ is a σ−additive on P.

b) I_{X} is bounded on R;

c) For every g ∈ Z, the real valued measure ⟨I_{X}, g⟩ is bounded on R;

d) For every g ∈ Z, the real valued measure ⟨I_{X}, g⟩ is σ−additive and bounded on R.

Proof. The proof will be done as follows: b) ⟺ c) ⟺ d) and a) ⟺ d). b) ⟹ c) and c) ⟹ b) can be proven in the same fashion as in [Din00].

c) ⟹ d) Assume c), and let g ∈ Z. The real valued measure ⟨I_{X}, g⟩ is defined on R by

⟨I_{X}, g⟩(A) = ⟨I_{X}(A), g⟩ = ∫ ⟨I_{X}(A), g⟩dP, for A ∈ R.

By assumption, ⟨I_{X}, g⟩ is bounded on R. We need to prove that the measure ⟨I_{X}, g⟩ is σ− additive. For that consider, as in [Din00], the real-valued process XG = (⟨X_{t}, G_{t}⟩)_{t≥0}, where G_{t} = E(g|F_{t}) for t ≥ 0. Then XG : ℝ_{+} × Ω → ℝ is a cadlag, adapted process and it can be proven, using the same techniques as in [Din00] that it is a quasimartingale.

Now, for each n, define the stopping time

T_{n}(ω) = inf{t : |X_{t}| > n}.

Then T_{n} ↑ ∞ and |X_{t}| ≤ n on [0, T_{n}). Since XG is a quasimartingale on (0, ∞], we know that (XG)_{T_{n}} ∈ L^{1} (Proposition A 3.5 in [BD87]: XG is a quasimartingale on (0, ∞] iff XG is a quasimartingale on (0, ∞) and sup_{t} ∥XG∥_{1} < ∞.)

Moreover,

|(XG)_{t}^{T_{n}}| = |(XG)_{t}|1_{{t<T_{n}}} + |(XG)_{T_{n}}|1_{{t≥T_{n}}}  (2)
≤ |X_{t}||G_{t}|1_{{t<T_{n}}} + |(XG)_{T_{n}}|1_{{t≥T_{n}}}
≤ n|G_{t}|1_{{t<T_{n}}} + |(XG)_{T_{n}}|1_{{t≥T_{n}}}.

Besides, since G_{t} = E(g|F_{t}) it follows that G is a uniformly integrable martingale.

Next we prove that the family {(XG)_{T}^{T_{n}}, T simple stopping time} is uniformly integrable.

In fact, note that by inequality (2) we have

∫_{{|(XG)_{T}^{Tn}|>p}} |(XG)_{T}^{T_{n}}|dP

≤ ∫_{{|(XG)_{T}^{Tn}|>p}∩{T <T_{n}}} n|(XG)_{T}^{T_{n}}|dP + ∫_{{|(XG)_{T}^{Tn}|>p}∩{T ≥T_{n}}} |(XG)_{T_{n}}|dP (3)

Now observe that

{|(XG)_{T} | > p} ∩ {T < T_{n}} = {|⟨X_{T} , G_{T} ⟩| > p} ∩ {T < T_{n}} ⊂ {|X|_{T} |G|_{T} > p} ∩ {T < T_{n}} ⊂ {p < n|G_{T} |} ∩ {T < T_{n}} ⊂ {p < nG_{T} }

Since G is a uniformly integrable martingale, it is a martingale of class D; from n|G_{t}|1_{{t<T_{n}}} ≤ n|G_{t}| we deduce that n|G_{t}|1_{{t<T_{n}}} is a martingale of class (D):

lim_{p→∞} ∫_{{n|G_{t}|1_{{t<Tn}}>p}} n|G_{t}|1_{{t<T_{n}}}dP ≤ lim_{p→∞} ∫_{{n|G_{t}|>p}} n|G_{t}|dP

= n lim_{p→∞} ∫_{\{|G_{t}|> \frac{p}{n} \}} n|G_{t}|dP = lim_{\frac{p}{n} →∞} ∫_{\{n|G_{t}|>p\}} n|G_{t}|dP = 0.

