The Extended Stochastic Integral in Linear Spaces with Differentiable Measures and Related Topics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Basic notions. Itô integral in C[0,1]
  • Multiple stochastic integral
  • Hermite polynomials and multiple stochastic integrals

• Itô's formula for stochastic integral

• Bibliographical remarks

The following sections are included:

• Basic definitions

• Local properties of stochastic derivative and ESI

• Extended stochastic integral with variable upper limit

• Itô's formula for the extended stochastic integral

• Bibliographical remarks

The following sections are included:

• Introduction

• Randomized ESI and its continuity

• Quadratic variation

• Itô's formula
  • Ltô's formula for randomized ESI without jumps
  • Itô's formula for jump discontinuous randomized ESI
  • Special cases of Itô's formula and some examples
  • Multidimensional Itô's formula
  • The probabilistic solution of some integro-differential equation

• Formula for Brownian partial derivatives

• Bibliographical remarks

The following sections are included:

• Measures in locally convex spaces

• Definition of differentiable measure

• Differentiability of Gaussian measure

• The extended stochastic integral for the Wiener measure as the differentiable measure

• Bibliographical remarks

The following sections are included:

• Introduction

• Itô–Wiener decomposition for Gaussian measure

• Sobolev spaces of Wiener functionals

• The vector extended stochastic integral

• The ESI in Hilbert space of generalized functions

• Quadratic variation of extended stochastic integral in abstract Wiener space

• Bibliographical remarks

The following sections are included:

• Introduction

• Main results

• Applications
  • The image of differentiable measure
  • “Gauss-Ostrogradsky” formula

• Some formulas for differential forms of finite codegree

• Bibliographical remarks

The following sections are included:

• Introduction

• Mixtures of Gaussian measures

• The direct integral of Fock spaces

• Itô–Wiener decomposition for mixtures of Gaussian measures

• Logarithmic derivatives of the mixtures of Gaussian measures

• Extended stochastic integral for mixtures of Gaussian measures

• Ornstein–Uhlenbeck semi-group

• Commutation relations

• The ESI in Hilbert space of distributions

• Bibliographical remarks

The following sections are included:

• The measure derivatives and transition measures

• Surface measures

• Algebraic properties of mapping μ ↦ μ^Ω

• Differentiation of surface measures

• Approximation of surface measures by volume measures

• On integrability of second order surfaces in Banach space

• Bibliographical remarks

The following sections are included:

• Definition of heat potentials

• Differential properties of heat potentials

• Analyticity of heat potentials

• Continuity of heat potentials

• The theorem on the jumps of heat potentials

• The heat potentials and (quasi)characteristic operators
  • The definition of heat motion on Banach space
  • Heat potentials and (quasi) characteristic operators

• Integral equations in theory of heat potentials

• Bibliographical remarks

The following sections are included:

• Bibliography

• Index