Informal Definition of Ito Integral
Overview
The Ito integral is built using the
Ito Process,
which is defined informally as
{% \Delta log S = \mu \Delta t + \sigma N(0,\Delta t) %}
Then the Ito integral is defined as
{% \int \mu dt + \sigma dW = lim_{\Delta t \rightarrow \infty} \sum (\mu \Delta t + \sigma N(0,\Delta t)) %}
where {% N (\mu, \sigma^2) %} is the normal distribution with mean {% \mu %} and variance {% \sigma^2 %}.
Convergence
The above definition looks like the
Riemann Stieltjes Integral,
however, it can be shown that it does not converge in the Riemann Stieltjes sense, because of the
quadrative variation of the brownian motion term.
However, the integral does converge in the mean square sense.
{% lim_{h \rightarrow 0} \mathbb{E}[\sum_{i=0}^{n-1} \Delta x_{t+1}^2 - \int_0^T (dx_t)^2]^2 = 0 %}
