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International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 217372, 7 pages
Stochastic Integration in Abstract Spaces
Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA
Received 2 June 2010; Accepted 7 July 2010
Academic Editor: Andrew Rosalsky
Copyright © 2010 J. K. Brooks and J. T. Kozinski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We establish the existence of a stochastic integral in a nuclear space setting as follows. Let , , and 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 into . If is an integrable, -valued predictable process and is an -valued square integrable martingale, then there exists a -valued process 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.
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 , , and be nuclear spaces which satisfy the special conditions listed in Section 3.1, and suppose that there is a continuous bilinear mapping of into . Assume that is an -valued square integrable martingale.
If is a bounded -valued predictable process, then there exists a -valued process , called the stochastic integral of with respect to , which is a square integrable martingale.
If we further assume that has a countable basis of seminorms, then the above conclusion holds when is a predictable -valued process, which is integrable with respect to (in this case, is, in general, unbounded).
This result extends the theory of nuclear stochastic integration of Ustunel  in several directions. In  it is assumed that is the strong dual of and is the real number field, and furthermore is assumed to be bounded. To develop our theory, we modify the vector bilinear integral developed in  for Banach spaces. After defining the space , 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.
2. Bilinear Vector Integration Theory
2.1. The Banach Setting
In this subsection, assume , , and are Banach spaces over the reals , with norms denoted by . Let be a -field of subsets of a set , and assume is a -additive measure. We will assume that there is a continuous bilinear mapping of into , which, in turn, yields a continuous linear map , where is the space of bounded linear operators from into .
The semivariation of relative to , , , , denoted by is defined on as follows: where the supremum is extended over all finite collections of elements in the unit ball of and over all finite disjoint collections of sets in which are contained in . We are only interested in the case when in order to develop an integration theory of -valued integrands. Sometimes we will write as . Note that we write in place of .
One can show that, for each , , where the supremum is taken over , the unit ball of the dual of , and is defined by , for . The total variation measure of is denoted by . Let . Thus, is a bounded collection of positive -additive measures. If (e.g., if is a Hilbert space), then one can show that is relatively weakly compact in the Banach space consisting of real-valued measures, with total variation norm. In this case, there exists a positive control measure such that is uniformly absolutely continuous with respect to . A set is -negligible if it is contained in a set such that .
The advantage of modifying the bilinear integration theory in  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 the collection of -valued simple functions. We say that is measurable if there exists a sequence from which converges pointwise to . For such , define where the supremum is taken over . Let be the collection of all such with finite. Then set to be the closure of in . The space with the seminorm is our Lebesgue space.
There are different, but equivalent ways to define for . We select one which yields more information (hence more usefulness) regarding the defining components. If , one can show that there exists a determining sequence of elements in — that is, the sequence is Cauchy in , and converges in -measure, namely, for each . Define the integral of in the obvious manner. A determining sequence for has the property that is uniformly absolutely continuous with respect to . Also in . The setwise limit , , exists and defines a -additive measure on . Denote this limit by . This limit is independent of the choice of the determining sequence for . We refer to as the space of integrable functions.
Theorem 2.1 (Vitali). Let be a sequence of integrable functions. Let be an -valued measurable function. Then and in if and only if(1) in -measure,(2) is uniformly absolutely continuous with respect to .
Theorem 2.2 (Lebesgue). Let , and let be a sequence of functions from . If in -measure and for each , then and in .
Theorem 2.3. If is relatively weakly compact, then contains the bounded measurable functions.
