Ito Integral

Overview


The Ito integral gives meaning to the following integrals.
{% \int dX(t,W(t)) = \int u(t) dt + \sigma(t) dW(t) %}
or
{% 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.

Riemann Integral


The first integral in the decomposition is
{% X(t) = \int u(t, X_t)dt %}
which can be seen to be a normal Riemann integral

Stochastic Term Integral


The second integral involves the Brownian motion W(t)
{% \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

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