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Introduction to Stochastic Calculus - Duke University

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. More Properties of Random Walks317.

i) The gambler’s ruin problem We play the following game: We start with 3$ in our pocket and we ip a coin. If the result is tail we loose one dollar, while if the result is positive we win one dollar. We stop when we have no money to bargain, or when we reach 9$. We may ask: what is the probability that I end up broke?

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