# Stochastic Integration in Abstract Spaces

Authors: J. K. Brooks; J. T. Kozinski
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

We establish the existence of a stochastic integral in a nuclear space setting
as follows. LetE, 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.


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## Body

## 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 Section3.1 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.
LetE, 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.
IfH 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 thatG 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 LG2, 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 Section2 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, assumeE, 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.Thesemivariation of m relative to ϕ, E, F, G, denoted by m̂ is defined on Σ as follows:(2.1)m̂(A)=sup|Σeim(Ai)|,
where the supremum is extended over all finite collections of elements ei in the unit ball E1 of E and over all finite disjoint collections of sets Ai in Σ which are contained in A. We are only interested in the case when m̂(T)<∞ in order to develop an integration theory of E-valued integrands. Sometimes we will write m̂ as m̂E,G. Note that we write e in place of ϕ(e).One can show that, for eachA∈Σ, m̂(A)=sup|mz|(A), where the supremum is taken over z∈G1′, the unit ball of the dual G′ of G, and mz:Σ→E′ is defined by mz(A)e=〈z,em(A)〉, for e∈E. The total variation measure of mz is denoted by |mz|. Let mE,G={|mz|:z∈G1′}. Thus, mE,G is a bounded collection of positive σ-additive measures. If co⊄G (e.g., if G is a Hilbert space), then one can show that mE,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 mE,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 by𝒮=𝒮E 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, define(2.2)N(h)=sup∫|h|d|mz|,
where the supremum is taken over z∈G1′. Let ℱ̃=ℱ̃(mE,G) be the collection of all such h with N(h) finite. Then set ℒ=ℒ(mE,G) to be the closure of 𝒮 in ℱ̃. 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{hn} of elements in 𝒮— that is, the sequence is Cauchy in ℒ, and {hn} converges in m-measure, namely, m̂(|h-hn|>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(hn1(·))}n is uniformly absolutely continuous with respect to m̂. Also hn→h in ℒ. The setwise limit ∫Ahndm, 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{hn} be a sequence of integrable functions. Let h be an E-valued measurable function. Then h∈ℒ and hn→h in ℒ if and only if(1)
hn→h in m-measure,(2)
{N(hn1(·))}n is uniformly absolutely continuous with respect to m̂.Theorem 2.2 (Lebesgue).
Letg∈ℒ, and let {hn} be a sequence of functions from ℒ. If hn→h in m-measure and |hn(·)|≤|g(·)| for each n, then h∈ℒ and hn→h in ℒ.Theorem 2.3.
IfmE,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 onE, 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. LF2(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 (ℱt)t≥0 is a filtration which satisfies the usual conditions. Suppose X:ℝ+×Ω→F is a cadlag adapted process, with Xt∈LF2 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 (=IX) be the additive LF2-valued measure first defined on the predictable rectangles by m((s,t]×A)=1A(Xt-Xs), A∈ℱs, and m(0A)=1AX0, A∈ℱ0. We regard E as being continuously embedded into L(LF2,LG2) 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 co⊄F, then m can be extended uniquely to a σ-additive LF2-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, m̂E,LG2(ℝ+×Ω)<∞. As a result, we can construct the stochastic integral (∫HdX)t, which is a process such that ∫0tHdX∈LG2, and this process is a G-valued square integrable martingale. If we still denote the extension of m to 𝒫 by m, then ∫0tHdX is defined to be ∫H1[0,t]dm, where H is integrable with respect to m, that is, H∈ℒ(mE,LG2), and the Hilbert spaces involved in the bilinear theory are E, LF2, and LG2. This integral will be used to define the stochastic integral in nuclear spaces.


