Continuous semi-Markov processes by Boris Harlamov

By Boris Harlamov

This name considers the distinctive of random strategies referred to as semi-Markov procedures. those own the Markov estate with admire to any intrinsic Markov time comparable to the 1st go out time from an open set or a finite new release of those instances. the category of semi-Markov methods comprises robust Markov strategies, LГ©vy and Smith stepped semi-Markov procedures, and a few different subclasses. broad insurance is dedicated to non-Markovian semi-Markov methods with non-stop trajectories and, specifically, to semi-Markov diffusion methods. Readers seeking to increase their wisdom on Markov techniques will locate this publication a helpful source.

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21. Covering of a compact set We designate K a set of all compact subsets of a set X (see proof of the previous theorem). 2. Let δ = (Δ1 , Δ2 , . ) ∈ DS and (∀n ∈ N) Δn = X. Then (∀K ∈ K) (∃r > 0) (∀x ∈ K) (∀n ∈ N) (∃k > n) B(x, r) ⊂ Δk . Proof. Let δ = (Δ1 , Δ2 , . ), where (∀n ∈ N) Δn = X and (∃x ∈ X) (∀ε > 0) (∃n ∈ N) (∀k ≥ n) B(x, ε)\Δk = Ø. Then, obviously, there exists a sequence (rn )∞ 1 (rn > 0, rn ↓ 0) such that (∀n ∈ N) (∀k ≥ n) B x, rn \Δk = Ø. We consider a sequence (xn )∞ 0 , where x0 ∈ X, x1 ∈ B x, rk1 \Δk1 , .

Let us designate ρtD (ξ1 , ξ2 ) a distance in the Skorokhod metric on an interval [0, t]: ρtD ξ1 , ξ2 = inf λ∈Λ[0,t] sup ρ ξ1 (s), ξ2 λ(s) s≤t + sup s − λ(s) . s≤t Similarly for ξ1 , ξ2 ∈ C ρtC ξ1 , ξ2 = sup ρ ξ1 (s), ξ2 (s) . s≤t 37. 5. The following estimates are fair: (1) let ξ1 , ξ2 ∈ D and t be a continuity point of both functions; then ρD ξ1 , ξ2 ≤ 4 t π ρ ξ1 , ξ2 + − arctan t; π D 2 (2) for any ξ1 , ξ2 ∈ C and t > 0 ρC ξ1 , ξ2 ≤ ρtC ξ1 , ξ2 + π − arctan t. 2 Proof. (1) Let a(s) = (2/π) arctan s (s ∈ R+ ).

It means τ1 , τ2 ∈ T (T1 ) ⇒ τ1 +τ2 ∈ T (T1 ). 15. 8. 10. The following properties are fair: (1) σΔ , σr ∈ Tb ∩ MT (r > 0, Δ ∈ A); (2) σΔ , σ[r] ∈ Tb ∩ MT+ (r > 0, Δ ∈ A), where σ[r] (ξ) = σ[B(X0 (ξ),r)] (ξ). Proof. (1) In Gihman and Skorokhod [GIH 73, p. 194] it is proved that for any Δ ∈ A σΔ ∈ MT. It is similarly proved that σr ∈ MT. So we have to prove that (∀t ∈ R+ ) σΔ ≤ t = αt−1 σΔ < ∞ , σr ≤ t = αt−1 σr < ∞ . 8), (∀t ∈ R+ ) {σΔ ≤ t} ⊂ αt−1 {σΔ < ∞}. Let σΔ αt (ξ) < ∞. Then (αt ◦ ξ)−1 (X\Δ) = Ø.

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