2.2. Application to the Stochastic Integral in Banach Spaces
We retain the assumptions on , , as stated in Section 2.1. The stochastic setting is as follows (definitions and terminology are found in ). Let be a probability space. is a space of -measurable, -valued functions such that , endowed with norm . Assume is a filtration which satisfies the usual conditions. Suppose is a cadlag adapted process, with for each . Let be the ring of subsets of generated by the predictable rectangles; thus , the predictable -field. Let () be the additive -valued measure first defined on the predictable rectangles by , , and , . We regard as being continuously embedded into in the obvious manner. The theory of  for Banach stochastic integration can be shown to apply in a parallel fashion to this setting, and we state a few pertinent results. If , then can be extended uniquely to a -additive -valued measure if and only if 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 is a square integrable martingale. In this case, . As a result, we can construct the stochastic integral , which is a process such that , and this process is a -valued square integrable martingale. If we still denote the extension of to by , then is defined to be , where is integrable with respect to , that is, , and the Hilbert spaces involved in the bilinear theory are , , and . This integral will be used to define the stochastic integral in nuclear spaces.
2.3. The Definition of , Locally Convex
In this subsection, assume is a measure space, is real-valued and -additive. Let be a complete locally convex space, and let be a basis of seminorms defining the topology of . A function is measurable if it is the pointwise limit of simple -valued measurable functions in . For and being measurable, let . Let be the space of measurable functions such that for each . Then is a locally convex space with being a basis of seminorms. Define , the space of integrable functions, to be the closure of in .
It can be shown that is the set of measurable functions which have a determining sequence , that is, the sequence satisfies for each , as , and for each and , we have as . In this case, , , is unambiguously defined for each determining sequence (the definition of is the obvious one).
The bounded measurable functions are in , and the Vitali and the Lebesgue dominated convergence theorem hold. Moreover, we have the following theorem.
Theorem 2.4. Let be a complete locally convex space with a countable basis of seminorms. Then is complete.
2.4. A Remark on the Bilinear Mapping
Suppose and are locally convex spaces with and denoting their respective bases of defining seminorms. Assume is a Hilbert space and is a continuous bilinear mapping that induces . Using the continuity of , observe that for each , there exists a such that , where and are the closed balls induced by and . If we define to be the infimum over all for which the above inclusion holds, it turns out that is a seminorm and is the closed convex balanced hull of , where the union is taken over those in the above infimum. Also , where the supremum is taken over (, ). Call the seminorm associated with and . Note that is isometrically embedded in , where is the Banach space consisting of equivalence classes modulo , completed under the norm induced by ; is similarly defined.
3. The Nuclear Setting. The Construction of the Stochastic Integral
3.1. Square Integrable Martingales in Nuclear Spaces
and are as in Section 2.2. Let denote a nuclear space which is reflexive, complete, bornological, and such that its strong dual satisfies the same conditions. We say satisfies the special conditions. These special conditions are the hypotheses of Ustunel, who established fundamental results for square integrable martingales in this setting. Let be such a space. Then for and there exist neighborhood bases of zero, , and , respectively, such that for each , the space is a separable Hilbert space over the reals, and its separable dual is identified with the Hilbert space as defined in , where is the polar of . Also, and are bases of closed, convex, balanced bounded sets in , , respectively. For , we denote by the continuous canonical map from onto . If and , then is the canonical mapping of onto .
Let be a probability space with being a filtration satisfying the usual conditions. The set is called a projective system of square integrable martingales if for each , we have that is an -valued square integrable martingale, and if whenever and , then and are indistinguishable. We also assume is cadlag for each . One says that has a limit in if there exists a weakly adapted mapping on into such that is a modification of for each .
The next theorem is crucial for defining the stochastic integral. Ustunel [1, Section II.4] assumed the existence of a limit in for . This hypothesis was removed in . We now state the theorem and provide a brief sketch of the proof, which uses a technique of Ustunel.
Theorem 3.1. Let be a projective system of square integrable martingales. Then there exists a limit in of which is strongly cadlag in , and for which is a modification of for each . Moreover, there exists a such that takes its values in .
Let denote the space of real-valued square integrable martingales. Define a mapping by , where is chosen in so that . Argue that T is well defined and linear. If in for some , then hence converges to in , and thus in . Consequently, T is continuous on . Since is bornological, T is continuous on . As a result, is a nuclear map of the form where , is equicontinuous in , and is bounded in . Choose such that all . Define the process by , where we choose to be a cadlag version. Then is the desired process.
