An introduction to stochastic integration with respect to continuous semimartingales

Transcription

1 A itroductio to stochastic itegratio with respect to cotiuous semimartigales Alexader Sool Departmet of Mathematical Scieces Uiversity of Copehage

2 Departmet of Mathematical Scieces Uiversity of Copehage Uiversitetspare 5 DK-21 Copehage Copyright 214 Alexader Sool ISBN

3 Cotets Preface v 1 Cotiuous-time stochastic processes Measurability ad stoppig times Cotiuous-time martigales Square-itegrable martigales Local martigales Exercises Stochastic itegratio Cotiuous semimartigales Costructio of the stochastic itegral Itô s formula Coclusio Exercises A Appedices 67 A.1 Aalysis ad measure theory A.2 Covergece results ad uiform itegrability A.3 Discrete-time martigales A.4 Browia motio ad the usual coditios A.5 Exercises B Hits ad solutios for exercises 93 B.1 Hits ad solutios for Chapter B.2 Hits ad solutios for Chapter B.3 Hits ad solutios for Appedix A Bibliography 133

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5 Preface This moograph cocers itself with the theory of cotiuous-time martigales with cotiuous paths ad the theory of stochastic itegratio with respect to cotiuous semimartigales. To set the scee for the theory to be developed, we cosider a example. Assume give a probability space (Ω, F, P ) edowed with a Browia motio W, ad cosider a cotiuous mappig f : [, ) R. We would lie to uderstad whether it is possible to defie a itegral f(s) dw s i a fashio aalogous to ordiary Lebesgue itegrals. I geeral, there is a correspodece betwee bouded siged measures o [, t] ad mappigs of fiite variatio o [, t]. Therefore, if we see to defie the itegral with respect to Browia motio i a pathwise sese, that is, by defiig f(s) dw (ω) s for each ω separately, by referece to ordiary Lebesgue itegratio theory, it is ecessary that the sample paths W (ω) have fiite variatio. However, Browia motio has the property that its sample paths are almost surely of ifiite variatio o all compact itervals. Our coclusio is that the itegral of f with respect to W caot i geeral be defied pathwisely by immediate referece to Lebesgue itegratio theory. We are thus left to see a alterate maer of defiig the itegral. A atural startig poit is to cosider Riema sums of the form 2 =1 f(t )(W t W t ), where t = t2, ad attempt to prove their covergece i some sese, say, i L 2. By the completeess of L 2, it suffices to prove that the sequece of Riema sums costitute a Cauchy sequece i L 2. I order to show this, put η(β) = sup{ f(s) f(u) s, u [, t], s u β}. η(β) is called the modulus of cotiuity for f over [, t]. As f is cotiuous, f is uiformly cotiuous o [, t], ad therefore η(β) teds to zero as β teds to zero. Now ote that for m, with ξ m = f(t m ) f(t 1 ) where is such that t 1 tm < t, we fid by the idepedet

6 vi CONTENTS ad ormally distributed icremets of Browia motio that E = E ( 2 m f(t m )(W t m W t m ) =1 ( 2 m =1 ξ m (W t m W t m )) 2 = 2 =1 2 m =1 m 2 η(t2 ) 2 E(W t m W t m ) 2 = tη(t2 ) 2, =1 f(t )(W t W t ) ) 2 ( ) 2 E ξ m (W t m W t m ) which teds to zero as teds to ifiity. We coclude that as m ad ted to ifiity, the L 2 distace betwee the correspodig Riema sums ted to zero, ad so the sequece of Riema sums 2 =1 f(t )(W t W t ) is a Cauchy sequece i L2. Therefore, by completeess, the sequece coverges i L 2 to some limit. Thus, while we caot i geeral obtai pathwise covergece of the Riema sums, we ca i fact obtai covergece i L 2, ad may the defie the stochastic itegral f(s) dw s as the limit. Our coclusio from the above deliberatios is that we caot i geeral defie the stochastic itegral with respect to a Browia motio usig ordiary Lebesgue itegratio theory, but i certai circumstaces, we may defie the itegral usig a alterate limitig procedure. This provides evidece that a theory of stochastic itegratio may be feasible. I the followig chapters, we will develop such a theory. The structure of what is to come is as follows. I Chapter 1, we will develop the basic tools of cotiuous-time martigale theory, as well as develop the geeral cocepts used i the theory of cotiuous-time stochastic processes. Usig these results, we will i Chapter 2 defie the stochastic itegral H s dx s for all processes such that X belogs to the class of processes ow as cotiuous semimartigales, which i particular icludes cotiuous martigales ad processes with cotiuous paths of fiite variatio, ad H is a process satisfyig certai measurability ad itegrability coditios. I this chapter, we also prove some basic properties of the stochastic itegral, such as the domiated covergece theorem ad Itô s formula, which is the stochastic versio of the fudametal theorem of aalysis. As regards the prerequisites for this text, the reader is assumed to have a reasoable grasp of basic aalysis, measure theory ad discrete-time martigale theory, as ca be obtaied through the boos Carothers (2), Ash (2) ad Rogers & Williams (2a).

7 CONTENTS vii Fially, may warm thas go to Niels Richard Hase, Bejami Faleborg ad Peter Uttethal for their editorial commets o the origial mauscript. Also, I would lie i particular to tha Erst Hase ad Marti Jacobse for teachig me probability theory. Alexader Sool Købehav, September 214

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9 Chapter 1 Cotiuous-time stochastic processes I this chapter, we develop the fudametal results of stochastic processes i cotiuous time, coverig mostly some basic measurability results ad the theory of cotiuous-time cotiuous martigales. Sectio 1.1 is cocered with stoppig times ad various measurability properties for processes i cotiuous time. I Sectio 1.2, we itroduce cotiuous martigales i cotiuous time. We defie the spaces cm, cm u ad cm b cosistig of cotiuous martigales, uiformly itegrable martigales ad bouded martigales, respectively, all with iitial value zero. Maily by discretizatio ad referece to the classical results from discrete-time martigale theory, we show that the mai theorems of discrete-time martigale theory carry over almost verbatim to the cotiuous-time case. I Sectio 1.3, we itroduce the space cm 2 of cotiuous martigales bouded i L 2 with iitial value zero. Aalogously to the special properties of L 2 amog the spaces L p for p 1, the space cm 2 has some particularly pleasat properties. We prove results o covergece properties of martigales i cm 2, we show a completeess property of cm 2 ad a type of Riesz represetatio theorem for cm 2, ad we use these results to demostrate the existece of a process, the quadratic variatio process, for elemets of cm b, which will be essetial to our developmet of the stochastic itegral.

