Dear Colleagues,
There is a famous formula of replacing variables in a stochastic differential equation (SDE), so called Itô's lemma. If we have SDE dx = a(x,t) dt + b(x,t) δW then the differential of a smooth function f(x,t) can be expressed as df = (f't+a f'x+b2f''xx/2)dt + bf'xδW.
Is there an expression for the second-order differential d2f ? What does it look like and does it have any meaning?
-Formally, Itô's lemma tells us that the property "Itô process" can be preserved by a twice-continuously differentiable function f. That is, if Xt is an Itô process, then f(t,Xt) is also an Itô process. Keeping this in mind, you can apply Itô's lemma recursively to compute dnf(t,Xt), for any positive integer n greater or equal to 2.
-The reason a second (or higher)-order version of Itô's lemma is less encountered and less practical, is the fact than any higher-order stochastic differential equation can be equivalently expressed as a system of first-order stochastic differential equations, for which the standard first-order Itô's lemma can be applied.
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