Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales Prakash Balachandran Department of Mathematics Duke University April 2, 2008 1 Review of Discrete-Time Martingales We assume the reader is familiar with the basic results from probability theory, particularly those con- cerning discrete-time martingales. Here, we briefly review the major theorems and definitions. For more details, see Patrick Billingsley’s Probability and Measure or Richard Durrett’s Probability: Theory and Examples. Throughout, (Ω; F;P ) is a probability space, and B(X) denotes the Borel σ-algebra generated by a topological space X. 1 Definition 1 A filtration is a collection of σ-algebras, fFngn=1 such that Fn ⊆ Fn+1. 1 Definition 2 Let fFngn=1 be a filtration. A stopping time T is a random variable T : (Ω; F) ! (N [ f1g; B(N)) such that f! 2 Ω: T (!) = ng 2 Fn for all n < 1. Definition 3 Let I be an index set, and fXigi2I be a collection of random variables. The collection fXigi2I is said to be uniformly integrable if lim sup E [jXij : jXij > M] < 1 M!1 i2I Definition 4 If f; g are random variables, then f ^ g is the random variable defined by (f ^ g)(!) = minff(!); g(!)g. Definition 5 If T is a stopping time and X is a random variable, then XT is the random variable defined by XT (!) = XT (!)(!). 1 Definition 6 A collection of random variables fXngn=0 is called a discrete-time martingale (resp. discrete-time submartingale) if 1 1 1. There exists a filtration fFngn=0. 2. Xn is Fn-adapted; i.e. Xn is Fn-measurable. 1 3. Xn 2 L (Ω; F;P ), 8n ≥ 0. 4. E[Xn+1jFn] = Xn (resp. E[Xn+1jFn] ≥ Xn), 8n ≥ 0. Now we need to review the notion of an upcrossing. Let [α; β] be an interval (α < β), and let X1;:::;Xn be random variables. Inductively define τ1; τ2;::: by: τ1(!) is the smallest j such that 1 ≤ j ≤ n and Xj(!) ≤ α, and is n if there is no such j. τk(!) for even k is the smallest j such that τk−1(!) < j ≤ n and Xj(!) ≥ β and is n if there is no such j. τk(!) for odd k exceeding 1 is the smallest j such that τk−1(!) < j ≤ n and Xj(!) ≤ α and is n if there is no such j. The number Un(!) of upcrossings of [α; β] by X1;:::;Xn is the largest j such that Xτ2j−1 ≤ α < β ≤ Xτ2j : Now, recall the basic results from discrete time martingales: 1 Theorem 1 (Doob’s Inequality): If fXngn=0 is a submartingale and λ > 0, then, + + E[Xn ] P ! 2 Ω : max Xm(!) ≥ λ ≤ : 0≤m≤n λ 1 Theorem 2 (The Upcrossing Inequality): If fXmgm=0 is a submartingale, then the number of up- crossings Un on [a; b] satisfies E[(X − a)+] − E[(X − a)+] E[U ] ≤ n 0 : n b − a 1 Theorem 3 (Doob’s Maximal Inequality): If fXngn=0 is a submartingale, then for 1 < p < 1, p p + p + p E max Xm ≤ E[(Xn ) ]: 0≤m≤n p − 1 1 + Theorem 4 (The Martingale Convergence Theorem): If fXngn=0 is a submartingale such that sup E[Xn ] < 1 for all n, then as n ! 1, Xn converges a.s. to a limit X with E[jXj] < 1. 1 Theorem 5 (The Optional Stopping Theorem): If L ≤ M are stopping times and fYM^ngn=0 is a uniformly integrable submartingale, then E[YL] ≤ E[YM ] and YL ≤ E[YM jFL] where FL = fA 2 F : A \ f! 2 Ω: L(!) ≤ kg 2 Fk; 1 ≤ k < 1g: 2 2 Continuous-Time Martingales We would like to extend the definitions and theorems in (1) to similar definitions and theorems involving continuous time martingales. Definition 7 A filtration is a collection of σ-algebras, fFtgt≥0 such that Fs ⊆ Ft for s < t. We recall the interpretation of σ-algebras as containing information: Ft is the information available to the adapted process fXtgt≥0 up to (and including) time t. Thus, it’s reasonable to interpret ! [ Ft− = σ Fs s<t as the σ-algebra of events strictly prior to t > 0 and \ Ft+ = Ft+ >0 as the σ field of events immediately after t ≥ 0. We take the definition F0− = F0 and say that the filtration fFtgt≥0 is right- (resp. left-) continuous if Ft = Ft+ (resp. Ft = Ft−) holds for every t ≥ 0. Definition 8 Let fFtgt≥0 be a filtration. A stopping time T is a random variable T : (Ω; F) ! ([0; 1]; B([0; 1])) such that f! 2 Ω: T (!) ≤ tg 2 Ft for every t ≥ 0. A random variable T , T : (Ω; F) ! ([0; 1]; B([0; 