Stochastic Integration in Abstract Spaces

Abstract

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let 𝐸, 𝐹, and 𝐺 be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of 𝐸×𝐹 into 𝐺. If 𝐻 is an integrable, 𝐸-valued predictable process and 𝑋 is an 𝐹-valued square integrable martingale, then there exists a 𝐺-valued process (∫𝐻𝑑𝑋)_𝑡 called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.

1. Introduction

In this note, we announce the existence of a stochastic integral in a nuclear space setting. The nuclear spaces are assumed to have special properties which are given in Section 3.1 below. Our main result will now be stated. All definitions and pertinent concepts will be given in Sections 2 and 3, as well as a presentation of the construction.

Theorem 1.1 .1. Let 𝐸, 𝐹, and 𝐺 be nuclear spaces which satisfy the special conditions listed in Section 3.1, and suppose that there is a continuous bilinear mapping of 𝐸×𝐹 into 𝐺. Assume that 𝑋 is an 𝐹-valued square integrable martingale.
If 𝐻 is a bounded 𝐸-valued predictable process, then there exists a 𝐺-valued process (∫𝐻𝑑𝑋)_𝑡, called the stochastic integral of 𝐻 with respect to 𝑋, which is a square integrable martingale.
If we further assume that 𝐺 has a countable basis of seminorms, then the above conclusion holds when 𝐻 is a predictable 𝐸-valued process, which is integrable with respect to 𝑋 (in this case, 𝐻 is, in general, unbounded).

This result extends the theory of nuclear stochastic integration of Ustunel [1] in several directions. In [1] it is assumed that 𝐹 is the strong dual of 𝐸 and 𝐺 is the real number field, and furthermore 𝐻 is assumed to be bounded. To develop our theory, we modify the vector bilinear integral developed in [2] for Banach spaces. After defining the space 𝐿²_𝐺, 𝐺 locally convex, the above bilinear integration theory will be applied when we use the property that a complete nuclear space is a projective limit of a family of Hilbert spaces.

In Section 2 we will present the underlying integration theory, and apply this, in Section 3, to construct the stochastic integral.

We omit the proofs in some of the integration theorems since they follow along the usual lines, with appropriate modifications necessary in a general setting (see [2, 3]).

2. Bilinear Vector Integration Theory

2.1. The Banach Setting

In this subsection, assume 𝐸, 𝐹, and 𝐺 are Banach spaces over the reals ℝ, with norms denoted by |⋅|. Let Σ be a 𝜎-field of subsets of a set 𝑇, and assume 𝐦∶Σ→𝐹 is a 𝜎-additive measure. We will assume that there is a continuous bilinear mapping Φ of 𝐸×𝐹 into 𝐺, which, in turn, yields a continuous linear map 𝜙∶𝐸→𝐿(𝐹,𝐺), where 𝐿(𝐹,𝐺) is the space of bounded linear operators from 𝐹 into 𝐺.

The semivariation of 𝐦 relative to 𝜙, 𝐸, 𝐹, 𝐺, denoted by 𝐦 is defined on Σ as follows:𝐦(𝐴)=sup|Σ𝑒_𝑖𝐦(𝐴_𝑖)|, (2.1) where the supremum is extended over all finite collections of elements 𝑒_𝑖 in the unit ball 𝐸₁ of 𝐸 and over all finite disjoint collections of sets 𝐴_𝑖 in Σ which are contained in 𝐴. We are only interested in the case when 𝐦(𝑇)<∞ in order to develop an integration theory of 𝐸-valued integrands. Sometimes we will write 𝐦 as 𝐦_𝐸,𝐺. Note that we write 𝑒 in place of 𝜙(𝑒).

