fi'. THESIS i frco Ill/III i xiii/Wiiiiiii 3 1293 02048 9112 I!!! Iii/Iii LIBRARY Michigan State University This is to certify that the dissertation entitled On Purely Discontinuous Martingaies presented by Jan Hannig has been accepted towards fulfillment of the requirements for Ph.D. degreein Statistics _d MM M a jobrprofessoy Date May 20, 2000_ MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 , M _ PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE moo chlRCIDateDuopfib—p.“ On purely discontinuous martingales By Jan Hannig A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 2000 Abstract ON PURELY DIS(_YONTINUOI_.TS MARTINGALES By J an Hannig Even though the general theory of stochastic processes is a rather well developed field, there is surprisingly little knowledge about the analytical properties of filtrations. In this dissertation we explore connections between purely discontinuous martingales and their filtrations. We are particularly interested in describing the conditions under which there are no non-constant continuous martingales adapted to our filtration. General martingale theory shows that every martingale can be decomposed into continuous and purely discontinuous parts. In the first part of this dissertation we give a necessary condition on a filtration .73 implied when the continuous part of the decomposition is 0 (1.8. for any .7, martingale. In the second part of the dissertation we give examples showing that our condition is not sufficient. We also prove various sufficient conditions. To my wife Marie iii ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor Professor Skorokhod for pa- tience and continuous guidance during the work on this dissertation. I also want to thank the co-chairman of my guidance committee Professor Salehi, who gave me a lot of helpful advice and encouragement during the years I spent at MSU. I really appre- ciate it. Similarly, I owe my sincere thanks to the other members of my committee Professor Huebner and Professor Gilliland. I would like to thank the department as a whole for giving me such a great opportunity and much support through the years. Especially, I want to mention Professor Hannan from whom I learned a lot about statistics, mathematics, and who introduced me to the pleasures of physical exercise, Professor Page who was the perfect director of the Statistical Consulting Service, and Professor Levental who taught me probability and mathematical finance. My thanks also goes to my good friend, Dr. White, who was extremely helpful in correcting my English in this work and Professor Stepan from Charles University, Prague, who first introduced me to the beauty of probability and guided my masters thesis. I also want to thank my wife who was very courageous to marry a mathematician and is very supportive of my work. I also owe a sincere thanks to my friends at Vineyard Christian Fellowship of Lansing, who became our family away from home. Finally I would like to thank to my LORD, Jesus Christ, who saved me and gave my life direction and purpose. iv Contents List of Figures List of Notation Introduction 0.1 Historical Remarks ............. 0.2 Theory of Stochastic Processes —— Overview 1 Main Theorem 2 Examples 3 Necessary and Sufficient Condition 4 Sufficient Conditions Bibliography vi vii 16 30 36 4O 57 List of Figures 1.1 2.1 4.1 Simulated sample path of a martingale with accumulation of jumps everywhere (2 different resolutions) .................... 28 Simulated times of jumps for Poisson processes with different A. . . . 32 Simulated sample paths of Brownian motion and the associated Azéma martingale. ................................ 49 vi List of Notation We will use standard notation introduced in many probability textbooks (e.g. [13], [34], [16]). The following is a list of some notation used in this dissertation: P(X = Y < 00) probability of the set {w E Q : X(w) = Y(w) < 00} V .73" the smallest filtration (resp. a-algebra) containing all f" /\ .7" the biggest filtration (resp. o-algebra) contained in all f" X;. 1gp X, for a cadlag process AX jump process (X; — Xt_) [7'] graph of the stopping time 7' a V b maximum of {a, b} a /\ b minimum of {a, b} X II Y X and Y are independent X 11;, Y X and Y are conditionally independent given H P’~P P’<>P P’ = p - P measure P’ defined by P’ (4) = fA de N set of all natural numbers BL [03 00) vii Introduction Even though the general theory of stochastic processes is a rather well developed field, there is surprisingly little knowledge about the analytical properties of filtrations. In this dissertation we explore connections between purely discontinuous martingales and their filtrations. We are particularly interested in describing conditions under which there are no non-constant continuous martingales adapted to our filtration. We assume throughout the whole dissertation that the filtrations satisfy the usual conditions (right continuity and completeness). When we define a filtration we always augment the filtration to the one satisfying the usual conditions without implicitly stating it. Similarly, all martingales are assumed to be in their ccidlcig version. In the rest of this chapter we explore the historical background and motivation of our problem. Then we give a list of important definitions, theorems, and notation used in this dissertation. Chapter 1 contains the main theorem of this dissertation. Under the assumption of quasi-left-continuity we prove that if a filtration ft is purely discontinuous (Le. any ft-adapted martingale is purely discontinuous), then ft=0{Afl{T§t}; 146.75}, TET}, where T is a countable collection of totally inaccessible stopping times that exhaust all possible jumps of martingales adapted to .75. The intuitive meaning of this theo- rem is that the information contained in our filtration came only from jumps of the martingales. The main difficulty we had to overcome in the proof of this theorem was related to the fact that we allowed infinitely many