### 2.3. The Definition ofLG2, 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 𝒮G. For r∈𝒢 and h being measurable, let Nr(h)=(∫r(h)2d|m|)1/2. Let ℱ̃G be the space of measurable functions h such that Nr(h)<∞ for each r∈𝒢. Then ℱ̃G is a locally convex space with {Nr:r∈𝒢} being a basis of seminorms. Define LG2, the space of integrable functions, to be the closure of 𝒮G in ℱ̃G.It can be shown thatLG2 is the set of measurable functions h which have a determining sequence(hn)⊂𝒮G, that is, the sequence satisfies for each r∈𝒢, Nr(hn-hm)→0 as n,m→∞, and for each ϵ>0 and r∈𝒢, we have |m|(r(hn-h)>ϵ)→0 as n→∞. In this case, ∫Ahdm=lim∫Ahndm, A∈Σ, is unambiguously defined for each determining sequence (the definition of ∫hndm is the obvious one).The bounded measurable functions are inLG2, and the Vitali and the Lebesgue dominated convergence theorem hold. Moreover, we have the following theorem.Theorem 2.4.
LetG be a complete locally convex space with a countable basis of seminorms. Then LG2 is complete.


### 2.4. A Remark on the Bilinear MappingE×F→G

SupposeE 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(Up,F1)⊂Ur, where Up and Ur 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 Up(r) is the closed convex balanced hull of ∪pUp, 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∈Ur∘ (ze:f→〈z,ef〉, f∈F). Call p(r) the seminorm associated with r and Φ. Note that E(Up(r)) is isometrically embedded in L(F,G(Ur)), where E(Up(r)) is the Banach space consisting of equivalence classes modulo kerp(r), completed under the norm induced by p(r); G(Ur) is similarly defined.


## 2.1. The Banach Setting

In this subsection, assumeE, 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.Thesemivariation of m relative to ϕ, E, F, G, denoted by m̂ is defined on Σ as follows:(2.1)m̂(A)=sup|Σeim(Ai)|,
where the supremum is extended over all finite collections of elements ei in the unit ball E1 of E and over all finite disjoint collections of sets Ai in Σ which are contained in A. We are only interested in the case when m̂(T)<∞ in order to develop an integration theory of E-valued integrands. Sometimes we will write m̂ as m̂E,G. Note that we write e in place of ϕ(e).One can show that, for eachA∈Σ, m̂(A)=sup|mz|(A), where the supremum is taken over z∈G1′, the unit ball of the dual G′ of G, and mz:Σ→E′ is defined by mz(A)e=〈z,em(A)〉, for e∈E. The total variation measure of mz is denoted by |mz|. Let mE,G={|mz|:z∈G1′}. Thus, mE,G is a bounded collection of positive σ-additive measures. If co⊄G (e.g., if G is a Hilbert space), then one can show that mE,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 mE,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 by𝒮=𝒮E 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, define(2.2)N(h)=sup∫|h|d|mz|,
where the supremum is taken over z∈G1′. Let ℱ̃=ℱ̃(mE,G) be the collection of all such h with N(h) finite. Then set ℒ=ℒ(mE,G) to be the closure of 𝒮 in ℱ̃. 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{hn} of elements in 𝒮— that is, the sequence is Cauchy in ℒ, and {hn} converges in m-measure, namely, m̂(|h-hn|>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(hn1(·))}n is uniformly absolutely continuous with respect to m̂. Also hn→h in ℒ. The setwise limit ∫Ahndm, 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{hn} be a sequence of integrable functions. Let h be an E-valued measurable function. Then h∈ℒ and hn→h in ℒ if and only if(1)
hn→h in m-measure,(2)
{N(hn1(·))}n is uniformly absolutely continuous with respect to m̂.Theorem 2.2 (Lebesgue).
Letg∈ℒ, and let {hn} be a sequence of functions from ℒ. If hn→h in m-measure and |hn(·)|≤|g(·)| for each n, then h∈ℒ and hn→h in ℒ.Theorem 2.3.
IfmE,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 onE, 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. LF2(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 (ℱt)t≥0 is a filtration which satisfies the usual conditions. Suppose X:ℝ+×Ω→F is a cadlag adapted process, with Xt∈LF2 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 (=IX) be the additive LF2-valued measure first defined on the predictable rectangles by m((s,t]×A)=1A(Xt-Xs), A∈ℱs, and m(0A)=1AX0, A∈ℱ0. We regard E as being continuously embedded into L(LF2,LG2) 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 co⊄F, then m can be extended uniquely to a σ-additive LF2-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, m̂E,LG2(ℝ+×Ω)<∞. As a result, we can construct the stochastic integral (∫HdX)t, which is a process such that ∫0tHdX∈LG2, and this process is a G-valued square integrable martingale. If we still denote the extension of m to 𝒫 by m, then ∫0tHdX is defined to be ∫H1[0,t]dm, where H is integrable with respect to m, that is, H∈ℒ(mE,LG2), and the Hilbert spaces involved in the bilinear theory are E, LF2, and LG2. This integral will be used to define the stochastic integral in nuclear spaces.