Hence there is a p_{1ϵ} such that for any p ≥ p_{1ϵ} and any simple stopping time T we have

∫_{\{|(XG)_{T}^{Tn}|>p\}∩\{T <T_{n}\}} n|(XG)_{T}^{T_{n}}|dP ≤ ∫_{\{n|G_{t}|>p\}} n|G_{t}|dP < \frac{ϵ}{2}  (4)

We look now at the second term of the right hand side of the inequality (3).

∫_{{|(XG)_{T}^{Tn}|>p}∩{T ≥T_{n}}} |(XG)_{T_{n}}|dP ≤ ∫_{{|(XG)_{Tn}|>p}} |(XG)_{T_{n}}|dP

Since (XG)_{T_{n}} ∈ L^{1}, for every ϵ > 0 there is a p_{2ϵ} > 0 such that for every p ≥ p_{2ϵ} we have

∫_{\{|(XG)_{T}^{Tn}|>p\}} |(XG)_{T_{n}}|dP < \frac{ϵ}{2}  (5)

If we put (4) and (5) together we deduce that for every ϵ > 0 there is a p_{ϵ} = max(p_{1ϵ}, p_{2ϵ}) such that for any p > p_{ϵ} and any T simple stopping time we have

∫_{\{|(XG)_{T}^{Tn}|>p\}} |(XG)_{T}^{T_{n}}|dP < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ,

which proves the fact that (XG)^{T_{n}} is a quasimartingale of class (D). From Theorem 14.2 of [Din00] we deduce that the Doléans measure μ_{(XG)^{Tn}} associated to the process (XG)^{T_{n}} is σ−additive and has bounded variation on R, hence it can be extended to a σ-additive measure with bounded variations on P (Theorem 7.4 b) of [Din00]).

Next we show that for any B ∈ P we have

μ_{(XG)^{Tn}}(B) = μ_{XG}(B ∩ [0, T_{n}]).

In fact, for A ∈ F_{0} we have

μ_{(XG)^{Tn}}({0} × A) = μ_{XG}(({0} × A) ∩ [0, T_{n}]).

and for (s, t] × A with A ∈ F_{s} we have

μ_{(XG)^{Tn}}((s, t] × A) = E(1_{A}((XG)_{t}^{T_{n}} − (XG)_{s}^{T_{n}})) = μ_{XG}(((s, t] × A) ∩ [0, T_{n}]), which proves our equality. Hence the measure μ_{XG} is σ-additive on the σ-ring P ∩ [0, T_{n}] for each n, hence it is σ-additive on the ring

B = ⋃_{1≤n<∞} P ∩ [0, T_{n}].

Next we observe that μ_{XG} is bounded on R, therefore it has bounded variation on R which implies that the measure defined on B ∩ R is σ−additive and has bounded variation. Since B ∩ R generates P, by Theorem 7.4 b) of [Din00], μ_{XG} can be extended to a σ-additive measure with bounded variation on P.

Since ⟨I_{X}, g⟩ = μ_{XG}, it follows that ⟨I_{X}, g⟩ is bounded and σ-additive on R, thus d) holds. The implication d)⟹c) is evident.

a) ⟹ d) is evident since for each g ∈ Z, the measure ⟨I_{X}, g⟩ is σadditive on P and since any σ−additive measure on a σ−algebra is bounded we conclude that for g ∈ Z, the measure ⟨I_{X}, g⟩ is bounded on P hence on R.

Next we prove d)⟹a). Assume d) is true. Then the real valued measure ⟨I_{X}, g⟩ is σ−additive and bounded on R. Since we proved that b) ⟺ c) ⟺ d) we deduce from (1) that

|⟨I_{X}, g⟩(A)| ≤ M∥g∥ for all A ∈ R

where M = sup{|I_{X}(A)| : A ∈ R}. By Proposition 2.16 of [Din00] it follows that

the measure ⟨I_{X}(·), g⟩ has bounded variation |⟨I_{X}, g⟩|(·) satisfying

|⟨I_{X}, g⟩|(A) ≤ 2M∥g∥, for A ∈ R.