jumps on finite intervals. Chapter 2 contains several examples showing that the behavior of a purely dis- continuous filtration can be sometimes non-intuitive. In particular we show that the necessary condition proved in Chapter 1 is not sufficient. We also show that a subfil- tration of a purely discontinuous filtration does not have to be purely discontinuous. In Chapter 3 we prove that a filtration is purely discontinuous if and only if our probability measure is an extreme point in the set of all probabilities that preserve all compensators. The proof of this theorem is almost identical to the proof of the characterization theorem for the weak representation property. The main drawback of this condition is that it is usually very hard to verify. Finally, Chapter 4 contains several sufficient conditions for a filtration to be purely discontinuous. These conditions, derived as analytical properties of the filtration, are easier to verify than the condition of Chapter 3. We also give several examples of purely discontinuous filtrations, the most important of them being a filtration generated by a purely discontinuous Levy process. The last theorem of this chapter deals with discrete time sequences and gives a result on an equivalent change of the probability measure under which the sequence becomes quasi-Markov. We hope to apply this theorem to get another sufficient condition for purely discontinuous filtrations in the future. The main problem that still remains Open is to find a necessary and sufficient condition using the analytical properties of the filtration. 0. 1 Historical Remarks Martingales are one of the most important objects in the modern theory of probability. The term martingale (originally denoting part of horse’s harness and later used for a special gambling system) was introduced into the probability theory in the first half of 20th century as a natural generalization of sums of independent random variables, by Bernstein (1927, 1937), Lévy (1937), and Ville (1939). The basic regularity theorems for continuous time martingales first appear in a paper by Doob (1951). Increasing sequences of o-algebra have been already used by Doob around 1940. They were also used in the work of ltd (1944, 1946) and others. In their full generality, filtrations first appear in the famous book, Stochastic Processes, by Doob (1953). The idea of systematically extending to the stopping times results that are valid for fixed times was inspired by the strong Markov property, first mentioned by Doob (1942) in a paper on Markov chains. We also owe a lot of useful results on stopping times to the school of Dynkin. Systematic study of stopping times and their associated a-algebras was initiated by a paper of Chung and Doob (1965). Predictability is a very clear concept in discrete time but it is somewhat unin- tuitive in continuous time. Predictable stopping times appear implicitly in works of Blumethal and Hunt (1957—1958). The theory of predictable and optional a-algebras was developed into the modern theory of stochastic processes by Meyer (1966), Del- laeherie (1972) and others. Let us note that this theory was motivated by Markov processes. The importance of predictable o-algebra became clear after Doléans (1967) had proved the equivalence between natural and predictable increasing processes, thereby establishing the ultimate version of Doob-Meyer decomposition. A compensator of a stopping time can be defined as the predictable part of Doob- Meyer’s decomposition of the process M79}. Compensators for more general objects, such as random measures, were obtained by Jacod (1975). The compensators for random measures can be derived using the theory of dual predictable projections. In 1976, Ymurp proved the existence of an orthogonal decomposition of martin- gales into continuous and purely discontinuous parts. Meyer (1976) then proved that the purely discontinuous part is a sum of compensated jumps. The term, purely discontinuous martingale, is misleading. It does not refer to a piecewise constant process. In fact there is a purely discontinuous martingale that has non-constant continuous trajectories with non-zero probability. Rather the term, purely discon- tinuous martingale, denotes the “non-continuous” part of the orthogonal decompo- sition. Incidentally, any purely discontinuous martingale is orthogonal (in the sense of quadratic variation) to all continuous martingales. In particular, any martingale with locally bounded variation is purely discontinuous. The main object of my dissertation is to characterize filtrations for which the continuous part of the decomposition is constant almost surely. The problem was motivated by an effort to generalize a notion of jumping filtration to admit any purely discontinuous martingale. A filtration ft is called jumping if it is generated by an increasing step process. Equivalently, .7:¢=0{Afl{rk gt}; A677,, kEN}, where 7,, ——> 00 is an increasing sequence1 of stopping times (the times of jumps of the step process). The main feature of this filtration is that it is constant on [Tk,7’k+1). A well known example of a jumping filtration is the natural filtration of a Poisson process. Many interesting analytical properties of jumping filtrations were studied in a se- ries of papers by He and Wang in the early 19805. An excellent summary of the results on jumping filtrations can be found in their book, Semimartingales and Stochastic Calculus (1992). An important characterization that directly motivated my research is due to Jacod and Skorokhod (1994). They prove, in full generality, that a filtra- tion is jumping if and only if any adapted martingale is a process of locally bounded variation. Thus, jumping filtrations support no non-constant continuous martingales. The limiting feature of jumping filtrations is that they allow only for a finitely many jumps on finite intervals. It is well-known that there exist purely discontinu- ous martingales that have infinitely