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.

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