10 2 Cotiuous-time stochastic processes Fially, i Sectio 1.4, we itroduce the space cm l of cotiuous local martigales with iitial value zero. We prove the basic stability properties of the space of local martigales ad exted the otio of quadratic variatio ad quadratic covariatio to cotiuous local martigales. 1.1 Measurability ad stoppig times We begi by reviewig basic results o cotiuous-time stochastic processes. We will wor i the cotext of a filtered probability space (Ω, F, (F t ), P ). Here, Ω deotes some set, F is a σ-algebra o Ω, P is a probability measure o (Ω, F) ad (F t ) t is a family of σ-algebras such that F s F t wheever s t ad such that F t F for all t. We refer to (F t ) t as the filtratio of the probability space. We defie F = σ( t F t ). We will require that the filtered probability space satisfies certai regularity properties give i the followig defiitio. Recall that a P ull set of F is a set F Ω with the property that there exists G F with P (G) = such that F G. Defiitio A filtered probability space (Ω, F, (F t ) t, P ) is said to satisfy the usual coditios if it holds that the filtratio is right-cotiuous i the sese that F t = s>t F s for all t, ad for all t, F t cotais all P ull sets of F. I particular, all P ull sets of F are F measurable. We will always assume that the usual coditios hold. Note that because of this permaet assumptio, our results a priori oly hold for such filtered probability spaces. Therefore, we also eed to esure that the usual coditios may be assumed i practical cases, for example whe dealig with Browia motio. These issues are cosidered i Sectio A.4. A stochastic process is a family (X t ) t of R-valued radom variables. The sample paths of the stochastic process X are the fuctios t X t (ω) for ω Ω. We refer to X as the iitial value of X. I particular, we say that X has iitial value zero if X is zero. I the followig, B deotes the Borel-σ-algebra o R. We put R + = [, ) ad let B + deote the Borel-σ-algebra o R +, ad we let B t deote the Borel-σ-algebra o [, t]. We say that two processes X ad Y are versios of each other if P (X t = Y t ) = 1 for all t. I this case, we say that Y is a versio of X ad vice versa. We say that two processes X ad Y are idistiguishable if their sample paths are almost surely equal, i the sese that the set where X ad Y are ot equal is a ull set, meaig that the set {ω Ω t : X t (ω) Y t (ω)}

11 1.1 Measurability ad stoppig times 3 is a ull set. We the say that X is a modificatio of Y ad vice versa. We call a process evaescet if it is idistiguishable from the zero process, ad we call a set A B + F evaescet if the process 1 A is evaescet. We say that a result holds up to evaescece, or up to idistiguishability, if it holds except perhaps o a evaescet set. We have the followig three measurability cocepts for stochastic processes. Defiitio Let X be a stochastic process. We say that X is adapted if X t is F t measurable for all t. We say that X is measurable if (t, ω) X t (ω) is B + F measurable. We say that X is progressive if X [,t] Ω, the restrictio of X to [, t] Ω, is B t F t measurable for t. If a process X has sample paths which are all cotiuous, we say that X is cotiuous. Note that we require that all paths of X are cotiuous, ot oly that X has cotiuous paths almost surely. Next, we itroduce the progressive σ-algebra Σ π ad cosider its basic properties. Lemma Let Σ π be the family of sets A B + F such that A [, t] Ω B t F t for all t. The Σ π is a σ-algebra, ad a process X is progressive if ad oly if it is Σ π measurable. Proof. We first show that Σ π is a σ-algebra. It holds that Σ π cotais R + Ω. If A Σ π, we have A [, t] Ω B t F t for all t. As A c [, t] Ω = ([, t] Ω) \ (A [, t] Ω), A c [, t] Ω is the complemet of A [, t] Ω relative to [, t] Ω. Therefore, as B t F t is stable uder complemets, we fid that A c [, t] Ω is i B t F t as well for all t. Thus, Σ π is stable uder taig complemets. Aalogously, we fid that Σ π is stable uder taig coutable uios, ad so Σ π is a σ-algebra. As regards the statemet o measurability, we first ote for ay A B the equality {(s, ω) R + Ω X(s, ω) A} [, t] Ω = {(s, ω) [, t] Ω X [,t] Ω (s, ω) A}. Now assume that X is progressive. Fix a set A B. From the above, we the obtai {(t, ω) R + Ω X(t, ω) A} [, t] Ω B t F t, so that X is Σ π measurable. I order to obtai the coverse implicatio, assume that X is Σ π measurable. The above the shows {(t, ω) [, t] Ω X [,t] Ω (t, ω) A} B t F t. Thus, beig progressive is equivalet to beig Σ π measurable. Lemma i particular shows that beig progressive is the same as beig measurable with respect to the progressive σ-algebra, which we also refer to as beig progressively measurable.

12 4 Cotiuous-time stochastic processes Lemma Let X be adapted. If X has left-cotiuous paths, the X is progressive. If X has right-cotiuous paths, the X is progressive. I particular, if X is a cotiuous, adapted process, the X is progressive. Proof. First cosider the case where X is adapted ad has left-cotiuous paths. I this case, X t = lim X t poitwise, where X is the process X t = = X 2 1 [2,(+1)2 )(t). Therefore, usig the result from Lemma that beig progressive meas measurability with respect to the σ-algebra Σ π, we fid that i order to show the result, it suffices to show that the process t X 2 1 [2,(+1)2 )(t) is progressive for ay 1 ad, sice i this case, X iherits measurability with respect to Σ π as a limit of Σ π measurable maps. I order to show that t X 2 1 [2,(+1)2 )(t) is progressive, let A B with / A. For ay t, we the have {(s, ω) [, t] Ω X 2 (ω)1 [2,(+1)2 )(s) A} = [2, ( + 1)2 ) [, t] (X 2 A). If 2 > t, this is empty ad so i B t F t, ad if 2 t, this is i B t F t as a product of a set i B t ad a set i F t. Thus, i both cases, we obtai a elemet of B t F t, ad from this we coclude that the restrictio of t X 2 1 [2,(+1)2 )(t) to [, t] Ω is B t F t measurable, demostratig that the process is progressive. This shows that X is progressive. Next, cosider the case where X is adapted ad has right-cotiuous paths. I this case, we fix t ad defie, for s t, Xs = X 1 {} (s) = X t(+1)2 1 (t2,t(+1)2 ](t). By right-cotiuity, X s = lim Xs poitwise for s t. Also, each term i the sum defiig X is B t F t measurable, ad therefore, X is B t F t measurable. As a cosequece, the restrictio of X to [, t] Ω is B t F t, ad so X is progressive. This cocludes the proof. Lemma Let X be cotiuous. If X t is almost surely zero for all t, X is evaescet. Proof. We claim that {ω Ω t : X t (ω) = } = q Q+ {ω Ω X q (ω) = }. The iclusio towards the right is obvious. I order to show the iclusio towards the left, assume that ω is such that X q (ω) is zero for all q Q +. Let t R +. Sice Q + is dese i R +, there is a sequece (q ) i Q + covergig to t. As X has cotiuous paths, X(ω) is cotiuous, ad so X t (ω) = lim X q (ω) =. This proves the iclusio towards the left. Now, as a coutable itersectio of almost sure sets agai is a almost sure set, we fid that q Q+ {ω Ω X q (ω) = } is a almost sure set. Therefore, {ω Ω t : X t (ω) = } is a almost sure set, showig that X is evaescet.

