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Alan
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What is Ito's lemma?

January 5th, 2003, 5:30 pm

QuoteOriginally posted by: OmarEric, This observation was first made, as far as I'm aware, in Barry Simon's book, <u>Functional Integration and Quantum Physics</u>, 1979, page 155. The book is out of print, but I'm sure you can find it.Omar, you reminded me what an interesting and deep book this is --I hope to understand more of it some day... Here's Barry Simon's take on the basic Ito lemma, where my onlychange in his words is to report just the scalar case. (Note that heuses b for a Brownian motion process). From p. 170 of his book:--------------------------------------------------------------------------------------------- A *Stochastic integral* is a random function c(b,t) obeying(16.1) c(b,t) = c(b,0) + Integral{0,t}[ f (b, s) db] + Integral{0,t} [ g(b,s) ds],where f and g are nonanticipatory functionals with suitable L^p propertiesand c(b, 0) is independent of b. One writes (16.1) in the shorthand: dc = f db + g dsThe following result is critical.Theorem 16.1 (Ito's lemma). If c is a stochastic integral and u is a C^2-functionon R x [0, Infinity] with some mild restrictions on growth at infinity, then x = u(c, t)is a stochastic integral, and dx = du/dy dc + du/dt ds + (1/2) d^2 u/dy^2 dc^2where dc^2 = f^2 ds (dc = f db + g ds); i.e., db^2 = ds-------------------------------------------------------------------------------------------------------------