Vector integration and stochastic integration in Banach spaces

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Vector integration and stochastic integration in Banach spaces / Nicolae Dinculeanu

資料種別:: 図書
出版情報:: New York : Wiley, c2000
形態:: xv, 424 p. ; 25 cm
シリーズ名:: Pure and applied mathematics <BA00007488>
著者名:: Dinculeanu, N. (Nicolae) <DA02698794>
ISBN:: 9780471377382 [0471377384]
書誌ID:: BA46443524

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注記:

Bibliography: p. 413-418
Includes indexes

主題:

Banach spaces; Stochastic integrals; Vector spaces

言語:

英語

目次情報:

Preface

Vector Integration / Chapter 1：

Preliminaries / 1.：

Banach spaces / A.：

Classes of sets / B.：

Measurable functions / C.：

Simple measurability of operator-valued functions / D.：

Weak measurability / E.：

Integral of step functions / F.：

Totally measurable functions and the immediate integral / G.：

The Riesz representation theorem / H.：

The classical integral / I.：

The Bochner integral / J.：

Convergence theorems / K.：

Measures with finite variation / 2.：

The variation of vector measures

Boundedness of [sigma]-additive measures

Variation of real-valued measures

Integration with respect to vector measures with finite variation

The indefinite integral

Integration with respect to gm

The Radon-Nikodym theorem

Conditional expectations

[sigma]-additive measures / 3.：

[sigma]-additive measures on [sigma]-rings

Uniform [sigma]-additivity

Uniform absolute continuity and uniform [sigma]-additivity

Weak [sigma]-additivity

Uniform [sigma]-additivity of indefinite integrals

Weakly compact sets in L[superscript 1 subscript F] ([mu])

Measures with finite semivariation / 4.：

The semivariation

Semivariation and norming spaces

The semivariation of [sigma]-additive measures

The family m[subscript F,Z] of measures

Integration with respect to a measure with finite semivariation / 5.：

Measurability with respect to a vector measure

The seminorm m[subscript F,G](f)

The space of integrable functions

The integral

Properties of the space F[subscript D] (B, m[subscript F,G])

Relationship between the spaces F[subscript D](m)

The indefinite integral of measures with finite semivariation

Strong additivity / 6.：

Extension of measures / 7.：

Applications / 8.：

Integral representation of continuous linear operations on L[superscript p]-spaces

Random Gaussian measures

The Stochastic Integral / Chapter 2：

Summable processes / 9.：

Notations

The measure I[subscript X]

Computation of I[subscript X] for predictable rectangles

Computation of I[subscript X] for stochastic intervals

The stochastic integral / 10.：

The space F[subscript D] [characters not reproducible]

The integral [function of] HdI[subscript X]

A convergence theorem

The stochastic integral H - X

The stochastic integral and stopping times / 11.：

Stochastic integral of elementary processes

Stopping the stochastic integral

Summability of stopped processes

The jumps of the stochastic integral

The completeness of the space L[superscript 1 subscript F,G](X) / 12.：

The Uniform Convergence Theorem

The Vitali and the Lebesgue Convergence Theorems

The stochastic integral of [sigma]-elementary and of caglad processes as a pathwise Stieltjes integral

Summability of the stochastic integral / 13.：

Summability criterion / 14.：

Quasimartingales and the Doleans measure

The summability criterion

Local summability and local integrability / 15.：

Definitions

Basic properties

Additional properties

Martingales / Chapter 3：

Stochastic integral of martingales / 16.：

Square integrable martingales / 17.：

Extension of the measure I[subscript M]

Summability of square integrable martingales

Properties of the space F[subscript F,G](M)

Isometrical isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]])

Processes with Finite Variation / Chapter 4：

Functions with finite variation and their Stieltjes integral / 18.：

Functions with finite variation

The variation function |g|

The measure associated to a function

The Stieltjes integral

Processes with finite variation / 19.：

Definition and properties

Optional and predictable measures

The measure [mu subscript X]

Summability of processes with integrable variation

The stochastic integral as a Stieltjes integral

The pathwise stochastic integral

Semilocally summable processes

Processes with Finite Semivariation / Chapter 5：

Functions with finite semivariation and their Stieltjes integral / 20.：

Functions with finite semivariation

The Stieltjes integral with respect to a function with finite semivariation

Processes with finite semivariation / 21.：

The semivariation process

The measure [mu]x

Summability of processes with integrable semivariation

Dual projections / 22.：

Dual projection of measures

Dual projections of processes

Existence of dual projections

Processes with locally integrable variation or semivariation

Examples of processes with locally integrable variation or semivariation

Decomposition of local martingales

The Ito Formula / Chapter 6：

The Ito formula / 23.：

Preliminary results

The vector quadratic variation

The quadratic variation

The process of jumps

Ito's formula

Stochastic Integration in the Plane / Chapter 7：

Order relation in R[superscript 2] / 24.：

The increment [Delta subscript zz], g

Right continuity

The filtration

The predictable [Sigma]-algebra

Stopping times

Stochastic processes

Extension of processes from R[superscript 2 subscript +] [times] [Omega] to R[superscript 2] [times] [Omega]

The seminorm I[subscript X] and the space F[subscript F,G](X) / 25.：

Properties of the stochastic integral / 26.：

Extension of I[subscript X] to P([infinity])

Existence of left limits of X in L[superscript p subscript E]

Some properties of the integral [function of] HdI[subscript X]

Two-Parameter Martingales / Chapter 8：

A decomposition theorem / 27.：

The measures [characters not reproducible] and [mu subscript [M]]

Summability of the square integrable martingales in Hilbert spaces

The space F[subscript F,G](I[subscript M])

Isometric isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]])

Two-Parameter Processes with Finite Variation / Chapter 9：

Functions with finite variation in the plane / 29.：

Monotone functions

Partitions

Variation corresponding to a partition

Variation of a function on a rectangle

Limits of the variation

Functions vanishing outside a quadrant

Variation of real-valued functions

Lateral limits

Measures associated to functions

[sigma]-additivity of the measure m[subscript g] / L.：

Processes with integrable variation / M.：

Two-Parameter Processes with Finite Semivariation / Chapter 10：

Functions with finite semivariation in the plane / 31.：

The Stieltjes integral for functions with finite semivariation in R[superscript 2]

Processes with finite semivariation in the plane / 32.：

References

Preface

Vector Integration / Chapter 1：

Preliminaries / 1.：

Banach spaces / A.：

Classes of sets / B.：

Measurable functions / C.：

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