By Boris Harlamov
This name considers the exact of random tactics often called semi-Markov strategies. those own the Markov estate with recognize to any intrinsic Markov time corresponding to the 1st go out time from an open set or a finite generation of those instances. the category of semi-Markov tactics contains robust Markov techniques, LГ©vy and Smith stepped semi-Markov tactics, and a few different subclasses. large assurance is dedicated to non-Markovian semi-Markov techniques with non-stop trajectories and, specifically, to semi-Markov diffusion strategies. Readers trying to improve their wisdom on Markov strategies will locate this booklet a precious source.
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Extra info for Continuous semi-Markov processes
The following properties are fair: Sequences of First Exit Times 61 (1) (∀t ∈ R+ ) (∀ε > 0) (∃δ > 0) ρD ξ1 , ξ2 < δ =⇒ ∃λ ∈ Λ[0, ∞) sup ρ ξ1 (s), ξ2 λ(s) s≤t + sup s − λ(s) < ε; s≤t (2) (∀ξ1 , ξ2 ∈ C) (∀t ∈ R+ ) (∀ε > 0) (∃δ > 0) ρC ξ1 , ξ2 < δ =⇒ ρtD ξ1 , ξ2 < ε. Proof. (1) If ρD (ξ1 , ξ2 ) < δ, then (∃λ ∈ Λ[0, 1]) sups<1 |s − λ(s)| < δ and sup(1 − s) ∧ 1 − λ(s) arctan ρ ξ1 a−1 (s) , ξ2 a−1 λ(s) < δ, s<1 where a(t) = (2/π) arctan t (t ∈ R+ ). Let t ∈ R+ and δ < (1 − a(t))/(1 + 2/π). Then sups≤t |a(s) − λ(a(s))| < δ, and since |s − r| , 1 + (s ∨ r)2 | arctan s − arctan r| ≥ sups≤t |s − a−1 λ(a(s))| < (π/2)δ(1 + tan2 (π/2)(a(t) + δ)), where s ∨ a−1 λ a(s) = a−1 a(s) ∨ λ a(s) < a−1 a(t) + δ π = tan a(t) + δ < ∞.
3) It follows from an obvious relation: (∀ξ ∈ D) t2 ∈ t1 , t1 + σr θt1 (ξ) =⇒ ∀t ∈ t1 , t1 + σr θt1 (ξ) ρ ξ t2 , ξ(t) ≤ ρ ξ t2 , ξ t1 + ρ ξ t1 , ξ(t) < 2r. 17. Deducing sequence Let us pay attention to a difference between sequences (σrn ) and (σδn ). The former is strictly increasing in n (up to the ſrst n, when it heads towards inſnity), while the latter is increasing at the step n only if Xσn−1 belongs to the next set Δn , otherwise δ the next term of the sequence is equal to the preceding one.
One more sigma-algebra of preceding events For τ ∈ MT+ let Fτ = σ(ατ , τ ). e. the new deſnition is coordinated with old one (see item 8). 54 Continuous Semi-Markov Processes 26. One more representation of a sigma-algebra Let us denote by R+ the all rational set t ∈ R+ (this is used in the proof of the previous proposition). 3. For any τ1 , τ2 ∈ MT+ we have Fτ2 . Fτ1 +˙ τ2 = σ Fτ1 , θτ−1 1 ˙ τ2 ). We have Proof. 14). 16] is correct with replacement Ta by Ta . Hence ατ2 ◦ θτ1 = θτ1 ◦ ατ1 +˙ τ2 ∈ σ ατ1 +˙ τ2 / F ∩ τ1 < ∞ , ˙ τ2 − τ1 ∈ σ ατ +˙ τ , τ1 + ˙ τ2 /B R+ .