Question

Q4. Prove or disprove that τ is a stopping time (with respect to the Markov chain {Xn|n ≥ 0} iff {τ > n} ∈ σ(X0, · · · , Xn), ∀ n ≥ 0 Q5. Prove or disprove that τ is a stopping time (with respect to the Markov chain {Xn|n ≥ 0}) iff {τ ≥ n} ∈ σ(X0, · · · , Xn), ∀ n ≥ 0 Q6. Let {Xn|n ≥ 0} be a Markov chain and A ⊂ S = {1, 2, · · · } such that A 6= S, 1 ∈ A. Define (i) τ1 = min{n ≥ 0|Xn+1 ∈ A}, . (ii) τ2 = min{n ≥ τ1|Xn = 1} Are τ1, τ2 stopping times? Justify your answer. Q7. Let τ be a stopping time with respect to a Markov chain {Xn : n ≥ 0}. For i0, i1, i2, i3 ∈ S, show that {X0 = i0, X1 = i1, X2 = i2, X3 = i3, τ = 3} = {X0 = i0, X1 = i1, X2 = i2, X3 = i3} or ∅. Q8. Let τ1, τ2 be stopping times with respect to a Markov chain. Show that τ1 ∧ τ2 is a stopping time. Here ∧ denote min. Q9. Let τ be a stopping time with respect to a Markov chain. Is τ + 1 a stopping time? 1

Answer 1