STOCHASTIC CALCULUS OF VARIATIONS

For Jump Processes

Yasushi Ishikawa (Ehime University, Matsuyama, Japón)

Berlín, Alemania. Walter de GRUYTER. ISBN: 9783110377767. X, 278 págs.  Marzo de 2016. Encuadernado.

PVP EUR 137,00 (4% IVA incluido)

Les ofrecemos el volumen nº 54 de la serie DE GRUYTER STUDIES IN MATHEMATICS. Podemos enviarles información sobre el resto de volúmenes publicados y disponibles.

Esta monografía es una introducción concisa del cálculo estocástico de variaciones (también conocido como cálculo variacional de Malliavin).

El autor ofrece varios resultados de cálculos sobre esta materia y también se puede aplicar a ecuaciones diferenciales estocásticas (SDEs).

Como en todos los títulos de Walter de GRUYTER, podemos ofrecer las siguientes ediciones:

-          Encuadernado. ISBN. 9783110377767. EUR 137,00 (4% IVA incluido).

-          eBook. ISBN. 9783110378078. EUR 158,00 (21% IVA incluido).

-          Papel y eBook: ISBN. 9783110378085. EUR 208,00 (4% IVA incluido).

 

Extracto del índice:

Preface|V

Preface to the second edition|VII

0 Introduction|1

1 Lévy processes and Itô calculus|5

1.1 Poisson random measure and Lévy processes|5

1.1.1 Lévy processes|5

1.1.2 Examples of Lévy processes|8

1.1.3 Stochastic integral for a finite variation process|11

1.2 Basic materials for SDEs with jumps|13

1.2.1 Martingales and semimartingales|13

1.2.2 Stochastic integral with respect to semimartingales|15

1.2.3 Doléans’ exponential and Girsanov transformation|22

1.3 Itô processes with jumps|25

2 Perturbations and properties of the probability law|33

2.1 Integration-by-parts on Poisson space|33

2.1.1 Bismut’s method|35

2.1.2 Picard’s method|45

2.1.3 Some previous methods|51

2.2 Methods of finding the asymptotic bounds (I)|58

2.2.1 Markov chain approximation|59

2.2.2 Proof of Theorem 2.3|63

2.2.3 Proof of lemmas|69

2.3 Methods of finding the asymptotic bounds (II)|75

2.3.1 Polygonal geometry|76

2.3.2 Proof of Theorem 2.4|77

2.3.3 Example of Theorem 2.4 – easy cases|87

2.4 Summary of short time asymptotic bounds|94

2.4.1 Case that μ(dz) is absolutely continuous with respect to the m-dimensional Lebesgue measure dz|94

2.4.2 Case that μ(dz) is singular with respect to dz|95

2.5 Auxiliary topics|97

2.5.1 Marcus’ canonical processes|97

2.5.2 Absolute continuity of the infinitely divisible laws|100

2.5.3 Chain movement approximation|105

2.5.4 Support theorem for canonical processes|107

3 Analysis of Wiener–Poisson functionals|111

3.1 Calculus of functionals on the Wiener space|111

3.1.1 Definition of the Malliavin–Shigekawa derivative Dt |113

3.1.2 Adjoint operator δ = D∗ |117

3.2 Calculus of functionals on the Poisson space|119

3.2.1 One-dimensional case|119

3.2.2 Multidimensional case|122

3.2.3 Characterisation of the Poisson space|125

3.3 Sobolev space for functionals over the Wiener–Poisson space|129

3.3.1 The Wiener space|129

3.3.2 The Poisson Space|130

3.3.3 The Wiener–Poisson space|137

3.4 Relation with the Malliavin operator|144

3.5 Composition on the Wiener–Poisson space (I) – general theory|146

3.5.1 Composition with an element in S

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