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# evans_SDE_course - AN INTRODUCTION TO STOCHASTIC...

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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, Itˆ o’s formula Chapter 5: Stochastic diFerential equations Chapter 6: Applications Appendices Exercises References 1

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PREFACE These notes survey, without too many precise details, the basic theory of probability, random diFerential equations and some applications. Stochastic diFerential 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 diFerential equations, and partial diFerential equations as well. But as an experiment I tried to design these lectures so that starting graduate students (and maybe really strong undergraduates) can 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 con±rming 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 my class with several lectures on ±nancial applications, and to ²raydoun 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 many readers who have found errors, especially Robert Piche, who provided me with an extensive list of typos and suggestions that I have incorporated into this latest version of the notes. 2
CHAPTER 1: INTRODUCTION A. MOTIVATION Fix a point x 0 R n and consider then the ordinary diferential equation: ( ODE ) ½ ˙ x ( t )= b ( x ( t )) ( t> 0) x (0) = x 0 , where b : R n R n is a given, smooth vector ±eld and the solution is the trajectory x ( · ):[0 , ) R n . x(t) x 0 Trajectory of the differential equation Notation. x ( t )isthe state of the system at time t 0, ˙ x ( t ):= d dt x ( t ). ¤ In many applications, however, the experimentally measured trajectories o² systems modeled by (ODE) do not in ²act behave as predicted: X(t) x 0 Sample path of the stochastic differential equation Hence it seems reasonable to modi²y (ODE), somehow to include the possibility o² random efects disturbing the system. A formal way to do so is to write: (1) ½ ˙ X ( t b ( X ( t )) + B ( X ( t )) ξ ( t )( 0) X (0) = x 0 , where B : R n M n × m (= space o² n × m matrices) and ξ ( · m -dimensional “white noise”.

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## This note was uploaded on 11/08/2009 for the course MATH 236 taught by Professor Papanicolaou during the Winter '08 term at Stanford.

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evans_SDE_course - AN INTRODUCTION TO STOCHASTIC...

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