- 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

and

**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.

## Answers

**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 [1].

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

**Ans 4 **Apply a Taylor approximation

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

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