13 1.1 Measurability ad stoppig times 5 Lemma Let X be progressive. The X is measurable ad adapted. Proof. By defiitio Σ π B + F. Therefore, as X is progressive, we have for ay A B that {(t, ω) R + Ω X t (ω) A} Σ π B + F. This proves that X is measurable. To show that X is adapted, ote that whe X is progressive, X [,t] Ω is B t F t measurable, ad therefore ω X t (ω) is F t measurable. Next, we defie stoppig times i cotiuous time ad cosider their iterplay with measurability cocepts o R + Ω. A stoppig time is a radom variable T : Ω [, ] such that (T t) F t for ay t. We say that T is fiite if T maps ito R +. We say that T is bouded if T maps ito a bouded subset of R +. If X is a stochastic process ad T is a stoppig time, we deote by X T the process Xt T = X T t ad call X T the process stopped at T. Furthermore, we defie the stoppig time σ-algebra F T of evets determied at T by puttig F T = {A F A (T t) F t for all t }. Clearly, F T is a σ-algebra, ad if T is costat, the stoppig time σ-algebra is the same as the filtratio σ-algebra, i the sese that {A F A (s t) F t for all t } = F s. Our first goal is to develop some basic results o stoppig times ad their iterplay with stoppig time σ-algebras. I the followig, we use the otatio that S T = mi{s, T } ad S T = max{s, T }. Lemma The followig statemets hold about stoppig times: 1. Ay costat i [, ] is a stoppig time. 2. A oegative variable T is a stoppig time if ad oly if (T < t) F t for t. 3. If S ad T are stoppig times, so are S T, S T ad S + T. 4. If T is a stoppig time ad F F T, the T F = T 1 F + 1 F c is a stoppig time as well. 5. If S T, the F S F T. Proof. Proof of (1). Let c be a costat i R +. The (c t) is either or Ω, both of which are i F t for ay t. Therefore, ay costat c i R + is a stoppig time. Proof of (2). Assume first that T is a stoppig time. The (T < t) = =1(T t 1 ) F t, sice (T t 1 ) F t 1. Coversely, assume (T < t) F t for all t. We the obtai

14 6 Cotiuous-time stochastic processes (T t) = = (T < t + 1 ) for all. This shows (T t) F t+ 1 for all 1. Sice (F t) is icreasig ad is arbitrary, we fid (T t) =1F t+ 1 = =1 s t+ 1 F s = s>t F s. By right-cotiuity of the filtratio, F t = s>t F s, so we coclude (T t) F t, provig that T is a stoppig time. Proof of (3). Assume that S ad T are stoppig times ad let t. We the have (S T t) = (S t) (T t) F t, so S T is a stoppig time. Liewise, we obtai (S T t) = (S t) (T t) F t, so S T is a stoppig time as well. Fially, cosider the sum S + T. Let 1 ad fix ω. If S(ω) ad T (ω) are fiite, there are q, q Q + such that q S(ω) q + 1 ad q T (ω) q + 1. I particular, q + q S(ω) + T (ω) ad S(ω) + T (ω) q + q + 2. Next, if S(ω) + T (ω) t, it holds i particular that both S(ω) ad T (ω) are fiite. Therefore, with Θ t = {q, q Q + q + q t}, we fid (S + T t) = =1 (q,q ) Θ t (S q + 1 ) (T q + 1 ). Now, the sequece of sets (q,q ) Θ t (S q+ 1 ) (T q + 1 ) is decreasig i, ad therefore we have for ay 1 that (S + T t) = = (q,q ) Θ t (S q + 1 ) (T q + 1 ) F t+ 1. I particular, (S + T t) F s for ay s > t, ad so, by right-cotiuity of the filtratio, (S + T t) s>t F s = F t, provig that S + T is a stoppig time. Proof of (4). Let T be a stoppig time ad let F F T. The (T F t) = (T t) F F t, as was to be prove. Proof of (5). Let A F S, so that A (S t) F t for all t. Sice S T, we have (T t) (S t) ad so A (T t) = A (S t) (T t) F t, yieldig A F T. For the ext results, we recall that for ay A R, it holds that if A < t if ad oly if there is s A such that s < t. Lemma Let (T ) be a sequece of stoppig times, the sup T ad if T are stoppig times as well. Proof. Assume that T is a stoppig time for each. Fix t, we the the have that (sup T t) = =1(T t) F t, so sup T is a stoppig time as well. Liewise, usig the secod statemet of Lemma 1.1.7, we fid (if T < t) = =1(T < t) F t, so if T is a stoppig time as well. Lemma Let X be a cotiuous adapted process, ad let U be a ope set i R. Defie T = if{t X t U}. The T is a stoppig time.

15 1.1 Measurability ad stoppig times 7 Proof. Note that if s ad (s ) is a sequece covergig to s, we have by cotiuity that X s coverges to X s ad so, sice U is ope, (X s U) =1(X s U). Usig that Q + is dese i R +, we fid (T < t) = ( s R + : s < t ad X s U) = ( s Q + : s < t ad X s U) = s Q+,s<t(X s U), ad sice X is adapted, we have (X s U) F s F t wheever s < t, provig that (T < t) F t. By Lemma 1.1.7, this implies that T is a stoppig time. Lemma Let X be a cotiuous adapted process, ad let F be a closed set i R. Defie T = if{t X t F }. The T is a stoppig time. Proof. Defie U = {x R y F : x y < 1 }. We claim that U is ope, that U decreases to F ad that U +1 U, where U +1 deotes the closure of U +1. As we have that U = y F {x R x y < 1 }, U is ope as a uio of ope sets. We have F U for all, ad coversely, if x =1U, we have that there is a sequece (y ) i F such that y x 1 for all. I particular, y teds to x, ad as F is closed, we coclude x F. Thus, F = =1U. Furthermore, if x is i the closure U +1, there is a sequece (x ) i U +1 such that x x 1, ad there is a sequece (y ) i F such that x y < 1 +1, showig that x y < Taig so large that , we see that U +1 U. Now ote that wheever t >, we have (T < t) = ( s R + : s < t ad X s F ) = =1( s R + : s t 1 ad X s F ), so by Lemma 1.1.7, it suffices to prove that ( s R + : s t ad X s F ) is F t measurable for all t >. We claim that ( s R + : s t ad X s F ) = =1( q Q + : q t ad X q U ). To see this, first cosider the iclusio towards the right. If there is s R + with s t ad X s F, the we also have X s U for all. As U is ope, there is ε > such that the ball of size ε aroud X s is i U. I particular, there is q Q + with q t such that X q U. This proves the iclusio towards the right. I order to obtai the iclusio towards the left, assume that for all, there is q Q + with q t such that X q U. As [, t] is compact, there exists s R + with s t such that for some subsequece, lim m q m = s. By cotiuity, X s = lim m X qm. As X qm U m, we have for ay i that X qm U i for m large eough. Therefore, we coclude X s i=1 U i i=1 U i = F, provig the other iclusio. This shows the desired result.

