Elect. Comm. in Probab. 15 (2010), 99–105 ELECTRONIC COMMUNICATIONS in PROBABILITY THE MEASURABILITY OF HITTING TIMES RICHARD F. BASS1 Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 email: [email protected] Submitted January 18, 2010, accepted in final form March 8, 2010 AMS 2000 Subject classification: Primary: 60G07; Secondary: 60G05, 60G40 Keywords: Stopping time, hitting time, progressively measurable, optional, predictable, debut theorem, section theorem Abstract Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems for optional and predictable sets are easy corollaries of the proof. 1 Introduction A fundamental theorem in the foundations of stochastic processes is the one that says that, under very general conditions, the first time a stochastic process enters a set is a stopping time. The proof uses capacities, analytic sets, and Choquet’s capacibility theorem, and is considered hard. To the best of our knowledge, no more than a handful of books have an exposition that starts with the definition of capacity and proceeds to the hitting time theorem. (One that does is [1].) The purpose of this paper is to give a short and elementary proof of this theorem. The proof is simple enough that it could easily be included in a first year graduate course in probability. In Section 2 we give a proof of the debut theorem, from which the measurability theorem follows. As easy corollaries we obtain the section theorems for optional and predictable sets. This argument is given in Section 3. 2 The debut theorem Suppose (Ω, , P) is a probability space. The outer probability P∗ associated with P is given by F P∗(A) = inf P(B) : A B, B . f ⊂ 2 F g A set A is a P-null set if P(A) = 0. Suppose t is a filtration satisfying the usual conditions: fF g ">0 t+" = t for all t 0, and each t contains every P-null set. Let π : [0, ) Ω Ω be defined\ F by π(Ft, !) = !. ≥ F 1 × ! 1RESEARCH PARTIALLY SUPPORTED BY NSF GRANT DMS-0901505 99 100 Electronic Communications in Probability Recall that a random variable taking values in [0, ] is a stopping time if (T t) t for all t; we allow our stopping times to take the value infinity.1 Since the filtration satisfies≤ 2 F the usual conditions, T will be a stopping time if (T < t) t for all t. If Ti is a finite collection or countable collection of stopping times, then supi Ti 2and F infi Ti are also stopping times. Given a topological space , the Borel σ-field is the one generated by the open sets. Let [0, t] S B denote the Borel σ-field on [0, t] and [0, t] t the product σ-field. A process X taking values B × F in a topological space is progressively measurable if for each t the map (s, !) Xs(!) from S ! [0, t] Ω to is measurable with respect to [0, t] t , that is, the inverse image of Borel × S B × F subsets of are elements of [0, t] t . If the paths of X are right continuous, then X is easily seen to beS progressively measurable.B × The F same is true if X has left continuous paths. A subset of [0, ) Ω is progressively measurable if its indicator is a progressively measurable process. 1 × If E [0, ) Ω, let DE = inf t 0 : (t, !) E , the debut of E. We will prove ⊂ 1 × f ≥ 2 g Theorem 2.1. If E is a progressively measurable set, then DE is a stopping time. 0 Fix t. Let (t) be the collection of subsets of [0, t] Ω of the form K C, where K is a compact K × × 0 subset of [0, t] and C t . Let (t) be the collection of finite unions of sets in (t) and let 2 F K K δ(t) be the collection of countable intersections of sets in (t). We say A [0, t] t is K K 2 B × F t-approximable if given " > 0, there exists B δ(t) with B A and 2 K ⊂ P∗(π(A)) P∗(π(B)) + ". (2.1) ≤ Lemma 2.2. If B δ(t), then π(B) t . If Bn δ(t) and Bn B, then π(B) = nπ(Bn). 