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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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18 Applications of Itô’s formula

Itô’s formula has many applications and we restrict ourselves to a few of them. We will use it to obtain a characterization of a Brownian motion as a martingale (Lévy’s theorem 18.5) and to describe the structure of ‘Brownian’ martingales (Theorems 18.11, 18.13 and 18.16). In some sense, this will show that a Brownian motion is both a very particular martingale and the typical martingale. Girsanov’s theorem 18.8 allows us to change the underlying probability measure which will become important if we want to solve stochastic differential equations. Finally, the Burkholder–Davis–Gundy inequalities, Theorem 18.17, provide moment estimates for stochastic integrals.

Throughout this section (Bt, ℱt)t≥0 is a BM with ...

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