## 2.3. The Definition ofLG2, 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 𝒮G. For r∈𝒢 and h being measurable, let Nr(h)=(∫r(h)2d|m|)1/2. Let ℱ̃G be the space of measurable functions h such that Nr(h)<∞ for each r∈𝒢. Then ℱ̃G is a locally convex space with {Nr:r∈𝒢} being a basis of seminorms. Define LG2, the space of integrable functions, to be the closure of 𝒮G in ℱ̃G.It can be shown thatLG2 is the set of measurable functions h which have a determining sequence(hn)⊂𝒮G, that is, the sequence satisfies for each r∈𝒢, Nr(hn-hm)→0 as n,m→∞, and for each ϵ>0 and r∈𝒢, we have |m|(r(hn-h)>ϵ)→0 as n→∞. In this case, ∫Ahdm=lim∫Ahndm, A∈Σ, is unambiguously defined for each determining sequence (the definition of ∫hndm is the obvious one).The bounded measurable functions are inLG2, and the Vitali and the Lebesgue dominated convergence theorem hold. Moreover, we have the following theorem.Theorem 2.4.
LetG be a complete locally convex space with a countable basis of seminorms. Then LG2 is complete.


## 2.4. A Remark on the Bilinear MappingE×F→G

SupposeE 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(Up,F1)⊂Ur, where Up and Ur 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 Up(r) is the closed convex balanced hull of ∪pUp, 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∈Ur∘ (ze:f→〈z,ef〉, f∈F). Call p(r) the seminorm associated with r and Φ. Note that E(Up(r)) is isometrically embedded in L(F,G(Ur)), where E(Up(r)) is the Banach space consisting of equivalence classes modulo kerp(r), completed under the norm induced by p(r); G(Ur) 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 (ℱt)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 (ℱt)t≥0 being a filtration satisfying the usual conditions. The set X={XU:U∈𝒰} is called a projective system of square integrable martingales if for each U, we have that XU is an E(U)-valued square integrable martingale, and if whenever U,V∈𝒰 and V⊂U, then 𝒦(U,V)XV and XU are indistinguishable. We also assume XU is cadlag for each U. One says that X has a limit in E if there exists a weakly adapted mapping X̃ on ℝ+×Ω into E such that 𝒦(U)X̃ is a modification of XU 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.
LetX be a projective system of square integrable martingales. Then there exists a limit X̃ in E of X which is strongly cadlag in E, and for which 𝒦(U)X̃ is a modification of XU for each U∈𝒰. Moreover, there exists a V∈𝒰′ such that X̃ takes its values in E[V∘].Let𝔐2 denote the space of real-valued square integrable martingales. Define a mapping T:E′→𝔐2 by T(e′)=〈e′,XU〉, where U is chosen in 𝒰 so that e′∈E′[U∘]. Argue that T is well defined and linear. If en′→e′ in E′[U∘] for some U∈𝒰, then(3.1)|T(en′)∞-T(e′)∞|≤∥X∞U∥E(U)∥en′-e′∥E′[U∘];
hence {T(en′)∞} converges to (T(e′)∞) in L2(𝒫)=L2, and thus T(en′)→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 form(3.2)T(e′)=∑λi<ei,e′>Mi,
where {λi}∈l1, {ei} is equicontinuous in E, and (Mi) is bounded in 𝔐2. Choose V∈𝒰′ such that all ei∈V∘. Define the process X̃ by Xt̃=∑λieiMti, where we choose (Xt̃) to be a cadlag version. Then X̃ is the desired process.From now on, we identifyX and X̃, and we assume that X takes its values in the Hilbert space E[V∘].