16 8 Cotiuous-time stochastic processes Lemma Let X be ay cotiuous adapted process with iitial value zero. Defiig T = if{t X t > }, (T ) is a sequece of stoppig times icreasig poitwise to ifiity, ad the process X T is bouded by. Proof. By Lemma 1.1.9, (T ) is a sequece of stoppig times. We prove that X T is bouded by. If T is ifiite, X t for all t, so o (T = ), X T is bouded by. If T is fiite, ote that for all ε >, there is t with T t < T + ε such that X t >. Therefore, by cotiuity, X T. I particular, as X has iitial value zero, T caot tae the value zero. Therefore, there is t with t < T. For all such t, X t. Therefore, agai by cotiuity, X T, ad we coclude that i this case as well, X T is bouded by. Note that we have also show that X T = wheever T is fiite. It remais to show that T coverges almost surely to ifiity. To obtai this, ote that as X is cotiuous, X is bouded o compacts. If for some samle path we have that T a for all, we would have X T = for all ad so X would be ubouded o [, a]. This is a cotradictio, sice X has cotiuous sample paths ad therefore is bouded o compact sets. Therefore, (T ) is ubouded for every sample path. As T is icreasig, this shows that T coverges to ifiity poitwise. Lemma Let X be progressively measurable, ad let T be a stoppig time. The X T 1 (T < ) is F T measurable ad X T is progressively measurable. Proof. We first prove that the stopped process X T is progressively measurable. Fix t, we eed to show that X T [,t] Ω is B t F t measurable, which meas that we eed to show that the mappig from [, t] Ω to R give by (s, ω) X T (ω) s (ω) is B t F t -B measurable. To this ed, ote that wheever s t, {(u, ω) [, t] Ω T (ω) u s} = ([, t] (T s)) ([, s] Ω) B t F t, so the mappig from [, t] Ω to [, t] give by (s, ω) T (ω) s is B t F t -B t measurable. Ad as the mappig from [, t] Ω to Ω give by (s, ω) ω is B t F t -F t measurable, we coclude that the mappig from [, t] Ω to [, t] Ω give by (s, ω) (T (ω) s, ω) is B t F t -B t F t measurable, sice it has measurable coordiates. As X is progressive, the mappig from [, t] Ω to R give by (s, ω) X s (ω) is B t F t -B measurable. Therefore, the composite mappig from [, t] Ω to R give by (s, ω) X T (ω) s (ω) is B t F t -B measurable. This shows that X T is progressively measurable. I order to prove that X T is F T measurable, we ote that for ay B B, we have that (X T 1 (T < ) B) (T t) = (X T t B) (T t). Now, X T t is F t measurable sice X T is

17 1.2 Cotiuous-time martigales 9 progressive ad therefore adapted by Lemma 1.1.6, ad (T t) F t sice T is a stoppig time. Thus, (X T 1 (T < ) B) (T t) F t, ad we coclude (X T 1 (T < ) B) F T. Lemma Let T ad S be stoppig times. Assume that Z is F S measurable. It the holds that both Z1 (S<T ) ad Z1 (S T ) are F S T measurable. Proof. We first show (S < T ) F S T. To prove the result, it suffices to show that the set (S < T ) (S T t) is i F t for all t. To this ed, we begi by otig that (S < T ) (S T t) = (S < T ) (S t). Cosider some ω Ω such that S(ω) < T (ω) ad S(ω) t. If t < T (ω), S(ω) t < T (ω). If T (ω) t, there is some q Q [, t] such that S(ω) q < T (ω). We thus obtai (S < T ) (S T t) = q Q [,t] {t} (S q) (q < T ), which is i F t, showig (S < T ) F S T. We ext show that Z1 (S<T ) is F S T measurable. Let B B with B ot cotaiig zero. As this type of sets geerate B, it will suffice to show that (Z1 (S<T ) B) (S T t) F t for all t. To obtai this, we rewrite (Z1 (S<T ) B) (S T t) = (Z B) (S < T ) (S T t) = (Z B) (S < T ) (S t). Sice Z is F S measurable, (Z B) (S t) F t. Ad by what we have already show, (S < T ) F S, so (S < T ) (S t) F t. Thus, the above is i F t, as desired. Fially, we show that Z1 (S T ) is F S T measurable. Let B B with B ot cotaiig zero. As above, it suffices to show that for ay t, (Z1 (S T ) B) (S T t) F t. To obtai this, we first write (Z1 (S T ) B) (S T t) = (Z B) (S T ) (S T t) = (Z B) (S t) (S T ) (S T t). Sice Z F S, we fid (Z B) (S t) F t. Ad sice we ow (T < S) F T S, (S T ) = (T < S) c F S T, so (S T ) (S T t) F t. This demostrates (Z1 (S T ) B) (S T t) F t, as desired. 1.2 Cotiuous-time martigales I this sectio, we cosider cotiuous martigales i cotiuous time. We say that a process M is a cotiuous-time martigale if for ay s t, E(M t F s ) = M s almost surely. I the same maer, if for ay s t, E(M t F s ) M s almost surely, we say that M is a

18 1 Cotiuous-time stochastic processes supermartigale, ad if for ay s t, E(M t F s ) M s almost surely, we say that M is a submartigale. We are iterested i trasferrig the results ow from discrete-time martigales to the cotiuous-time settig, maily the criteria for almost sure covergece, L 1 covergece ad the optioal samplig theorem. The classical results from discrete-time martigale theory are reviewed i Appedix A.3. We will for the most part oly tae iterest i martigales M whose iitial value is zero, M =, i order to simplify the expositio. We deote the space of cotiuous martigales i cotiuous time with iitial value zero by cm. By cm u, we deote the elemets of cm which are uiformly itegrable, ad by cm b, we deote the elemets of cm u which are bouded i the sese that there exists c > such that M t c for all t. Clearly, cm ad cm b are both vector spaces, ad by Lemma A.2.4, cm u is a vector space as well. We begi by presetig our most basic example of a cotiuous martigale i cotiuous time, the p-dimesioal F t Browia motio. Recall from Appedix A.4 that a p-dimesioal Browia motio is a cotiuous process W with values i R p such that the icremets are idepedet over disjoit itervals, ad for s t, W t W s follows a p-dimesioal ormal distributio with mea zero ad variace (t s)i p, where I p is the idetity matrix of order p. Furthermore, as i Defiitio A.4.4, a p-dimesioal F t Browia motio is a process W with values i R p adapted to (F t ) such that for ay t, the distributio of s W t+s W t is a p-dimesioal Browia motio idepedet of F t. The differece betwee a plai p-dimesioal Browia motio ad a p-dimesioal F t Browia motio is that the p-dimesioal F t Browia motio possesses a certai regular relatioship with the filtratio. The followig basic result shows that the martigales associated with ordiary Browia motios reoccur whe cosiderig F t Browia motios. Theorem Let W be a p-dimesioal F t Browia motio. For i p, W i ad (Wt i ) 2 t are martigales, where W i deotes the i th coordiate of W. For i, j p with i j, Wt i W j t is a martigale. Proof. Let i p ad let s t. W i is the a F t Browia motio, so W i t W i s is ormally distributed with mea zero ad variace t s ad idepedet of F s. Therefore, we obtai E(W i t F s ) = E(W i t W i s F s ) + W i s = E(W i t W i s) + W i s = W i s, provig that W i is a martigale. Furthermore, we fid E((W i t ) 2 t F s ) = E((W i t W i s) 2 (W i s) 2 + 2W i sw i t F s ) t = E((W i t W i s) 2 F s ) (W i s) 2 + 2W i se(w i t F s ) t = (W i s) 2 s, so (W i t ) 2 t is a martigale. Next, let i, j p with i j. We the obtai that for s t,

