Brownian Motion, 2nd Edition

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

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 (B_t)_t≥0 be a BM¹ and ()_t≥0 an admissible, right-continuous complete filtration, e. g. , cf. Theorem 6.21. Itô’s formula is the stochastic ...