Brownian Motion and Stochastic CalculusSpringer Science & Business Media, Dis 6, 2012 - 470 mga pahina Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property. * This book is written for readers who are acquainted with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in their continuous time context. It has been our goal to write a systematic and thorough exposi tion of this subject, leading in many instances to the frontiers of knowledge. At the same time, we have endeavored to keep the mathematical prerequisites as low as possible, namely, knowledge of measure-theoretic probability and some familiarity with discrete-time processes. The vehicle we have chosen for this task is Brownian motion, which we present as the canonical example of both a Markov process and a martingale. We support this point of view by showing how, by means of stochastic integration and random time change, all continuous-path martingales and a multitude of continuous-path Markov processes can be represented in terms of Brownian motion. This approach forces us to leave aside those processes which do not have continuous paths. Thus, the Poisson process is not a primary object of study, although it is developed in Chapter 1 to be used as a tool when we later study passage times and local time of Brownian motion. |
Mga Nilalaman
1 | |
CHAPTER 2 | 21 |
Brownian Motion | 27 |
CHAPTER 3 | 128 |
Brownian Motion and Partial Differential Equations | 239 |
CHAPTER 5 | 281 |
xvii | 397 |
CHAPTER 6 | 399 |
12 | 403 |
30 | 409 |
49 | 415 |
59 | 442 |
447 | |
459 | |
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assume B₁ Borel-measurable bounded Brownian family Brownian motion Brownian path coefficients constant continuous function continuous semimartingale convergence theorem Corollary d-dimensional Brownian motion define Definition denote Dynkin system Exercise exists F-measurable filtration F finite finite-dimensional distributions fixed follows formula holds implies independent inequality inf{t initial distribution Itô Itô's rule Lemma Lévy local martingale M₁ Markov property martingale problem nondecreasing nonnegative o-field obtain one-dimensional Brownian motion optional Poisson probability measure probability space progressively measurable progressively measurable process proof of Theorem Proposition quadratic variation random variable Remark representation result right-continuous right-hand side sample paths satisfies the usual Section semimartingale sequence space Q stochastic differential equation stochastic integral strong solution submartingale subsection Suppose t₁ vector W₁ weak solution X₁ xe Rd Y₁ Z₁ zero