# Stochastic Integration: A Quick Summary

What follows is a heuristic derivation of the Stochastic Integral, Stochastic Differential Equations and Itô’s Formula.

First note that for $(B_t: t \geq 0)$ a standard Brownian motion argue that, for all $T$ and for $\delta$ sufficiently small and positive,

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. From this it is reasonable to expect that

and

The first sum, above, 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 $\left(B_{t+\delta} - B_t\right)^2\approx \delta$. This is, very roughly, how a stochastic integral is defined.

We can also define stochastic differential equations. If we inductively define $X_t$ by the recursion

then, by summing over values of $t\in \{ 0, \delta,....,T-\delta\}$, we expect $X_t$ to approximately obey an equation of the form

This gives a Stochastic Differential Equation.

Often in differential and integration, we apply chain rule, $\frac{d f(x_t)}{dt} = f'(x_t) \frac{dx_t}{dt}$. Ito’s the analogous result for Stochastic Integrals. Let $X_t$ be as above. For a twice continuously differentiable function $f$ and $\delta>0$ small, we can apply a Taylor approximation

In the last equality we use that $(B_{t+\delta}-B_t)=o(\delta^{1/2})$ (which follows from ). Thus we see that

Consequently we expecrt that $f(X_t)$ obeys the following Stochastic Differential Equation:

This is Ito’s formula.