Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

/pdf/hitchhiker-s-guide-to-the-fractional-sobolev-spaces-irm2sao95g.pdf

Hitchhiker's guide to the fractional Sobolev spaces

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

/pdf/levy-processes-and-stochastic-calculus-2rcj1efu2c.pdf

Lévy Processes and Stochastic Calculus

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Financial Modelling with Jump Processes

Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis- often with different definitions and under different names. Among the synonyms are `Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions' or `operator monotone functions'. This monograph- now in its second revised and extended edition- offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

/pdf/bernstein-functions-theory-and-applications-1xeexh2387.pdf

Bernstein Functions: Theory and Applications

Prelude Dependence chart Prologue 1. The pleasures of counting 2. sigma-algebras 3. Measures 4. Uniqueness of measures 5. Existance of measures 6. Measurable mappings 7. Measurable functions 8. Integration of positive functions 9. Integrals of measurable functions and null sets 10. Convergence theroems and their applications 11. The function spaces 12. Product measures and Fubini's theorem 13. Integrals with respect to image measures 14. Integrals of images and Jacobi's transformation rule 15. Uniform integrability and Vitali's convergence theorem 16. Martingales 17. Martingale convergence theorems 18. The Radon-Nikodym theorem and other applications of martingales 19. Inner product spaces 20. Hilbert space 21. Conditional expectations in L2 22. Conditional expectations in Lp 23. Orthonormal systems and their convergence behaviour Appendix A. Lim inf and lim supp Appendix B. Some facts from point-set topology Appendix C. The volume of a parallelepiped Appendix D. Non-measurable sets Appendix E. A summary of the Riemann integral Further reading Bibliography Notation index Name and subject index.

Measures, Integrals and Martingales

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that Cc∞(ℝn)⊂D(A) and A|Cc∞(ℝn) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c0Rep(x,ξ). We show that the associated Feller process {Xt}t≥0 on ℝn is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞x:={λ>0:lim|ξ|→∞|x−y|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞x:={λ>0:liminf|ξ|→∞|x−y|≤2/|ξ||e|≤1|p(y,|ξ|e)|/|ξ|λ=0}, and obtain a.s. (ℙx) that lim t→0t−1/λs≤t|Xs−x|=0 or ∞ according to λ>β∞x or λ<δ∞x. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Levy) case considered by W. Pruitt [27].

Growth and Hölder conditions for the sample paths of Feller processes

A Primer on Feller Semigroups and Feller Processes- Feller Generators and Symbols- Construction of Feller Processes- Transformations of Feller Processes- Sample Path Properties- Global Properties- Approximation- Open Problems- References- Index

René L. Schilling

Papers

Financial Modelling with Jump Processes

Bernstein Functions: Theory and Applications

Measures, Integrals and Martingales

Growth and Hölder conditions for the sample paths of Feller processes

Lévy-type processes : construction, approximation and sample path properties