Stochastic Integrals
Overview
One of the motivating examples of stochastic integration comes from
population growth modeling.
Consider the following equation modeling the population N at a given time.
{% \frac{dN}{dt} = (r + noise) \times N %}
where the noise term is a
stochastic process
such as
Brownian Motion.
If we were using the normal rules of calculus, the solution would be an integral such as
{% N(t) = \int_{t_0}^t (r + noise) N dt %}
However, this integral does not converge when the noise is a Brownian motion.
The theory of stochastic integration was developed to tackle the definition of an integral that is well defined
and would solve problems as the one posed above.
Stochastic Integrals
There are two widely used definitions for the stochastic integral:
Definitions, Theoremns and Additional Topics
