This note of stochastic calculus covers mathematical descriptions of information and conditioning.
In dynamic hedging, the position of underlying security at each future time is contigent on how the uncertainty between the present time and that future time is resolved. In order to make contingency plans, we need a way to mathematically model the information on which future decisions can be based.
A discrete example:
- We imagine some random experiment is performed. We denote as the set of all possible outcomes, and denote as one particular outcome.
- Suppose we are given some information. Such information is not enough to tell the exact value of , but is enough for us to narrow down the possibilities. Specifically, based on such given information, we can construct a list of sets of outcomes, called the sets resolved by the information, within which we know what sets are sure to contain and what other sets are sure not to contain it.
- At each time, the list of sets resolved from the given information form a σ-algebra, so we obtain a series of σ-sigma, , indexed by time. As time moves forward, we obtain finer resolution, meaning that contains more information than for any .
- Above collection of σ-algebra, , is an example of a filtration.
Below is the definition of infiltration in the continuous-time sense. Read more Stochastic Calculus Notes 2: Information and Conditioning