Transcription of Stochastic Calculus: An Introduction with Applications
1 Stochastic calculus : An Introduction withApplicationsGregory F. Lawlerc 2014, Gregory F. LawlerAll rights reservediiContents1 Martingales in discrete Conditional expectation .. Martingales .. Optional sampling theorem .. Martingale convergence theorem and Polya s urn .. Square integrable martingales .. Integrals with respect to random walk .. A maximal inequality .. Exercises .. 272 Brownian Limits of sums of independent variables .. Multivariate normal distribution .. Limits of random walks .. Brownian motion .. Construction of Brownian motion .. Understanding Brownian motion .. motion as a continuous martingale .. motion as a Markov process.
2 Motion as a Gaussian process .. motion as a self-similar process .. Computations for Brownian motion .. Quadratic variation .. Multidimensional Brownian motion .. Heat equation and generator .. One dimension .. Expected value at a future time .. Exercises .. 75iiiivCONTENTS3 Stochastic What is Stochastic calculus ? .. Stochastic integral .. of Riemann integration .. of simple processes .. of continuous processes .. It o s formula .. More versions of It o s formula .. Diffusions .. Covariation and the product rule .. Several Brownian motions .. Exercises .. 1144 More Stochastic Martingales and local martingales.
3 An example: the Bessel process .. Feynman-Kac formula .. Binomial approximations .. Continuous martingales .. Exercises .. 1365 Change of measure and Girsanov Absolutely continuous measures .. Giving drift to a Brownian motion .. Girsanov theorem .. Black-Scholes formula .. Martingale approach to Black-Scholes equation .. Martingale approach to pricing .. Martingale representation theorem .. Exercises .. 1726 Jump L evy processes .. Poisson process .. Compound Poisson process .. Integration with respect to compound Poisson processes .. Change of measure .. Generalized Poisson processes I .. Generalized Poisson processes II.
4 The L evy-Khinchin characterization .. Integration with respect to L evy processes .. Symmetric stable process .. Exercises .. 2197 Fractional Brownian Definition .. Stochastic integral representation .. Simulation .. 2278 Harmonic Dirichlet problem .. Time changes .. Complex Brownian motion .. Exercises .. 240viCONTENTSI ntroductory commentsThis is an Introduction to Stochastic calculus . I will assume that the readerhas had a post- calculus course in probability or statistics. For much of thesenotes this is all that is needed, but to have a deep understanding of thesubject, one needs to know measure theory and probability from that per-spective. My goal is to include discussion for readers with that backgroundas well.
5 I also assume that the reader can write simple computer programseither using a language like C++ or by using software such as Matlab advanced mathematical comments that can be skipped bythe reader will be indented with a different font. Comments here willassume that the reader knows that language of measure-theoreticprobability will discuss some of the Applications to finance but our main focuswill be on the mathematics. Financial mathematics is a kind of appliedmathematics, and I will start by making some comments about the use ofmathematics in the real world . The general paradigm is as follows. A mathematical model is made of some real world phenomenon. Usu-ally this model requires simplification and does not precisely describethe real situation.
6 One hopes that models arerobustin the sense thatif the model is not very far from reality then its predictions will alsobe close to accurate. The model consists of mathematicalassumptionsabout the real world. Given these assumptions, one does mathematical analysis to see whatthey imply. The analysis can be of various types: Rigorous derivations of Derivations that are plausible but are not mathematically rigor-ous. Approximations of the mathematical model which lead totractable calculations. Numerical calculations on a computer. For models that include randomness,Monte Carlo simulationsusing a (pseudo) random number generator. If the mathematical analysis is successful it will make predictions aboutthe real world. These are then checked.
7 If the predictions are bad, there are two possible reasons: The mathematical analysis was faulty. The model does not sufficiently reflect user of mathematics does not always need to know the details ofthe mathematical analysis, but it is critical to understand theassumptionsin the model. No matter how precise or sophisticated the analysis is, if theassumptions are bad, one cannot expect a good 1 Martingales in discrete timeA martingale is a mathematical model of a fair game. To understand the def-inition, we need to defineconditional expectation. The concept of conditionalexpectation will permeate this Conditional expectationIfXis a random variable, then its expectation,E[X] can be thought of asthe best guess forXgiven no information about the result of the trial.
8 Aconditional expectation can be considered as the best guess given some butnot total ,X2,..be random variables which we think of as a time serieswith the data arriving one at a time. At timenwe have viewed the valuesX1,..,Xn. IfYis another random variable, thenE(Y|X1,..,Xn) is thebest guess forYgivenX1,..,Xn. We will assume thatYis anintegrablerandom variable which meansE[|Y|]< . To save some space we willwriteFnfor the information contained inX1,..,Xn andE[Y| Fn] forE[Y|X1,..,Xn]. We viewF0as no information. The best guess shouldsatisfy the following properties. If we have no information, then the best guess is the expected other words,E[Y|F0] =E[Y]. The conditional expectationE[Y| Fn] should only use the informa-tion available at timen.
9 In other words, it should be a function ofX1,..,Xn,E[Y|Fn] = (X1,..,Xn).We say thatE[Y|Fn] 1. MARTINGALES IN DISCRETE TIMEThe definitions in the last paragraph are certainly vague. Wecan use measure theory to be precise. We assume that the randomvariablesY,X1,X2,..are defined on a probability space( ,F,P).HereFis a -algebraor -fieldof subsets of , that is, a collectionof subsets satisfying F; A Fimplies that \A F; A1,A2,.. Fimplies that n=1An informationFnis the smallest sub -algebraGofFsuch thatX1,..,XnareG-measurable. The latter statement means that forallt R, the event{Xj t} Fn. The no information -algebraF0is the trivial -algebra containing only and .The definition of conditional expectation is a little tricky, so let us tryto motivate it by considering an example from undergraduate probabilitycourses.
10 Suppose that (X,Y) have a joint densityf(x,y),0< x,y < ,with marginal densitiesf(x) = f(x,y)dy, g(y) = f(x,y) conditional densityf(y|x) is defined byf(y|x) =f(x,y)f(x).This is well defined provided thatf(x)>0, and iff(x) = 0, thenxis an impossible value forXto take. We can writeE[Y|X=x] = y f(y|x) can use this as the definition of conditional expectation in this case,E[Y|X] = y f(y|X)dy= y f(X,y)dyf(X). CONDITIONAL EXPECTATION5 Note thatE[Y|X] is a random variable which is determined by the valueof the random variableX. Since it is a random variable, we can take itsexpectationE[E[Y|X]] = E[Y|X=x]f(x)dx= [ y f(y|x)dy]f(x)dx= y f(x,y)dy dx=E[Y].This calculation illustrates a basic property of conditional we are interested in the value of a random variableYand we aregoing to be given dataX1.
