Stochastic Integration – a Heuristic view


Heuristic derivation of

  • the Stochastic Integral
  • Stochastic Differential Equations
  • Ito’s Formula

 

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 (B_t: t \geq 0) a standard Brownian motion argue that, for all T and for \delta 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 \delta. 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 \left(B_{t+\delta} - B_t\right)^2\approx \delta [1].

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

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

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

Ans 4. Apply a Taylor approximation

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

Ex 5. [Continued] Argue that

This is Itô’s formula.

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

 

 

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