Applying Proposition 4.15 in [Din00] we deduce that Ĩ_{X}_{ℝ,E} is bounded. By Theorem 3.7 b) of [BD01] we conclude that the measure I_{X} : R → L_{E}^{p} has an additive extension I_{X} : P → Z^{∗∗} to P such that for each g ∈ Z, the real valued measure ⟨I_{X}, g⟩ is a σ−additive on P which is Assertion a).

3 Examples of Additive Summable Processes

Definition 25. Let X : ℝ_{+}×Ω → E be an E-valued process. We say that X has finite variation, if for each ω ∈ Ω, the path t ↦ X_{t}(ω) has finite variation on each interval [0, t]. If 1 ≤ p < ∞, the process X has p-integrable variation if the total variation |X|_{∞} = var(X, ℝ_{+}) is p-integrable.

Definition 26. We define the variation process |X| by

|X|_{t}(ω) = var(X.(ω), (−∞, t]), for t ∈ ℝ and ω ∈ Ω,

where X_{t} = 0 for t < 0.

Noting that if m : D → E ⊂ L(F, G) is a σ-additive measure then for each z ∈ G^{∗}, the measure m_{z} : D → F^{∗} is σ−additive, we deduce that, if the process X is summable, then it is also additive summable. Hence the following theorem is a direct consequence of Theorem 19.13 in [Din00]

Theorem 27. Let X : ℝ_{+} × Ω → E be a cadlag, adapted process with integrable variation |X|. Then X is 1–additive summable relative to any embedding E ⊂ L(F, G).

Proof. If m : D → E ⊂ L(F, G) is a σ-additive measure then for each z ∈ G^{∗}, the measure m_{z} : D → F^{∗} is σ−additive. We deduce that, if the process X is summable, then it is additive summable. Hence applying Theorem 19.13 b) in [Din00] we conclude our proof.

3.1 Processes with Integrable Semivariation

Definition 28. We define the semivariation process of X relative to (F, G) by

X̃_{t}(ω) = svar_{F,G}(X.(ω), (−∞, t]), for t ∈ ℝ and ω ∈ Ω,

where X_{t} = 0 for t < 0.

Definition 29. The total semivariation of X is defined by

X̃_{∞}(ω) = sup_{t≥0} X̃_{t}(ω) = svar_{F,G}(X.(ω), ℝ), for ω ∈ Ω.

Definition 30. Let X : ℝ_{+} × Ω → E ⊂ L(F, G). The process X is said to have finite semivariation relative to (F, G), if for every ω ∈ Ω, the path t ↦ X_{t}(ω) has finite semivariation relative to (F, G) on each interval (−∞, t]. The process X is said to have p-integrable semivariation X̃_{F,G} if the total semivariation (X̃_{F,G})_{∞} belongs to L^{p}.

Remark: If X : ℝ_{+} × Ω → E ⊂ L(F, G) is a process and z ∈ G^{∗} we define, the process X_{z} : ℝ_{+} × Ω → F^{∗} by

⟨x, (X_{z})_{t}(ω)⟩ = ⟨X_{t}(ω)x, z⟩, for x ∈ F, t ∈ ℝ_{+} and ω ∈ Ω.

For fixed t ≥ 0, we consider the function X_{t} : ω ↦ X_{t}(ω) from Ω into E ⊂ L(F, G) and for z ∈ G^{∗} we define (X_{t})_{z} : Ω → F^{∗} by the equality

⟨x, (X_{t})_{z}(ω)⟩ = ⟨X_{t}(ω)x, z⟩, for ω ∈ Ω, and x ∈ F.

It follows that

(X_{t})_{z}(ω) = (X_{z})_{t}(ω), for t ∈ ℝ_{+} and ω ∈ Ω.

The semivariation X̃ can be computed in terms of the variation of the processes X_{z}:

X̃_{t}(ω) = sup_{z∈G_{1}^{∗}} |X_{z}|_{t}(ω).

If X has finite semivariation X̃, then each X_{z} has finite variation |X_{z}|.

The following theorem is an improvement over the Theorem 21.12 in [Din00], where it was supposed that c_{0} ∉ E and c_{0} ∉ G.

Theorem 31. Assume c_{0} ⊄ G. Let X : ℝ_{+} × Ω → E ⊂ L(F, G) be a cadlag, adapted process with p-integrable semivariation relative to (ℝ, E) and relative to (F, G). Then X is p-additive summable relative to (F, G)

Proof. First we present the sketch of the proof, after which we prove all the details.

