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E-raamat: Stochastic Calculus of Variations: For Jump Processes

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Stochastic Calculus of Variations: For Jump ProcessesSuurem pilt
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This monograph is a concise introduction to the stochastic calculus of variations (also known as Malliavin calculus) for processes with jumps. It is written for researchers and graduate students who are interested in Malliavin calculus for jump processes. In this book "processes with jumps" includes both pure jump processes and jump-diffusions. The author provides many results on this topic in a self-contained way; this also applies to stochastic differential equations (SDEs) "with jumps". The book also contains some applications of the stochastic calculus for processes with jumps to the control theory and mathematical finance. Namely, asymptotic expansions functionals related with financial assets of jump-diffusion are provided based on the theory of asymptotic expansion on the WienerPoisson space. Solving the HamiltonJacobiBellman (HJB) equation of integro-differential type is related with solving the classical Merton problem and the Ramsey theory. The field of jump processes is nowadays quite wide-ranging, from the Lévy processes to SDEs with jumps. Recent developments in stochastic analysis have enabled us to express various results in a compact form. Up to now, these topics were rarely discussed in a monograph.

Contents: Preface Preface to the second edition Introduction Lévy processes and Itô calculus Perturbations and properties of the probability law Analysis of WienerPoisson functionals Applications Appendix Bibliography List of symbols Index
Preface v
Preface to the second edition vii
0 Introduction
1(4)
1 Levy processes and ltd calculus
5(28)
1.1 Poisson random measure and Levy processes
5(8)
1.1.1 Levy processes
5(3)
1.1.2 Examples of Levy processes
8(3)
1.1.3 Stochastic integral for a finite variation process
11(2)
1.2 Basic materials for SDEs with jumps
13(12)
1.2.1 Martingales and semimartingales
13(2)
1.2.2 Stochastic integral with respect to semimartingales
15(7)
1.2.3 Doleans' exponential and Girsanov transformation
22(3)
1.3 Ito processes with jumps
25(8)
2 Perturbations and properties of the probability law
33(78)
2.1 Integration-by-parts on Poisson space
33(25)
2.1.1 Bismut's method
35(10)
2.1.2 Picard's method
45(6)
2.1.3 Some previous methods
51(7)
2.2 Methods of finding the asymptotic bounds (I)
58(17)
2.2.1 Markov chain approximation
59(4)
2.2.2 Proof of Theorem 2.3
63(6)
2.2.3 Proof of lemmas
69(6)
2.3 Methods of finding the asymptotic bounds (II)
75(19)
2.3.1 Polygonal geometry
76(1)
2.3.2 Proof of Theorem 2.4
77(10)
2.3.3 Example of Theorem 2.4 -- easy cases
87(7)
2.4 Summary of short time asymptotic bounds
94(3)
2.4.1 Case that μ(dz) is absolutely continuous with respect to the m-dimensional Lebesgue measure dz
94(1)
2.4.2 Case that μ(dz) is singular with respect to dz
95(2)
2.5 Auxiliary topics
97(14)
2.5.1 Marcus' canonical processes
97(3)
2.5.2 Absolute continuity of the infinitely divisible laws
100(5)
2.5.3 Chain movement approximation
105(2)
2.5.4 Support theorem for canonical processes
107(4)
3 Analysis of Wiener--Poisson functionals
111(84)
3.1 Calculus of functionals on the Wiener space
111(8)
3.1.1 Definition of the Malliavin--Shigekawa derivative Dt
113(4)
3.1.2 Adjoint operator δ = D*
117(2)
3.2 Calculus of functionals on the Poisson space
119(10)
3.2.1 One-dimensional case
119(3)
3.2.2 Multidimensional case
122(3)
3.2.3 Characterisation of the Poisson space
125(4)
3.3 Sobolev space for functionals over the Wiener--Poisson space
129(15)
3.3.1 The Wiener space
129(1)
3.3.2 The Poisson Space
130(7)
3.3.3 The Wiener--Poisson space
137(7)
3.4 Relation with the Malliavin operator
144(2)
3.5 Composition on the Wiener--Poisson space (I) -- general theory
146(12)
3.5.1 Composition with an element in S'
147(6)
3.5.2 Sufficient condition for the composition
153(5)
3.6 Smoothness of the density for Ito processes
158(34)
3.6.1 Preliminaries
158(3)
3.6.2 Big perturbations
161(4)
3.6.3 Concatenation (I)
165(7)
3.6.4 Concatenation (II) -- the case that (D) may fail
172(6)
3.6.5 More on the density
178(14)
3.7 Composition on the Wiener--Poisson space (II) -- Ito processes
192(3)
4 Applications
195(66)
4.1 Asymptotic expansion of the SDE
195(34)
4.1.1 Analysis on the stochastic model
198(21)
4.1.2 Asymptotic expansion of the density
219(4)
4.1.3 Examples of asymptotic expansions
223(6)
4.2 Optimal consumption problem
229(32)
4.2.1 Setting of the optimal consumption
229(3)
4.2.2 Viscosity solutions
232(19)
4.2.3 Regularity of solutions
251(4)
4.2.4 Optimal consumption
255(3)
4.2.5 Historical sketch
258(3)
Appendix 261(4)
Bibliography 265(10)
List of symbols 275(2)
Index 277
Yasushi Ishikawa, Ehime University, Matsuyama, Japan.
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