By Boris Harlamov
This name considers the distinctive of random strategies referred to as semi-Markov techniques. those own the Markov estate with recognize to any intrinsic Markov time akin to the 1st go out time from an open set or a finite generation of those instances. the category of semi-Markov procedures contains robust Markov procedures, Lévy and Smith stepped semi-Markov strategies, and a few different subclasses. wide insurance is dedicated to non-Markovian semi-Markov methods with non-stop trajectories and, specifically, to semi-Markov diffusion strategies. Readers trying to increase their wisdom on Markov strategies will locate this ebook a priceless source.
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Proof. (1) and (2): see Blumenthal and Getoor [BLU 68, p. 34]; (3), (4), (5): it is obvious. 11. 7. For any τ ∈ MT the following representation is fair: Fτ = ατ−1 F. Proof. For any t ≥ 0 let us check the equality Ft = αt−1 F. For this aim we use representation of an indexed sigma-algebra: for any measurable map M it is true that M −1 σ(B : B ∈ A) = σ M −1 B : B ∈ A , where A is a family of sets [NEV 69]. From here αt−1 σ Xs , s < ∞ = σ Xs ◦ αt , s < ∞ = σ Xs∧t , s < ∞ = σ Xs , s ≤ t . The condition B ∈ αt−1 F means that (∃B ∈ F) B = αt−1 B .
6 can be applied to prove the statement: (∀k ∈ N) (∀r > 0) the function βσrk is continuous on Πk (r). 5. Time of regeneration Semi-Markov processes, which will be deſned in the next chapter, are based on the concept of a regeneration time. It is a Markov time with respect to which the given family of measures possesses the Markov property. ). ). The term “time of Markov interference of chance”, which is also sometimes used, does not seem to us to be good enough, generally because of its length.
K ). 62 Continuous Semi-Markov Processes Proof. Let Δ ∈ A, σΔ (ξ) = ∞, ξ ∈ Π(Δ) and ρD (ξn , ξ) → 0 (n → ∞). 17] s≤t sup s − λn (s) < ε. 18] s≤t to be fulſlled for λn ∈ Λ[0, ∞]). Let ε > 0 such that σΔ−ε (ξ) > t. Then σΔ (ξn ◦ λn ) > t and, hence, λ−1 n σΔ (ξn ) > t, σΔ (ξn ) > λn (t) and σΔ (ξn ) > t − ε. From here, because of arbitrary choice t and ε, we obtain σΔ ξn −→ ∞, βσΔ ξn −→ βσΔ (ξ). 19] Let σΔ (ξ) < ∞ and ρD (ξn , ξ) → 0 (n → ∞). 18]. Let us take ε and t such that σΔ+ε (ξ) ≤ t. Then σΔ−ε (ξ) ≤ σΔ (ξn ◦ λn ) ≤ σΔ+ε (ξ).