1])) is called an optional time of the filtration if f! 2 Ω: T (!) < tg 2 Ft for every t ≥ 0. We have the following immediate result: Proposition 1 Every stopping time is optional, and the two concepts coincide if the filtration is right- continuous. Proof: 1 If T is a stopping time, then T ≤ t − 2 F 1 ⊆ Ft for all n ≥ 1, n 2 Z: So: n t− n 1 [ 1 [T < t] = T ≤ t − 2 F : n t n=1 Now, suppose that T is an optional time, and fFtgt≥0 is a right continuous filtration. Fix > 0, and notice that for η < rational, Ft+η ⊆ Ft+: This implies [T < t + η] 2 Ft+η ⊆ Ft+ since T is an optional time, so that \ [T ≤ t] = [T < t + η] 2 Ft+: 0<η< rational 3 Since > 0 was arbitrary, this implies that [T ≤ t] 2 Ft+ for every > 0, so that \ [T ≤ t] 2 Ft+ = Ft+ = Ft >0 since the filtration is right continuous. Definition 9 Let I be an index set, and fXigi2I be a collection of random variables. The collection fXigi2I is said to be uniformly integrable if lim sup E [jXij : jXij > M] < 1 M!1 i2I Definition 10 If f; g are random variables, then f ^ g is the random variable defined by (f ^ g)(!) = minff(!); g(!)g. Definition 11 If T is a stopping time and fXtgt≥0 is a stochastic process, then XT is the random variable defined by XT (!) = XT (!)(!). Definition 12 Let T be a stopping time of the filtration fFtgt≥0. The σ-algebra FT of events deter- mined prior to the stopping time T consists of those events A 2 F for which A \ f! 2 Ω: T (!) ≤ tg 2 Ft for every t ≥ 0. Clearly, if fXtgt≥0 is adapted to fFtgt≥0, then XT is FT measurable. Definition 13 Let T be an optional time of the filtration fFtgt≥0. The σ-field FT + of events determined immediately after the optional time T consists of those events A 2 F for which A \ fT ≤ tg 2 Ft+ for every t ≥ 0. 1 Definition 14 A collection of random variables fXngn=0 is called a continuous-time martingale (resp. continuous-time submartingale) if 1. There exists a filtration fFtgt≥0. 2. Xt is Ft-adapted; i.e. Xt is Ft-measurable. 1 3. Xt 2 L (Ω; F;P ) for each t ≥ 0. 4. E[XsjFt] = Xt (resp. E[XsjFt] ≥ Xt) for 0 ≤ t < s < 1. Now we need to define the notion of an upcrossing. Let fXtgt≥0 be a real-valued stochastic process. Consider two numbers α < β, and a finite subset F of [0; 1). We define the number of upcrossings UF (α; β; X(!)) of the interval [α; β] by the restricted sample path fXt(!); t 2 F g as follows. Set τ1(!) = minft 2 F ; Xt(!) ≤ αg 4 and define recursively for j = 1; 2;::: σj(!) = minft 2 F ; t ≥ τj(!);Xt(!) > βg τj+1(!) = minft 2 F ; t ≥ σj(!);Xt(!) < αg: We define the minimum of the empty set to be 1 and let UF (α; β; X(!)) to be the largest number j such that σj(!) < 1. If I ⊂ [0; 1) is not necessarily finite, we define UI (α; β; X(!)) = supfUF (α; β; X(!)); F ⊆ I; F finiteg: Theorem 6 Let fXtgt≥0 be a submartingale w.r.t. fFtgt≥0, whose every path is right continuous. Let [σ; τ] be a subinterval of [0; 1) and let α < β, λ > 0 be real numbers. We have the following results: 1. (Doob’s Inequality): + E[Xτ ] P ! 2 Ω : sup Xt(!) ≥ λ ≤ : σ≤t≤τ λ 2. (The Upcrossing Inequality): E [X+] + jαj E[U (α; β; X(!))] ≤ τ : [σ,τ] β − α 3. (Doob’s Maximal Inequality): p p p p E sup Xt ≤ E[Xt ] σ≤t≤τ p − 1 p for p > 1, provided Xt ≥ 0 a.s. for every t ≥ 0 and Xτ 2 L (Ω; F;P ). Proof: 1) Consider the enumeration σ = t0 < t1 < ··· Sj 1 of the numbers in fσ; τg [ ([σ; τ] \ Q). Define Fj = i=0 tj [ fτg. Clearly, fFjgj=0 is an increasing sequence of finite sets such that 1 ~ [ F = Fj = fσ; τg [ ([σ; τ] \ Q) : j=0 n o Let Aj = ! 2 Ω : supt2Fj Xt(!) ≥ λ . Then, Fj ⊆ Fj+1 ) sup Xt(!) ≤ sup Xt(!) ) Aj ⊆ Aj+1: t2Fj t2Fj+1 5 Using continuity from below on ( ) 1 [ A1 = ! 2 Ω : sup Xt(!) ≥ λ = Aj t2F~ j=0 (note that fAjg are measurable) we have P (A1) = lim P (An): n!1 + E[X ] + max Fj E[Xτ ] By the discrete-time Doob inequality: P fAjg ≤ λ = λ where max Fj denotes the largest 1 element in the set Fj, which is τ by construction of fFjgj=0; so that + E[Xτ ] P (A1) = lim P (Aj) ≤ : j!1 λ Claim 1 supt2F~ Xt(!) = supt2[σ,τ] Xt(!) Proof of Claim 1: ~ Since F ⊆ [σ; τ], supt2F~ Xt(!) ≤ supt2[σ,τ] Xt(!).