many jumps on finite intervals. We will call this phenomena accumulation of jumps. An example of such a martingale is a Gamma process (a particular purely discontinuous Levy process). The main theorem of this dissertation gives a necessary condition for a filtration to support only purely dis- continuous martingales. This condition is very similar to the definition of jumping filtration. We are able to accommodate all types of purely discontinuous processes 1It is possible that. P(‘r;c = 00) > 0. by allowing for the accumulation of jumps. Sufficient conditions for a filtration to be purely discontinuous (i.e. support no non—constant continuous martingale) are closely related to the weak predictable repre- sentation property for martingales. A process X has weak predictable representation property if any local martingale can be written in the form M=Mo+H-XC+W*,1, where X C is the continuous part of X, [i is the compensated random measure associ- ated with jumps of X, and the integrators are predictable. Hence if X c = 0 then any martingale is a stochastic integral with respect to the compensated jump measure it and therefore it is purely discontinuous. Random measures and the associated integrals were introduced by ltd (1951). They were studied by Skorokhod (1965) in the case when the compensator is deter- ministic. The general theory was developed by Jacod (1976). The notion of pre- dictable representation property is due to Jacod (1977), where he also proved that the process X has a weak predictable representation property if and only if the prob- ability measure P is an extreme point in a certain set of probability measures. We apply the techniques used to prove this theorem in Chapter 3. 0.2 Theory of Stochastic Processes — Overview The purpose of this section is to give a reader an overview of the main definitions and results that were used in this dissertation. 0.2.1 Filtrations, stopping times, martingales Let (9,]: , P) be a probability space. Unless stated otherwise, all objects will be defined on this probability space. A filtration (Rhem is a family of increasing sub- o-algebras of the o-algebra 7-". Denote foo = V,>0 .7}. A filtration is right-continuous if, for any t, E = flKs .73. A filtration is P- complete if .70 contains all P-null sets. We assume throughout this dissertation that the filtrations at hand are right-continuous and complete (called the usual conditions). A random variable T is a stopping time if {TS t} E f}, for alltZ 0. A stepping time T is predictable if there is an non-decreasing sequence of stopping times Tn —> T, such that Tn < T on the set {T > 0}. The sequence Tn is called an announcing sequence. Stopping time T is totally inaccessible if P(T = H < 00) = 0 for any predictable stopping time H. Let T be a stopping time. Define o—algebras fT:{AEfmiAfl{TSt}Eft}, fT_=0'{Afl{T>t}I/l€ft}. Clearly FT- C Pr, and o-algebra TT is, intuitively, the knowledge at time T. Simi- larly, f}- is the knowledge immediately preceding time T. It can be shown that if Tn I T then f} = /\ 7%”. Similarly, if T is a predictable stopping time and Tn is its announcing sequence then 7-}- = Van A filtration J: is quasi-left-continuous if .73}- = 7-} for any predictable stopping time T. A process M, is adapted if M, is .7, measurable. An adapted process is called a martingale if E [Mt ] .733] =-- M, for any 8 < t. Similarly, an adapted process is called a submartingale if E [Mt I .75] 2 Ms. Notice that the notion of martingale depends on the underlying filtration. It is possible to have a process that is adapted to two different filtrations, but is a martingale with respect to one and is not a martingale with respect to the other. However, the following is clearly true. Proposition 0.1. Let X be a martingale for the filtration th. Let Q be a subfiltra- tion of CE (i.e. Q C E), such that X is adapted to 9,. Then X is a martingale for Q:- A process M, is a local martingale if there is an increasing sequence of stopping times Tn T 00 such that Mum, is a martingale for each n. Such a sequence of stopping times is called localizing sequence. A local martingale does not have to be a martingale. Locally bounded variation, locally integrable, etc. are defined similarly. Proposition 0.2. Let Alt be a submartingale. Then IV, has a cadlag (right contin- uous, left limits) version if and only if the function t —> EX, is right continuous. For a proof see [34] Theorem 6.27. (Remember that the filtration is assumed to 8 satisfy the usual conditions.) We assume that all martingales are in their cadlag version. Proposition 0.3. If flit is a continuous local martingale of locally finite variation, then M = Julio (1.8. For a proof see [34] Proposition 15.2. Proposition 0.4. Filtration .7: is quasi-left-continuous if and only if AM, 2: 0 as. for any martingale [VI and any predictable stopping time T. For a proof see [34] Proposition 22.19. The last notion we introduce in this section is predictable o-algebra. Let P be a o-algebra in R, x 9 generated by all continuous adapted processes. The sets in P are called predictable sets, and the P measurable functions on R+ x Q are called predictable processes. Lemma 0.5. For any stopping time T and predictable process Xt, the random vari- able X,1{T IR such that E|f(C)|p < 00 for somep 21. Then n”l ZflOkC) —+ E[f(() [ Q] a.s. and in L1". k T(w),v E T}, where T is a stopping time, and At(w) : sup{T(w); T(w) < t,T E T}, 16 where t is a deterministic time. Note that S, is a stopping time, while At is not. The random variables 5, and A, will be often referred to as “first jump after T” and “last jump before t” respectively. This comes from the observation that if the set T is the set of all possible jumps of adapted martingales then any 5‘] martingale does not have jumps on the interval (T, 5,) and has at least one jump on [ST, S, +5) for all e > 0. An analogous statement is true for At. Now we can state the main theorem of this dissertation. Theorem 1.1. Let E be a