COUNTING PROCESSES AND MARTINGALES F, B. Dolivo and Frederick J. Beutler Computer, Information and Control Engineering Program The University of Michigan Ann Arbor, Michigan 48104 This research was sponsored by the Air Force Office of Scientific Research, AFSC, USAF, under Grant No. AFOSR-70-1920C, and the National Science Foundation under Grant No. GK-20385.

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ABSTRACT. Counting Processes (CP), which are stochastic processes having right-continuous sample paths except for randomly located positive jumps of size one, are examined here in the light of a new notion resulting from the Doob-Meyer decomposition for supermartingales: the Integrated Conditional Rate (ICR). It is shown elsewhere ([4], [15]) that this ICR is particularly pertinent to the solution of the problems of filtering and detection for CP's. The terminology ICR is motivated by the fact that when a CP(N ) satisfies tt some sufficient conditions its ICR takes on the form (f X ds) where (\t) is a 0 t nonnegative process called the conditional rate, satisfying Xt lim E[hNt+h - Nt)l VK]. Our approach, however, requires only the weak assumption that N is a. s finite for each t; there always exists an ICR while in general a conditional rate cannot be defined. Sufficient conditions for the existence of a conditional rate are presented. Based on the character (e. g., totally inaccessible) of the stopping times defined by its jumps any CP is shown to be uniquely decomposable into the sum of a regular CP and an accessible CP. It is also demonstrated that each class is completely characterized by continuity properties of the ICR. CP's with independent increments are uniquely distinguished by a property of their ICR's: they are deterministic and given by the mean of the CP. 1. INTRODUCTION AND SUMMARY 1. 1 INTRODUCTION. We are interested here in a description of Counting Processes which is appropriate to study the problems of detection and filtering. By a ounting Process we mean

2 Definition 1. 1. 1: A Counting Process (N ) (hereafter abbreviated CP) is a stochastic process having sample paths which are zero at the time origin, right-continuous step functions with positive jumps of size one. CP's have been examined in terms of counting properties (i. e. properties related to the number of jumps falling within specified subsets or more generally in terms of random measures) or in terms of interval properties (i. e., relative spacing between points, see e. g. [2]); but none of these approaches are specific~ally tailored to treat the problems of detection and filtering; hence we propose here a new approach. The solution to these problems, which involves the computation of conditional expectations, is very much dependent on the information available to the observer or, in mathematical terms, on an increasing family of cr-algebras: the family of observation o- algebras ( t). Let (Nt) be a CP adapted to ( t) and with the sole assumption that (i) The random variable Nt is a. s. finite for each t. Then the Doob-Meyer decomposition for supermartingales associates to (N ) a unique natural increasing process (At) dependent on the family ('t). This process plays a central part in solving the detection and filtering problems (see [5], [15]). We call it the Integrated Conditional Rate (ICR) because under sufficient conditions (given t in section 5. 0) it takes on the form (J \ ds) where (X ) is a nonnegative process calledtherate, satisfin lim E[(N)h Thipaperisconcalled the rate, satisfying Xt = lim E[(N -N )h- 1\t ]. This paper is conh t+h t t cerned with a study of this notion of ICR. It should be strongly emphasized that an ICR always exists, while in general a conditional rate cannot be defined. Hence this study also generalizes and puts in the proper mathematical context

3 previous works on modelling CP's by conditional rates (see e.g. [12], [13], [14], [3], [11], [1]). 1.2 SUMMARY. Let (Nt) be a CP satisfying assumption (i) and adapted to an increasing right-continuous family of a-algebras (I t). It is shown in Section 3. 0 (Section 2. 0 is concerned with preliminaries) that the Doob-Meyer decomposition for supermartingales associates to the CP (Nt) a unique natural increasing process (At) which makes the process (M Nt - At) a local martingale with respect to ( t). This decomposition (Mt t t (Nt = M + At) is intuitively a decomposition into the part (Mt) which is not predictable and the ICR (At) which can be perfectly predicted. We refer to that as the separating property of the Doob-Meyer decomposition for CP's (see Section 4. 3). Properties and examples of ICR's are exhibited. In Section 4. 0 three classes, regular, accessible and predictable CP's, are defined, these latter constituting a subclass of accessible CP's. We show that any counting process can be uniquely decomposed into the sum of two counting processes which are respectively regular and accessible. Regular counting processes have, loosely speaking, totally unexpected times of jump. Poisson processes are of this type. On the contrary, the times of jump of an accessible counting process can be predicted with some chance of success. A counting process which jumps with some positive probability at given fixed times is an example of this kind of processes. It is also demonstrated that each class is completely characterized by continuity properties of the ICR. Finally in Section 5.0 we give sufficient conditions for the existence of conditional rates.