One can show that, for each 𝐴∈Σ, 𝐦(𝐴)=sup|𝐦_𝑧|(𝐴), where the supremum is taken over 𝑧∈𝐺^′₁, the unit ball of the dual 𝐺^′ of 𝐺, and 𝐦_𝑧∶Σ→𝐸^′ is defined by 𝐦_𝑧(𝐴)𝑒=⟨𝑧,𝑒𝐦(𝐴)⟩, for 𝑒∈𝐸. The total variation measure of 𝐦_𝑧 is denoted by |𝐦_𝑧|. Let 𝐦_𝐸,𝐺={|𝐦_𝑧|∶𝑧∈𝐺^′₁}. Thus, 𝐦_𝐸,𝐺 is a bounded collection of positive 𝜎-additive measures. If 𝑐_𝑜⊄𝐺 (e.g., if 𝐺 is a Hilbert space), then one can show that 𝐦_𝐸,𝐺 is relatively weakly compact in the Banach space 𝑐𝑎(Σ) consisting of real-valued measures, with total variation norm. In this case, there exists a positive control measure 𝜆 such that 𝐦_𝐸,𝐺 is uniformly absolutely continuous with respect to 𝜆. A set 𝑄⊂𝑇 is 𝐦-negligible if it is contained in a set 𝐴∈Σ such that |𝐦|(𝐴)=0.

The advantage of modifying the bilinear integration theory in [2] to the case where the integrand is operator-valued rather than the measure being operator-valued will become apparent when the nuclear stochastic integral is studied. This modification changes some of the results in the previous theory, but we are still able to construct the desired Lebesgue space of integrable functions and establish convergence theorems. We now sketch this theory.

Denote by 𝒮=𝒮_𝐸 the collection of 𝐸-valued simple functions. We say that ℎ∶𝑇→𝐸 is measurable if there exists a sequence from 𝒮 which converges pointwise to ℎ. For such ℎ, define𝑁(ℎ)=sup|ℎ|d|𝐦_𝑧|, (2.2) where the supremum is taken over 𝑧∈𝐺^′₁. Let ℱ=ℱ(𝐦_𝐸,𝐺) be the collection of all such ℎ with 𝑁(ℎ) finite. Then set ℒ=ℒ(𝐦_𝐸,𝐺) to be the closure of 𝒮 in ℱ. The space ℒ with the seminorm 𝑁 is our Lebesgue space.

There are different, but equivalent ways to define ∫ℎd𝐦 for ℎ∈ℒ. We select one which yields more information (hence more usefulness) regarding the defining components. If ℎ∈ℒ, one can show that there exists a determining sequence {ℎ_𝑛} of elements in 𝒮— that is, the sequence is Cauchy in ℒ, and {ℎ_𝑛} converges in 𝐦-measure, namely, 𝐦(|ℎ−ℎ_𝑛|>𝜖)→0 for each 𝜖>0. Define the integral of ℎ∈𝒮 in the obvious manner. A determining sequence for ℎ has the property that {𝑁(ℎ_𝑛1_(⋅))}_𝑛 is uniformly absolutely continuous with respect to 𝐦. Also ℎ_𝑛→ℎ in ℒ. The setwise limit ∫_𝐴ℎ_𝑛d𝐦, 𝐴∈Σ, exists and defines a 𝜎-additive measure on Σ. Denote this limit by ∫_𝐴ℎd𝐦. This limit is independent of the choice of the determining sequence for ℎ. We refer to ℒ as the space of integrable functions.

Theorem 2.1 (Vitali). Let {ℎ_𝑛} be a sequence of integrable functions. Let ℎ be an 𝐸-valued measurable function. Then ℎ∈ℒ and ℎ_𝑛→ℎ in ℒ if and only if(1)ℎ_𝑛→ℎ in 𝐦-measure,(2){𝑁(ℎ_𝑛1_(⋅))}_𝑛 is uniformly absolutely continuous with respect to 𝐦.

Theorem 2.2 (Lebesgue). Let 𝑔∈ℒ, and let {ℎ_𝑛} be a sequence of functions from ℒ. If ℎ_𝑛→ℎ in 𝐦-measure and |ℎ_𝑛(⋅)|≤|𝑔(⋅)| for each 𝑛, then ℎ∈ℒ and ℎ_𝑛→ℎ in ℒ.

Theorem 2.3. If 𝐦_𝐸,𝐺 is relatively weakly compact, then ℒ contains the bounded measurable functions.