purely discontinuous filtration. Then ft=0{AO{TSt}; A613, 7'67"}. (1-1) where T is a countable collection of totally inaccessible stopping times with disjoint graphs. The intuitive meaning of this theorem is that the information contained in our filtration came only from jumps of the martingales. To prove it we will need the following lemma. Lemma 1.2. The following is true under the assumptions of Theorem 1.1: Let H S S be two stopping times. If any ft martingale is continuous on the interval (H, S) then .7, : TH on {H S t < S} (i.e. for every A E 3 there is A’ 6 IF” such that An{Hgt t} and the Ng‘ = 0 on the set {S g t}. Hence M, =/ Hud(X -— Y), (1.4) o where H,, = —l\l,f_1{,_>_,,} is a bounded predictable process. Formula (1.4) then implies: sAt N: = N3 +/ N,,_d}';, as if H g t < s. O 18 Now observe that Y, = 0 for s g H and therefore N? = N;}£(Y),M if H S t < S, where 8 (y) denotes the Doléans" exponential of Y (see Theorem 0.15). Since A was an arbitrary set we get NSQ : N§8(Y),M if H S t < S. This implies NglNQ = MON; as. on {H g t < S}. (1.5) Define an TH measurable set A’ = {N2 2 Ni} > 0}. The relation (1.5), NA = 1A0{Hgt T:_1;[All’[[ 6 (TI, fi]}. Set TM 2 {T,:‘; 7",? 75 00 as}. The definition of 7']? assures that if T, v 6 TM are two different stopping times, then the set {T = v < oo} = 0. Since the filtration is quasi-left-continuous, the stopping times T: are totally in- accessible. It is known that (see Theorem 0.9) M = Z Airings, — C,,, (1.6) T 6 TM where the right-hand-side of (1.6) is a compensated sum of jumps and the compen- sators Cm are continuous functions of finite variation. 19 Since foo is countably generated, we can find a countable set of square integrable martingales {77"} such that (7771377111) : 0 and each M E .M“2 starting at 0 can be written as M 2 Zn f Vndnn (see Theorem 0.11). We define T = U" T n. Because T is countable, we can order the elements of T to form a sequence {Tn}. The jumping process N, 2 Zn 2_"1{,n5,} is bounded. Therefore the martingale AI, = N, —— C, is square integrable and the compensator C, is continuous. Thus [AM] 2 [AN] = UTEflT]. Define T 2 TM. Clearly T contains stopping times with mutually disjoint graphs and UT,Ef[T'] = UT€T[7']. Let Q, : o{A F) {T g t};A E .7:T,T E T}. It is easy to see that Q C .7. To prove Theorem 1.1 we need to prove {7,} = {9,}. To begin with we prove that for any totally inaccessible f, stopping time v the filtrations g, = .73, on the set {v < 00}. If v E T, the assertion follows from the definition. Let us assume that v ¢ T. The f, martingale X, = 1{vSt} — C, is square integrable, hence X : E: f Vndnn as. It follows that AX = Z V}, - Ann a.s.. From this we get [v] = U{T6T}{T = v} 0 [T]. Thus for any finite t 6 IR, and for any A E .73., An{vgt}= U An{T=v}n{Tgt}eg,. {TET} It follows that v is a stopping time with respect to g, and Q, = .7-"v on the set {v < 00}. This restriction arises from the definition of go, 2 V9, 2 o{A F) {T < OO};A E f,,T E T}. The following simple observations are valid for any sequence of stopping times 20 {Tn}. If f," = g," on {Tn < 00} and Tn I T, then r. = /\f.,, = /\g.,, = g. 0“ UV" < 00} I {T < ool‘ Similarly if Tn T T, then .75, : VT," 2V9,” :9, on fl{Tn < oo} = {T < 00}. The latter statement is true because the filtrations are quasi-left-continuous. Recall that in the Definition 1.2 we have defined: S, = inf{v{v>,}; v E T}, and A,(w) = sup{T(w); T(w) < t,T E T}. The random variable S, is a stopping time, while A, is not. However A, is a g, measurable random variable. This follows from the relation: {At S3}:{Ss 2t}€g,. Since T is a countable set, it follows from the previous statements that f5, = 9,, on the set {5, < 00}. Similarly, if C is a predictable stopping time and the set {C = A, g t} has a non-zero probability, .7} = Q, on the set containing {C = 44t}~ 21 To prove the latter note that .70 = g, is a trivial o-algebra. We assume without loss of generality that C > 0 as, and C, < C is a sequence announcing C. It follows from the definition of A, that S(n(w) < A,(w) g t on the set {C = A,}. The sequence S,“ is nondecreasing, so we can define S = lim 8,". Noticing that S is a g, stopping time we deduce that .735 = g, on the set {S < 00}. The statement is implied by the fact that {C = A,} C {C = S < 00}. To finish the proof it will be enough to prove f}, = 9,0 for a fixed to. I will do it separately on three different 9,, measurable sets. Following for a moment the proof of Proposition 22.4 in [34] let p = supP (LJ{A,0 = T: < oo}) , n where the supremum extends over all possible sequences of predictable stopping times. Combining sequences such that the probability on the right-hand-side approaches p we construct a sequence of predictable stopping times for which the supremum is attained. Let {T,";’} be this sequence. (Note that if p = 0 this sequence is empty.) Define the following sets: 81 :{14t0 I: to} U {Ste 2 f0} 32 = (UiAto = 5} U U {Ato = Tnal) \B, n TMET 8329\(BIUBQ) These sets are 9,0 measurable. This is clearly true for B,. The set 82 is 9,0 measurable 22 because all the stopping times involved in its definition can be taken to be 9, stopping times. It follows from the previous discussion that .30 = g,, on the set 3,. Denote the sequence of stopping times that was involved in definition of B2 by {14,}. We have established, that we can choose this sequence so that P(I/n = A,0 < 00) > 0, and .72," = 9,, on the set {14, < 00} D {14, 2 Am}. (The statement is true for both totally inaccessible and predictable stopping times.) As mentioned before no .7, martingale has jumps between times on and Sun for any fixed n. It follows from Lemma 1.2 that if B E .7}, there is B’ 6 7'), = G," such that Bfl{1/,,St 3} D {A > s} = {T /\ A > 8}. Since {A > s} is an atom in ’H,, there is D, E .7, such that C, = D, H {A > s}. We define Ds: U n DQ'Z' (11>3 02391 0160 02912 The right-continuity of the filtrations involved gives D, E 7,, and the definition of C, gives C, = D, 0 {-1 > 3}. Define T’(w) = sup{t : w E D,}. It is a .7, stopping time and T = T’ on the set {T < A}. The fact that A is a totally inaccessible .7, stopping time follows directly. In a similar way we prove that for any T E T, .7 _ = .7,_. Namely, let t < to and B e 73,, Bfl{T>t}=(Bfl{t t}. As a next step we want to prove that 7~ = 9:; on the set {A < 00}. For any B E 7,, T E T Bfl{T t}, B e 13",} :a{Bfl{A>T>t}, reT,B€7t} =o{Dfl{A>t}, DEfT—} C9,; ~ Since on the same set fA— = 3,;_ C 9‘, C 74, it is enough to prove 7,;_ = 7;. To do it we will use a rather unusual feature of our enlargement. Notice {A33}:{Agt}fl{A,Ss} fors§tz}a (1-8) 25 where El21{.