19 1.2 Cotiuous-time martigales 11 usig idepedece ad the martigale property, E(W i t W j t F s ) = E(W i t W j t W i sw j s F s ) + W i sw j s = E(W i t W j t W i t W j s + W i t W j s W i sw j s F s ) + W i sw j s = E(W i t (W j t W j s ) W j s (W i t W i s) F s ) + W i sw j s = E(W i t (W j t W j s ) F s ) + W i sw j s = E((W i t W i s)(w j t W j s ) F s ) + E(W i s(w j t W j s ) F s ) + W i sw j s = W i sw j s, where we have used that the variables E(W j s (W i t W i s) F s ), E((W i t W i s)(w j t W j s ) F s ) ad E(W i s(w j t W j s ) F s ) are all equal to zero, because t W t+s W s is idepedet of F s ad has the distributio of a p-dimesioal Browia motio. Thus, W i W j is a martigale. Furthermore, whe W is a p-dimesioal F t Browia motio, W i has the distributio of a Browia motio, so all ordiary distributioal results for Browia motios trasfer verbatim to F t Browia motios, for example that the followig results hold almost surely: lim sup t W i t 2t log log t = 1, lim if t Wt i Wt i = 1, lim 2t log log t t t =. After itroducig this cetral example, we will i the remaider of this sectio wor o trasferrig the results of discrete-time martigale theory to cotiuous-time martigale theory. The mai lemma for doig so is the followig. Lemma Let M be a cotious-time martigale, supermartigale or submartigale, ad let (t ) be a icreasig sequece i R +. The (F t ) 1 is a discrete-time filtratio, ad the process (M t ) 1 is a discrete-time martigale, supermartigale or submartigale, respectively, with respect to the filtratio (F t ) 1. Proof. This follows immediately from the defiitio of cotiuous-time ad discrete-time martigales, supermartigales ad submartigales. Lemma (Doob s upcrossig lemma). Let Z be a cotiuous supermartigale bouded i L 1. Defie U(Z, a, b) = sup{ s 1 < t 1 < s < t : Z s < a, Z t > b, } for ay a, b R with a < b. We refer to U(Z, a, b) as the umber of upcrossigs from a to b by Z. The U(Z, a, b) is F measurable ad it holds that EU(Z, a, b) a + sup t E Z t. b a

20 12 Cotiuous-time stochastic processes Proof. We will prove the result by reducig to the case of upcrossigs relative to a coutable umber of timepoits ad applyig Lemma ad the discrete-time upcrossig result of Lemma A.3.1. For ay D R, we defie U(Z, a, b, D) = sup{m s 1 < t 1 < s m < t m : s i, t i D, Z si < a, Z ti > b, i m}, ad we refer to U(Z, a, b, D) as the umber of upcrossigs from a to b at the timepoits i D. Defie D + = {2, 1}, we refer to D + as the dyadic oegative ratioals. It holds that D + is dese i R +. Now as Z is cotiuous, we fid that for ay fiite sequece s 1 < t 1 < s m < t m such that s i, t i R + with Z si < a ad Z ti > b for i m, there exists p 1 < q 1 < p m < q m such that p i, q i D + with Z pi < a ad Z qi > b for i m. Therefore, U(Z, a, b) = U(Z, a, b, D + ). I other words, it suffices to cosider upcrossigs at dyadic oegative ratioal timepoits. I order to use this to prove that U(Z, a, b) is F measurable, ote that for ay m 1, we have ( s 1 < t 1 < s m < t m : s i, t i D +, Z si < a, Z ti > b, i m) = {(Z si < a, Z ti > b for all i m) s 1 < t 1 < s m < t m : s i, t i D + }, which is i F, as (Z si < a, Z ti > b for all i m) is F measurable, ad all subsets of =1D + are coutable. Here, D + deotes the -fold product of D +. From these observatios, we coclude that the set ( s 1 < t 1 < s m < t m : s i, t i D +, Z si < a, Z ti > b, i m) is F measurable. Deote this set by A m, we the have U(Z, a, b)(ω) = sup{m N ω A m }, so that i particular (U(Z, a, b) m) = =m A F ad so U(Z, a, b) is F measurable. It remais to prove the boud for the mea of U(Z, a, b). Puttig t = 2 ad defiig D = {t }, we obtai D + = =1D. We the have sup{m s 1 < t 1 < s m < t m : s i, t i D +, Z si < a, Z ti > b, i m} = sup =1{m s 1 < t 1 < s m < t m : s i, t i D, Z si < a, Z ti > b, i m} = sup sup{m s 1 < t 1 < s m < t m : s i, t i D, Z si < a, Z ti > b, i m}, so U(Z, a, b, D + ) = sup U(Z, a, b, D ). Now fix N. As (t ) is a icreasig sequece, Lemma shows that (Z t ) is a discrete-time supermartigale with respect to the filtratio (F t ). As (Z t ) t is bouded i L 1, so is (Z t ). Therefore, Lemma A.3.1 yields EU(Z, a, b, D ) a + sup E Z t b a a + sup t E Z t. b a As (D ) is icreasig, U(Z, a, b, D ) is icreasig, so the mootoe covergece theorem ad

