Stochastic processes - Bass, Richard F. (Cambridge : Cambridge University Press, 2011)
'This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black-Scholes formula for the pricing of derivatives in financial mathematics, the Kalman-Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature'--
'In a first course on probability one typically works with a sequence of random variables X1,X2, ... For stochastic processes, instead of indexing the random variables by the non-negative integers, we index them by t G [0, oo) and we think of Xt as being the value at time t. The random variable could be the location of a particle on the real line, the strength of a signal, the price of a stock, and many other possibilities as well. We will also work with increasing families of s -fields {J-t}, known as filtrations. The s -field J-t is supposed to represent what we know up to time t. 1.1 Processes and s -fields Let (Q., J-, P) be a probability space. A real-valued stochastic process (or simply a process) is a map X from [0, oo) x Q. to the reals. We write Xt = Xt = X(t,?). We will impose stronger measurability conditions shortly, but for now we require that the random variables Xt be measurable with respect to J- for each t 0. A collection of s -fields J-t such that J-t C J- for each t and J-s C J-t if s t is called a filtration. Define J-t+ = P\e0J-t+e. A filtration is right continuous if J-t+ = J-t for all t 0'--
1. Basic notions -- 2. Brownian motion -- 3. Martingales -- 4. Markov properties of Brownian motion -- 5. The Poisson process -- 6. Construction of Brownian motion -- 7. Path properties of Brownian motion -- 8. The continuity of paths -- 9. Continuous semimartingales -- 10. Stochastic integrals -- 11. It{u3BE7}s formula -- 12. Some applications of It{u3BE7}s formula -- 13. The Girsanov theorem -- 14. Local times -- 15. Skorokhod embedding -- 16. The general theory of processes -- 17. Processes with jumps -- 18. Poisson point processes -- 19. Framework for Markov processes -- 20. Markov properties -- 21. Applications of the Markov properties -- 22. Transformations of Markov processes -- 23. Optimal stopping -- 24. Stochastic differential equations -- 25. Weak solutions of SDEs -- 26. The Ray-Knight theorems -- 27. Brownian excursions -- 28. Financial mathematics -- 29. Filtering -- 30. Convergence of probability measures -- 31. Skorokhod representation -- 32. The space C[0, 1] -- 33. Gaussian processes -- 34. The space D[0, 1] -- 35. Applications of weak convergence -- 36. Semigroups -- 37. Infinitesimal generators -- 38. Dirichlet forms -- 39. Markov processes and SDEs -- 40. Solving partial differential equations -- 41. One-dimensional diffusions -- 42. Ly processes -- Appendices: A. Basic probability; B. Some results from analysis; C. Regular conditional probabilities; D. Kolmogorov extension theorem.
Includes bibliographical references and index.
Keywords: Stochastic analysis