-igt}lftl Z _ Z _ ma— P[A SM]. and 52v)— ElZI{.§>1}lftl P[A > tlf,] ' Observe that the process E[Z1{‘,,<,}|.7,] is a submartingale and the function t —> E[Z1{,,S,}] is continuous, hence there is a cadlag modification of E[Z1{AS,}I7,]. Thus we can assume without loss of generality that the processes {,2 (t) and {22 (t) are .7, adapted cadlag processes. This immediately implies that if, = out 2 e 2) = out 01—). z e 21c 1%... since any .7, adapted cadlag process is as. continuous at the time A. Lemma 1.2 implies that .7, is constant on any interval [8, S,), e.g. for any t > s and B E .7, we have 8’ E .7, such that B H {S, > t} = B’ F) {S, > t}. A similar statement is true for o-algebra ’H, and consequently2 for 7,. To finish the proof we will closely follow the proof that appears in section 2 (page 22) of [29]. Let M, be any uniformly integrable .7, martingale such that M, is 0 on [0,/I] and constant on [t0,oo). To prove that M, is 0 on [0,00), we define M,” 2 MMS, - M,,(,. Note that {S, < to} C {S, < A}, and therefore the martingale M," E 0 on the set {S, < to}, so M; = Mflgosga}. The statement established in the previous paragraph implies that for any t > 3 there is a .7, measurable random variable N, such that N, 2 M: on the set {3 S t < S,}. Call G a regular version of the law of the pair (5,, Mg) conditional on 7,, and C”(t) = C((t, 00] x R0 [to, 00] x R)). 2Using (1.8) we can actually prove that the filtration .7, is quasi-left-continuous. 26 Ift 2 s, we have the following string of as. equalities (see [29] for justification): ~ NtG"(t) = Eli\711{1<8.}1{tosssl ljil = E[.-lI,31{,,Sg,}1{,,}G(du,dx). (1.9) The functions G”(t) and f x 1{u>,}C(du, dx) (taken as a function oft) are as. constant on the interval [0,t0). The fact that the second function is constant follows from C((t,t0) x (R \ {0})) = 0 as. To conclude that M: = 0 as. on the interval [0,t0] notice that the set {0”(0) = 0} is .7, measurable, and more important {C”(0) = 0} C {S, < to}. (The continuity of M" at the point to is implied by 7,- D 7,_ = 7, = 7,.) Since 3 was arbitrary, we get M, = 0 for t E [0,t0]. From here we finally obtain ~ fro = 7;, = Q; C 9,, on the set {A < oo} : B3, Cl The following two examples demonstrate that it is indeed possible to have P(Bl) or P(B3) bigger then 0. The filtration defined in each example is purely discontinuous. We postpone the proof thereof untill the last chapter. (Example 1.1 is covered by Corollary 4.3 and Example 1.2 is covered by Example 4.3.) Example 1.1. Let Tn be sequence of independent random variables with exponential distribution (A = 1). Define 7, = o{{T,, g s}; s 5 t} (i.e. 7, is the filtration generated by the sequence of processes {1{,ns,}}). It is easy to see that 7, is a quasi-left-continuous filtration satisfying the conclusion of Theorem 1.1, and for any i E (0,1), A, = S, = t. Thus P(B,)=1. See Figure 1.1 for a simulated sample path of the compensated sum of jumps 27 0.75 0.5 _ , ........... 0.25 ' -0.25 , -0.5 ,P’\ -O.75 0.46 t' 0.45 NV?“ 0.44 I 0.43 \ A A m AAAAAAAAAAAAAAAAAAAAAA Figure 1.1: Simulated sample path of a martingale with accumulation of jumps ev- erywhere (2 different resolutions). 28 Zn ”L, (1{TnSt} — C7,") . The simulations were performed using MATHEIVIATICA 4.0 on iMac DVD owned by the author. To draw the picture we used first 1000 terms of the sum. Example 1.2. Let B(t) be a Brownian motion and W, be its natural filtration. Let Z, = inf{s > t: B(s) : 0}, and D, = sup{s < t: B(s) = 0}. Note that D, < t < Z, as. It follows easily from the strong Markov property of Brownian motion that for any W-stopping time 1/ we have P(u = D,) = 0. Denote the natural filtration of the process sign(B,) by 7,. We will prove in the Example 4.3 that this filtration is purely discontinuous and the set T = {Z, : q E Q} is the set of all totally inaccessible 7,-stopping times3. Thus A, = D, and P(B3) = 1. 3In the sense that for any totally inaccessible stopping time T, [T] C U,,E7.[v]. 29 Chapter 2 Examples In this chapter we will present several examples that constitute a negative answer to some rather interesting questions. First we will prove that the necessary condition (1.1) in Theorem 1.1 is not a sufficient condition. Then we will go on and prove that a subfiltration of a purely discontinuous filtration does not have to be purely discontinuous. In the previous chapter we have proved a necessary condition for filtration to be purely discontinuous. However, as the following examples show the condition is not sufficient. Namely we find filtrations that are defined in agreement with the formula (1.1) of the main theorem, but are not purely discontinuous. Example 2.1. Let {H?} be a sequence of independent Poisson processes with inten- sities A, = n, W, be a Brownian motion independent of {IIZ‘}, and 0 < 9 < 1 be an increasing, continuously differentiable function. Denote the k-th jump of II" by T2. Define 7, as the smallest o-algebra for which the processes {g(W,:)1{,:S,}} are all adapted. Then 7, satisfies the conclusion of Theorem 1.1 with T 2 {T3, n, k E N}, 30 and W, is an adapted continuous martingale. Proof. Since the Poisson processes involved are mutually independent the stopping times in T have disjoint graphs. It is a well-known fact that times of jumps of Poisson process are totally inaccessible with respect to its natural filtration. Since the processes II" and IV are mutually independent, the compensator of 7’: calculated under the natural filtration of II" is equal to a compensator calculated under 7,. Hence the compensator is continuous and the stopping time 7"? is totally inaccessible with respect to the filtration 7,. Define T" = 220:, rg'1,,,.s,<,,+l}. Then 7",, and g(I’V,n) are 7,-measurable random variables. We need to prove that T" ——> t as. Calculate m—l {t—T">€}C U{T,?