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

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17 Itô’s formula

An important consequence of the fundamental theorem of integral and differential calculus is the fact every differentiation rule has an integration counterpart. Consider, for example, the chain rule (f ◦ g)′(t) = f′(g(t)) · g′(t). Then

e9783110307290_i3679.jpg

which is just the substitution or change-of variable rule for integrals. Moreover, it provides a useful calculus formula if we want to evaluate integrals.

Let (Bt)t≥0 be a BM1 and (e9783110307290_i3680.jpg)t≥0 an admissible, right-continuous complete filtration, e. g. , cf. Theorem 6.21. Itô’s formula is the stochastic ...

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