21 1.2 Cotiuous-time martigales 13 our previous results yield EU(Z, a, b) = EU(Z, a, b, D + ) = E sup U(Z, a, b, D ) This cocludes the proof of the lemma. = E lim U(Z, a, b, D ) = lim EU(Z, a, b, D ) a + sup t E Z t. b a Theorem (Doob s supermartigale covergece theorem). Let Z be a cotiuous supermartigale. If Z is bouded i L 1, the Z is almost surely coverget to a itegrable limit. If Z is uiformly itegrable, the Z also coverges i L 1, ad the limit Z satisfies that for all t, E(Z F t ) Z t almost surely. If Z is a martigale, the iequality may be exchaged with a equality. Proof. Assume that Z is bouded i L 1. Fix a, b Q with a < b. By Lemma 1.2.3, the umber of upcrossigs from a to b made by Z has fiite expectatio, i particular it is almost surely fiite. As Q is coutable, we coclude that it almost surely holds that the umber of upcrossigs from a to b made by Z is fiite for ay a, b Q. Therefore, Lemma A.1.16 shows that Z is almost surely coverget to a limit i [, ]. Usig Fatou s lemma, we obtai E Z = E lim if t Z t lim if t E Z t sup t E Z t, which is fiite, so we coclude that the limit Z is itegrable. Assume ext that Z is uiformly itegrable. I particular, Z is bouded i L 1, so Z t coverges almost surely to some variable Z. The Z t also coverges i probability, so Lemma A.2.5 shows that Z t coverges to Z i L 1. We the fid that for ay t that, usig Jese s iequality, E E(Z F t ) E(Z s F t ) E Z Z s, so E(Z s F t ) teds to E(Z F t ) i L 1 as s teds to ifiity, ad we get E(Z F t ) = lim s E(Z s F t ) Z t. This proves the results o supermartigales. I order to obtai the results for the martigale case, ext assume that Z is a cotiuous submartigale bouded i L 1. The Z is a cotiuous supermartigale bouded i L 1. From what we already have proved, Z is almost surely coverget to a fiite limit, yieldig that Z is almost surely coverget to a fiite limit. If Z is uiformly itegrable, so is Z, ad so we obtai covergece i L 1 as well for Z ad therefore also for Z. Also, we have E( Z F t ) Z t, so E(Z F t ) Z t. Fially, assume that Z is a martigale. The Z is both a supermartigale ad a submartigale, ad the result follows. Theorem (Uiformly itegrable martigale covergece theorem). Let M cm. The followig are equivalet:

22 14 Cotiuous-time stochastic processes 1. M is uiformly itegrable. 2. M is coverget almost surely ad i L There is some itegrable variable ξ such that M t = E(ξ F t ) almost surely for t. I the affirmative, with M deotig the limit of M t almost surely ad i L 1, we have for all t that M t = E(M F t ) almost surely, ad M = E(ξ F ), where F = σ( t F t ). Proof. We show that (1) implies (2), that (2) implies (3) ad that (3) implies (1). Proof that (1) implies (2). Assume that M is uiformly itegrable. By Lemma A.2.3, M is bouded i L 1, ad Theorem shows that M coverges almost surely. I particular, M is coverget i probability, ad so Lemma A.2.5 allows us to coclude that M is coverget i L 1. Proof that (2) implies (3). Assume ow that M is coverget almost surely ad i L 1. Let M be the limit. Fix F F s for some s. As M t coverges to M i L 1, 1 F M t coverges to 1 F M i L 1 as well, ad we the obtai E1 F M = lim t E1 F M t = lim t E1 F E(M t F s ) = E1 F M s, provig that E(M F s ) = M s almost surely for ay s. Proof that (3) implies (1). Fially, assume that there is some itegrable variable ξ such that M t = E(ξ F t ). By Lemma A.2.6, M is uiformly itegrable. It remais to prove that i the affirmative, with M deotig the limit, it holds that for all t, M t = E(M F t ) almost surely, ad M = E(ξ F ). By what was already show, i the affirmative case, M t = E(M F t ). We thus have E(M F t ) = E(ξ F t ) almost surely for all t. I particular, for ay F t F t, we have EM 1 F = EE(ξ F )1 F. Now let D = {F F EM 1 F = EE(ξ F )1 F }. We the have that D is a Dyi class cotaiig t F t, ad t F t is a geeratig class for F, stable uder itersectios. Therefore, Lemma A.1.19 shows that F D, so that EM 1 F = EE(ξ F )1 F for all F F. Sice M is F measurable as the almost sure limit of F measurable variables, this implies M = E(ξ F ) almost surely, provig the result. Lemma If Z is a cotiuous martigale, supermartigale or submartigale, ad c, the the stopped process Z c is also a cotiuous martigale, supermartigale or

23 1.2 Cotiuous-time martigales 15 submartigale, respectively. Z c is always coverget almost surely ad i L 1 to Z c. I the martigale case, Z c is a uiformly itegrable martigale. Proof. Fix c. Z c is adapted ad cotiuous. Let s t ad cosider the supermartigale case. If c s, we also have c t ad the adaptedess of Z allows us to coclude E(Z c t F s ) = E(Z t c F s ) = E(Z c F s ) = Z c = Z c s. If istead c s, the supermartigale property yields E(Z c t F s ) = E(Z t c F s c ) Z s c = Z c s. This shows that Z c is a supermartigale. From this, it follows that the submartigale ad martigale properties are preserved by stoppig at c as well. Also, as Z c is costat from a determiistic poit owards, Z c coverges almost surely ad i L 1 to Z c. If Z is a martigale, Theorem shows that Z is uiformly itegrable. Theorem (Optioal samplig theorem). Let Z be a cotiuous supermartigale, ad let S ad T be two stoppig times with S T. If Z is uiformly itegrable, the Z is almost surely coverget, Z S ad Z T are itegrable, ad E(Z T F S ) Z S. If Z is oegative, the Z is almost surely coverget as well ad E(Z T F S ) Z S. If istead S ad T are bouded, E(Z T F S ) Z S holds as well, where Z S ad Z T are itegrable. Fially, if Z is a martigale i the uiformly itegrable case or the case of bouded stoppig times, the iequality may be exchaged with a equality. Proof. Assume that Z is a supermartigale which is coverget almost surely ad i L 1, ad let S T be two stoppig times. We will prove E(Z T F S ) Z S i this case ad obtai the other cases from this. To prove the result i this case, we will use a discretizatio procedure alog with Lemma ad Theorem A.3.5 to obtai the result. First, defie a mappig S by puttig S = wheever S =, ad S = 2 whe ( 1)2 S < 2. We the fid (S t) = =(S = 2 ) (2 t) = =(( 1)2 S < 2 ) (2 t), which is i F t, as (()2 S < 2 ) is i F t whe 2 t. Therefore, S is a stoppig time. Furthermore, we have S S with S covergig dowwards to S, i the sese that S is decreasig ad coverges to S. We defie (T ) aalogously, such that (T ) is a sequece of stoppig times covergig dowwards to T, ad (T = 2 ) = (( 1)2 T < 2 ). We the obtai that (T = 2 ) = (( 1)2 T < 2 ) (S < 2 ) = (S 2 ), from which we coclude S T.