+1—T,?>5}U{T,’,',——t <—E——=————,—, (m )— (lm nl n l—(I111_1_t)2 (m_nt)2 for m/n > t by Chebyshev’s inequality. Choose m = n.2 and calculate for n > t m P t—T" >5 < rne_"‘+——, :n2e_"5+——. ( ) 7 (m — nt)‘2 (n — 1‘.)2 Since the right-hand side of (2.1) is summable, we conclude by Borel-Cantelli’s lemma that P(t — T" > e, i. o.) = 0. Hence T" —> t 0.3., and consequently g(l:V,n) —-> g(ll"',) a.s.. Thus W, is an 7, adapted process. Finally the independence of II" and W ensures that IV is a Brownian motion with respect to the filtration 7,. El 31 100 --- - - - - - time 5 Figure 2.1: Simulated times ofjumps for Poisson processes with different A. We will prove in the last chapter that the o-algebra generated by the sequence of independent Poisson processes is purely discontinuous (see Theorem 4.2). That means that we “smuggled in” the continuous martingale into the filtration using the size ofjumps. A natural question arises whether filtrations generated by jumps only are purely discontinuous. The next example shows that there is a filtration generated by jumps of size 1 that still supports continuous martingale. Example 2.2. Let U), be a sequence of independent random variables with uniform distribution on (0,1). Let ll" be a Brownian motion independent of the sequence {II/,7}. Denote W', the natural filtration of It". There is a. strictly increasing continuous process 6, 0 S G S 1 that generated W,. Define1 v—l . Tl: = CUMMIW): ( ' [V [\D V 1The symbol Gite) is the inverse function to the function G.(w). If (It-(w) Z Gm(w), then Tk(w‘) = 00. 32 and 7, = o{{T,c S s}: s S t, k E N}. (2.3) Then 7, satisfies the conclusion of Theorem 1.1, but W, is an adapted continuous martingale. Proof. First we find the suitable G. Let G,(w) 2/0 e—3g(I'V,(w))ds, where g(t) 2: fiarctanfi) + This process is an increasing, W, adapted, continuous .1. 2. process, and 0 < C, < 1 as. The continuity of the entities involved implies Gt — Gt_,— 1 t = — r W, -‘ W. 5 [_e ,( >—>e g( a Hence IV is adapted to the natural filtration of C. It follows from formula (2.2) that {Tk} is a stationary sequence of random variables and Woo is its shift invariant o-algebra. Moreover P[n, g ill/V0,] 2 PW g G, | W00] 2 G, (2.4) Thus the well-known ergodic theorem (see Theorem 0.17) implies 1 E 2:1{Tkgt}_) E[1{713t} [W00] 2 Gt 0.8. 33 We proved that It", is 7, adapted. However, we still have to check that W, is a martingale with respect to this bigger o-algebra. Fix t < s and denote C, = G,,(,. Then T), /\t = C,_1(U,,) /\t = C{1(U,,) /\ t. However C and C“ are W, x B[0,t] measurable, whence 7'), /\t is independent of W, — 11”,. Thus IV is a Brownian motion under the filtration 7. To finish the proof notice that the random variables 7'), have continuous distribu- tion, T1 “Woo f (7'2,T3, . . .) for any measurable function f, and deduce that 7'], are totally inaccessible stopping times. C] The following simple example shows that a subfiltration of a purely discontinuous filtration is not necessarily purely discontinuous. Example 2. 5’. Let W, be a Brownian motion and T be a random variable uniformly distributed on (0, 1) independent of W. Denote H, the smallest filtration that makes both If", and 1{,S,} adapted. Define filtration 7,=o{Afl{TS t} : AEHI} (2-5) and a process X, : WM, — WTVUM). Let g, denote the natural filtration of X. Then .7, is a purely discontinuous filtration, X is a continuous g,-martingale, and Q, C .7, for each t. Proof. The filtration 7, was generated by one jump. It is a purely discontinuous filtration by Theorem 0.12. The process X is a continuous martingale with respect to both ”H, and Q, by Proposition 0.1. Finally, gt C 7,, because X, is ’H, measurable 34 for any t Z 0, and X, = 0 for t S T. E] Notice that X, is an 7,-adapted process, but it is not an 7,-martingale. This was accomplished by “moving the information” in the filtration ’H, forward. 35 Chapter 3 Necessary and Sufficient Condition In this chapter we prove a general necessary and sufficient condition for a filtration to be purely discontinuous. Our condition is similar to the condition discovered by J. Jacod and M. Yor (1977), [31], that claims that a martingale M has a strong representation property if and only if the original probability is an extreme point in the set of all probability measures that keep M a martingale. The big drawback of our condition is that it is usually very hard to verify. Denote C ,7 ’6 the compensator of £1{,S,}, where T is a totally inaccessible stopping time and C is a bounded 7, measurable random variable. Theorem 3.1. Let 7, be a quasi-left-continuous filtration. Put P’ is a probability on 700 such that F = P’: CI’é is a compensator ofCI{,S,} under P’ Then 7 is purely discontinuous and 70 is a trivial o-algebra if and only if P is an 36 extreme point of F. The proof is a an adaptation of the proof of a result similar to the theorem mentioned in the first paragraph of this chapter (Theorem 13.21 from [16]). Proof. Assume that P is an extreme point of F. Let C be bounded .70 measurable random variable with EC 2 0 and N be a bounded continuous martingale with N0 = 0. Denote 11/! = C + N. Assume without loss of generality that [.Ml S 1. Define M00 111,, .p.-(.,_,_), dp.-(.-_,-). Since E Mo = 0, P1, P2 are probability measures equivalent to P and 1 1 P = —P —P« . 