2.2. Application to the Stochastic Integral in Banach Spaces

We retain the assumptions on 𝐸, 𝐹, 𝐺 as stated in Section 2.1. The stochastic setting is as follows (definitions and terminology are found in [4]). Let (Ω,ℱ,𝐏) be a probability space. 𝐿²_𝐹(𝐏) is a space of ℱ-measurable, 𝐸-valued functions such that 𝐸(|𝑓|²)=∫|𝑓|²𝑑𝐏<∞, endowed with norm |𝑓|=𝐸(|𝑓|²)^1/2. Assume (ℱ_𝑡)_𝑡≥0 is a filtration which satisfies the usual conditions. Suppose 𝑋∶ℝ₊×Ω→𝐹 is a cadlag adapted process, with 𝑋_𝑡∈𝐿²_𝐹 for each 𝑡. Let ℛ be the ring of subsets of ℝ₊×Ω generated by the predictable rectangles; thus 𝜎(ℛ)=𝒫, the predictable 𝜎-field. Let 𝐦 (=𝐼_𝑋) be the additive 𝐿²_𝐹-valued measure first defined on the predictable rectangles by 𝐦((𝑠,𝑡]×𝐴)=1_𝐴(𝑋_𝑡−𝑋_𝑠), 𝐴∈ℱ_𝑠, and 𝐦(0_𝐴)=1_𝐴𝑋₀, 𝐴∈ℱ₀. We regard 𝐸 as being continuously embedded into 𝐿(𝐿²_𝐹,𝐿²_𝐺) in the obvious manner. The theory of [3] for Banach stochastic integration can be shown to apply in a parallel fashion to this setting, and we state a few pertinent results. If 𝑐_𝑜⊄𝐹, then 𝐦 can be extended uniquely to a 𝜎-additive 𝐿²_𝐹-valued measure if and only if 𝐦 is bounded on ℛ. For our purposes in this paper, we will be interested only in the case when all the spaces are Hilbert spaces and 𝑋 is a square integrable martingale. In this case, 𝐦_𝐸,𝐿²_𝐺(ℝ₊×Ω)<∞. As a result, we can construct the stochastic integral (∫𝐻𝑑𝑋)_𝑡, which is a process such that ∫^𝑡₀𝐻𝑑𝑋∈𝐿²_𝐺, and this process is a 𝐺-valued square integrable martingale. If we still denote the extension of 𝐦 to 𝒫 by 𝐦, then ∫^𝑡₀𝐻𝑑𝑋 is defined to be ∫𝐻1_[0,𝑡]𝑑𝐦, where 𝐻 is integrable with respect to 𝐦, that is, 𝐻∈ℒ(𝐦_𝐸,𝐿²_𝐺), and the Hilbert spaces involved in the bilinear theory are 𝐸, 𝐿²_𝐹, and 𝐿²_𝐺. This integral will be used to define the stochastic integral in nuclear spaces.

2.3. The Definition of 𝐿²_𝐺, 𝐺 Locally Convex

In this subsection, assume (𝑇,Σ,𝐦) is a measure space, 𝐦 is real-valued and 𝜎-additive. Let 𝐺 be a complete locally convex space, and let 𝒢 be a basis of seminorms defining the topology of 𝐺. A function 𝑓∶𝑇→𝐺 is measurable if it is the pointwise limit of simple 𝐺-valued measurable functions in 𝒮_𝐺. For 𝑟∈𝒢 and ℎ being measurable, let 𝑁_𝑟(ℎ)=(∫𝑟(ℎ)²𝑑|𝐦|)^1/2. Let ℱ_𝐺 be the space of measurable functions ℎ such that 𝑁_𝑟(ℎ)<∞ for each 𝑟∈𝒢. Then ℱ_𝐺 is a locally convex space with {𝑁_𝑟∶𝑟∈𝒢} being a basis of seminorms. Define 𝐿²_𝐺, the space of integrable functions, to be the closure of 𝒮_𝐺 in ℱ_𝐺.