From now on, we identify and , and we assume that takes its values in the Hilbert space .
3.2. Construction of the Stochastic Integral
Assume that , , and are nuclear spaces over the reals satisfying the special conditions set forth in Section 3.1. Also assume that is a continuous bilinear mapping. The neighborhood bases of zero in and are denoted by and . Let be a square integrable martingale. By Theorem 2.4, we may assume is Hilbert space valued. As a result, we may now assume is a real Hilbert space. The bilinear map induces a continuous linear map , which in turn induces the continuous linear map , where is the space constructed in Section 2.3.
Since , the stochastic measure () first defined on the predictable rectangles can be extended to a -additive measure, still denoted by , . Note that if and are Hilbert spaces, then has finite semivariation with respect to every continuous linear embedding of into .
If , we define by , for . Given any , if , then (where is the seminorm associated with relative to the mapping ) by In fact, , relative to the mapping . Let . Then , where the supremum is extended over . Observe that is the semivariation of relative to , which arises from the isometric mapping of into . One can show that is isometrically embedded in the Hilbert space and, as a result, has a finite semivariation relative to each of these embeddings; thus is finite for each , and is relatively weakly compact in .
A process is a predictable process, or simply measurable, if it is the pointwise limit of processes from , the simple predictable -valued processes. For such a measurable process , define, for , where the supremum is extended over . Let be the space of measurable functions such that for each . Then is a locally convex space containing . Let denote the closure of in . One can show that for each there exists a determining sequence from such that is mean Cauchy in (), for each , and for each and .
Now assume has a countable basis of seminorms, that is, is now a nuclear Fréchet space. Thus there exists a positive measure such that for each . Since is complete and, for , we have , where the integral is defined in the obvious way, then for general with determining sequence , we can define The completeness of ensures that is a function in . Define the process by , called the stochastic integral of with respect to . We say is integrable with respect to if . If , one can show that is a -valued square integrable martingale. By means of using determining sequences, the general stochastic integral enjoys this property.
Next, assume that just satisfies the special conditions (no longer nuclear Fréchet). Let be a bounded measurable -valued process; hence the range of is contained in a closed, bounded, convex, balanced set , where is a Hilbert space. By the continuity of , it follows that is contained in a bounded set having the same properties as , and is a Hilbert space.
Algebraically, induces which is bilinear, and since for every , is continuous. As a result, this induces a continuous linear map , which in turn induces the continuous linear map . Hence we can define as before, which is -additive and has finite semivariation relative to .
Since is measurable, it is the pointwise limit of functions from , and thus if , . This implies that for any open subset of the reals. By the reflexivity of , since we have chosen . Let ; then , and for , it follows that , that is, . As a consequence, is weakly measurable, and since is separable, by the Pettis theorem we conclude that is bounded and measurable as an -valued function.
We now use the integration theory in Section 2.1. There exists a control measure in this setting, since ; hence it follows that the space of integrable functions, relative to the map , contains the bounded measurable functions. Thus , and the process defines the stochastic integral; note that this process is a square integrable martingale. Since the norm on is stronger than any , one can show that is continuously injected in .
Remarks 3.2. (1) When we assumed was a nuclear Fréchet space, we constructed the stochastic integral for every integrable with respect to . In particular, if is bounded, the stochastic integral agrees with the one constructed by means of using .
(2) Suppose is nuclear Fréchet and is integrable relative to . For each seminorm on , there is a seminorm which induces the isometric embedding of into , where is a Hilbert space since it is isometrically embedded in . Thus each is integrable relative to and gives rise to the stochastic integral defined by , which is a square integrable martingale. The projective system of square integrable martingales has a limit in , and this limit is , .
Since there is a control measure for , one can show that .
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