4 2. 0 PRELIMINARIES. Let (2,,P) be a complete probability space. By (Xt) we denote a real valued stochastic process defined on PR, the positive real line. By a right-(resp. left-) continuous process we mean a process with right- (resp. left-) continuous sample paths. The process (Xt) is a modification of the process (Y) if X =Y a. s. for each t e R+. We do not distinguish here between modifications of the same process. If two right(resp. left-) continuous processes (Xt) and (Yt) are modifications of each other t t then we also have P{Xt = Yt, t E sR} = 1 so that we can safely use the simplified notation X = Yt a. s. t t Let { t) be an increasing, right-continuous family of (-subalgebras of j, with?O containing all the P negligible sets. In particular we often consider the family (o-(N, 0 < u < t)), denoted ( t), generated by a CP(Nt) up to and at time t. The notions of stopping times and martingales are assumed known. The basic references for this material are [9j and [7], We recall simply sme-definitions and facts useful in the context of this study. - (a) The family (7 t) is said to be free of times of discontinuity if for every increasing sequence (S ) of ( st) stopping times n S lim S n n n - (b) With respect to the family (iZ ), a stopping time T is said to be t If the family (d t) is not right-continuous, we consider then the family (t+ _ sf Os) (see [8]). s>t

5 totally inaccessible if T is not a. s. infinite and if for every increasing sequence (S ) of stopping times majorized by T we have P {lim S = T, S < T < oo for every n = 0 n n n -inaccessible if there exists a totally inaccessible ( St) stopping time Ssuchthat P{T =S<oo}>0 - accessible if it is not inaccessible - predictable if there exists an increasing sequence (S ) of stopping n times which converges a. s. to T and such that for every n one has S < T on the set {T > 0} -(c) An increasing process (A ) is a stochastic process adapted to the t family ( t) with (1) sample paths which are a. s. zero at t=0, increasing and right-continuous with (2) At integrable for each t. The increasing process (At) is integrable if sup E A < oo. t t t t (d) A (9t) supermartingale (Xt) admits a Doob-Meyer decomposition t t X = Mt -A where (Mt) is a ( t) martingale and (At) an increasing process if and only if (Xt) belongs to the class (DL). - (e) This decomposition is unique if the increasing process (At) is natural - (f) The natural increasing process (At) is continuous if and only if the supermartingale (Xt) is regular. t

6 - (g) If the supermartingale (Xt) is of class (D) (hence uniformly integrable) with a unique Doob-Meyer decomposition (Xt = M - A) then M = E(A + X It) and (At) is also the unique natural t oo00 00 t t increasing process which generates the potential Pt = X - E(X lt). t t 0o t - (h) If the supermartingale (Xt) is bounded by a constant c (i. e., t IXtj Ic a. s.) then sup E M < co (for necessary and sufficient t t conditions - see [5], Lemma 2. 2. 2). - (i) A ( t) martingale with sup E Mt < o is called a square integrable t martingale (see [9], Chapter VIII, Section 3). - (j) A () local martingale (Xt) is a process such that there exists a sequence of (7 t) stopping times (T) increasing a. s. to oo which - n makes each process (XtT ) a uniformly integrable martingale. n If furthermore (Xt ) is square integrable then (Xt) is a tAT t n square integrable local martingale. - (k) We denote by c ({t) the space of (h ) local martingale which are zero a. s. at the time origin. - (1) A sequence of stopping times (T ) reduces the local martingale (Xt) if (XtT ) is a uniformly integrable martingale. - (m) As a consequence of the Doob-Mayer decompositionwe can associate to every square integrable ({ft) local martingale (Xt) a unique natural increasing process ft X >) such that (Xt - <X>) i ( t) (see [9], Chapter VIII Section 3 [7]).

