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 aG -valued process [subscript](∫HdX)t[/subscript] 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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Recommended by Andrew Rosalsky

Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA

Received 2 June 2010; Accepted 7 July 2010

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 _{(∫H dX)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 ^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 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 ... , with norms denoted by |·| . Let Σ be a σ -field of subsets of a set T , and assume m:Σ[arrow right]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 [varphi]:E[arrow right]L(F,G) , where L(F,G) is the space of bounded linear operators from F into G .

The semivariation of m relative to [varphi] , E , F , G , denoted by m... is defined on Σ as follows: [figure omitted; refer to PDF] 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 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 [varphi](e) .

One can show that, for each A∈Σ , m...(A)=sup |_mz |(A) , where the supremum is taken over z∈^{G1[variant prime]} , the unit ball of the dual ^{G[variant prime]} of G , and _mz :Σ[arrow right]^{E[variant prime]} 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[variant prime]} } . 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 by ...AE;=_...AE;E the collection of E -valued simple functions. We say that h:T[arrow right]E is measurable if there exists a sequence from ...AE; which converges pointwise to h . For such h , define [figure omitted; refer to PDF] where the supremum is taken over z∈^{G1[variant prime]} . Let ......=......(_mE,G ) be the collection of all such h with N(h) finite. Then set [Lagrangian (script capital L)]=[Lagrangian (script capital L)](_mE,G ) to be the closure of ...AE; in ...... . The space [Lagrangian (script capital L)] with the seminorm N is our Lebesgue space.

There are different, but equivalent ways to define ∫h dm for h∈[Lagrangian (script capital L)] . We select one which yields more information (hence more usefulness) regarding the defining components. If h∈[Lagrangian (script capital L)] , one can show that there exists a determining sequence {_hn } of elements in ...AE; -- that is, the sequence is Cauchy in [Lagrangian (script capital L)] , and {_hn } converges in m -measure, namely, m...(|h-_hn |>...)[arrow right]0 for each ...>0 . Define the integral of h∈...AE; in the obvious manner. A determining sequence for h has the property that _{{N(hn1(·) )}n} is uniformly absolutely continuous with respect to m... . Also _hn [arrow right]h in [Lagrangian (script capital L)] . The setwise limit _∫Ahn dm , A∈Σ , exists and defines a σ -additive measure on Σ . Denote this limit by _∫A h dm . This limit is independent of the choice of the determining sequence for h . We refer to [Lagrangian (script capital 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∈[Lagrangian (script capital L)] and _hn [arrow right]h in [Lagrangian (script capital L)] if and only if

(1) _hn [arrow right]h in m -measure,

(2) {N(_hn1(·) )_}n is uniformly absolutely continuous with respect to m... .

Theorem 2.2 (Lebesgue).

Let g∈[Lagrangian (script capital L)] , and let {_hn } be a sequence of functions from [Lagrangian (script capital L)] . If _hn [arrow right]h in m -measure and |_hn (·)|≤|g(·)| for each n , then h∈[Lagrangian (script capital L)] and _hn [arrow right]h in [Lagrangian (script capital L)] .

Theorem 2.3.

If _mE,G is relatively weakly compact, then [Lagrangian (script capital 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,...,P) be a probability space. ^LF2 (P) is a space of ... -measurable, E -valued functions such that E(^|f|2 )=∫^|f|2 dP<∞ , endowed with norm |f|=E^{(|f|2 )1/2} . Assume _{(...t )t≥0} is a filtration which satisfies the usual conditions. Suppose X:_...+ ×Ω[arrow right]F is a cadlag adapted process, with _Xt ∈^LF2 for each t . Let ... be the ring of subsets of _...+ ×Ω generated by the predictable rectangles; thus σ(...)=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∈_...s , and m(_0A )=_1AX0 , A∈_...0 . 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 ... . 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,^LG2 (_...+ ×Ω)<∞ . As a result, we can construct the stochastic integral (∫H dX_)t , which is a process such that ^∫0t H dX∈^LG2 , and this process is a G -valued square integrable martingale. If we still denote the extension of m to P by m , then ^∫0t H dX is defined to be ∫H_1[0,t] dm , where H is integrable with respect to m , that is, H∈[Lagrangian (script capital 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 of ^LG2 , 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 ...A2; be a basis of seminorms defining the topology of G . A function f:T[arrow right]G is measurable if it is the pointwise limit of simple G -valued measurable functions in _...AE;G . For r∈...A2; and h being measurable, let _Nr (h)=^{(∫r(h)2 d|m|)1/2} . Let _......G be the space of measurable functions h such that _Nr (h)<∞ for each r∈...A2; . Then _......G is a locally convex space with {_Nr :r∈...A2;} being a basis of seminorms. Define ^LG2 , the space of integrable functions, to be the closure of _...AE;G in _......G .