The prove goes as follows:

1) First we will show that

I_{X}(A)(ω) = m_{X(ω)}(A(ω)), for A ∈ R and ω ∈ Ω,  (*)

where A(ω) = {t; (t, ω) ∈ A} and X(ω) is X_{·}(ω). For the definition of the measure m_{X(ω)} see Section 2.2.

2) Then we will prove that the measure m_{X(ω)} has an additive extension to B(ℝ_{+}), with bounded semivariation relative to (F, G) and such that for every g ∈ G^{∗} the measure (m_{X(ω)})_{g} is σ−additive.

3) Next we prove that the function ω ↦ m_{X(ω)}(M(ω)) belongs to L_{E}^{p} for all M ∈ P.

4) Then we show that the extension of the measure I_{X} to P has bounded semivariation relative to (F, L_{G}^{p}).

5) Finally we show that for each z ∈ (L_{G}^{p})^{∗} the measure (I_{X})_{z} : P → F^{∗} is σ−additive.

6) We conclude that the process X is p−additive summable.

Now we prove each step in detail.

1) First we prove (*) for predictable rectangles. Let A = {0} × B with B ∈ F_{0}. Then we have

I_{X}({0} × B)(ω) = 1_{B}(ω)X_{0}(ω) = ∫ 1_{{0}×B}(s, ω)dX_{s}(ω) = m_{X(ω)}(A(ω)).

Now let A = (s, t] × B with B ∈ F_{s}. In this case we also obtain

I_{X}((s, t]×B)(ω) = 1_{B}(ω)(X_{t}(ω)−X_{s}(ω)) = ∫ 1_{(s,t]×B}(p, ω)dX_{p}(ω) = m_{X(ω)}(A(ω)).

Since both I_{X}(A)(ω) and m_{X(ω)}(A(ω)) are additive we conclude that the equality (*) is true for A ∈ R.

2) Since X has p-integrable semivariation relative to (F, G) we infer that (X̃_{F,G})_{∞}(ω) < ∞ a.s. If we redefine X_{t}(ω) = 0 for those ω for which (X̃_{F,G})_{∞}(ω) = ∞ we obtain a process still denoted X with bounded semivariation. In this case for each ω ∈ Ω the function t ↦ X_{t}(ω) is right continuous and with bounded semivariation. By Theorem ?? we deduce that the measure m_{X(ω)} can be extended to an additive measure m_{X(ω)} : B(ℝ_{+}) → E ⊂ L(F, G), with bounded semivariation relative to (F, G) and such that for every g ∈ G^{∗} the measure (m_{X(ω)})_{g} : B(ℝ_{+}) → F^{∗} is σ−additive.

3) Since X has p-integrable semivariation relative to (F, G), for each t ≥ 0 we have X_{t} ∈ L_{E}^{p}. Hence, by step 1, the function ω ↦ m_{X(ω)}(M(ω)) belongs to L_{E}^{p} for all M ∈ R. To prove that ω ↦ m_{X(ω)}(M(ω)) belongs to L_{E}^{p} for all M ∈ P we will use the Monotone Class Theorem. We will prove that the set P_{0} of all sets M ∈ P for which the affirmation is true is a monotone class, containing R, hence equal to P. In fact, let M_{n} be a monotone sequence from P_{0} converging to M. By assumption, for each n the function ω ↦ m_{X(ω)}(M_{n}(ω)) belongs to L_{E}^{p} and for each ω the sequence (M_{n}(ω)) is monotone in B(ℝ_{+}) and has limit M(ω). Moreover |m_{X(ω)}(M_{n}(ω))| ≤ m̃_{X(ω)}(ℝ_{+} × Ω) = X̃_{∞}(ω), which is p-integrable. By Lebesgue’s Theorem we deduce that the mapping ω ↦ m_{X(ω)}(M(ω)) belongs to L_{E}^{p}, hence M ∈ P_{0}. Therefore P_{0} is a monotone class.

4) We use the equality (*) to extend I_{X} to the whole P, by

I_{X}(A)(ω) = m_{X(ω)}(A(ω)), for A ∈ P.