2 K 2 F 2 K # \ The hypothesis that the Bn be in δ(t) is important. For example, if Bn = [1 (1=n), 1) Ω, then K − × π(Bn) = Ω but π( nBn) = . This is why the proof given in [2, Lemma 6.18] is incorrect. \ ; Proof. If B = K C, where K is a nonempty subset of [0, t] and C t , then π(B) = C t . Therefore π B × if B 0 t . If B m A with A 0 t , then2 Fπ B m π A 2 F. ( ) t ( ) = i=1 i i ( ) ( ) = i=1 ( i) t For each ! and2 each F set C2, Klet [ 2 K [ 2 F S(C)(!) = s t : (s, !) C . (2.2) f ≤ 2 g If B δ(t) and Bn B with Bn (t) for each t, then S(Bn)(!) S(B)(!), so S(B)(!) is compact.2 K # 2 K # Now suppose B δ(t) and take Bn B with Bn δ(t). S(Bn)(!) is a compact subset of 2 K # 2 K [0, t] for each n and S(Bn)(!) S(B)(!). One possibility is that nS(Bn)(!) = ; in this case, # \ 6 ; if s nS(Bn)(!), then (s, !) Bn for each n, and so (s, !) B. Therefore ! π(Bn) for each 2 \ 2 2 2 n and ! π(B). The other possibility is that nS(Bn)(!) = . Since the sequence S(Bn)(!) 2 \ ; c is a decreasing sequence of compact sets, S(Bn)(!) = for some n, for otherwise S(Bn)(!) ; f g would be an open cover of [0, t] with no finite subcover. Therefore ! = π(Bn) and ! = π(B). We 2 2 conclude that π(B) = nπ(Bn). \ Finally, suppose B δ(t) and Bn B with Bn (t). Then π(B) = nπ(Bn) t . 2 K # 2 K \ 2 F Proposition 2.3. Suppose A is t-approximable. Then π(A) t . Moreover, given " > 0 there exists 2 F B δ(t) such that P(π(A) π(B)) < ". 2 K n Proof. Choose An δ(t) with An A and P(π(An)) P∗(π(A)). Let Bn = A1 An and 2 K ⊂ ! [···[ let B = nBn. Then Bn δ(t), Bn B, and P(π(Bn)) P(π(An)) P∗(π(A)). It follows that [ 2 K " ≥ ! π(Bn) π(B), and so π(B) t and " 2 F P(π(B)) = lim P(π(Bn)) = P∗(π(A)). Hitting times 101 For each n, there exists Cn such that π(A) Cn and P(Cn) P∗(π(A)) + 1=n. Setting 2 F ⊂ ≤ C = nCn, we have π(A) C and P∗(π(A)) = P(C). Therefore π(B) π(A) C and P(π(B)) = \ ⊂ ⊂ ⊂ P∗(π(A)) = P(C). This implies that π(A) π(B) is a P-null set, and by the completeness assumption, n π(A) = (π(A) π(B)) π(B) t . Finally, n [ 2 F lim P(π(A) π(Bn)) = P(π(A)) π(B)) = 0. n n n We now prove Theorem 2.1. Proof of Theorem 2.1. Fix t. Let = A [0, t] t : A is t-approximable . M f 2 B × F g We show is a monotone class. Suppose An , An A. Then π(An) π(A). By Proposition M 2 M " " 2.3, π(An) t for all n, and therefore π(A) t and P(π(A)) = limn P(π(An)). Choose n large 2 F 2 F so that P(π(A)) < P(π(An)) + "=2. Then choose Kn δ(t) such that Kn An and P(π(An)) < 2 K ⊂ P(π(Kn)) + "=2. This shows A is t-approximable. Now suppose An and An A. Choose Kn δ(t) such that Kn An and P(π(An) π(Kn)) < n+1 2 M # 2 K ⊂ n "=2 . Let Ln = K1 Kn, L = nKn. Since each Kn δ(t), so is each Ln, and hence \···\ \ 2 K L δ(t). Also Ln L and L A. By Lemma 2.2, π(Ln) π(L), hence P(π(Ln)) P(π(L)). 2 K # ⊂ # ! Therefore P(π(Ln)) < P(π(L)) + "=2 if n is large enough. We write P(π(A)) P(π(An)) P(π(An) π(Ln)) + P(π(Ln)) ≤ n π ≤A π K n π L P( i=1( ( i) ( i))) + P( ( n)) ≤ n [ n X P(π(Ai) π(Ki)) + P(π(Ln)) ≤ i=1 n < " + P(π(L)) if n is large enough. Therefore A is t-approximable. 0 If (t) is the collection of sets of the form [a, b) C, where a < b t and C t , and (t) 0 isI the collection of finite unions of sets in (t), then× (t) is an algebra≤ of sets.2 F We noteI that I 0 I (t) generates the σ-field [0, t] t . A set in (t) of the form [a, b) C is the union of sets 0 inI (t) of the form [a, bB (1=m)]× F C, and itI follows that every set in× (t) is the increasing unionK of sets in (t). Since− is a monotone× class containing (t), thenI contains (t).