7 - (n) For a local martingale (X ) (not necessarily square integrable) with sample paths of bounded variation on every finite interval, the quadratic variation process [X] is defined by ~ [x], _ t s<t and the process (X2 - [X]t) X X (see [4]). 3.0 INTEGRATED CONDITIONAL RATE. The points in time at which a CP (Nt) jumps are basic to this study: Definition 3. 0. 1: The stopping time: inf {t: N > n} Jn = n th. o if the above set is empty is called the time of the n jump of the CP (N). The fact that J is a stopping time with respect to any family ( t) to which the CP (Nt) is adapted can be easily verified: the set {J < t} = {Nt>n} belongs to t for every t. 3. 1 DOOB-MEYER DECOMPOSITION FOR COUNTING PROCESSES. As a direct application to CPIs of the Doob-Meyer decomposition of supermartingales into the sum of a martingale and an increasing process we have (see [91, Theorem 31-VII; [6]) For a right-continuous function f with left-hand limits A f denotes the jump ft - f t

8 THEOREM 3. 1. 1: (Doob-Mayer Decomposition for CPLs). Let (Nt) be a CP adapted to an increasing family ( t). (a) If for each t iR+, Nt is a. s. finite then there exists a unique natural increasing process (At) such that the process (Mt = N - A) is a square integrable (,i t) local martingale. The unique decomposition (N = M + At) is called the Doob-Meyer decomposition for the CP (Nt) with respect to the family (7 ). (b) If furthermore EN is finite for each t then the process (M = N - At) is a ($ t) martingale. ~t~~~ ~th Proof: (a) Let J be the time of the n jump of the CP (Nt) and define n t (Nt = NtJ ). By assumption N is a.s. finite for each t. Hence the se~~~n quence of stopping times (J) increases a. s. to infinity. Also by construction the stopped process (N ) is bounded by n. For t > s we obviously have E(-N i ) < - N. Thus (-Nt) is a bounded (t) supermartingale and by the Doob-Meyer decomposition we can obtain the unique decomposition: (3.1) Nn Mn + A t t t n n where (Mt) is a square integrable (q ) martingale (see section 2. 0, (h)) and t t (At) a natural integrable increasing process. Now for n <m the unique n 1n n+ Am Doob-Meyer decomposition of (N ) with respect to (t ) is also given by (3.2) N M Am Therefore comparing (3. 1) and (3.2) we get Mm = Mt a.s. and A =A a, s. t tJ nt t A J n n nl n

9 Hence we can uniquely define for all t an increasing natural process (At) and a n n M n square integrable local martingale (M ) by At A and M M for t < J L t t t t t n and we clearly have Nt = M + At a. s.; this proves part (a). (b) If EN is finite for each t then the process (-Nt) is a right-continuous t t negative supermartingale. By Theorem 19-VI of [9], this supermartingale belongs to the class (DL). Then result (b) follows directly from the Doob-Meyer decomposition (Theorem 31-VII of [93). Remark: If the random variable N is not a. s. finite for each t then the sequence (J ) of times of jump of (Nt) does not converge a. s. to infinity. Define J = lim J. By Theorem 42-IV of [9], J is a stopping time. For t >:, n N = oo and the best we can do in this case is to consider what is happening on the stochastic interval [0, J) only. If now a local martingale (Xt) is redefined as being a process such that there exists a sequence of stopping times (R ) increasing to J a. s. (instead of co) which makes the stopped process (Xt R) t AR n a uniformly integrable martingale, then as above we can associate a unique Doob-Meyer decomposition to the CP(N ) on the stochastic interval [0, J). t When speaking of a CP (N ) we always assume that the random variable (N) is a. s. finite for each t since this is clearly the weakest condition under t which the unique Doob-Meyer decomposition of (Nt) is defined on the entire positive real line. Note that this assumption is very weak as it is violated only if the times of jump of the CP (Nt) considered converge with some positive probability to a finite time, or, in other words, that the point process associated with the CP (Nt) contains with some positive probability a point of accumulation, an unlikely situation in practice.

