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By Morters P., Peres Y.

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Extra resources for Brownian motion

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If Tn ↑ T is an increasing sequence of stopping times, then T is also a stopping time, as {T ≤ t} = ∞ {Tn ≤ t} ∈ F + (t) . n=1 • Suppose H is a closed set, for example a singleton. Then T = inf{t ≥ 0 : B(t) ∈ H} is a stopping time. Indeed, let G(n) = {x ∈ Rd : ∃y ∈ H with |x − y| < 1/n} so that H = G(n). Then Tn := inf{t ≥ 0 : B(t) ∈ G(n)} are stopping times, which are increasing to T . • Let T be a stopping time. Define stopping times Tn by Tn = (m + 1)2−n if m2−n ≤ T < (m + 1)2−n . In other words, we stop at the first time of the form k2−n after T .

While, as seen above, {M (t) − B(t) : t ≥ 0} is a Markov process, it is important to note that the maximum process {M (t) : t ≥ 0} itself is not a Markov process. However the times when new maxima are achieved form a Markov process, as the following theorem shows. 33 For any a ≥ 0 define the stopping times Ta = inf{t ≥ 0 : B(t) = a}. Then {Ta : a ≥ 0} is an increasing Markov process with transition kernel given by the densities a 2π(s−t)3 p(a, t, s) = √ exp − a2 2(s−t) 1{s > t}, for a > 0. This process is called the stable subordinator of index 12 .

We denote by Gn the σ-algebra generated by the random variables Xn , Xn+1 , . .. Then G∞ := ∞ k=1 Gk ⊂ · · · ⊂ Gn+1 ⊂ Gn ⊂ · · · ⊂ G1 . e. that almost surely, Xn = E Xn−1 Gn for all n ≥ 2 . 37. Indeed, if s ∈ (t1 , t2 ) is the inserted point we apply it to the symmetric, independent random variables B(s) − B(t1 ), B(t2 ) − B(s) and denote by F the σ-algebra generated by (B(s) − B(t1 ))2 + (B(t2 ) − B(s))2 . Then E B(t2 ) − B(t1 ) 2 F = B(s) − B(t1 ) 2 2 + B(t2 ) − B(s) , and hence E B(t2 ) − B(t1 ) 2 2 − B(s) − B(t1 ) − B(t2 ) − B(s) 2 F = 0, which implies that {Xn : n ∈ N} is a reverse martingale.

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