[A First Course In Probability So
Jamar Lizarraga
A First Course in Stochastic Calculus is a complete guide for advanced undergraduate students to take the next step in exploring probability theory and for master's students in mathematical finance who would like to build an intuitive and theoretical understanding of stochastic processes. This book is also an essential tool for finance professionals who wish to sharpen their knowledge and intuition about stochastic calculus.
Louis-Pierre Arguin offers an exceptionally clear introduction to Brownian motion and to random processes governed by the principles of stochastic calculus. The beauty and power of the subject are made accessible to readers with a basic knowledge of probability, linear algebra, and multivariable calculus. This is achieved by emphasizing numerical experiments using elementary Python coding to build intuition and adhering to a rigorous geometric point of view on the space of random variables. This unique approach is used to elucidate the properties of Gaussian processes, martingales, and diffusions. One of the book's highlights is a detailed and self-contained account of stochastic calculus applications to option pricing in finance.
Undergraduate and graduate students interested in advanced probability and the applications of stochastic calculus to finance. Finance professionals who want to develop their knowledge and intuition of stochastic calculus.
Congratulations to both the author for writing this valuable book and the AMS for its publication as a volume in the prestigious series "Pure and Applied Undergraduate Texts." There are all good reasons to strongly recommended the book to the thousands of students worldwide studying stochastic calculus, in particular to students following MSc and PhD programs in the area of 'mathematical finance.' Teachers of courses in stochastic calculus can efficiently combine this book with other sources.
The book is quite concise and very well-written, with many illustrative figures. A nice detail is that in almost all chapters, the topic is taken a little further than usual: gambler's ruin, Tanaka formula, Dirichlet problem, martingale representation, Feynman-Kac formula, and Heston model! But more than this, the book has several nice and original details. Many exercises allow the reader to delve deeper into topics beyond the basic points, hinting at the paths to deeper levels of understanding of the theory. The historical notes are certainly interesting. And above all, the so-called numerical projects, proposed at every chapter, make the book clearly numerical and computational aspects oriented, a fact that is very important in the training of future quantitative analysts, and useful to progress into stochastic calculus without a significant measure-theoretic background. In short, it is a book of stochastic calculus applied to finance, relatively standard, but very up-to-date and with many interesting details. The numerical and computational approach is undoubtedly the most original aspect of the book. A strongly recommend-able book!
Louis-Pierre Arguin's masterly introduction to stochastic calculus seduces the reader with its quietly conversational style; even rigorous proofs seem natural and easy. Full of insights and intuition, reinforced with many examples, numerical projects, and exercises, this book by a prize-winning mathematician and great teacher fully lives up to the author's reputation. I give it my strongest possible recommendation.
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.
The aim of the course is to provide an introduction to the mathematical modelling of random experiments. The emphasis is on methods applicable to problems in engineering, economy and natural sciences. After completing the course the student should have the knowledge and skills required to:
- identify experimental situations where random factors may affect the results.
- construct relevant probabilistic models for simple randomexperiments.
- describe the basic concepts and theorems of probability theory, such as random variable, distribution function and the law of total probability.
- compute important quantities in probabilistic models, e.g., probabilities and expectations.
- follow a basic course in statistical theory.
Sample space, events and probabilities. Combinatorics. Conditional probabilities and independent events. Discrete and continuous random variables. Distribution functions, probability mass functions, probability density functions. Conditional distributions and independent random variables. Functions of random variables. Expectation, variance, standard deviation, covariance, correlation coefficient. Particular distributions, e.g., Gaussian, exponential, uniform, binomial and Poisson distributions. Law of large numbers and the central limit theorem. The Poisson process.
A syllabus has been established for each course. The syllabus specifies the aim and contents of the course, and the prior knowledge that a student must have in order to be able to benefit from the course.
Courses are timetabled after a decision has been made for this course concerning its assignment to a timetable module. A central timetable is not drawn up for courses with fewer than five participants. Most project courses do not have a central timetable.
Courses with few participants (fewer than 10) may be cancelled or organised in a manner that differs from that stated in the course syllabus. The board of studies is to deliberate and decide whether a course is to be cancelled or changed from the course syllabus.
Written and oral examinations are held at least three times a year: once immediately after the end of the course, once in August, and once (usually) in one of the re-examination periods. Examinations held at other times are to follow a decision of the board of studies.
The examination schedule is based on the structure of timetable modules, but there may be deviations from this, mainly in the case of courses that are studied and examined for several programmes and in lower grades (i.e. 1 and 2).
In order to take an examination, a student must register in advance at the Student Portal during the registration period, which opens 30 days before the date of the examination and closes 10 days before it. Candidates are informed of the location of the examination by email, four days in advance. Students who have not registered for an examination run the risk of being refused admittance to the examination, if space is not available.
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