Ito Integral

Overview


The Ito integral gives meaning to the following integrals.
{% \displaystyle \int dX(t,W(t)) = \int u(t) dt + \sigma(t) dW(t) %}
or
{% \displaystyle X(t) = X(0) + \int u(t, X_t)dt + \int \sigma(t, X_t) dW(t) %}
The integral can be seen as the sum of two terms.

  1. The first integral in the decomposition is
    {% \displaystyle X(t) = \int u(t, X_t)dt %}
    which can be seen to be a normal Riemann integral
  2. The second integral involves the Brownian motion W(t)
    {% \displaystyle \int f(t, X_t) dW(t) %}
    This integral does not converge in the Riemann sense. Instead, this integral is defined by the following:

    1. Integral over Simple Functions
    2. Approximate the integrand by simple functions and take the Limit