24 16 Cotiuous-time stochastic processes We would lie to apply the discrete-time optioal samplig theorem to the stoppig times S ad T. To this ed, first ote that with t = 2, we obtai that by Lemma 1.2.2, (Z t ) is a discrete-time supermartigale with respect to the filtratio (F t ). As Z is coverget almost surely ad i L 1, so is (Z t ), ad the Lemma A.2.5 shows that (Z t ) is uiformly itegrable. Therefore, (Z t ) satisfies the requiremets i Theorem A.3.5. Furthermore, it holds that Z t coverges to Z. Puttig K = S 2, K taes its values i N { } ad (K ) = (S 2 ) F t, so K is a discrete-time stoppig time with respect to (Ft ). As regards the discrete-time stoppig time σ-algebra, we have F t K = {F F F (K ) F t for all } = {F F F (S t ) F t for all } = {F F F (S t) F t for all t } = F S, where F t K deotes the stoppig time σ-algebra of the discrete filtratio (F t ). Puttig L = T 2, we fid aalogous results for the sequece (L ). Also, sice S T, we obtai K L. Therefore, we may ow apply Theorem A.3.5 with the uiformly itegrable discrete-time supermartigale (Z t ) to coclude that Z S ad Z T are itegrable ad that E(Z T F S ) = E(Z t L F t K ) Z t K = Z S. Next, we show that Z T coverges almost surely ad i L 1 to Z T. This will i particular show that Z T is itegrable. As before, (Z t +1) is a discrete-time supermartigale satisfyig the requiremets i Theorem A.3.5. Also, (2L ) = (2T 2 ) = (T 2 (+1) ), which is i F t +1, so 2L is a discrete-time stoppig time with respect to (F t +1), ad L +1 = T T 2 +1 = 2L. Therefore, applyig Theorem A.3.5 to the stoppig times 2L ad L +1, we obtai E(Z T F T+1 ) = E(Z t +1 F 2L t +1 ) Z L t +1 = Z T+1. Iteratig this relatioship, we fid that for, Z T E(Z T F T ). Thus, (Z T ) is a +1 L +1 bacwards submartigale with respect to (F T ). Therefore, ( Z T ) is a bacwards supermartigale. Furthermore, as Z T E(Z T F T ) for, we have EZ T EZ T1, so E( Z T ) E( Z T1 ). This shows that sup 1 E( Z T ) is fiite, ad so we may apply Theorem A.3.6 to coclude that ( Z T ), ad therefore (Z T ), coverges almost surely ad i L 1. By cotiuity, we ow that Z T also coverges almost surely to Z T. By uiqueess of limits, the covergece is i L 1 as well, which i particular implies that Z T is itegrable. Aalogously, Z S coverges to Z S almost surely ad i L 1. Now fix F F S. As S S, we have F S F S. Usig the covergece of Z T to Z T ad Z S to Z S i L 1, we fid that 1 F Z T coverges to 1 F Z T ad 1 F Z S coverges to 1 F Z S i L 1, so that E1 F Z T = lim E1 F Z T = lim E1 F E(Z T F S ) lim E1 F Z S = E1 F Z S, ad therefore, we coclude E(Z T F S ) Z S, as desired.

25 1.2 Cotiuous-time martigales 17 This proves that the optioal samplig result holds i the case where Z is a supermartigale which is coverget almost surely ad i L 1 ad S T are two stoppig times. We will ow obtai the remaiig cases from this case. If Z is a uiformly itegrable supermartigale, it is i particular coverget almost surely ad i L 1, so we fid that the result holds i this case as well. Next, cosider the case where we merely assume that Z is a supermartigale ad that S T are bouded stoppig times. Lettig c be a boud for S ad T, Lemma shows that Z c is a supermartigale, ad it is coverget almost surely ad i L 1. Therefore, as Z T = Z c T, we fid that Z T is itegrable ad that E(Z T F S ) = E(Z c T F S) Z c S = Z S, provig the result i this case as well. Fially, cosider the case where Z is oegative ad S T are ay two stoppig times. We the fid that E Z t = EZ t EZ, so Z is bouded i L 1. Therefore, Theorem shows that Z is almost surely coverget ad so Z T is well-defied. From what we already have show, Z T is itegrable ad E(Z T F S ) Z S. For ay F F S, we fid F (S ) F S for ay by Lemma Therefore, we obtai E1 F Z T = E1 F 1 (S ) Z T + E1 F 1 (S>) Z T E1 F 1 (S ) Z S + E1 F 1 (S>) Z S = E1 F Z S, ad so, by Lemma A.1.14, E(Z T F S ) Z S. Applyig Fatou s lemma for coditioal expectatios, we obtai E(Z T F S ) = E(lim if Z T F S ) lim if E(Z T F S ) lim if Z S = Z S, as was to be show. We have ow proved all of the supermartigale statemets i the theorem. The martigale results follow immediately from the fact that a martigale is both a supermartigale ad a submartigale. Lemma Let T be a stoppig time. If Z is a supermartigale, the Z T is a supermartigale as well. I particular, if M cm, the M T cm as well, ad if M cm u, the M T cm u as well. Proof. Let a supermartigale Z be give, ad let T be some stoppig time. Fix two timepoits s t, we eed to prove E(Z T t F s ) Z T s almost surely, ad to this ed, it suffices to show that E1 F Z T t E1 F Z T s for ay F F s. Let F F s be give. By Lemma , F (s T ) is F s T measurable, ad so Theorem applied with the two bouded stoppig

26 18 Cotiuous-time stochastic processes times T s ad T t yields E1 F Z T t = E1 F (s T ) Z T t + E1 F (s>t ) Z T t E1 F (s T ) Z T s + E1 F (s>t ) Z T t = E1 F (s T ) Z T s + E1 F (s>t ) Z T s = E1 F Z T s. Thus, E(Zt T F s ) Zs T ad so Z T is a supermartigale. From this it follows i particular that if M cm, it holds that M T cm as well. Ad if M cm u, we fid that M T cm from what was already show. The, by Theorem 1.2.5, Mt T = E(M F T t ), so by Lemma A.2.6, M T is uiformly itegrable, ad so M T cm u. We ed the sectio with two extraordiarily useful results, first a criterio for determiig whe a process is a martigale or a uiformly itegrable martigale, ad secodly a result showig that a particular class of cotiuous martigales cosists of the zero process oly. Lemma (Komatsu s lemma). Let M be a cotiuous adapted process with iitial value zero. It holds that M cm if ad oly if M T is itegrable with EM T = for ay bouded stoppig time T. If the limit lim t M t exists almost surely, it holds that M cm u if ad oly if M T is itegrable with EM T = for ay stoppig time T. Proof. We first cosider the case where we assume that the limit lim t M t exists almost surely. By Theorem 1.2.7, we have that if M cm u, M T is itegrable ad EM T = for ay for ay stoppig time T. Coversely, assume that M T is itegrable ad EM T = for ay stoppig time T. We will prove that M t = E(M F t ) for ay t. To this ed, let F F t ad ote that by Lemma 1.1.7, t F is a stoppig time, where t F = t1 F + 1 F c, taig oly the values t ad ifiity. We obtai EM tf = E1 F M t + E1 F cm, ad we also have EM = E1 F M + E1 F cm. By our assumptios, both of these are zero, ad so E1 F M t = E1 F M. As M t is F t measurable by assumptio, this proves M t = E(M F t ). From this, we see that M is i cm, ad by Theorem 1.2.5, M is i cm u. Cosider ext the case where we merely assume that M is a cotiuous process with iitial value zero. If M cm, Theorem shows that M T is itegrable with EM T = for ay bouded stoppig time. Assume istead that M T is itegrable ad EM T = for ay bouded stoppig time T. From what we already have show, we the fid that M t is i cm u for ay t ad therefore, M cm. For the statemet of the fial result, we say that a process X is of fiite variatio if it has