10 3.2 INTEGRATED CONDITIONAL RATE: DEFINITION. For every CP(Nt) with Nt a. s. finite for each t and adapted to a family (t) the uniqueness of the Doob-Meyer decomposition for this CP (Nt) allows us to propose: Definition 3.2. 1: We will call Integrated Conditional Rate (hereafter abbreviated ICR) with respect to the family (1 t) the unique natural increasing process which appears in the Doob-Meyer decomposition of (N) with respect to the family (e t). The terminology Integrated Conditional Rate is motivated by the following: when (N ) satisfies some sufficiency conditions given in section 5. 0 the ICR t t takes on the form (J \ ds) where (\t) is a nonnegative process called the S conditional rate as it satisfies N -N = lim E( lt). t ~h h -0 The terminology ICR will be used even when, as may be the case, a conditional rate does not exist. Note that if (Nt) is a nonhomogeneous Poisson process then the notion of conditional rate with respect to the family of cr-algebras (# t) generated by the process itself reduces to the usual notion of rate. Let (Nt) be a CP and denote by f7t) the family of (r-algebras generated th by (Nt). Let J be the time of n jump. Clearly for each n the stopped process (N ) is a submartingale with respect to any family of (4r such t AJ t n that Mt ) At. Hence we can define an ICR, say (A), with respect to any such family (/t) The process (N - At), that we will systematically denote by (Mt), is in the general case a square integrable ( t) local martingale, and a (Zf ) martingale when the mean ENt is finite for each t. By definition

11 the ICR is a natural process. This last property is dependent on the family (07 t) chosen so that the ICR (At) is varies according to the family (t) considered. For emphasis we therefore speak of a "(7 ) ICR. " Given a CP (Nt) and its ICR's with respect to two distinct families (4 t) and ( t) such that io < 7,t' it is natural to ask how these two ICR's are related. This is what we examine now. Assume that the CP (Nt) has a finite mean; even in this case there is no simple useful answer to this problem. Denote respectively by (A ) and (A) the ICR's of (Nt) with respect to the families (t) and ( t), and by (C) a right-continuous modification of (E(A4 I| a )). It is easy to check that (a) The process (Xt = N - C ) is a ( ) martingale; t t t t (b) The process (Ct) is not necessarily increasing or natural so that t (Ct) is generally not the (t) ICR of (Nt); t (c) The process (Ct) is a (t) submartingale of class (DL) which has a Doob-Meyer decomposition (Ct = Y + B ) where (Yt) is a (t) t t t t martingale and (Bt) a natural increasing process. t (d) The relation between (A ) and (At ) is then t t AL= B = E(A )-Yt t t t t It is also clear that if A )) is in fact adapted to the family (t) then A -A t 4t In conclusion there is no simple way to relate the two ICR's (At) and (At ) in the general case. But when conditional rates with respect to the two families (t) iand (a t) exist then these two conditional rates are simply related (see Theorem 5. 0. 2).

12 We now demonstrate two simple propositions. The first one shows the intuitive result that a. s. no jump occurs in an interval on which the ICR is a. s. a constant as a function of time. THEOREM 3. 2. 2: Suppose (Nt) is a CP adapted to a family ( t) which has an ICR with respect to this family that is a. s. constant as a function of time on the stochastic interval [T, S] (T and S are stopping times, finite or not such that T < S a. s). Then (N ) is a. s. constant as a function of time for t i [T, S]. Proof: Let (R ) be a sequence of stopping times reducing the local martn ingale (M N - At) where (At) is the ICR of (Nt). We have E(N -IN ) = 0. t t t t t SAR TAR n n But the random variable N - NT is a. s. nonnegative n n so that N = N a. s. and the result follows by taking the limit on n. SAR TAR THEOREM 3. 2. 3: Let (At) be the (ft) ICR of a CP (N). Then t t t ENt < oo ifand only if EA <oo and EN = EAt. Proof: If EN < oo then by Theorem 3. 1. 1 (b) the process (M Nt-A) t t t t is a zero mean ( ) martingale so that EA = EN < oo. Conversely if J is t t t n th the time of the n jump of (Nt) then the process (N - A ) is a zero t ttAJ t Ai n n mean martingale so that EN =EAt J and the result follows by the n n monotone convergence theorem. We close this section with identification of a special class of ICR: THEOREM 3. 2. 4; Let (Nt) be a CP of independent increments with a finite mean m for each t. Then the (7Zt) ICR (At) is given by A = mt. t t tI \ t

