Nicolae Dinculeanu

Vector Integration and Stochastic Integration in Banach Spaces

Vector Integration. The Stochastic Integral. Martingales. Processes with Finite Variation. Processes with Finite Semivariation. The Ito Formula. Stochastic Integration in the Plane. Two-Parameter Martingales. Two-Parameter Processes with Finite Variation. Two-Parameter Processes with Finite Semivariation. References.

Strong additivity, absolute continuity and compactness in spaces of measures

In this paper we shall examine strong additivity, absolute continuity, and compactness (weak and strong) in the space of vector measures and discuss various relationships among these concepts. Many of the theorems cast new light on the structure of measures, even in the scalar case. Some of the results presented here have been announced in Brooks and Dinculeanu [S, 91. Variations on the above themes are also treated in detail in Brooks [7]. In Section 1, the main concept of uniform strong additivity of a family of vector measures X is introduced, namely, m(E,)-+O uniformly for m E -X, when (EJ is a disjoint sequence of sets. This notion is inextricably tied up with weak and strong compactness of %” in different topological settings. The existence of a positive control measure p such that X < p, when X is uniformly strongly additive is presented in Section 2. A local control measure is constructed in Section 3 by means of establishing a “synthesis theorem” which allows us to piece together locally equivalent families of positive measures. This theorem is also used to prove the existence of a local control measure for relatively weakly compact sets in the space of vector measures with local finite variation (Section 4). Criteria for weak compactness in this locally convex space (Theorem 4.2) extends the work of Dieudonne [14], who considered a special case, viz, the space of locally integrable functions on a locally compact space. Conditions concerning compactness (weak and strong) with respect to the quasivariation norm are presented in Sections 5

Weak compactness and uniform convergence of operators in spaces of Bochner integrable functions

The classical characterization of weak compactness in the space L’(p) over a measure space (X, C, p), states that a bounded set Kc L’(p) is relatively weakly compact iff it is uniformly o-additive (or, equivalently, uniformly inegrable, or uniformly p-absolutely continuous, in case P-l(X) < 00 1. In this paper we give a new characterization of weak compactness in L’(p) which is completely different from the classical one. It is stated in terms of uniform weak convergence of certain “admissible” sequences of operators (Theorem 6). It is interesting to make a comparative analysis of these two characterizations of weak compactness.

Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

In [6, theorem IV.8.18], relatively norm compact sets K in Lp([mu]) are characterized by means of strong convergence of conditional expectations, E[pi]f --> f in Lp([mu]), uniformly for f [set membership, variant] K, where (E[pi]) is the family of conditional expectations corresponding to the net of all finite measurable partitions. In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces of vector valued pth power Bochner integrable functions (and spaces M([Sigma], E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in by means of uniform strong convergence E[pi]f --> f, as well as relatively weak compact sets Kby means of uniform weak convergence E[pi]f --> f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.

Projections and regularity of abstract processes

In this note we shall prove the existence of optional and predictable projections of stochastic processes X taking values in a Banach space E. Furthermore, if the range of X is contained in a compact set and if X is cadlag (respectively caglad), then the optional (respectively predictable) projection possesses the same property. Finally, we shall prove that every E-valued martingale has a cadlag modification

Bimeasures in Banach spaces

In this paper we study integration with respect to a bimeasure with finite semivariation. The bimeasures as well as the functions to be integrated, take on their values in Banach spaces.

Vector Valued Stochastic Processes III Projections and Dual Projections

In the first part of this paper we study the optional and predictable projections of vector valued processes with separable range. The existence of the projections of measurable processes is proved in [1]. Moreover, in [1] we proved that the property of being right (respectively left) continuous is inherited by the optional (respectively predictable) projection, provided that the process has relatively compact range. In this paper we prove the existence of “weak projections” of weakly measurable processes and show that the weak right (resp. left) continuity is inherited by the weak optional (resp. predictable) projection, without restrictions on the range.

CHAPTER 8 – Vector Integration in Banach Spaces and Application to Stochastic Integration

This chapter discusses the theory of vector integration in Banach spaces and its application to stochastic integration. This theory reduces to integration with respect to vector measures with finite variation, which reduces to the Bochner integral with respect to a positive measure. The Bochner integral is based upon the classical integral of real-valued functions with respect to a positive measure. The Bochner integral is obtained by extending by continuity the integral of step functions. The topology defined by the seminorm is called “the topology of convergence in the mean.” The property that asserts the uniform σ-additivity and the uniform absolute continuity of the indefinite integrals of functions in a Cauchy sequence is analyzed in the chapter. Integrability with respect to a measure with finite variation reduces to Bochner integrability with respect to the variation measure. The Radon–Nikodym-type theorem for vector measures with finite variation is also presented in the chapter.

Additive summable processes and their stochastic integral

We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B].We relax the definition of a summable processesX:Ω×ℝ+→E⊂L(F, G) by asking for the associated measureIX to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (IX)z, forz∈(LGp)*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ+→E⊂L(F, G) with integrable semivariation ifc0 ∋G.

Stochastic processes with finite semivariation in Banach spaces and their stochastic integral

In this Paper we study a new class of Banach-valued Processes which are summable: the Processes with integrable semivariation. One can define the Stochastic Integral for such processes, which can be computed pathwise, as a Stieltjes integral with respect to a function with finite semivariation (rather than finite variation).