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Unformatted text preview: 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 differential equations Chapter 6: Applications Appendices Exercises References 1 PREFACE These notes survey, without too many precise details, the basic theory of prob ability, 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 partial dif ferential 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 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 my 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 illus trating 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 ∈ R n and consider then the ordinary differential equation: ( ODE ) braceleftbigg ˙ x ( t ) = b ( x ( t )) ( t > 0) x (0) = x , where b : R n → R n is a given, smooth vector field and the solution is the trajectory x ( · ) : [0 , ∞ ) → R n . X78X28X74X29 X78 X30 Trajectory of the differential equation Notation. x ( t ) is the state of the system at time t ≥ 0, ˙ x ( t ) := d dt x ( t ). square In many applications, however, the experimentally measured trajectories of systems modeled by (ODE) do not in fact behave as predicted: X58X28X74X29 X78 X30 Sample path of the stochastic differential equation Hence it seems reasonable to modify (ODE), somehow to include the possibility of random effects disturbing the system. A formal way to do so is to write: (1) braceleftbigg ˙ X ( t ) = b ( X ( t )) + B ( X ( t )) ξ ( t ) ( t > 0) X (0) = x , where B : R n → M n × m (= space of...
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 Spring '10
 323
 Probability theory

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