2 1 + 2 2 For arbitrary but fixed T and C denote X, = Cl{,s,} — CZ’f. The Girsanov’s Theorem (see Theorem 0.16) implies that the P semimartingale 1 X’zX——-ZX .1 z_ < 3 >1 (3 ) where Z = (1 + Moo/2), is a P1 martingale. However, (Z, X) = 0 and therefore P, E F. Hence P1 = P and subsequently M E 0. Assume that P is not an extreme point of F, namely P 2 AP, + (1 — A)P2 for 37 some suitable P1, P2 6 F. Clearly P1 << P. Define dP Z = 21—31" 2, = E[Z|.7,]. If AZ gé 0 we can find a totally inaccessible stopping time T ¢ 00 such that [T] C [AZ # 0], and AZ, is bounded. The process X, = AZ,1{,S,} —C,T’AZ' is a martingale and the Girsanov’s theorem implies that 1 X’=X——- Z,X Z_( > is a P1 martingale. However the predictable process of bounded variation —Z—1_- - (Z, X) is not a constant P, as, and therefore the compensator of AZ,1{,S,} under P1 is not CZ’AZ’. Thus AZ 2 0. El Notice that the proof really used only probabilities absolutely continuous with respect to P. Hence we can reformulate the theorem in the following way. Corollary 3.2. Let 7, be a quasi-left-continuous filtration. Put P’ << P is a probability on 700 such that F’ = P’ : C,“ is a compensator of{1{,s,} under P’ Then 7 is purely discontinuous if and only if F’ = {P}. Remark 3.1. As noted earlier the condition of Theorem 3.1 is rather difficult to verify. However it can be verified in the case of o-algebra generated by a purely discontinuous Levy process. It is a known fact that the probability measure is uniquely determined 38 by the predictable characteristics of the Levy process and all the compensators are uniquely determined by the Levy measure. 39 Chapter 4 Sufficient Conditions In this chapter we will investigate various sufficient conditions for purely discontinuous filtrations as well as some auxiliary results. These conditions will be easier to verify than the condition of Theorem 3.1. We will refer to the following sets of assumptions corresponding to the conclusion of Theorem 1.1. Assumptions. A1. Let 7,, 2 0 be a sequence of random variables with disjoint graphs, 6,, be a sequence of random variables, and .75} be the filtration generated by processes {6.1mm}. AQ. We assume A1 and 6,, : 1. R1. In addition to A1 we assume that the filtration .73, is quasi-left-continuous, the stopping times 7,, that generated the filtration are all totally inaccessible, and 40 if r’ is a totally inaccessible stopping time, then [7'] c Um]. (4.1) R2. We assume R1 and {n = 1. Clearly, if a filtration satisfies assumption A2 (resp. R2) it also satisfies assump— tion A1 (resp. R1). We will also use the following notation. Definition 4.1. Let filtration ft satisfy Assumption Al we define 7:," as filtration defined by processes {Enlhngt}: n g k}, and .7?" as filtration defined by processes {€n1{7ngt}, n > k}. We say that the filtration E is tail-free if the o-algebra wzny keN is trivial for t z 00. The following examples show that in the assumption R1 the quasi-left—continuity and the relation (4.1) does not hold automatically and should be assumed. Example 4 .1 . Let W be a Brownian motion starting at 1. Define Zq : inf{t : W, = 0, t > q}. It is a well-known fact that P[q < Z, < oo] = 1. Let {q,,};,'°:1 = Q 0 (0,00), q0 = 0. 41 Since the random variables {Zq} do not have disjoint graphs, define 00 on AZUlc:(l){Zn:ZQk}9 N: H 9n an on Q \ A. Denote by f, the filtration generated by the processes {1 {2’0} }. Then the properties of Brownian motion assure that Z, are totally inaccessible stopping times. We have defined Z, in such a way that Z, = ZO for any rational q > 0. Furthermore Z0 was not involved in the definition of the filtration .7,. However, 1. Zo Z inf Zq qE‘QNOJ) is a totally inaccessible stopping time with respect to the filtration .75}. Example 4.2. Let {ek} be a sequence of i.i.d. exponential (A z 1) variables defined on a probability space (Ql,fl,P1). Let ((22 = {0,1},.7:2 = {(0, {0}, {1}, {0,1}},P2 = (1/2,1/2)) be another probability space. Define the probability space (9,}— , P) as the product space of 91 and 92. Also define random variables 00 for (.12 = 1, Tk(w‘1,w2) = (4.2) ek(w1) +1 for wg = 0, and filtration f} = o{{r,c g s} : s g t, k E N}. Clearly the filtration ft is of the form of Assumption R2, perhaps short of being quasi-left-continuous. We will indeed prove it is not quasi-left-continuous. Notice that the o-algebra .733 is trivial for s < 1. Hence fl_ is also trivial. 42 However, the right-continuity of the filtration implies that the set AzijHngsyzmx{m 1 T a.s., then £_=£. We now proceed to the first theorem of this section. Theorem 4.1. Let filtration f} satisfy assumption A1. Additionally, for each It there is nk, such that for any E E .7750, EKUH=HMHW- M& Then the filtration .75} is purely discontinuous. We say that a filtration is quasi-Markov if it satisfies the assumptions of Theo- rem 4.1. Proof. Let fit be a continuous, square integrable .7,-martingale satisfying £0 = 0. There is a sequence 5’“ of square integrable fife-measurable random variables such 43 that gk L2; goo. (4.4) Define {f = E[£" IE] 2 E[€"" [ft]. The last equality is the assumption of the theorem. The filtration f," is generated by a finite number of stopping times and it is jumping. The Theorem 0.12 implies that {f is a compensated sum ofjumps, whence (€717 g) = 0 (45) This and equation 4.4 imply if = 0 as. E] The Assumption 1 in Theorem 4.1 was used only to establish the fact that the filtrations f," are purely discontinuous. Thus we have already proved the following theorem. Theorem 4.2. Let 7:," be an increasing sequence of purely discontinuous filtrations. And .7: = V .7". Assume that for each k there is nk, such that for any g 6 7:0, Elélfal = ElélfT"l- (4.6) Then the filtration .7 is purely discontinuous. We will now state two interesting examples. 44 Corollary 4.3. Let 7,, Z 0 be a sequence of independent, non-atomic random vari- ables, and ft be the filtration generated by the processes {Tn g t}. Then 1. Tn are totally inaccessible stopping times with respect to f}. 