### 3.2. Construction of the Stochastic Integral

Assume thatE, 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 𝒰E and 𝒰G. 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(LF2,LG2), where LG2 is the space constructed in Section 2.3.Sinceco⊄F, the stochastic measure m (=IX) first defined on the predictable rectangles can be extended to a σ-additive measure, still denoted by m, m:𝒫→LF2. 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(LF2,K2).Ifz∈(LG2)′, we define mz:𝒫→E′ by mz(A)e=〈z,em(A)〉, for e∈E. Given any r∈𝒢, if z∈UNr∘, then mz:𝒫→E′[Up(Nr)∘]=E(Up(Nr))′ (where p(Nr) is the seminorm associated with Nr relative to the mapping E→L(LF2,LG2)) by(3.3)mz(A)[e]p(Nr)=〈z,[e]p(Nr)m(A)〉=〈z,em(A)〉.
In fact, p(Nr)=p(r), relative to the mapping E→L(F,G). Let mr={|mz|:z∈UNr∘}. Then m̂r(A)=sup|mz|(A), where the supremum is extended over z∈UNr∘. Observe that m̂r is the semivariation of m relative to E(Up(r)), LG2(UNr) which arises from the isometric mapping of E(Up(r)) into L(LF2,LG2(UNr)). One can show that LG2(UNr) is isometrically embedded in the Hilbert space LG(Ur)2 and, as a result, m has a finite semivariation relative to each of these embeddings; thus m̂r is finite for each r∈𝒢, and mr is relatively weakly compact in ca(𝒫).A processH:ℝ+×Ω→E is a predictable process, or simply measurable, if it is the pointwise limit of processes from 𝒮E, the simple predictable E-valued processes. For such a measurable process H, define, for r∈𝒢,(3.4)𝔑r(H)=sup∫p(r)(H)d|mz|,
where the supremum is extended over z∈UNr∘. Let ℱ̃=ℱ̃(mE,LG2) be the space of measurable functions H such that 𝔑r(H)<∞ for each r∈𝒢. Then ℱ̃ is a locally convex space containing 𝒮E. Let ℒ=ℒ(mE,LG2) denote the closure of 𝒮E in ℱ̃. One can show that for each H∈ℒ there exists a determining sequence (Hn) from 𝒮E such that (Hn) is mean Cauchy in ℒ (𝔑r(Hn-Hm)→n,m0), for each r∈𝒢, and m̂r(p(r)(Hn-H)>ϵ)→n0 for each ϵ>0 and r∈𝒢.Now assumeG has a countable basis of seminorms, that is, G is now a nuclear Fréchet space. Thus there exists a positive measure λ such that m̂r≪λ for each r∈𝒢. Since LG2 is complete and, for H∈𝒮E, we have Nr(∫Hdm)≤𝔑r(H), where the integral is defined in the obvious way, then for general H∈ℒ with determining sequence (Hn), we can define(3.5)∫(·)Hdm=lim∫(·)Hndm∈LG2.
The completeness of LG2 ensures that ∫AHdm is a function in LG2. Define the process (∫HdX)t=∫0tHdX by ∫0tHdX=∫H1[0,t]dm, called the stochastic integral ofHwith respect toX. We say H is integrable with respect to X if H∈ℒ. If H∈𝒮E, 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 thatG 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 ϕ̃0:E[B1]→L(LF2,LG[B]2). Hence we can define m=IX:𝒫→LF2 as before, which is σ-additive and has finite semivariation relative to ϕ̃0.SinceH is measurable, it is the pointwise limit of functions from 𝒮E, and thus if x′∈E′, x′Hn→x′H. This implies that (x′H)-1(𝒪)∈𝒫 for any open subset 𝒪 of the reals. By the reflexivity of E,(3.6)E[B1]′=E[B1]=[E′(B1∘)]′=E′(B1∘),
since we have chosen B1=V∘∈𝒰′. Let e′∈E[B1]′; then e′=[x′]B1∘, and for e∈E[B1], it follows that 〈e′,e〉=〈[x′]B1∘,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 Section2.1. There exists a control measure λ in this setting, since co⊄G[B]; hence it follows that the space of integrable functions, relative to the map ϕ̃0, contains the bounded measurable functions. Thus ∫HdX=∫Hdm∈LG[B]2, 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 LG[B]2 is continuously injected in LG2.Remarks 3.2.
(1) When we assumedG 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 LG[B]2.
(2) SupposeG is nuclear Fréchet and H is integrable relative to ϕ:E→L(LF2,LG2). For each seminorm Nr on LG2, there is a seminorm p(r)∈ℰ which induces the isometric embedding ϕ̃ of E(Up(r)) into L(LF2,LG2(UNr)), where LG2(UNr) is a Hilbert space since it is isometrically embedded in LG(Ur)2. Thus each [H]p(r):ℝ+×Ω→E(Up(r)) is integrable relative to ϕ̃ 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 formE,LG2, one can show that E(M∞∣ℱt)=Mt.