It can be shown that 𝐿²_𝐺 is the set of measurable functions ℎ which have a determining sequence (ℎ_𝑛)⊂𝒮_𝐺, that is, the sequence satisfies for each 𝑟∈𝒢, 𝑁_𝑟(ℎ_𝑛−ℎ_𝑚)→0 as 𝑛,𝑚→∞, and for each 𝜖>0 and 𝑟∈𝒢, we have |𝐦|(𝑟(ℎ_𝑛−ℎ)>𝜖)→0 as 𝑛→∞. In this case, ∫_𝐴ℎ𝑑𝐦=lim∫_𝐴ℎ_𝑛𝑑𝐦, 𝐴∈Σ, is unambiguously defined for each determining sequence (the definition of ∫ℎ_𝑛𝑑𝐦 is the obvious one).

The bounded measurable functions are in 𝐿²_𝐺, and the Vitali and the Lebesgue dominated convergence theorem hold. Moreover, we have the following theorem.

Theorem 2.4. Let 𝐺 be a complete locally convex space with a countable basis of seminorms. Then 𝐿²_𝐺 is complete.

2.4. A Remark on the Bilinear Mapping 𝐸×𝐹→𝐺

Suppose 𝐸 and 𝐺 are locally convex spaces with ℰ and 𝒢 denoting their respective bases of defining seminorms. Assume 𝐹 is a Hilbert space and Φ∶𝐸×𝐹→𝐺 is a continuous bilinear mapping that induces 𝜙∶𝐸→𝐿(𝐹,𝐺). Using the continuity of Φ, observe that for each 𝑟∈𝒢, there exists a 𝑝∈ℰ such that Φ(𝑈_𝑝,𝐹₁)⊂𝑈_𝑟, where 𝑈_𝑝 and 𝑈_𝑟 are the closed balls induced by 𝑝 and 𝑟. If we define 𝑝(𝑟) to be the infimum over all 𝑝 for which the above inclusion holds, it turns out that 𝑝(𝑟) is a seminorm and 𝑈_𝑝(𝑟) is the closed convex balanced hull of ∪_𝑝𝑈_𝑝, where the union is taken over those 𝑝 in the above infimum. Also 𝑝(𝑟)(𝑒)=sup|𝑧𝑒|_𝐹′, where the supremum is taken over 𝑧∈𝑈^∘_𝑟 (𝑧𝑒∶𝑓→⟨𝑧,𝑒𝑓⟩, 𝑓∈𝐹). Call 𝑝(𝑟) the seminorm associated with 𝑟 and Φ. Note that 𝐸(𝑈_𝑝(𝑟)) is isometrically embedded in 𝐿(𝐹,𝐺(𝑈_𝑟)), where 𝐸(𝑈_𝑝(𝑟)) is the Banach space consisting of equivalence classes modulo ker𝑝(𝑟), completed under the norm induced by 𝑝(𝑟); 𝐺(𝑈_𝑟) is similarly defined.

3. The Nuclear Setting. The Construction of the Stochastic Integral

3.1. Square Integrable Martingales in Nuclear Spaces

(Ω,ℱ,𝐏) and (ℱ_𝑡)_𝑡≥0 are as in Section 2.2. Let 𝐹 denote a nuclear space which is reflexive, complete, bornological, and such that its strong dual 𝐹^′ satisfies the same conditions. We say 𝐹 satisfies the special conditions. These special conditions are the hypotheses of Ustunel, who established fundamental results for square integrable martingales in this setting. Let 𝐸 be such a space. Then for 𝐸 and 𝐸^′ there exist neighborhood bases of zero, 𝒰, and 𝒰^′, respectively, such that for each 𝑈∈𝒰, the space 𝐸(𝑈) is a separable Hilbert space over the reals, and its separable dual is identified with the Hilbert space 𝐸^′[𝑈^∘] as defined in [5], where 𝑈^∘ is the polar of 𝑈. Also, {𝑈^∘∶𝑈∈𝒰} and {𝑉^∘∶𝑉∈𝒰^′} are bases of closed, convex, balanced bounded sets in 𝐸^′, 𝐸, respectively. For 𝑈∈𝒰, we denote by 𝒦(𝑈) the continuous canonical map from 𝐸 onto 𝐸(𝑈). If 𝑈,𝑉∈𝒰 and 𝑉⊂𝑈, then 𝒦(𝑈,𝑉) is the canonical mapping of 𝐸(𝑉) onto 𝐸(𝑈).

