3. Markov Processes and Their Generators
Thomas G. Kurtz · Society for Industrial and Applied Mathematics eBooks · 1981
In studying stochastic processes it is sometimes helpful to think in terms of observing the process X evolve in time. If our observations and our memory are perfect, then at time t we will know all the values X(s) for s≦t and perhaps some additional information as well. If our memory is perfect the information we have will increase with t (at least it will not decrease). Assume the process X is defined on some probability space (Ω, ℱ, P) and has values in (E, r), a complete, separable metric space. What we observe are the values X(t, ω) for some fixed (but unknown) ω ∈ Ω. We formalize the idea of the information known at time t by identifying this information with a σ-algebra ℱt⊂ℱ . In particular, if A∈ℱt then we “know” whether or not ω ∈ A. Since we know whether or not X(s, ω) ∈ Γ for s≦t , we know whether or not ω ∈{X(s) ∈ Γ}. Therefore {X(s)∈Γ}∈ℱt for s≦t (we also require Γ ∈ ℬ(E)). This, of course, is the same as saying that X(s) is ℱt -measurable for s≦t The family of σ-algebras {ℱt} is increasing, that is ℱs⊆ℱt for s≦t . Such an increasing family is called a filtration.