Introductory Stochastic Analysis for Finance and Insurance

CHAPTER 3

DISCRETE-TIME STOCHASTIC PROCESSES

3.1   STOCHASTIC PROCESSES AND INFORMATION STRUCTURES

Let be a probability space. Recall that is a collection of all possible events and represents all the information contained in the probability space. Imagine that a series of experiments is performed at times t = t₀, t₁, t₂, ··· . For simplicity we assume in this chapter that t_i = i, i.e., t = 0, 1, 2, ··· , unless otherwise specified. Let be the collection of all possible events in that may occur before or at time t. Thus represents the information up to time t. Obviously,

(i)  is an information structure coarser than , i.e., ⊆ , since it contains no more information than ;

(ii)  If