Let (Ω,ℱ,𝒫) be a probability space with (ℱ_𝑡)_𝑡≥0 being a filtration satisfying the usual conditions. The set 𝑋={𝑋^𝑈∶𝑈∈𝒰} is called a projective system of square integrable martingales if for each 𝑈, we have that 𝑋^𝑈 is an 𝐸(𝑈)-valued square integrable martingale, and if whenever 𝑈,𝑉∈𝒰 and 𝑉⊂𝑈, then 𝒦(𝑈,𝑉)𝑋^𝑉 and 𝑋^𝑈 are indistinguishable. We also assume 𝑋^𝑈 is cadlag for each 𝑈. One says that 𝑋 has a limit in 𝐸 if there exists a weakly adapted mapping 𝑋 on ℝ₊×Ω into 𝐸 such that 𝒦(𝑈)𝑋 is a modification of 𝑋^𝑈 for each 𝑈∈𝒰.

The next theorem is crucial for defining the stochastic integral. Ustunel [1, Section II.4] assumed the existence of a limit in 𝐸 for 𝑋. This hypothesis was removed in [6]. We now state the theorem and provide a brief sketch of the proof, which uses a technique of Ustunel.

Theorem 3.1. Let 𝑋 be a projective system of square integrable martingales. Then there exists a limit 𝑋 in 𝐸 of 𝑋 which is strongly cadlag in 𝐸, and for which 𝒦(𝑈)𝑋 is a modification of 𝑋^𝑈 for each 𝑈∈𝒰. Moreover, there exists a 𝑉∈𝒰^′ such that 𝑋 takes its values in 𝐸[𝑉^∘].

Let 𝔐² denote the space of real-valued square integrable martingales. Define a mapping 𝐓∶𝐸^′→𝔐² by 𝐓(𝑒^′)=⟨𝑒^′,𝑋^𝑈⟩, where 𝑈 is chosen in 𝒰 so that 𝑒^′∈𝐸^′[𝑈^∘]. Argue that T is well defined and linear. If 𝑒^′_𝑛→𝑒^′ in 𝐸^′[𝑈^∘] for some 𝑈∈𝒰, then|𝐓(𝑒^′_𝑛)_∞−𝐓(𝑒^′)_∞|≤‖𝑋^𝑈_∞‖_𝐸(𝑈)‖𝑒^′_𝑛−𝑒^′‖_{𝐸′[𝑈∘]}; (3.1) hence {𝐓(𝑒^′_𝑛)_∞} converges to (𝐓(𝑒^′)_∞) in 𝐿²(𝒫)=𝐿², and thus 𝐓(𝑒^′_𝑛)→𝑇(𝑒^′) in 𝔐². Consequently, T is continuous on 𝐸^′[𝑈^∘]. Since 𝐸^′ is bornological, T is continuous on 𝐸^′. As a result, 𝐓∶𝐸^′→𝔐² is a nuclear map of the form𝐓(𝑒^′)=𝜆_𝑖<𝑒_𝑖,𝑒^′>𝑀^𝑖, (3.2) where {𝜆_𝑖}∈𝑙¹, {𝑒_𝑖} is equicontinuous in 𝐸, and (𝑀^𝑖) is bounded in 𝔐². Choose 𝑉∈𝒰^′ such that all 𝑒_𝑖∈𝑉^∘. Define the process 𝑋 by 𝑋_𝑡=∑𝜆_𝑖𝑒_𝑖𝑀^𝑖_𝑡, where we choose (𝑋_𝑡) to be a cadlag version. Then 𝑋 is the desired process.

