Itō’s formula is certainly the most important and useful formula of stochastic calculus. It is the change of variable formula for stochastic integrals. It is a very simple formula whose specificity is the appearance of a quadratic variation term, that reflects the fact that semimartingales have a finite quadratic variation.
Due to its importance, we first provide a heuristic argument on how to derive Itō ‘s formula. Let be a smooth function and be a path . We have the following heuristic computation:
This suggests, by summation, the following correct formula:
Let us now try to consider a Brownian motion instead of the smooth path and let us try to adapt the previous computation to this case. Since Brownian motion has quadratic variation which is not zero, , we need to go at the order 2 in the Taylor expansion of . This leads to the following heuristic computation:
By summation, we are therefore led to the formula
which is, as we will see it later perfectly correct.
In what follows, we consider a filtered probability space that satisfies the usual conditions. Our starting point to prove Itō’s formula is the following formula which is known as the integration by parts formula for semimartingales:
Proposition. (Integration by parts formula)
Let and be two continuous semimartingales, then the process is a continuous semimartingale and we have:
Proof. By bilinearity of the multiplication, we may assume . Also by considering, if needed, instead of , we may assume that . Let . For every sequence such that
By letting , we therefore obtain the following identity which yields the expected result:
We are now in position to prove Itō’s formula in its simpler form.
Theorem. (Itō’s formula I) Let be a continuous and adapted semimartingale and let be a function which is twice continuously differentiable. The process is a semimartingale and the following change of variable formula holds:
Proof. We assume that the semimartingale is bounded. If it is not, we may apply the following arguments to the semimartingale , where and then let . Let be the set of two times continuously differentiable functions for which the formula given in the statement of the theorem holds holds. It is straightforward that is a vector space. Let us show that is also an algebra, that is also let stable by multiplication. Let . By using the integration by parts formula with the semimartingales and , we obtain
The terms of the previous sum may be separately treated in the following way. Since , we get:
We deduce that .
As a conclusion, is an algebra of functions. Since contains the function , we deduce that actually contains every polynomial function. Now in order to show that every function which is twice continuously differentiable is actually in , we first observe that since is assumed to be bounded, it take its values in a compact set. It is then possible to find a sequence of polynomials such that, on this compact set, uniformly converges toward , uniformly converges toward and uniformly converges toward
As a particular case of the previous formula, if we apply this formula with as a Brownian motion, we get the formula that was already pointed out at the beginning of the section: If is twice continuously differentiable function, then
It is easy to derive the following variations of Itō’s formula:
Theorem: (Itō’s formula II) Let be a continuous and adapted semimartingale, and let be an adapted bounded variation process. If is a function that is once continuously differentiable with respect to its first variable and that is twice continuously differentiable with respect to its second variable, then for :
Theorem. (Itō’s formula III) Let ,…, be adapted and continuous semimartingales and let be a twice continuously differentiable function. We have:
Exercise. Let be a function that is once continuously differentiable with respect to its first variable and twice continuously differentiable with respect to its second variable that satisfies
Show that if is a continuous local martingale, then is a continuous local martingale. Deduce that for , the process is a local martingale.
Exercise. The Hermite polynomial of order is defined as
- Compute .
- Show that if is a Brownian motion, then the process is a martingale.
- Show that