An Introduction To Stochastic Differential Equations – LawrenceCEvans (1)

Next we study the term C: mn−1  k=0 (W(tn k+1) −W(τ n k ))(W(τ n k ) −W(tn k))]2) = mn−1  k=0 E([W(tn k+1) −W(τ n k )]2)E([W(τ n k ) −W(tn k)]2) (independent increments) = mn−1  k=0 (1 −λ)(tn k+1 −tn k)λ(tn k+1 −tn k) ≤λ(1 −λ)T|P n | →0. Hence C →0 in L2(Ω) as n →∞.

We combine the limiting expressions for the terms A, B, C, and thereby establish the Lemma. □ It turns out that Itˆo’s definition (later, in §B) of 0 W dW corresponds to the choice λ = 0. That is, W dW = W 2(T) and, more generally,  r s W dW = W 2(r) −W 2(s) −(r −s) for all r ≥s ≥0. This is not what one would guess offhand.

An alternative definition, due to Stratonovich, takes λ = 1 2; so that W ◦dW = W 2(T) (Stratonovich integral). See Chapter 6 for more. More discussion. What are the advantages of taking λ = 0 and getting W dW = W 2(T) 2 ? First and most importantly, building the Riemann sum approximation by evaluating the integrand at the left-hand endpoint τ n k = tn k on each subinterval [tn k, tn k=1] will ultimately permit the definition of G dW for a wide class of so-called “nonanticipating” stochastic processes G(·).

Exact definitions are later, but the idea is that t represents time, and since we do not know what W(·) will do on [tn k, tn k+1], it is best to use the known value of G(tn k) in the approximation. Indeed, G(·) will in general depend on Brownian motion W(·), and we do not know at time tn k its future value at the future time τ n k = (1 −λ)tn k + λtn k+1, if λ > 0.

□ Let W(·) be a 1-dimensional Brownian motion defined on some probability space (Ω, U, P). DEFINITIONS. (i) The σ-algebra W(t) := U(W(s) | 0 ≤s ≤t) is called the history of the Brownian motion up to (and including) time t. (ii) The σ-algebra W+(t) := U(W(s)−W(t) | s ≥t) is the future of the Brownian motion beyond time t. □ DEFINITION. A family F(·) of σ-algebras ⊆U is called nonanticipating (with respect to W(·)) if (a) F(t) ⊇F(s) for all t ≥s ≥0 (b) F(t) ⊇W(t) for all t ≥0 (c) F(t) is independent of W+(t) for all t ≥0.

We also refer to F(·) as a filtration. IMPORTANT REMARK.

Chapter 2: A crash course in basic probability theory Chapter 3: Brownian motion and “white noise” Chapter 4: Stochastic integrals, Itˆo’s formula Chapter 5: Stochastic differential equations Chapter 6: Applications Appendices Exercises References 1 PREFACE These are an evolving set 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. 2 CHAPTER 1: INTRODUCTION A. MOTIVATION Fix a point x0 ∈Rn and consider then the ordinary differential equation: (ODE)  ˙x(t) = b(x(t)) (t > 0) x(0) = x0, where b : Rn →Rn is a given, smooth vector field and the solution is the trajectory x(·) : [0, ∞) →Rn.


 Trajectory of the differential equation Notation.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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