## 3.1. Square Integrable Martingales in Nuclear Spaces

(Ω,ℱ,P) and (ℱt)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 (ℱt)t≥0 being a filtration satisfying the usual conditions. The set X={XU:U∈𝒰} is called a projective system of square integrable martingales if for each U, we have that XU is an E(U)-valued square integrable martingale, and if whenever U,V∈𝒰 and V⊂U, then 𝒦(U,V)XV and XU are indistinguishable. We also assume XU is cadlag for each U. One says that X has a limit in E if there exists a weakly adapted mapping X̃ on ℝ+×Ω into E such that 𝒦(U)X̃ is a modification of XU 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.
LetX be a projective system of square integrable martingales. Then there exists a limit X̃ in E of X which is strongly cadlag in E, and for which 𝒦(U)X̃ is a modification of XU for each U∈𝒰. Moreover, there exists a V∈𝒰′ such that X̃ takes its values in E[V∘].Let𝔐2 denote the space of real-valued square integrable martingales. Define a mapping T:E′→𝔐2 by T(e′)=〈e′,XU〉, where U is chosen in 𝒰 so that e′∈E′[U∘]. Argue that T is well defined and linear. If en′→e′ in E′[U∘] for some U∈𝒰, then(3.1)|T(en′)∞-T(e′)∞|≤∥X∞U∥E(U)∥en′-e′∥E′[U∘];
hence {T(en′)∞} converges to (T(e′)∞) in L2(𝒫)=L2, and thus T(en′)→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 form(3.2)T(e′)=∑λi<ei,e′>Mi,
where {λi}∈l1, {ei} is equicontinuous in E, and (Mi) is bounded in 𝔐2. Choose V∈𝒰′ such that all ei∈V∘. Define the process X̃ by Xt̃=∑λieiMti, where we choose (Xt̃) to be a cadlag version. Then X̃ is the desired process.From now on, we identifyX and X̃, and we assume that X takes its values in the Hilbert space E[V∘].


