Mathematics – Probability
Scientific paper
2010-05-24
Stoch. Anal. Appl. 24 (2006), no. 2, 367--380
Mathematics
Probability
10 pages, 9/9 papers from my 2000-2006 collection. I proved these results and more earlier in 2004. I generalize this theory i
Scientific paper
10.1080/07362990500522411
A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It\^o's integral calculus? From It\^o's definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.
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