Ito’s Formula: a heuristic derivation

• A heuristic look at the stochastic integral.
• heuristic derivation of Itô’s formula.

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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.

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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].