## 3.2. Construction of the Stochastic Integral

Assume thatE, 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 𝒰E and 𝒰G. 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(LF2,LG2), where LG2 is the space constructed in Section 2.3.Sinceco⊄F, the stochastic measure m (=IX) first defined on the predictable rectangles can be extended to a σ-additive measure, still denoted by m, m:𝒫→LF2. 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(LF2,K2).Ifz∈(LG2)′, we define mz:𝒫→E′ by mz(A)e=〈z,em(A)〉, for e∈E. Given any r∈𝒢, if z∈UNr∘, then mz:𝒫→E′[Up(Nr)∘]=E(Up(Nr))′ (where p(Nr) is the seminorm associated with Nr relative to the mapping E→L(LF2,LG2)) by(3.3)mz(A)[e]p(Nr)=〈z,[e]p(Nr)m(A)〉=〈z,em(A)〉.
In fact, p(Nr)=p(r), relative to the mapping E→L(F,G). Let mr={|mz|:z∈UNr∘}. Then m̂r(A)=sup|mz|(A), where the supremum is extended over z∈UNr∘. Observe that m̂r is the semivariation of m relative to E(Up(r)), LG2(UNr) which arises from the isometric mapping of E(Up(r)) into L(LF2,LG2(UNr)). One can show that LG2(UNr) is isometrically embedded in the Hilbert space LG(Ur)2 and, as a result, m has a finite semivariation relative to each of these embeddings; thus m̂r is finite for each r∈𝒢, and mr is relatively weakly compact in ca(𝒫).A processH:ℝ+×Ω→E is a predictable process, or simply measurable, if it is the pointwise limit of processes from 𝒮E, the simple predictable E-valued processes. For such a measurable process H, define, for r∈𝒢,(3.4)𝔑r(H)=sup∫p(r)(H)d|mz|,
where the supremum is extended over z∈UNr∘. Let ℱ̃=ℱ̃(mE,LG2) be the space of measurable functions H such that 𝔑r(H)<∞ for each r∈𝒢. Then ℱ̃ is a locally convex space containing 𝒮E. Let ℒ=ℒ(mE,LG2) denote the closure of 𝒮E in ℱ̃. One can show that for each H∈ℒ there exists a determining sequence (Hn) from 𝒮E such that (Hn) is mean Cauchy in ℒ (𝔑r(Hn-Hm)→n,m0), for each r∈𝒢, and m̂r(p(r)(Hn-H)>ϵ)→n0 for each ϵ>0 and r∈𝒢.Now assumeG has a countable basis of seminorms, that is, G is now a nuclear Fréchet space. Thus there exists a positive measure λ such that m̂r≪λ for each r∈𝒢. Since LG2 is complete and, for H∈𝒮E, we have Nr(∫Hdm)≤𝔑r(H), where the integral is defined in the obvious way, then for general H∈ℒ with determining sequence (Hn), we can define(3.5)∫(·)Hdm=lim∫(·)Hndm∈LG2.
The completeness of LG2 ensures that ∫AHdm is a function in LG2. Define the process (∫HdX)t=∫0tHdX by ∫0tHdX=∫H1[0,t]dm, called the stochastic integral ofHwith respect toX. We say H is integrable with respect to X if H∈ℒ. If H∈𝒮E, 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 thatG 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 ϕ̃0:E[B1]→L(LF2,LG[B]2). Hence we can define m=IX:𝒫→LF2 as before, which is σ-additive and has finite semivariation relative to ϕ̃0.SinceH is measurable, it is the pointwise limit of functions from 𝒮E, and thus if x′∈E′, x′Hn→x′H. This implies that (x′H)-1(𝒪)∈𝒫 for any open subset 𝒪 of the reals. By the reflexivity of E,(3.6)E[B1]′=E[B1]=[E′(B1∘)]′=E′(B1∘),
since we have chosen B1=V∘∈𝒰′. Let e′∈E[B1]′; then e′=[x′]B1∘, and for e∈E[B1], it follows that 〈e′,e〉=〈[x′]B1∘,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 Section2.1. There exists a control measure λ in this setting, since co⊄G[B]; hence it follows that the space of integrable functions, relative to the map ϕ̃0, contains the bounded measurable functions. Thus ∫HdX=∫Hdm∈LG[B]2, 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 LG[B]2 is continuously injected in LG2.Remarks 3.2.
(1) When we assumedG 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 LG[B]2.
(2) SupposeG is nuclear Fréchet and H is integrable relative to ϕ:E→L(LF2,LG2). For each seminorm Nr on LG2, there is a seminorm p(r)∈ℰ which induces the isometric embedding ϕ̃ of E(Up(r)) into L(LF2,LG2(UNr)), where LG2(UNr) is a Hilbert space since it is isometrically embedded in LG(Ur)2. Thus each [H]p(r):ℝ+×Ω→E(Up(r)) is integrable relative to ϕ̃ 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 formE,LG2, one can show that E(M∞∣ℱt)=Mt.



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*Source: 217372-2010-08-16.xml*