Stochastic Integration with Jumps (Encyclopedia of Mathematics and its Applications, Series Number 89)

Stochastic Processes With Jumps And Random Measures Are Importance As Drivers In Applications Like Financial Mathematics And Signal Processing. This 2002 Text Develops Stochastic Integration Theory For Both Integrators (semimartingales) And Random Measures From A Common Point Of View. Using Some Novel Predictable Controlling Devices, The Author Furnishes The Theory Of Stochastic Differential Equations Driven By Them, As Well As Their Stability And Numerical Approximation Theories. Highlights Feature Dct And Egoroff's Theorem, As Well As Comprehensive Analogs Results From Ordinary Integration Theory, For Instance Previsible Envelopes And An Algorithm Computing Stochastic Integrals Of Càglàd Integrands Pathwise. Full Proofs Are Given For All Results, And Motivation Is Stressed Throughout. A Large Appendix Contains Most Of The Analysis That Readers Will Need As A Prerequisite. This Will Be An Invaluable Reference For Graduate Students And Researchers In Mathematics, Physics, Electrical Engineering And Finance Who Need To Use Stochastic Differential Equations. Motivation: Stochastic Differential Equations -- Wiener Process -- The General Model -- Integrators And Martingales -- The Elementary Stochastic Integral -- The Semivariations -- Path Regularity Of Integrators -- Processes Of Finite Variation -- Martingales -- Extension Of The Integral -- The Daniell Mean -- The Integration Theory Of A Mean -- Countable Additivity In P-mean -- Measurability -- Predictable And Previsible Processes -- Special Properties Of Daniell's Mean -- The Indefinite Integral -- Functions Of Integrators -- Ito's Formula -- Random Measures -- Control Of Integral And Integrator -- Change Of Measure--factorization -- Martingale Inequalities -- The Doob-meyer Decomposition -- Semimartingales -- Previsible Control Of Integrators -- Levy Processes -- Stochastic Differential Equations -- Existence And Uniqueness Of The Solution -- Stability: Differentiability In Parameters -- Pathwise Computation Of The Solution -- Weak Solutions -- Stochastic Flows -- Semigroups, Markov Processes, And Pde -- Complements To Topology And Measure Theory -- Notations And Conventions -- Topological Miscellanea -- Measure And Integration -- Weak Convergence Of Measures -- Analytic Sets And Capacity -- Suslin Spaces And Tightness Of Measures -- The Skorohod Topology -- The L[superscript P]-spaces -- Semigroups Of Operators. Klaus Bichteler. Includes Bibliographical References (p. 477-482) And Indexes.

the Complete Theory Of Stochastic Differential Equations Driven By Jumps, Their Stability, And Numerical Approximation Theories.

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bichteler (mathematics, U. Of Texas At Austin) Aims To Present The Mathematical Underpinning Of Stochastic Analysis. Wiener Process Is Treated For Economics Students And Driving Terms With Jumps Are Covered To Give Mathematics Students The Background To Connect With The Literature And Discrete Time Martingales. This Leads To The Most General Lebesgue-stieltjes Integral. Bichteler Identifies The Useful Lebesgue-stieltjes Distribution Functions Among All Functions On The Line And Looks At Criteria For Process To Be Useful As Random Distribution Functions. Integration Theory Is Demonstrated To Be Useful For Finding These Criteria. Annotation C. Book News, Inc., Portland, Or (booknews.com)

"This book develops stochastic integration theory for both integrators (semimartingales) and random measures from a common point of view. Using some novel predictable controlling devices, the author furnishes the theory of stochastic differential equations driven by them, as well as their stability and numerical approximation theories. Highlights feature DCT and Egoroff's Theorem, as well as comprehensive analogs to results from ordinary integration theory."