Stochastic Integrals

Stochastic Integrals discusses one area of diffusion processes: the differential and integral calculus based upon the Brownian motion. The book reviews Gaussian families, construction of the Brownian motion, the simplest properties of the Brownian motion, Martingale inequality, and the law of the iterated logarithm. It also discusses the definition of the stochastic integral by Wiener and by Ito, the simplest properties of the stochastic integral according to Ito, and the solution of the simplest stochastic differential equation. The book explains diffusion, Lamperti's method, forward equation, Feller's test for the explosions, Cameron-Martin's formula, the Brownian local time, and the solution of dx=e(x) db + f(x) dt for coefficients with bounded slope. It also tackles Weyl's lemma, diffusions on a manifold, Hasminski's test for explosions, covering Brownian motions, Brownian motions on a Lie group, and Brownian motion of symmetric matrices. The book gives as example of a diffusion on a manifold with boundary the Brownian motion with oblique reflection on the closed unit disk of R squared. The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics. Content: Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Page ii Front Matter, Page iii Copyright, Page iv Dedication, Page v PREFACE, Pages vii-viii LIST OF NOTATIONS, Pages xi-xiii 1 - BROWNIAN MOTION, Pages 1-19 2 - STOCHASTIC INTEGRALS AND DIFFERENTIALS, Pages 20-49 3 - STOCHASTIC INTEGRAL EQUATIONS (d = 1), Pages 50-81 4 - STOCHASTIC INTEGRAL EQUATIONS (d ≥ 2), Pages 82-132 REFERENCES, Pages 133-138 SUBJECT INDEX, Pages 139-140