Introduction to Stochastic Calculus

1 Introduction to Stochastic CalculusMath 545 - Duke UniversityAndrea Agazzi, Jonathan C. MattinglyContentsChapter 1. Introduction51. Motivations52. Outline For a Course6 Chapter 2. Probabilistic Background91. Countable probability spaces92. Uncountable Probability Spaces123. General Probability Spaces and Sigma Algebras134. Distributions and Convergence of Random Variables20 Chapter 3. Brownian Motion and Stochastic Processes231. An Illustrative Example: A Collection of Random Walks232. General Stochastic Proceses243. Definition of Brownian motion (Wiener Process)254. Constructive Approach to Brownian motion285. Brownian motion has Rough Trajectories296.

2 More Properties of Random Walks317. More Properties of General Stochastic Processes328. A glimpse of the connection withpdes37 Chapter 4. It o Integrals391. Properties of the noise Suggested by Modeling392. Riemann- Stieltjes Integral403. A motivating example414. It o integrals for a simple class of step functions425. Extension to the Closure of Elementary Processes466. Properties of It o integrals487. A continuous in time version of the the It o integral498. An Extension of the It o Integral509. It o Processes51 Chapter 5. Stochastic Calculus531. It o s Formula for Brownian motion532. Quadratic Variation and Covariation563.

3 It o s Formula for an It o Process604. Full Multidimensional Version of It o Formula625. Collection of the Formal Rules for It o s Formula and Quadratic Variation66 Chapter 6. Stochastic Differential Equations691. Definitions692. Examples of SDEs703. Existence and Uniqueness for SDEs7334. Weak solutions to SDEs765. Markov property of It o diffusions78 Chapter 7. PDEs and SDEs: The connection791. Infinitesimal generators792. Martingales associated with diffusion processes813. Connection with PDEs834. Time-homogeneous Diffusions865. Stochastic Characteristics876. A fundamental example: Brownian motion and the Heat Equation88 Chapter 8.

4 Martingales and Localization911. Martingales & Optional stopping933. Localization954. Quadratic variation for martingales985. L evy-Doob characterization of Brownian motion996. Random time changes1017. Martingale inequalities1048. Martingale representation theorem106 Chapter 9. Girsanov s Theorem1091. An illustrative example1092. Tilted Brownian motion1103. Girsanov s Theorem forsdes111 Chapter 10. One Dimensionalsdes1171. Natural Scale and Speed measure1172. Existence of Weak Solutions1183. Exit From an Interval1184. Recurrence1195. Intervals with Singular End Points119 Bibliography121 Appendix A. Some Results from Analysis123 Appendix B.

5 Exponential Martingales and Hermite Polynomials1254 CHAPTER 1 Introduction1. MotivationsEvolutions in time with random influences/random the number of rabbits in some population or the price of a stock . Then one might want to make amodel of the dynamics which includes random influences . A (very) simple example isdNptqdt aptqNptqwhereaptq rptq` noise .( )Making sense of noise and learning how to make calculations with it is one of the principalobjectives of this course. This will allow us predict, in a probabilistic sense, the behavior of situations like the one introduced above are ubiquitous in nature:i)The gambler s ruin problemWe play the following game: We start with 3$ in ourpocket and we flip a coin.

6 If the result is tail we loose one dollar, while if the result ispositive we win one dollar. We stop when we have no money to bargain, or when we reach9$. We may ask: what is the probability that I end up broke?ii)Population dynamics/Infectious diseasesAs anticipated, ( ) can be used to modelthe evolution in the number of rabbits in some population. Similar models are used tomodel the number of genetic mutations an animal species. We may also think aboutNptqas the number of sick individuals in a population. Reasonable and widely appliedmodels for the spread of infectious diseases are obtained by modifying ( ), and observingits behavior. In all these cases, one may be interested in knowing if it is likely for thedisease/ mutation to take over the population, or rather to go )Stock pricesWe may think about a set ofMrisky investments ( a stock), where thepriceNiptqforiP t1.

7 Muper unit at timetevolves according to ( ). In this case, oneone would like to optimize his/her choice of stocks to maximize the total value Mi 1 iNiptqat a later with diffusion theory and exists a deep connection betweennoisy processes such as the one introduced above and the deterministic theory of partial differentialequations. This starling connection will be explored and expanded upon during the course, but weanticipate some examples below:i)Dirichlet problemLetupxqbe the solution to thepdegiven below with the notedboundary conditions. Here B2Bx2`B2By2. The amazing fact is the following: If we start aBrownian motion diffusing from a pointpx0,y0qinside the domain then the probabilitythat it first hits the boundary in the darker region is given byupx0, )Black Scholes EquationSuppose that at timet 0 the person iniii)is offered theright (without obligation) to buy one unit of the risky asset at a specified priceSand ata specified future timet T.

8 Such a right is called aEuropean call option. How muchshould the person be willing to pay for such an option? This question can be answered bysolving the famous Black Scholes equation, giving for any stock priceNptqthe right valueSof the European 112 u= 0u= 02. Outline For a CourseWhat follows is a rough outline of the class, giving a good indication of the topics to be covered,though there will be ) Weeks 1-2: Motivation and Introduction to Stochastic Process(a)Motivating Examples: Random Walks, Population Model with noise, Black-Scholes,Dirichlet problems(b) Themes: Direct calculation with Stochastic Calculus , connections withpdes(c) Introduction : Probability Spaces, Expectations, -algebras, Conditional expectations,Random walks and discrete time Stochastic processes.

9 Continuous time Stochastic pro-cesses and characterization of the law of a process by its finite dimensional distributions(Kolmogorov Extension Theorem). Markov Process and ) Weeks 3-4: Brownian motion and its Properties(a)Definitions of Brownian motion (BM) as a continuous Gaussian process with indepen-dent increments. Chapman-Kolmogorov equation, forward and backward Kolmogorovequations for BM. Continuity of sample paths (Kolmogorov Continuity Theorem).BM and more Markov process and Martingales.(b)First and second variation ( variation and quadratic variation) Application to BMiii) Week 5: Stochastic Integrals(a) The Riemann-Stieltjes integral. Why can t we use it ?

10 (b) Building the It o and Stratonovich integrals (Making sense of t0 dB. )(c) Standard properties of integrals hold: linearity, additivity(d) It o isometry:Ep fdBq2 E ) Week 6: It o s Formula and Applications(a) Change of variable(b) Connections withpdes and the Backward Kolmogorov equationv) Week 7: Stochastic Differential Equations(a) What does it mean to solve ansde?(b) Existence of solutions (Picard iteration), Uniqueness of solutionsvi) Week 8-9: Stopping Times(a)Definition. -algebra associated to stopping time. Bounded stopping times. Doob soptional stopping theorem(b) Dirichlet Problems and hitting probabilities(c) Localization via stopping timesvii) Week 10: Levy-Doob theorem and Girsonov s Theorem(a) How to tell when a continuous martingale is a Brownian motion(b) Random time changes to turn a Martingale into a Brownian motion6(c) Hermite Polynomials and the exponential martingale(d) Girsanov s Theorem, Cameron-Martin formula, and changes of measure(1) The simple example of Gaussian random variables shifted(2) Idea of Importance sampling and how to sample from tails(3) The shift of a Brownian motion(4) Changing the drift in a diffusionviii) Week 11.