13 Proof: It is easy to show that the process (Nt - mt) is a (7ft) martingale. Furthermore the increasing process mt is natural because it is deterministic so that (Nt) has the unique Doob-Meyer decomposition (N = (Nt-mt) + m). Finally the uniqueness requires mt to be the (7Lt) ICR. We will reexamine CP's of independent increments in a future paper and prove in particular a converse result to the above proposition: namely that if a P (Nt) has a deterministic ( /t) ICR then it is a process of independent increments. Hence CP's of independent increments are uniquely characterized by the fact that their ICR's are deterministic (see [5], Theorem 2. 6. 1). It can be shown (see the Remark following Corollary 4. 3. 3) that the ICR (At) of a CP (N) with respect to the family of oa-algebras ( t = l ) is t t t 00 given by A = Nt. Hence for a CP of independent increments the (/ t) ICR is given by the mean m and the (6t = A- ) ICR by (N). This illustrates t t 00 t the dependence of ICR's on the choice of family of conditioning a —algebras. 4. 0 REGULAR AND ACCESSIBLE COUNTING PROCESSES 4. 1 DEFINITION AND DECOMPOSITION. Let (Nt) be a CP adapted to a t family (t). Denote by J the time of nt jump. It is natural to classify CP's t. n in terms of the properties of their stopping times J Definition 4. 1. 1: A CP (N ) is called respectively regular, accessible t or predictable with respect to the family (7t) in accordance with the total inaccessibility, accessibility or predictability of its times of jump J with respect to this same faculty (see Section 0. 2(b) ).

14 While a process can be none of these, the next theorem will show that any CP (N t) can be decomposed uniquely into the sum of a regular CP and an accessible CP. Here again these definitions are dependent on the particular /2 family (7 t) chosen. We will see later on (below Theorem 4.2. 2) that a CP can be regular with respect to one family and predictable with respect to another. The term regular was originally used ([9], Definition 33-VII) to characterize a supermartingale (or submartingale ) (Xt) such that for any sequence of stopping times (S ) increasing to a bounded stopping time S we have lim EX = EX. The next proposition shows that our terminology is conS S' n n sistent. THEOREM 4. 1. 2: Let (Nt) be a CP. Then the three following statements are equivalent: (a) The CP (Nt) is regular in the sense of Definition 4. 1. 1. (b) For any stopping time S such that EN < oo the process (Nt ) is a regular submartingale in the sense of Definition 33-VII of [9]. ) (c) lim EN = EN for any sequence of stopping times increasing n n a. s. to R and such that EN < oo. R Proof: Let S be a stopping time such that EN < oo and (T ) any sequence S n of stopping times increasing to T a. s. If the relation (4. 1) lim N NT a. s T AS TAS n n However, Rubin [11] uses the term in a different sense: it loosely denotes a CP with a random rate which must possess numerous technical properties.

15 holds then by the monotone convergence theorem we have (4.2) lim EN = EN n T AS TAS" n n Conversely if relation (4.2) holds we have E(NTS - lim NT S) 0 n by the monotone convergence theorem. As the random variable N ^S lim N is positive, relation (4. 1) must be valid. Hence conditions (4. 1) T AS n n and (4. 2) are equivalent. We show now that (a) is equivalent to (b). If (a) is true then the times of jump of the submartingale (Nt A S) are totally inaccessible th (the time of n jump of (Nt S) is equal to J on the set {Jn < S} and to co otherwise) so that relation (4. 1) is valid and, being equivalent to (4. 2), (b) follows. Conversely if (b) is true, relation (4.2) is satisfied. Then (4. 1) holds which implies that the times of jump of (Nt S) are totally inaccessible (othert A S th wise we reach a contradiction). By taking S = J, the time of n jump of (Nt), we get that J is a totally inaccessible stopping time. This is true for each n so that (a) follows. The equivalence of (b) and (c) follows easily from the definition of a regular supermartingale (Definition 33-VII of [9]). Now the announced decomposition result: THEOREM 4. 1. 3: Let (Nt) be a CP adapted to a family (4). Then there r a exists two CP's, (N ) and (Nt) which are respectively regular and accessible t t with respect to the above family and such that r a N = N + N for every t t t t This decomposition is unique. Remark 4.1.4: The (4 ) ICR of (N) is given by r a t t t where (At) and (At) are reectively the ( ) ICR's of (Nt) and (N).