It can be shown that ^LG2 is the set of measurable functions h which have a determining sequence (_hn )⊂_...AE;G , that is, the sequence satisfies for each r∈...A2; , _Nr (_hn -_hm )[arrow right]0 as n,m[arrow right]∞ , and for each ...>0 and r∈...A2; , we have |m|(r(_hn -h)>...)[arrow right]0 as n[arrow right]∞ . In this case, _∫A hdm=lim _∫Ahn dm , A∈Σ , is unambiguously defined for each determining sequence (the definition of ∫_hn dm is the obvious one).

The bounded measurable functions are in ^LG2 , 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 ^LG2 is complete.

2.4. A Remark on the Bilinear Mapping E×F[arrow right]G

Suppose E and G are locally convex spaces with ... and ...A2; denoting their respective bases of defining seminorms. Assume F is a Hilbert space and Φ:E×F[arrow right]G is a continuous bilinear mapping that induces [varphi]:E[arrow right]L(F,G) . Using the continuity of Φ , observe that for each r∈...A2; , there exists a p∈... 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[variant prime]}} , where the supremum is taken over z∈^{Ur[composite function]} (ze:f[arrow right]...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 ker p(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,...,P) and _{(...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[variant prime]} 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[variant prime]} there exist neighborhood bases of zero, U , and ^{U[variant prime]} , 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[variant prime]} [^{U[composite function]} ] as defined in [5], where ^{U[composite function]} is the polar of U . Also, {^{U[composite function]} :U∈U} and {^{V[composite function]} :V∈^{U[variant prime]} } are bases of closed, convex, balanced bounded sets in ^{E[variant prime]} , E , respectively. For U∈U , we denote by ...A6;(U) the continuous canonical map from E onto E(U) . If U,V∈U and V⊂U , then ...A6;(U,V) is the canonical mapping of E(V) onto E(U) .

Let (Ω,...,P) be a probability space with _{(...t )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 ...A6;(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 X... on _...+ ×Ω into E such that ...A6;(U)X... 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.

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 ...A6;(U)X... is a modification of ^XU for each U∈U . Moreover, there exists a V∈^{U[variant prime]} such that X... takes its values in E[^{V[composite function]} ] .

Let ^...2 denote the space of real-valued square integrable martingales. Define a mapping T:^{E[variant prime]} [arrow right]^...2 by T(^{e[variant prime]} )=...^{e[variant prime]} ,^XU ... , where U is chosen in U so that ^{e[variant prime]} ∈^{E[variant prime]} [^{U[composite function]} ] . Argue that T is well defined and linear. If ^{en[variant prime]} [arrow right]^{e[variant prime]} in ^{E[variant prime]} [^{U[composite function]} ] for some U∈U , then [figure omitted; refer to PDF] hence {T(^{en[variant prime]}_)∞ } converges to (T_{(^{e[variant prime]} )∞} ) in ^L2 (P)=^L2 , and thus T(^{en[variant prime]} )[arrow right]T(^{e[variant prime]} ) in ^...2 . Consequently, T is continuous on ^{E[variant prime]} [^{U[composite function]} ] . Since ^{E[variant prime]} is bornological, T is continuous on ^{E[variant prime]} . As a result, T:^{E[variant prime]} [arrow right]^...2 is a nuclear map of the form [figure omitted; refer to PDF] where {_λi }∈^l1 , {_ei } is equicontinuous in E , and (^Mi ) is bounded in ^...2 . Choose V∈^{U[variant prime]} such that all _ei ∈^{V[composite function]} . Define the process X... by _Xt ...=∑_λiei^Mti , where we choose (_Xt ...) 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[composite function]} ] .

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[arrow right]G is a continuous bilinear mapping. The neighborhood bases of zero in E and G are denoted by _UE and _UG . Let X:_...+ ×Ω[arrow right]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 [varphi]:E[arrow right]L(F,G) , which in turn induces the continuous linear map [varphi]¯:E[arrow right]L(^LF2 ,^LG2 ) , where ^LG2 is the space constructed in Section 2.3.

Since _co ⊄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[arrow right]^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 ) .

If z∈(^{LG2)[variant prime]} , we define _mz :P[arrow right]^{E[variant prime]} by _mz (A)e=...z,em(A)... , for e∈E . Given any r∈...A2; , if z∈^{U_Nr [composite function]} , then _mz :P[arrow right]^{E[variant prime]} [^{Up(_Nr )[composite function]} ]=E^{(_{Up(Nr )} )[variant prime]} (where p(_Nr ) is the seminorm associated with _Nr relative to the mapping E[arrow right]L(^LF2 ,^LG2 ) ) by [figure omitted; refer to PDF] In fact, p(_Nr )=p(r) , relative to the mapping E[arrow right]L(F,G) . Let _mr ={|_mz |:z∈^{U_Nr [composite function]} } . Then _m...r (A)=sup |_mz |(A) , where the supremum is extended over z∈^{U_Nr [composite function]} . Observe that _m...r 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 _m...r is finite for each r∈...A2; , and _mr is relatively weakly compact in ca(P) .

