Stochastic Integration with Jumps
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Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. 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 integra...
Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. 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, 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.
Table of contents:
1. Introduction; 2. Integrators and martingales; 3. Extension of the integral; 4. Control of integral and integrator; 5. Stochastic differential equations; Appendix A. Complements to topology and measure theory; Appendix B. Answers to selected problems; References; Index.
This comprehensive reference for graduate students and researchers in mathematics, physics, electrical engineering and finance develops stochastic integration theory for semimartingales and random measures from a common point of view. Full proofs are given for all results, and motivation is stressed throughout. An appendix contains most of the analysis that readers will need as a prerequisite.
The complete theory of stochastic differential equations driven by jumps, their stability, and numerical approximation theories.
Table of contents:
1. Introduction; 2. Integrators and martingales; 3. Extension of the integral; 4. Control of integral and integrator; 5. Stochastic differential equations; Appendix A. Complements to topology and measure theory; Appendix B. Answers to selected problems; References; Index.
This comprehensive reference for graduate students and researchers in mathematics, physics, electrical engineering and finance develops stochastic integration theory for semimartingales and random measures from a common point of view. Full proofs are given for all results, and motivation is stressed throughout. An appendix contains most of the analysis that readers will need as a prerequisite.
The complete theory of stochastic differential equations driven by jumps, their stability, and numerical approximation theories.
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