- A heuristic look at the stochastic integral.
- heuristic derivation of Itô’s formula.
What follows is a heuristic proof of Itô’s Formula, which states that if
and if is a twice continuously differentiable function then
(Rigorous proofs of these exercises are not expected.)
Ex 1 [A Heuristic look at Stochastic Integration] For a standard Brownian motion argue that, for all and for sufficiently small and positive,
Ex 2 [Continued] Discuss why it is reasonable to expect that
Ex 3 [Continued] If we inductively define by the recursion
then discuss why we expect to approximately obey an equation of the form
Ex 4[Continued]Let be a twice differentiable function, argue that
Ex 5 [Continued]Argue that
This is Itô’s formula.
Ans 1 The 1st sum is an interpolating sum. By independent increments property of Brownian motion, the 2nd sum adds IIDRVs with each with mean . Thus the strong law of large numbers gives the approximation.
Ans 2 The first sum is approximation from a Riemann-Stieltjes integral, i.e.
So one might expect a integral limit. (This is unrigorous because Riemann-Stieltjes Integration only applies to functions with finite variation – while Brownian motion does not have finite variation.)
The second sum is a Riemann integral upon using the approximation .
Ans 3 Sum to gain and apply approximations from .
Ans 4 Apply a Taylor approximation
In the last equality we use that , cf. .
Ans 5 Apply an interpolating sum to  and then apply .