SWARTHMORE MATH 136 - MATH 136 LECTURE 17

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extbf {Last Time} extbf {A Motivating Problem: Gambler's Ruin} extbf {Discrete Time Stopping Time} extbf {Discrete Time Stopping Time: Examples} extbf {Optional Stopping Theorem: Motivation} extbf {Stopped Process} extbf {Doob's Optional Stopping Theorem} extbf {Gambler's Ruin: Probability of Ruin} extbf {Gambler's Ruin: Expected Number of Games}Last Time•Continuous time martingales•Right-continuous filtrationsToday’s lecture: Sections 4.3.1MATH136/STAT219 Lecture 17, November 3, 2008 – p. 1/9A Motivating Problem: Gambler’s Ruin•A gambler enters a casino with a dollars•He keeps playing a game until either he loses all his moneyor he wins b dollars•How long does the gambler play?•How much money does he have when he stops playing?MATH136/STAT219 Lecture 17, November 3, 2008 – p. 2/9Discrete Time Stopping Time•A random variable τ taking values in {0, 1, 2, . . .} ∪ {+∞} isastopping time with respect to filtration {Fn} if{τ ≤ n} = {ω : τ(ω) ≤ n} ∈ Fnfor all n = 0, 1, 2, . . .•τ is a discrete time stopping time for filtration {Fn} if andonly if{τ = n} ∈ Fnfor all n = 0, 1, 2, . . .•A random time τ is a stopping time if at any point in timeyou can determine whether or not the time τ has alreadyoccurred based on the information currently availableMATH136/STAT219 Lecture 17, November 3, 2008 – p. 3/9Discrete Time Stopping Time: Examples•Any non-random time is a stopping time•Hitting time: let {Xn} be a process adapted to {Fn}. For aBorel set B, define a random variableτ(ω) = inf{n ≥ 0 : Xn(ω) ∈ B}with τ(ω) = ∞ if Xn/∈ B for all n. Then τ is a stopping timefor {Fn}•Hitting time is the first time (if ever) the process takes avalue in the set B•If θ, τ, τ1, τ2,...are all stopping times for the same filtrationthen the following are also stopping times:min(θ, τ), max(θ, τ), θ + τ, supn≥1τn, infn≥1τnMATH136/STAT219 Lecture 17, November 3, 2008 – p. 4/9Optional Stopping Theorem: Motivation•Suppose {(Xn, Fn)} is a martingale•Then IE(Xn) = IE(X0) for any n•If τ is a stopping time for {Fn}, is it true thatIE(Xτ) = IE(X0)?•Equality not true in general•Optional stopping theorem provides sufficient conditions forabove equality to hold•Can use martingales and optional stopping theorem toobtain distributional properties of τMATH136/STAT219 Lecture 17, November 3, 2008 – p. 5/9Stopped Process•Suppose {Xn} is a SP and τ is a stopping time for filtration{Fn}•The stopped (at time τ ) process {Xn∧τ, n = 0, 1, 2, . . .} isdefined byXn∧τ=(Xnn ≤ τXτn > τ•If {(Xn, Fn)} is a sub-martingale and τ is an {Fn}-stoppingtime then {(Xn∧τ, Fn)} is a sub-martingale. In particular,IE(Xn∧τ) ≥ IE(X0).•If {(Xn, Fn)} is a martingale and τ is an {Fn}-stopping timethen {(Xn∧τ, Fn)} is a martingale. In particular,IE(Xn∧τ) = IE(X0).MATH136/STAT219 Lecture 17, November 3, 2008 – p. 6/9Doob’s Optional Stopping Theorem•Suppose {(Xn, Fn)} is a sub-martingale and τ is an{Fn}-stopping time. If◦τ < ∞ a.s., and◦{Xn∧τ, n = 0, 1, 2, . . .} is uniformly integrable (2)•Then IE(Xτ) ≥ IE(X0).•If instead {(Xn, Fn)} is a martingale then IE(Xτ) = IE(X0).•If any of the following is true then (2) is satisfied◦τ is a bounded stopping time: τ ≤ N a.s. for someconstant N < ∞◦There is a constant c < ∞ such that IE|Xn| ≤ c for all nMATH136/STAT219 Lecture 17, November 3, 2008 – p. 7/9Gambler’s Ruin: Probability of Ruin•Let ξ1, ξ2, . . . be i.i.d. with IP (ξk= 1) = IP (ξk= −1) = 1/2.Let S0= 0 and Sn=Pnk=1ξk.•Fix integers a, b > 0.•Define τ = inf{n ≥ 0 : Sn= b or Sn= −a}•Then:◦τ is a stopping time with τ < ∞ a.s.◦{Sn∧τ} is uniformly integrable◦Probability of ruinIP (Sτ= −a) =ba + bMATH136/STAT219 Lecture 17, November 3, 2008 – p. 8/9Gambler’s Ruin: Expected Number of Games•Let ξ1, ξ2, . . . be i.i.d. with IP (ξk= 1) = IP (ξk= −1) = 1/2.Let S0= 0 and Sn=Pnk=1ξk.•Fix integers a, b > 0.•Define τ = inf{n ≥ 0 : Sn= b or Sn= −a}•Let Yn= S2n− n•Then:◦τ is a stopping time with τ < ∞ a.s.◦{Yn∧τ} is uniformly integrable◦Expected number of games played:IE(τ) = abMATH136/STAT219 Lecture 17, November 3, 2008 – p.


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