Let A ∈ P, (A_{i})_{i∈I} be a finite family of disjoint sets from P contained in A, and (x_{i})_{i∈I} a family of elements from F with |x_{i}| ≤ 1. Then we have

∥ ∑ I_{X}(A_{i})x_{i}∥_{p}^{p} = E(| ∑ I_{X}(A_{i})(ω)x_{i}|^{p})

= E(| ∑ m_{X(ω)}(A_{i}(ω))x_{i}|^{p}) ≤ E(|(m̃_{X(ω)})_{F,G}(A(ω))|^{p})

= ∥(m̃_{X(ω)})_{F,G}(A(ω))∥_{p}^{p} = ∥X̃_{F,G}(A(ω))∥_{p}^{p} ≤ ∥(X̃_{F,G})_{∞}∥_{p}^{p} < ∞.

Taking the supremum over all the families (A_{i}) and (x_{i}) as above, we deduce (Ĩ_{X})_{F,L_{G}^{p}} ≤ ∥(X̃_{F,G})∥_{p} < ∞.

5) Let z ∈ (L_{G}^{p})^{∗} and x ∈ F. Then z(ω) ∈ G^{∗} and for each set M ∈ P we have

⟨(I_{X})_{z}(M), x⟩ = ⟨I_{X}(M)x, z⟩ = E(⟨I_{X}(M)(ω)x, z(ω)⟩)
= E(⟨m_{X(ω)}(M(ω))x, z(ω)⟩) = E(⟨(m_{X(ω)})_{z(ω)}(M(ω)), x⟩). (3)

By step we conclude that the measure (I_{X})_{z} is σ−additive for each z ∈ (L_{G}^{p})^{∗}. 6) By the definition in step 4,

I_{X}(A)(ω) = m_{X(ω)}(A(ω)), for A ∈ P and ω ∈ Ω,

and by steps 2 and 3 we conclude that the measure I_{X} has an additive extension I_{X} : P → L_{E}^{p}. By step 5 the measure (I_{X})_{z} is σ−additive for each z ∈ (L_{G}^{p})^{∗}. By step 4 this extension has bounded semivariation. Therefore the process X is p-additive summable.

The following theorem gives sufficient conditions for two processes to be indistinguishable. For the proof see [Din00], Corollary 21.10 b’).

Theorem 32. ([Din00]21.10b’)) Assume c_{0} ⊄ E and let A, B : ℝ_{+} ×Ω → E be two predictable processes with integrable semivariation relative to (ℝ, E). If for every stopping time T we have E(A_{∞} − A_{T}) = E(B_{∞} − B_{T}), then A and B are indistinguishable.

The next theorem gives examples of processes with locally integrable variation or semivariation. For the proof see [Din00], Theorems 22.15 and 22.16.

Theorem 33. ([Din00]22.15,16) Assume X is right continuous and has finite variation |X| (resp. finite semivariation X̃_{F,G}). If X is either predictable or a local martingale, then X has locally integrable variation |X| (resp. locally integrable semivariation X̃_{F,G}).

Proposition 34. Let X : ℝ_{+} × Ω → E be a process with finite variation. If X has locally integrable semivariation X̃_{ℝ,E}, then X has locally integrable variation.

Proof. Assume X has locally integrable semivariation X̃ relative to (ℝ, E). Then there is an increasing sequence S_{n} of stopping times with S_{n} ↑ ∞ such that E(X̃_{S_{n}}) < ∞ for each n. For each n define the stopping times T_{n} by T_{n} = S_{n} ∧ inf{t| |X|_{t} ≥ n}. It follows that |X|_{T_{n}−} ≤ n. Since X has finite variation, by Proposition 6 we have Δ|X_{T_{n}}| = |ΔX_{T_{n}}| ≤ X̃_{T_{n}}. From Δ|X|_{T_{n}} = |X|_{T_{n}}−|X|_{T_{n}−} we deduce that |X|_{T_{n}} = |X|_{T_{n}−}+Δ|X_{T_{n}}| ≤ n+X̃_{T_{n}}; Therefore E(|X|_{T_{n}}) ≤ n + E(X̃_{T_{n}}) < ∞; hence X has locally integrable variation.

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