From now on, we identify 𝑋 and 𝑋, and we assume that 𝑋 takes its values in the Hilbert space 𝐸[𝑉^∘].

3.2. Construction of the Stochastic Integral

Assume that 𝐸, 𝐹, and 𝐺 are nuclear spaces over the reals satisfying the special conditions set forth in Section 3.1. Also assume that Φ∶𝐸×𝐹→𝐺 is a continuous bilinear mapping. The neighborhood bases of zero in 𝐸 and 𝐺 are denoted by 𝒰_𝐸 and 𝒰_𝐺. Let 𝑋∶ℝ₊×Ω→𝐹 be a square integrable martingale. By Theorem 2.4, we may assume 𝑋 is Hilbert space valued. As a result, we may now assume 𝐹 is a real Hilbert space. The bilinear map Φ induces a continuous linear map 𝜙∶𝐸→𝐿(𝐹,𝐺), which in turn induces the continuous linear map 𝜙∶𝐸→𝐿(𝐿²_𝐹,𝐿²_𝐺), where 𝐿²_𝐺 is the space constructed in Section 2.3.

Since 𝑐_𝑜⊄𝐹, the stochastic measure 𝐦 (=𝐼_𝑋) first defined on the predictable rectangles can be extended to a 𝜎-additive measure, still denoted by 𝐦, 𝐦∶𝒫→𝐿²_𝐹. Note that if 𝐾₁ and 𝐾₂ are Hilbert spaces, then 𝐦 has finite semivariation with respect to every continuous linear embedding of 𝐾₁ into 𝐿(𝐿²_𝐹,𝐾₂).

If 𝑧∈(𝐿²_𝐺)^′, we define 𝐦_𝑧∶𝒫→𝐸^′ by 𝐦_𝑧(𝐴)𝑒=⟨𝑧,𝑒𝐦(𝐴)⟩, for 𝑒∈𝐸. Given any 𝑟∈𝒢, if 𝑧∈𝑈^∘_𝑁𝑟, then 𝐦_𝑧∶𝒫→𝐸^′[𝑈^∘_{𝑝(𝑁𝑟)}]=𝐸(𝑈_{𝑝(𝑁𝑟)})^′ (where 𝑝(𝑁_𝑟) is the seminorm associated with 𝑁_𝑟 relative to the mapping 𝐸→𝐿(𝐿²_𝐹,𝐿²_𝐺)) by𝐦_𝑧(𝐴)[𝑒]_{𝑝(𝑁𝑟)}=⟨𝑧,[𝑒]_{𝑝(𝑁𝑟)}𝐦(𝐴)⟩=⟨𝑧,𝑒𝐦(𝐴)⟩. (3.3) In fact, 𝑝(𝑁_𝑟)=𝑝(𝑟), relative to the mapping 𝐸→𝐿(𝐹,𝐺). Let 𝐦_𝑟={|𝐦_𝑧|∶𝑧∈𝑈^∘_𝑁𝑟}. Then 𝐦_𝑟(𝐴)=sup|𝐦_𝑧|(𝐴), where the supremum is extended over 𝑧∈𝑈^∘_𝑁𝑟. Observe that 𝐦_𝑟 is the semivariation of 𝐦 relative to 𝐸(𝑈_𝑝(𝑟)), 𝐿²_𝐺(𝑈_𝑁𝑟) which arises from the isometric mapping of 𝐸(𝑈_𝑝(𝑟)) into 𝐿(𝐿²_𝐹,𝐿²_𝐺(𝑈_𝑁𝑟)). One can show that 𝐿²_𝐺(𝑈_𝑁𝑟) is isometrically embedded in the Hilbert space 𝐿²_{𝐺(𝑈𝑟)} and, as a result, 𝐦 has a finite semivariation relative to each of these embeddings; thus 𝐦_𝑟 is finite for each 𝑟∈𝒢, and 𝐦_𝑟 is relatively weakly compact in 𝑐𝑎(𝒫).