16 th A Proof: As usual, denote by J the time of n jump. By J we mean n n the stopping time J if o E A n jA = n oo otherwise for A E -r. By Theorem 44-VII of [9] there exists for each n an essentially n unique partition of the set {J < oo} into two sets of 43, A and R, such that A R the stopping times J and J are respectively accessible and totally inaccessn n a A A r I I >jRandy ible. The two CP's Nt t > J } and Nt { > clearly n n satisfy the conditions of the theorem. The uniqueness of this decomposition follows from the essential uniqueness of the partition of each set {Jn < oo. Example 4. 1.5: Take 2 = [0, 1] and P the Lebesgue measure defined on the Lebesgue side of [0, 1]. Let J be a random variable uniformly distributed on 2. Define the random variables J+l = J + n, for n > 1. Let n+l 1 (Nt) be the CP having J as time of nt jump, i. e., Nt = {t Jn. One (Nt bethe 1151) n h Z { n On n can show that (see [5]) for any CP (Nt) the time of the first jump J1 is totally inaccessible with respect to (t) if and only if P { J1 = a} = 0 for any nonnegative constant a. Hence the time of jump J1, being here uniformly distributed on 2, is totally inaccessible. It is easy to show that for n > 2, the times of r a of jump J are predictable. Thus the decomposition N = N + N with resn t t t r a pect to the family (7<t) is given by N = t >J } and N = {t > J } c t 1 {t>J. n>2 a In this very simple example the CP (Nt) is in fact predictable. This is not always the case. If we assume in the above example that jumps may be skipped independently of each other with a positive probability, then (Nt) and

17 (N ) are still given as above but the CP (N ) is no longer predictable (see t t Example 4.3.5). For clarity we outline now some of the results we are going to investigate. First regular, then accessible CP's are studied in detail. In particular we will see that a CP is regular with respect to a family (i t) if and only if its ( t) ICR is continuous (Theorem 4. 2. 2); when the family ( t) is free of times of discontinuity then accessible CP's are predictable (Theorem 4. 3. 1). Predictable CP's are uniquely characterized by the fact that their ICR is given by the CP itself (Corollary 4. 3. 3). In other words predictable CP's are natural processes. Combining these facts with the above decomposition for CP's (Theorem 4. 1. 3) gives, when the family (~t) is free of times of discontinuity, the separating property of the unique Doob-Meyer decomposition for CP's (see Corollary 4. 3. 4). The case where the family (It) does contain times of discontinuity is more complex. Most of these results are obtained by studying the different terms in the equation ANT = AMT + iAT in relation to the appropriate property bf the stopping time T (Theorem 4. 3. 2). 4.2 REGULAR COUNTING PROCESSES. Let (Nt) be a regular CP with respect to a family (7 t). By definition the times of jump J of (Nt) are n t totally inaccessible. This has the immediate consequence that the probability a jump occurs at time t is zero. Also,if T is a (I t) stopping time we cannot t make with a positive probability a prediction of any time of jump after T, the prediction being based on the information available up to and at time T. More precisely we have:

18 THEOREM 4. 2. 1: Let (Nt) be a regular CP with respect to a family ( t) and T a (t) stopping time. Assume W is a strictly positive XT measurable random variable. Then for each n P{T +W = Jn} = 0 n th where J is the time of n jump of (Nt). n t Proof: By contradiction. Assume that for n = n there exists W = W 0o a strictly positive ( V7T) measurable random variable, with p {T + W = J } = p > 0. The sequence of (4 ) stopping times (see [9], 0 n t 0 Theorems 37 and 38-IV) (Ti = T + (l-l/i)W ) is increasing and 0 th P {lim T. = J } = p > 0, i. e. the time of n jump is not totally inaccessible, i o a contradiction. The next theorem is a direct consequence of Theorem 4. 1. 2 and a result on the Doob-Meyer decomposition of regular supermartingales (-9], Theorem 37-VII). THEOREM 4. 2, 2: Let (Nt) be a CP adapted to a family (> ). Then the (t) ICR (At) of (Nt) is continuous if and only if the CP (Nt) is regular with respect to this family. th nLA Proof: Let J be the n time jump and define (N Nt ) n t AJ nn nJ (A = A ), (P _ t ) Note that by the uniqueness of the Doob(t t AJ t tAJ n n Meyer decomposition (At) is the n( t) ICR of (N ). By Theorem 37-VII of 19] the process (At) is continuous for each n if and only if the CP (Nt) is regular. The result follows then by taking the limit as the sequence (J ) increases to n n oo. One uses here the fact that on any interval [0, t ], At = At for sufficiently large n (depending on ca).

