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].