A process 𝐻∶ℝ₊×Ω→𝐸 is a predictable process, or simply measurable, if it is the pointwise limit of processes from 𝒮_𝐸, the simple predictable 𝐸-valued processes. For such a measurable process 𝐻, define, for 𝑟∈𝒢,𝔑_𝑟(𝐻)=sup𝑝(𝑟)(𝐻)𝑑|𝐦_𝑧|, (3.4) where the supremum is extended over 𝑧∈𝑈^∘_𝑁𝑟. Let ℱ=ℱ(𝐦_𝐸,𝐿²_𝐺) be the space of measurable functions 𝐻 such that 𝔑_𝑟(𝐻)<∞ for each 𝑟∈𝒢. Then ℱ is a locally convex space containing 𝒮_𝐸. Let ℒ=ℒ(𝐦_𝐸,𝐿²_𝐺) denote the closure of 𝒮_𝐸 in ℱ. One can show that for each 𝐻∈ℒ there exists a determining sequence (𝐻_𝑛) from 𝒮_𝐸 such that (𝐻_𝑛) is mean Cauchy in ℒ (𝔑_𝑟(𝐻_𝑛−𝐻_𝑚)→_𝑛,𝑚0), for each 𝑟∈𝒢, and 𝐦_𝑟(𝑝(𝑟)(𝐻_𝑛−𝐻)>𝜖)→_𝑛0 for each 𝜖>0 and 𝑟∈𝒢.

Now assume 𝐺 has a countable basis of seminorms, that is, 𝐺 is now a nuclear Fréchet space. Thus there exists a positive measure 𝜆 such that 𝐦_𝑟≪𝜆 for each 𝑟∈𝒢. Since 𝐿²_𝐺 is complete and, for 𝐻∈𝒮_𝐸, we have 𝑁_𝑟(∫𝐻𝑑𝐦)≤𝔑_𝑟(𝐻), where the integral is defined in the obvious way, then for general 𝐻∈ℒ with determining sequence (𝐻_𝑛), we can define_(⋅)𝐻𝑑𝐦=lim_(⋅)𝐻_𝑛𝑑𝐦∈𝐿²_𝐺. (3.5) The completeness of 𝐿²_𝐺 ensures that ∫_𝐴𝐻𝑑𝐦 is a function in 𝐿²_𝐺. Define the process (∫𝐻𝑑𝑋)_𝑡=∫^𝑡₀𝐻𝑑𝑋 by ∫^𝑡₀𝐻𝑑𝑋=∫𝐻1_[0,𝑡]𝑑𝐦, called the stochastic integral of 𝐻 with respect to 𝑋. We say 𝐻 is integrable with respect to 𝑋 if 𝐻∈ℒ. If 𝐻∈𝒮_𝐸, one can show that (∫𝐻𝑑𝑋)_𝑡 is a 𝐺-valued square integrable martingale. By means of using determining sequences, the general stochastic integral enjoys this property.

Next, assume that 𝐺 just satisfies the special conditions (no longer nuclear Fréchet). Let 𝐻 be a bounded measurable 𝐸-valued process; hence the range of 𝐻 is contained in a closed, bounded, convex, balanced set 𝐵₁, where 𝐸[𝐵₁] is a Hilbert space. By the continuity of Φ, it follows that Φ(𝐵₁,𝐹₁) is contained in a bounded set 𝐵 having the same properties as 𝐵₁, and 𝐺[𝐵] is a Hilbert space.

Algebraically, Φ induces Φ₀∶𝐸[𝐵₁]×𝐹→𝐺[𝐵] which is bilinear, and since Φ₀⁻¹(𝛼𝐵)⊃(𝛼𝐵₁)×𝐹 for every 𝛼∈ℝ, Φ₀ is continuous. As a result, this induces a continuous linear map 𝜙₀∶𝐸[𝐵₁]→𝐿(𝐹,𝐺[𝐵]), which in turn induces the continuous linear map 𝜙₀∶𝐸[𝐵₁]→𝐿(𝐿²_𝐹,𝐿²_𝐺[𝐵]). Hence we can define 𝐦=𝐼_𝑋∶𝒫→𝐿²_𝐹 as before, which is 𝜎-additive and has finite semivariation relative to 𝜙₀.