19 Examples of regular CP's with respect to the family ( t) are, by Proposition 3. 3. 2 and the above theorem, any CP's of independent increments with continuous mean, in particular Poisson processes. Note that these processes of independent increments with continuous mean are not regular but predictable if we take the family ( t = Zo) (see Proposition 3. 3. 1). For a regular CP (Nt) with ICR (At) we have just proved that all the jumps t t are contained in the local martingale (Mt = Nt - At But these jumps completely L t t determined the CP (Nt). This suggests that there is a direct relation between (Mt) and the ICR (At). This point is made clear in the following theorem. Recall that if (N = M + At) is the unique Doob-Meyer decomposition of (Nt) then (Mt) is a square integrable local martingale (Theorem 3. 1. 1) to which a unique natural increasing process (< M >t) can be associated (section 2. 0, (m)). THEOREM 4. 2. 3: Let (Nt) be a regular CP with respect to a family (I t) Denote by (At) its ( 7t) ICR and by (Mt) the square integrable local martingale (Nt At). [See section 2. 0 (h)]. We have (a) A =< M > t t 2 2 (b) If EN is finite then so is EMt, with EM < ENt Proof: (a) One has (4 3) N M +A t t t This shows that (Mt) is a martingale of bounded variation so that (see section 2. 0, (n)) the quadratic variation process of (Mt) is given by (4.4) [M] = (AM ) [ M~~~~~t

20 But (Nt) is a regular CP and by Theorem 4. 2.2 its ICR (At) is continuous so 2 2 that AM =AN. Now AN is either 0 or 1. Hence (AM ) = (AN) = AN s S S which implies by (4. 4) (4.5) [M]t = Nt 2 2 The two processes (Mt < M>) and (M -[M]t) are local martingales (see section 2. 0 (m) and (n)); thus so is their difference (X =[M] -<M >t) and by (4. 5) we get (4. 6) Nt = X +<M> Nt = Xt t where Xt e. The increasing process ( <M >t) is natural so that by uniqueness of the Doob-Meyer decomposition one must have, comparing (4. 3) and (4.6), A =<M> andX =M. t t t t1 2 (b) We have seen above that the process (Mt - [M]t) o or by (4.5) (M - N) i. Let (T ) be a sequence of stopping times reducing this local martingale i. e., the process (M N ) is a uniformly integrable MtAT tAT n n martingale. In particular 2 2 -E(M tAT o o n n Hence (4. 7) EM = EN tAT tAT n n Since Mt converges to Mt, Fatou's lemma implies tAT t 2 2 EM < lim (inf E M. t -- tAT n n and by (4. 7) and the monotone convergence theorem (Nt increases to N) tAT t we get E M(t < lim inf (E ) = li N ) = E N. n ^n n n

21 4.3 ACCESSIBLE COUNTING PROCESSES. Theorem 4.2. 2, which says that the ICR of a CP is continuous if and only if this CP is regular, implies that the ICR (At) of an accessible CP (Nt) is discontinuous. We could conjecture that the times of jump of the ICR (At) are the same as those of the accessible CP (Nt). As we will see this would be true, and we would have in fact (At Nt) but for the possible presence of times of discontinuity for the family (c t) considered (see Definitions 39 and 40-VII, [9]). Recall that an accessible (t t) stopping time which is not a time of discontinuity for the family ( t) is (ct) predictable (see Theorem 45-VII of r9]). This immediately gives us: THEOREM 4.3.1: An accessible CP (Nt) with respect to a family ( t) t t which is free of times of discontinuity is predictable. Let (Nt) be any CP with ICR (At). We examine now the jump LAT in relation to the property of the stopping time T. We already know that for a regular CP AAT = 0 for any stopping time T (Theorem 4. 2. 2). The next result will lead to a unique characterization of predictable CP's (Corollary 4. 3. 3) and the s eparating property of the unique Doob-Meyer decomposition for CP's (Corollary 4. 3. 4). THEOREM 4. 3. 2 Suppose (Nt) is any CP adapted to a family (t) Denote by (At) its (c t) ICR. (a) If T is ( t) predictable then TA = E(TNTIV T ) n n where (T ) is any sequence of stopping times increasing to T. In particular n O <AT < l, andA 1 (or 0)a.s. if and only ifLN = 1 (or 0)a.s.