A process H:_...+ ×Ω[arrow right]E is a predictable process, or simply measurable, if it is the pointwise limit of processes from _...AE;E , the simple predictable E -valued processes. For such a measurable process H , define, for r∈...A2; , [figure omitted; refer to PDF] where the supremum is extended over z∈^{U_Nr [composite function]} . Let ......=......(_mE,^LG2 ) be the space of measurable functions H such that _...r (H)<∞ for each r∈...A2; . Then ...... is a locally convex space containing _...AE;E . Let [Lagrangian (script capital L)]=[Lagrangian (script capital L)](_mE,^LG2 ) denote the closure of _...AE;E in ...... . One can show that for each H∈[Lagrangian (script capital L)] there exists a determining sequence (_Hn ) from _...AE;E such that (_Hn ) is mean Cauchy in [Lagrangian (script capital L)] (_...r (_Hn -_Hm )_{[arrow right]n,m} 0 ), for each r∈...A2; , and _m...r (p(r)(_Hn -H)>...)_{[arrow right]n} 0 for each ...>0 and r∈...A2; .

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∈...A2; . Since ^LG2 is complete and, for H∈_...AE;E , we have _Nr (∫H dm)≤_...r (H) , where the integral is defined in the obvious way, then for general H∈[Lagrangian (script capital L)] with determining sequence (_Hn ) , we can define [figure omitted; refer to PDF] The completeness of ^LG2 ensures that _∫A H dm is a function in ^LG2 . Define the process _{(∫H dX)t} =^∫0t H dX by ^∫0t H dX=∫H_1[0,t] dm , called the stochastic integral of H with respect to X . We say H is integrable with respect to X if H∈[Lagrangian (script capital L)] . If H∈_...AE;E , one can show that _{(∫H dX)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[arrow right]G[B] which is bilinear, and since ^Φ0-1 (αB)⊃(α_B1 )×F for every α∈... , _Φ0 is continuous. As a result, this induces a continuous linear map _[varphi]0 :E[_B1 ][arrow right]L(F,G[B]) , which in turn induces the continuous linear map _[varphi]...0 :E[_B1 ][arrow right]L(^LF2 ,^LG[B]2 ) . Hence we can define m=_IX :P[arrow right]^LF2 as before, which is σ -additive and has finite semivariation relative to _[varphi]...0 .

Since H is measurable, it is the pointwise limit of functions from _...AE;E , and thus if ^{x[variant prime]} ∈^{E[variant prime]} , ^{x[variant prime]}_Hn [arrow right]^{x[variant prime]} H . This implies that ^{(x[variant prime] H)-1} (...AA;)∈P for any open subset ...AA; of the reals. By the reflexivity of E , [figure omitted; refer to PDF] since we have chosen _B1 =^{V[composite function]} ∈^{U[variant prime]} . Let ^{e[variant prime]} ∈E^{[_B1 ][variant prime]} ; then ^{e[variant prime]} =_{[^{x[variant prime]} ]^{B1[composite function]}} , and for e∈E[_B1 ] , it follows that ...^{e[variant prime]} ,e...=..._{[^{x[variant prime]} ]^{B1[composite function]}} ,e...=...^{x[variant prime]} ,e... , that is, ^{x[variant prime]} H=^{e[variant prime]} H . As a consequence, H:_...+ ×Ω[arrow right]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 _co ⊄G[B] ; hence it follows that the space of integrable functions, relative to the map _[varphi]...0 , contains the bounded measurable functions. Thus ∫H dX=∫H dm∈^LG[B]2 , and the process _{(∫H dX)t} =∫H_1[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∈...A2; , one can show that ^LG[B]2 is continuously injected in ^LG2 .

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 ^LG[B]2 .

(2) Suppose G is nuclear Fréchet and H is integrable relative to [varphi]:E[arrow right]L(^LF2 ,^LG2 ) . For each seminorm _Nr on ^LG2 , there is a seminorm p(r)∈... which induces the isometric embedding [varphi]... 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) :_...+ ×Ω[arrow right]E(_Up(r) ) is integrable relative to [varphi]... 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∈...A2;} has a limit in G , and this limit is _{(∫H dX)t} :=_(M)t , _M∞ =∫H dX .

Since there is a control measure for _mE,^LG2 , one can show that E(_M∞ |"_...t )=_Mt .

References

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[3] J. K. Brooks, N. Dinculeanu, "Stochastic integration in Banach spaces," Seminar on Stochastic Processes, 1990 (Vancouver, BC, 1990) , vol. 24, pp. 27-115, Birkhäuser, Boston, Mass, USA, 1991.

[4] C. Dellacherie, P.-A. Meyer Probabilities and Potential , vol. 29, of North-Holland Mathematics Studies, pp. viii+189, North-Holland, Amsterdam, The Netherlands, 1978.

[5] F. Trèves Topological Vector Spaces, Distributions and Kernels , pp. xvi+624, Academic Press, New York, NY, USA, 1967.

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