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E-raamat: Stochastic Flows and Jump-Diffusions

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Stochastic Flows and Jump-DiffusionsSuurem pilt
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Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations.Researchers and graduate student in probability theory will find this book very useful.

This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps.
In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations.
Researchers and graduate student in probability theory will find this book very useful.

Arvustused

The presentation is self-contained, clear and precise. The book is definitely a must-read for researchers in the field of stochastic flows and stochastic differential equations. (G. V. Riabov, Mathematical Reviews, October, 2020)

1 Probability Distributions and Stochastic Processes
1(44)
1.1 Probability Distributions and Characteristic Functions
1(7)
1.2 Gaussian, Poisson and Infinitely Divisible Distributions
8(6)
1.3 Random Fields and Stochastic Processes
14(1)
1.4 Wiener Processes, Poisson Random Measures and Levy Processes
15(10)
1.5 Martingales and Backward Martingales
25(6)
1.6 Quadratic Variations of Semi-martingales
31(6)
1.7 Markov Processes and Backward Markov Processes
37(4)
1.8 Kolmogorov's Criterion for the Continuity of Random Field
41(4)
2 Stochastic Integrals
45(32)
2.1 Ito's Stochastic Integrals by Continuous Martingale and Wiener Process
45(4)
2.2 Ito's Formula and Applications
49(6)
2.3 Regularity of Stochastic Integrals Relative to Parameters
55(4)
2.4 Fisk--Stratonovitch Symmetric Integrals
59(5)
2.5 Stochastic Integrals with Respect to Poisson Random Measure
64(3)
2.6 Jump Processes and Related Calculus
67(6)
2.7 Backward Integrals and Backward Calculus
73(4)
3 Stochastic Differential Equations and Stochastic Flows
77(48)
3.1 Geometric Property of Solutions I; Case of Continuous SDE
77(4)
3.2 Geometric Property of Solutions II; Case of SDE with Jumps
81(5)
3.3 Master Equation
86(10)
3.4 Lp-Estimates and Regularity of Solutions; C∞-Flows
96(4)
3.5 Backward SDE, Backward Stochastic Flow
100(1)
3.6 Forward--Backward Calculus for Continuous C∞-Flows
101(3)
3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE
104(5)
3.8 Forward-Backward Calculus for C∞-Flows of Jumps
109(7)
3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps
116(3)
3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift
119(6)
4 Diffusions, Jump-Diffusions and Heat Equations
125(42)
4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov Equations
126(3)
4.2 Exponential Transformation and Backward Heat Equation
129(8)
4.3 Backward Diffusions and Heat Equations
137(3)
4.4 Dual Semigroup, Inverse Flow and Backward Diffusion
140(6)
4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps
146(9)
4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of Diffeomorphic Jumps
155(6)
4.7 Volume-Preserving Flows
161(3)
4.8 Jump-Diffusion on Subdomain of Euclidean Space
164(3)
5 Malliavin Calculus
167(78)
5.1 Derivative and Its Adjoint on Wiener Space
168(6)
5.2 Sobolev Norms for Wiener Functionals
174(9)
5.3 Nondegenerate Wiener Functionals
183(7)
5.4 Difference Operator and Adjoint on Poisson Space
190(6)
5.5 Sobolev Norms for Poisson Functionals
196(5)
5.6 Estimations of Two Poisson Functionals by Sobolev Norms
201(8)
5.7 Nondegenerate Poisson Functionals
209(5)
5.8 Equivalence of Nondegenerate Conditions
214(8)
5.9 Product of Wiener Space and Poisson Space
222(4)
5.10 Sobolev Norms for Wiener--Poisson Functionals
226(7)
5.11 Nondegenerate Wiener--Poisson Functionals
233(6)
5.12 Compositions with Generalized Functions
239(6)
6 Smooth Densities and Heat Kernels
245(58)
6.1 H-Derivatives of Solutions of Continuous SDE
246(4)
6.2 Nondegenerate Diffusions
250(3)
6.3 Density and Fundamental Solution for Nondegenerate Diffusion
253(6)
6.4 Solutions of SDE on Wiener--Poisson Space
259(6)
6.5 Nondegenerate Jump-Diffusions
265(8)
6.6 Density and Fundamental Solution for Nondegenerate Jump-Diffusion
273(4)
6.7 Short-Time Estimates of Densities
277(7)
6.8 Off-Diagonal Short-Time Estimates of Density Functions
284(4)
6.9 Densities for Processes with Big Jumps
288(7)
6.10 Density and Fundamental Solution on Subdomain
295(8)
7 Stochastic Flows and Their Densities on Manifolds
303(38)
7.1 SDE and Stochastic Flow on Manifold
303(8)
7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold
311(6)
7.3 Brownian Motion, Levy Process and Their Duals on Lie Group
317(4)
7.4 Smooth Density for Diffusion on Manifold
321(7)
7.5 Density for Jump-Diffusion on Compact Manifold
328(13)
Bibliography 341(6)
Symbol Index 347(2)
Index 349
Kunita was an invited speaker at the ICM 1986. 
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