Stochastic Integrals and Their Expectations
نویسنده
چکیده
Stochastic calculus is famous for providing the foundations for modern mathematical finance and is also used extensively in a large number of other areas of applied probability. The introductory text by Øksendal [3] strikes an excellent balance between theory and accessibility. Here we give a very brief review of the underlying concepts. A central notion for stochastic calculus is that of a (continuous) semimartingale: a random process X that can be written as the sum of a local martingale M (for example, Brownian motion) and a drift process V (a continuous process of locally bounded variation, typically the solution of some conventional differential equation). The decomposition X = X H0L + M + V is unique and can be thought of as a decomposition of X into signal V plus noise M. Fundamental to the theory of stochastic calculus is Itô’s lemma: if f HX L is a smooth function of the semimartingale X, then
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