2. Any ft adapted continuous martingale is constant as. Proof. To prove statement 1 it is enough to prove that the compensator of 1mg} is continuous. Define .7,“) as filtration defined by process {klmst}, and .731") as filtration defined by processes {§,,1{,n9}, n 76 k}. Calculate E1{s (12) is a local martingale under P’ and (X) P = (X’)P,. The process 2% ~ (X, Z) is a continuous process of bounded variation. Thus X’ is a non-constant continuous local martingale under P’. The statement of the lemma follows. [1 Remark 4.2. There is no disagreement with Theorem 3.1, because compensators cal- culated under P are not necessarily equal to the compensators calculated under P’. We will now investigate a particular change of measure. We need to do a few calculations and introduce some notation prior to stating next theorem. Consider the assumption R2. Assume that for any I g k the distribution of 7'), . . . ,Tk given f; is equivalent to the the measure Pb", which is the unconditional distribution of T), . . . , rk, with the density bounded away from 0 and 00. Formally we assume the following: For any measurable bounded f, E[f(Tli"'7TkaTk-+-la"')[jazz] :/f(n,...,Tk,Tk+1,...)p"k(T),...,Tk,Tk+1,...)P”k(d‘rl,...,d7'k), (4.13) where 0 < CM. < p”" < C1,), < 00 is a measurable function such that fpl'k(7'(,...,Tk,Tk+1,...)Pl‘k(dT1,...,di)Zl. 50 For n 2 k denote [32" = E[p”’c | $330]. Since [1’5" is a positive discrete time martin- gale, the martingale convergence theorem implies that )oigk —> p”" as. (4.14) We can choose versions of the conditional expectations in the definition of pig", so that the convergence in (4.14) is everywhere. Define ~lk ~ (,3 ri£k(Tk+1.....Tn) —— /piik(Tz,...,Tn)P”k(dT(,...,di) and p2": = —:k- 11 Since 0 < [32" < CM, the dominated convergence theorem implies that lim rfigk = fp”"(rl, . . .,rk,rk+1, . . . )P”’°(dr), . . . ,drk) 2 1. n—+oc Hence pig’“ ——> p"" and therefore (.1: . p 11m -l_k n—+oo pr; 2 1 as. (4.15) Let 5;, l 0 be a summable sequence. Choose a sequence nk in such a way that nozlandifk>1 nk—l ink p Pl. —1 “Vile—1.711: 7lk+1 > 5k} < 5k- (4.16) 51 Borel-Cantelli’s lemma implies that nk— l ’nk p Pl. le_1.7lk nk+1 —1 :> 5k ill] ==(). Thus pnk- l ink —1 Ms < C(w) < 00, "k- Ink pnk+1 a... H p—a whence nk- Ink p: H p——,,k_ 1m. converges as. (4.17) pnk+1 The theory of infinite products assures that we can find a modification of p such that 0 < p < 00. To simplify the notation we will use the following relabeling: TIC : (Tnk_17 ' ' '3T71k): k " '— a P (di) : Pnk ln"(d7'.nk_l, . . . ,dTnk), k “k __ “ “co __ “co .7: :fgo", f —f:oka f —‘7009 "k Ink "k 1"): kaP ", Pk=Pnk;l 52 Using the new notation the relation (4.13) transcribes to E[f(a,e,.+,,...)|fk]=/new“.....)p,.(e,.,a+1,...)Pk(d.ek). (4.18) pk(fkvfk+lv°') Pk(fkafk+ll ' Similarly p 2 H21, We want to change a probability measure in such a way that under the new measure the conditional distribution of 7",, given j?" depends only on n+1. Let us state the following theorem. Theorem 4.7. Assume R2, equations (4.13), (4.17), and use the notation above. Suppose there is a probability measure P’, such that for any bounded measurable f and for any k E’[f(a,a+,,...)|fi’°] =/f(a,a+1,...),5,,(a,ek+,)Pk(de,.), (4.19) and .7300 is a trivial o-algebra under both P and P’. Then P ~ P’ and dP = p . dP’. If dP’ = i - dP is a probability measure {it is automatically equivalent to P since 0 < p < 00), then the formula (4.19) is valid. We will first prove the following version of conditional Fatou lemma. Lemma 4.8. Suppose Xn Z 0, Y = limiann is integrable, and .75] 3 7:2 3 is a decreasing sequence of o-algebras. Denote .700 = fl" fn. Then lim infE[X,, | 75,] 2 E[lim inf Xn Ifoo] as. 53 Proof. Let 1;, : inkanXk. Since 0 S 1;, S X", I}, is integrable and 1",, T 1". For 2,, = Y —1',, and n 2 k, E[Zn | In] 3 E[Zk | In]. Hence 0 3 lim sup E[Zn | 77”] S E[Z)c I .700] —> E[0 l .700] as. 11-900 However 0 = lim sup E[Zn Ifn] = E[Y I foo] — liminfE[Y,, | 7"] a.s., n—>oo whence lim infE[X,, l 7"") Z liminfE[Y,, lfn] = E[limiann | 5:00] as. C] Proof of Theorem 4.7. Let {1 = f1(fl), . . . ,gN = fN(r'N) be bounded random vari- ables. Calculate using (4.18) and (4.19): E5152 ' "5N = E€2 - - -€NE[§1 [.731] : E52 . . '€NE'l£1% l f1] = E§3.-.§NE’[§2QE'[€1§1lj-IHjfl] p2 P1 = E43 - - -€~E'[ae@€3 (it?) PI P2 N =EE'[£1°°°€NH%|J-‘N] i=1 ' =EE’l€1"'€N glpl, i=1 ' for any n > N. 54 By Fatou lemma, Lemma 4.8, and the fact that 7:00 is trivial we get E(€i€2°'°&v) Z EE'léi ' ' '5” (H :1) A M [72-00] = E’(51"'§NP/\ M)» 1:1 ’ for all M > 0. It follows that HP S 1 and E(§1€2"'€N) Z E’(€1°°'€NP)- (4-20) The very same calculations with P and P’ exchanged yield , l E El and E€1€2"'€N2E€1"'£N;- 1 p Thus P ~ P’ and P = p - P’. The proof of the first part of the theorem is complete. Assume P = p - P’. We will first prove k—l A E,g(T?k,T:k+1,. . .) : E, (11%) g(fk,7?k+l, . . . ), (4.21) i=1 J for any k and measurable bounded function g. Notice that pk and pk are Pk‘l-adapted and calculate A 00 A E’g(Tk,Tk-+-1, ° ' ) : Eg(Tk77-k+la ' ' ')E[p—l lflln — 1 i=2 p.- =E9(Tk,?k+1a---)( ‘1)/A1(F1,?2)Pk(dfl) i=2 pi : Eg(T-*k,7:k+1, . . .) —1. i=1: ’0‘ 55 We proved k—l E’g(Fka?k+la---) I E’( %)g(fic,7—Jk+1,...). (4.22) i=1 ' The equation (4.19) is valid if and only if E’f(fk,?k+1,---)lsk+, = E’lsk+1 /f(?k,?k+1,---)fik(Fk.?k+1)Pk(d?k)a (4.23) for every Skid E .7”. However the same calculations as above yield: E’f(?k,fk+l,...)lsk+l k p- : E’ (H pl) 15“, /f(FkaPk+lao--)PkifkaFk+1)Pk(dFk) i=1 1 = E’lsk+1 /f(?ka7?k+la---)Pk(FkaFk+1)Pk(dFk)- The last equality follows from the formula (4.21), and the fact that the random variable 15”, f my, a“, . . . )p*,.(r,., ek+,)ek(da) is fk-adapted. E] The assumption that the density p”: is bounded away from 0 and 00 is technical and could be somewhat relaxed by assuming that certain entities are integrable. We have proved that the sequence (f1, f2, . . .) is a discrete time Markov process. Particularly: —o Tl? ' ° °aTk-1H"r'k Tk+laTk+29 ' ' ' 56 Bibliography [1] BERNSTEIN, SN. (1927) Sur l’extension du théorr‘ne limite du calcul des prob- abilités aux sommes de quantités dépendentes. Math. Ann, 97, 1—59. [2] BERNSTEIN, SN. 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