Solutions To Stochastic Calculus For Finance II...
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Solutions To Stochastic Calculus For Finance II...
"... a book that is a marvelous first step for the person wanting a rigorous development of stochastic calculus, as well as its application to derivative pricing. By focusing solely on Brownian motion, the reader is able to develop an intuition and a feel for how to go about solving problems as well as deriving results." --- Mark A. Cassano (see the full review from the Journal of Finance)
"...the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently." --- Martin Schweizer (Berlin) from the review in Zentralblatt fur Mathematik: (0962.60001) "I thoroughly enjoyed reading this book. The author is to be complimented for his efforts in providing many useful insights behind the various theories. It is a superb introduction to stochastic calculus and Brownian motion." --- Elias Shiu (from the review in JASA)
The book is primarily about the core theory of stochastic calculus, but it focuses on those parts of the theory that have really proved that they can "pay the rent" in practical applications. The intention is also to coach people toward honest mastery. This means that one must be selective in the topics that are treated, and one must engage those topics to some depth.
Eventually I plan to provide links here to the index and to provide complete solutions for some NEW problems. Over time I would like to make this more like a community page and less like a publisher's flyer. I've made a little progress in that direction with the Cauchy-Schwarz home page, but it takes time. In the meanwhile, I am thinking about a problem book on Brownian motion. This book would also have problems that are directed toward stochastic calculus.
It goes without saying that Rubenstein's book is unique. A grandmaster of the field looks at its most important papers and puts into clear prose what he sees as their main contributions. You could have sold me this book one page at a time for several bucks per page. There is not a ton of stochastic calculus in these books, but there certainly are some interesting connections that help explain how stochastic calculus found its place in the world.
We have opted to present a range of modules that will be most useful in quantitative finance albeit with an emphasis on stochastic calculus and stochastic processes. By choosing such modules we are necessarily leaving out many useful and interesting areas of mathematics.
Stochastic Analysis (or Stochastic Calculus) is the theory that underpins modern mathematical finance. It provides a natural framework for carrying out derivatives pricing. While quantitative finance is one of the main application areas of stochastic analysis, it also has a rich research history in the fields of pure mathematics, theoretical physics and engineering.
There are a wealth of textbooks on the stochastic calculus necessary for derivatives pricing. The most widely recommended introductory text with a reasonable level of mathematical rigour is by Shreve:
Within finance stochastic optimal control is used for optimal asset allocation decisions, as well as for pricing of American option contracts. It is also closely related to the machine learning field of Reinforcement Learning, itself famous for its recent successes in beating humans at both the ancient game of Go and real-time strategy video games.
Markov Processes are a well-studied tool in mathematical finance, since they form the basis of the Markov Chain Monte Carlo (MCMC) algorithm, which underpins computational Bayesian statistics. Another application for Markov Chains is in estimating parameters in stochastic volatility models. We have previously discussed Hidden Markov Models as a further application.
In probability theory and related fields, a stochastic (/stoʊˈkæstɪk/) or ran