Since 𝐻 is measurable, it is the pointwise limit of functions from 𝒮_𝐸, and thus if 𝑥^′∈𝐸^′, 𝑥^′𝐻_𝑛→𝑥^′𝐻. This implies that (𝑥^′𝐻)⁻¹(𝒪)∈𝒫 for any open subset 𝒪 of the reals. By the reflexivity of 𝐸,𝐸[𝐵₁]^′=𝐸[𝐵₁]=[𝐸^′(𝐵^∘₁)]^′=𝐸^′(𝐵^∘₁), (3.6) since we have chosen 𝐵₁=𝑉^∘∈𝒰^′. Let 𝑒^′∈𝐸[𝐵₁]^′; then 𝑒^′=[𝑥^′]_𝐵^∘₁, and for 𝑒∈𝐸[𝐵₁], it follows that ⟨𝑒^′,𝑒⟩=⟨[𝑥^′]_𝐵^∘₁,𝑒⟩=⟨𝑥^′,𝑒⟩, that is, 𝑥^′𝐻=𝑒^′𝐻. As a consequence, 𝐻∶ℝ₊×Ω→𝐸[𝐵₁] is weakly measurable, and since 𝐸[𝐵₁] is separable, by the Pettis theorem we conclude that 𝐻 is bounded and measurable as an 𝐸[𝐵₁]-valued function.

We now use the integration theory in Section 2.1. There exists a control measure 𝜆 in this setting, since 𝑐_𝑜⊄𝐺[𝐵]; hence it follows that the space of integrable functions, relative to the map 𝜙₀, contains the bounded measurable functions. Thus ∫𝐻𝑑𝑋=∫𝐻𝑑𝐦∈𝐿²_𝐺[𝐵], and the process (∫𝐻𝑑𝑋)_𝑡=∫𝐻1_[0,𝑡]𝑑𝐦 defines the stochastic integral; note that this process is a square integrable martingale. Since the norm on 𝐺[𝐵] is stronger than any 𝑟∈𝒢, one can show that 𝐿²_𝐺[𝐵] is continuously injected in 𝐿²_𝐺.

Remarks 3.2. (1) When we assumed 𝐺 was a nuclear Fréchet space, we constructed the stochastic integral for every 𝐻 integrable with respect to 𝑋. In particular, if 𝐻 is bounded, the stochastic integral agrees with the one constructed by means of using 𝐿²_𝐺[𝐵].
(2) Suppose 𝐺 is nuclear Fréchet and 𝐻 is integrable relative to 𝜙∶𝐸→𝐿(𝐿²_𝐹,𝐿²_𝐺). For each seminorm 𝑁_𝑟 on 𝐿²_𝐺, there is a seminorm 𝑝(𝑟)∈ℰ which induces the isometric embedding 𝜙 of 𝐸(𝑈_𝑝(𝑟)) into 𝐿(𝐿²_𝐹,𝐿²_𝐺(𝑈_𝑁𝑟)), where 𝐿²_𝐺(𝑈_𝑁𝑟) is a Hilbert space since it is isometrically embedded in 𝐿²_{𝐺(𝑈𝑟)}. Thus each [𝐻]_𝑝(𝑟)∶ℝ₊×Ω→𝐸(𝑈_𝑝(𝑟)) is integrable relative to 𝜙 and gives rise to the stochastic integral defined by (∫[𝐻]_𝑝(𝑟)1_[0,𝑡]𝑑𝑋)_𝑡≥0, which is a square integrable martingale. The projective system of square integrable martingales {[𝐻]_𝑝(𝑟)}_𝑟∈𝒢 has a limit in 𝐺, and this limit is (∫𝐻𝑑𝑋)_𝑡∶=(𝑀)_𝑡, 𝑀_∞=∫𝐻𝑑𝑋.
Since there is a control measure for 𝐦_𝐸,𝐿²_𝐺, one can show that 𝐸(𝑀_∞∣ℱ_𝑡)=𝑀_𝑡.