For finite-variation continuous processes *X* and functions *F* ∈ *C*^{1}(), recall the main formula of integration^{1}

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 Taylor^{2} formula with the remainder in some special Peano-type^{3} form.

LEMMA 5.1 (Taylor formula).– *Let F* ∈ *C*^{2}() *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* lim_{h↓0}*r*(*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 < ...

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