Ito’s Formula: a heuristic derivation

  • 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 f:{\mathbb R} \rightarrow {\mathbb R} is a twice continuously differentiable function then

(Rigorous proofs of these exercises are not expected.)

Ex 1 [A Heuristic look at Stochastic Integration] For (B_t: \geq 0) a standard Brownian motion argue that, for all T and for \delta sufficiently small and positive,

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

and

 

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

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

 

Ex 4[Continued]Let f 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 \delta. 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 \left(B_{t+\delta} - B_t\right)^2\approx \delta [1].

Ans 3  Sum to gain X_T-X_0 and apply approximations from [2].

Ans 4 Apply a Taylor approximation

In the last equality we use that (B_{t+\delta}-B_t)=o(\delta^{1/2}), cf. [1].

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

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