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.: |