For finite-variation continuous processes X and functions F ∈ C1(), recall the main formula of integration1
From example 4.10 we see that it does not hold for stochastic integrals. Its stochastic counterpart is the Itô formula which is proved below. For this, we need the Taylor2 formula with the remainder in some special Peano-type3 form.
LEMMA 5.1 (Taylor formula).– Let F ∈ C2() have a uniformly continuous second derivative (for example, F has a bounded third derivative F'"). Then
where, the remainder term satisfies the estimate |R(x, y)| r(|y − x|)(y − x)2 with an increasing function r: → such that limh↓0r(h) = 0.
Proof. We use the Taylor formula with the well-known Lagrange remainder term
where the point ξ = ξ(x, y) is between x and y (x < ξ < y or y < ξ < ...