AN INTRODUCTION TO STOCHASTIC
                       DIFFERENTIAL EQUATIONS
                             VERSION 1.2
                            Lawrence C. Evans
                        Department of Mathematics
                              UCBerkeley
        Chapter 1: Introduction
        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 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
               themineasycases(forsimplerandomvariablesorforstepfunctions), 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.
                 Mythanks 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.
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              CHAPTER1: INTRODUCTION
              A. MOTIVATION
              Fix a point x ∈ Rn and consider then the ordinary differential equation:
                         0
                                       ˙
            (ODE)                       x(t)=b(x(t)) (t>0)
                                        x(0) = x ,
                                               0
            where b : Rn → Rn is a given, smooth vector field and the solution is the trajectory
            x(·):[0,∞) → Rn.
                                               

              

                             Trajectory of the differential equation
                                                             ˙     d x(t).          

              Notation. x(t)isthestate of the system at time t ≥ 0, x(t):=dt
              In many applications, however, the experimentally measured trajectories of systems
            modeled by (ODE) do not in fact behave as predicted:
                                               
             

                       Sample path of the stochastic differential equation
            Hence it seems reasonable to modify (ODE), somehow to include the possibility of random
            effects disturbingthe system. A formal way to do so is to write:
                                ˙
            (1)                  X(t)=b(X(t))+B(X(t))ξ(t)(t>0)
                                 X(0) = x ,
                                         0
            where B : Rn → Mn×m (= space of n×m matrices) and
                                 ξ(·):=m-dimensional “white noise”.
            This approach presents us with these mathematical problems:
              • Define the “white noise” ξ(·) in a rigorous way.
                                                3
              • Define what it means for X(·) to solve (1).
              • Show (1) has a solution, discuss uniqueness, asymptotic behavior, dependence upon
            x , b, B, etc.
             0
              B. SOME HEURISTICS
              Let us first study (1) in the case m = n, x =0,b ≡ 0, and B ≡ I. The solution of
                                                  0
            (1) in this settingturns out to be the n-dimensional Wiener process,orBrownian motion,
            denoted W(·). Thus we may symbolically write
                                            ˙
                                           W(·)=ξ(·),
            thereby assertingthat “white noise” is the time derivative of the Wiener process.
              Now return to the general case of the equation (1), write d instead of the dot:
                                                              dt
                                  dX(t) = b(X(t))+B(X(t))dW(t),
                                    dt                     dt
            and finally multiply by “dt”:
            (SDE)               dX(t)=b(X(t))dt+B(X(t))dW(t)
                                  X(0) = x .
                                          0
              This expression, properly interpreted, is a stochastic differential equation. We say that
            X(·) solves (SDE) provided
            (2)     X(t)=x + tb(X(s))ds+ tB(X(s))dW        for all times t>0 .
                            0
                                0             0
            Now we must:
              • Construct W(·): See Chapter 3.
              • Define the stochastic integral t ···dW : See Chapter 4.
                                         0
              • Show (2) has a solution, etc.: See Chapter 5.
            And once all this is accomplished, there will still remain these modeling problems:
              • Does (SDE) truly model the physical situation?
              • Is the term ξ(·) in (1) “really” white noise, or is it rather some ensemble of smooth,
               but highly oscillatory functions? See Chapter 6.
              Aswewillseelaterthesequestionsaresubtle, and different answers can yield completely
            different solutions of (SDE). Part of the trouble is the strange form of the chain rule in
            the stochastic calculus:
                    ˆ
              C. ITO’S FORMULA
              Assume n = 1 and X(·) solves the SDE
            (3)                         dX =b(X)dt+dW.
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