27 1.2 Cotiuous-time martigales 19 sample paths which are fuctios of fiite variatio, see Appedix A.1 for a review of the properties of fuctios of fiite variatio. If the process X has fiite variatio, we deote the variatio over [, t] by (V X ) t, such that (V X ) t = sup =1 X t X t, where the supremum is tae over partitios = t < < t = t of [, t]. Lemma Let X be adapted ad cotiuous with fiite variatio. The the variatio process V X is adapted ad cotiuous as well. Proof. By Lemma A.1.4, V X is cotiuous. As for provig that V X is adapted, ote that from Lemma A.1.9, we have (V X ) t = sup =1 X q X q, where the supremum is tae over partitios of [, t] with elemets i Q + {t}. As =1(Q + {t}) is coutable, there are oly coutably may such partitios, ad so we fid that (V X ) t is F t measurable, sice X q is F t measurable wheever q t. Therefore, V X is adapted. Lemma Let X be adapted ad cotiuous with fiite variatio. The (V X ) T = V X T. Proof. Fix ω Ω. With the supremum beig over all partitios of [, T (ω) t], we have (V X ) T t (ω) = (V X ) T (ω) t (ω) = sup X t (ω) X t (ω) = sup =1 Xt T (ω) Xt T (ω) = (V X T ) t T (ω) (ω) (V X T ) t (ω). =1 Coversely, with the supremum beig over all partitios of [, t], we also have (V X T ) t (ω) = sup Xt T (ω) Xt T (ω) = sup =1 X t T (ω)(ω) X t T (ω)(ω) (V X ) t T (ω) (ω). =1 Combiig our coclusios, the result follows. Lemma If M cm with paths of fiite variatio, the M is evaescet. Proof. We first cosider the case where M cm b ad the variatio process V M is bouded, (V M ) t beig the variatio of M over [, t]. Fix t ad let t = t2. Now ote that by the martigale property, EM t (M t M t ) = EM t E(M t M t F t ) =, ad

28 2 Cotiuous-time stochastic processes by rearragemet, M 2 t M 2 t = 2M t (M t M t ) + (M t M t )2. Therefore, we obtai EM 2 t = E = E 2 =1 2 =1 (Mt 2 M 2 t ) = 2E 2 M t (M t M t ) + E =1 (M t M t ) 2 E(V M ) t max M t M 2 t. 2 =1 (M t M t ) 2 Now, as M is cotiuous, (V M ) t max M t M t teds poitwisely to zero as teds to ifiity. The boudedess of M ad V M the allows us to apply the domiated covergece theorem ad obtai EM 2 t lim E(V M ) t max 2 M t M t E lim (V M ) t max 2 M t M t =, so that M t is almost surely zero by Lemma A.1.15, ad so by Lemma 1.1.5, M is evaescet. I the case of a geeral M cm, defie T = if{t (V M ) t > }. By Lemma , (T ) is a sequece of stoppig times icreasig almost surely to ifiity, ad (V M ) T is bouded by. By Lemma A.1.6 ad Lemma , Mt T (V M T ) t = (V M ) T t for all t. Therefore, M T is a bouded martigale with bouded variatio, so our previous results show that M T is evaescet. Lettig ted to ifiity, T teds to ifiity, ad so we almost surely obtai M t = lim Mt T =, allowig us to coclude by Lemma that M is evaescet. 1.3 Square-itegrable martigales I this sectio, we cosider the properties of square-itegrable martigales, ad we apply these properties to prove the existece of the quadratic variatio process for bouded cotiuous martigales. We say that a cotiuous martigale M is square-itegrable if sup t EMt 2 is fiite. The space of cotiuous square-itegrable martigales with iitial value zero is deoted by cm 2. We ote that cm 2 is a vector space. For ay M cm 2, we put Mt = sup s t M s ad M = sup t M t. We use the otatioal covetio that M 2 t = (M t ) 2, ad liewise M 2 = (M ) 2. Theorem Let M cm 2. The, there exists a square-itegrable variable M such that M t = E(M F t ) for all t. Furthermore, M t coverges to M almost surely ad i L 2, ad EM 2 4EM 2.

29 1.3 Square-itegrable martigales 21 Proof. As M is bouded i L 2, M is i particular uiformly itegrable by Lemma A.2.4, so by Theorem 1.2.5, M t coverges almost surely ad i L 1 to some variable M, which is itegrable ad satisfies that M t = E(M F t ) almost surely for t. It remais to prove that M is square-itegrable, that we have covergece i L 2 ad that EM 2 4EM 2 holds. Put t = 2 for,. The (M t ) is a discrete-time martigale for with sup EM 2 t fiite. By Lemma A.3.4, M t coverges almost surely ad i L2 to some square-itegrable limit as teds to ifiity. By uiqueess of limits, the limit is M, so we coclude that M is square-itegrable. Lemma A.3.4 also yields E sup M 2 t 4EM 2. We the obtai by the mootoe covergece theorem ad the cotiuity of M that EM 2 = E lim sup Mt 2 = lim E sup Mt 2 4EM 2. This proves the iequality EM 2 4EM 2. It remais to show that M t coverges to M i L 2. To this ed, ote that as we have (M t M ) 2 (2M ) 2 = 4M 2, which is itegrable, the domiated covergece theorem yields lim t E(M t M ) 2 = E lim t (M t M ) 2 =, so M t also coverges i L 2 to M, as desired. Lemma Assume that M cm 2. The M T cm 2 as well. Proof. By Lemma 1.2.8, M T is a martigale. Furthermore, we have sup t E(M T ) 2 t E sup t (M T t ) 2 E sup Mt 2 t ad this is fiite by Theorem 1.3.1, provig that M T cm 2. = EM 2, Theorem Assume that (M ) is a sequece i cm 2 such that (M ) is coverget i L 2 to a limit M. The there is some M cm 2 such that for all t, M t = E(M F t ). Furthermore, E sup t (Mt M t ) 2 teds to zero. Proof. The difficulty i the proof lies i demostratig that the martigale M obtaied by puttig M t = E(M F t ) has a cotiuous versio. First ote that M M m cm 2 for all ad m, so for ay δ > we may apply Chebyshev s iequality ad Theorem to obtai, usig (x + y) 2 2x 2 + 2y 2, P ((M M m ) δ) δ 2 E(M M m ) 2 4δ 2 E(M M m ) 2 8δ 2 (E(M M ) 2 + E(M M m ) 2 ) 16δ 2 sup E(M M ) 2. m