What follows is a heuristic proof of Itô’s Formula. (Rigorous proofs of the 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,

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.

Ex 2. [Continued] Discuss why it is reasonable to expect that

and

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 [1].

Ex 3. [Continued] If we inductively define by the recursion

then discuss why we expect to approximately obey an equation of the form

Ans 3. Sum to gain and apply approximations from [2].

Ex 4. [Continued] Let be a twice differentiable function, argue that

Ans 4. Apply a Taylor approximation

In the last equality we use that , cf. [1].

Ex 5. [Continued] Argue that

This is Itô’s formula.

Ans 5. Apply an interpolating sum to [4] and then apply [2].