Chapter 9 - Stochastic Calculus And Ito Processes

Question 1

Describe the problem in evaluating Ito integrals

Answer 1

When attempting to develop a calculus for Brownian motion and other diffusions, one has to face the fact that their sample paths are nowhere differentiable.
A direct approach to stochastic integrals like int from 0 to t of Ys dBs is therefore doomed to failure.

Question 2

Give possible reasons why Brownian motion may be considered inadequate for representing financial data

Answer 2

There may be good reasons for believing that a Brownian model is inadequate to represent the data. For example, the data may appear to come from a stationary process, or may exhibit a tendency to revert to a mean value.
As long as there is no indication that the process being modelled is discontinuous, a diffusion model or an Ito process model can be applied. (For our purposes a diffusion or an Ito process are simply alternative representations of a single process).

Question 3

Describe what is involved in fitting a diffusion model and outline the problems involved

Answer 3

Fitting a diffusion model involves estimating the drift function μ(Xt) and the volatility or diffusion coefficient σ(Xt). Estimating arbitrary drift and diffusion functions is virtually impossible unless a veyr large quantity of data is to hand.
It is much more usual to specify a parametric form of the mean or the variance and to estimate the parameters.