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发布于：2015-12-11 09:53
AN INTRODUCTION TO STOCHASTIC
DIFFERENTIAL EQUATIONS VERSION 1.2 Lawrence C. Evans Department of Mathematics UC Berkeley Chapter 1: Introduction Chapter 2: A crash course in basic probability theory Chapter 3: Brownian motion and “white noise” Chapter 4: Stochastic integrals, It?o’s formula Chapter 5: Stochastic differential equations Chapter 6: Applications Appendices Exercises References 1 PREFACE These are an evolvingset of notes for Mathematics 195 at UC Berkeley. This course is for advanced undergraduate math majors and surveys without too many precise details random differential equations and some applications. Stochastic differential equations is usually, and justly, regarded as a graduate level subject. A really careful treatment assumes the students’ familiarity with probability theory, measure theory, ordinary differential equations, and perhaps partial differential equations as well. This is all too much to expect of undergrads. But white noise, Brownian motion and the random calculus are wonderful topics, too good for undergraduates to miss out on. Therefore as an experiment I tried to design these lectures so that strong students could follow most of the theory, at the cost of some omission of detail and precision. I for instance downplayed most measure theoretic issues, but did emphasize the intuitive idea of σ–algebras as “containing information”. Similarly, I “prove” many formulas by confirming them in easy cases (for simple random variables or for step functions), and then just stating that by approximation these rules hold in general. I also did not reproduce in class some of the more complicated proofs provided in these notes, although I did try to explain the guiding ideas. My thanks especially to Lisa Goldberg, who several years ago presented the class with several lectures on financial applications, and to Fraydoun Rezakhanlou, who has taught from these notes and added several improvements. I am also grateful to Jonathan Weare for several computer simulations illustrating the text. Thanks also to Robert Piche, who provided me with an extensive list of typos and suggestions that I have incorporated into this latest version of the notes 汐雷 
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