22 r5 (b) If T is (, t) accessible but not a time of discontinuity for (*t) then AAT = AN (c) If T is (ct ) totally inaccessible then AAT 0. t T th (d) Let J be the n time of jump of (N). Then n t iAj = 1 n if and only if J is a predictable ( st) stopping time. In particular AAJ = 1 n t J if J is accessible but not a time of discontinuity for the family ( t/!). n t th Proof: (a) (see [9], section 51-VII) Let J be the n time of jump of ~n an N ), and define (N t Nt ) (A At J We know (Theorem 3. 1. 1) tt Ntj (At t AJ n n that the process (M - N1 -A is a square integrable ( ) martingale. Thus for i > m and any set H e /T where -(T ) is a sequence of stopping m times increasing to T we have f(MT M )dP =f E(Mn Mn j )dP -0 H H Letting i increase to infinity one obtains, by the monotone convergence theorem T T.(T T. T fA -LAn=dP = f NTdP VHe H H m E(AT T ) = E(LN WT )a.s. n m m This implies rn rn

23 and taking the limit with respect to m, by the Lemma of [ 10] m m m m The process (A) is natural with respect to the family ( t) so that, by Theorem 49-VII of [9], the random variable AAT is (V XT )measurable. Thus the rn m m above relation gives AAT = E (AN V T ~~rT 1TV(T m m and by the bounded convergence theorem we get the desired result letting n go to oo. (b) By Theorem 45-VII of [9], T is predictable so that part (a) is applicable. Furthermore T = V CT (T is not a time of discontinuity of (Ht)) Hence m m AAT= E(NTI V ET ) = E(NTI XT)= ANT m m Part (c) is just a restatement of condition (b) of Theorem 49-VII of [9] and is given here for completeness. (d) (<=) J is predictable and ANj = 1 so that by part (a) AA = 1 n J J n n (>) Assume AA 1. Let C = Aj I6A = I J t J {t> J.} {t J}' n n - The process (C t) is natural because it satisfies the necessary and sufficient conditions (a) and (b) of Theorem 69-VII of [9] (if not then the natural process (At) would not satisfy these two conditions, a contradiction). By Theorem 52-VII of [9] J is then a predictable stopping time. n COROLLARY 4. 3.3: A CP (Nt) with ICR (At) with respect to (7t) is predictable with respect to this family if and only if (At = Nt).

24 Proof: (-=>) (Nt) is predictable so by (d) of Theorem 4. 3.2 AA -= 1 for each n J n th where J is the time of n jump of (Nt). This implies A > N a. s. But for n t t- t each n we also have E(Nt -A ) =0 and the relation A = N a. s. holds. tAJ tAJ t t n n (<=) If (Nt = At) then hAJ = 1 for each n and by (d) of Theorem n 4. 3. 2 J is a predictable stopping time for each n, i. e. (Nt) is predictable. n t Remark: It is clear that any (. t) stopping time is predictable with respect to the family ( t, = O). Hence the ( = r? ) ICR of a CP (Nt) t 00oo t o00 t is given by (Nt). COROLLARY 4.3.4: Let (Nt) be an CP with (c4 ) ICR (A ) and define A (Mt = N - At) Then if the family (vt) is free of times of discontinuity (a) The local martingale (M ) has jumps of size one taking place only at (c t) totally inaccessible stopping times. (b) The (<It) ICR (A ) has jumps of size one only at (7 ) predictable t t t stopping times. Remarks: In other words, (M ) represents the part of (N ) which is t t unexpected and the ICR (At) the one which can be perfectly predicted. This t is what we have called the separating property of the Doob-Meyer decomposition for CP's. Proof: Let r a (4. 9) N Nt+Nt Nt ~~~~t t~~ t denote the unique decomposition of Theorem 4. 1.3 where (Nt) is a regular CP a a and (Nt) an accessible CP. and (N ) an accessible CP. Let respectively t). By Theorem 4. ) is (At) and (A) be the (Vt) ICR of (N) and (N ). By Theorem 4. 2. 2, (A )is

25 continuous so that the local martingale (4.10) Mt N -A t t t has only jumps of size one taking place at totally inaccessible stopping times (namely the times of jump of (N )). By assumption the family ( 7 t) is free of t' times of discontinuity so that by Theorem 4. 3. 1 (N ) is a predictable CP t and by Corollary 4. 3. 3 a a (4. 11) A = N t t Introducing (4. 10) in (4. 9) one gets r +r a N = M+ (A +N) t t t t which is a unique Doob-Meyer decomposition of (Nt) as, by (4. 11), r a r A (A + N = + ) is a natural increasing process. But (N =M + ) t t t t t t t is also such a unique decomposition so that one must have M = Mr t t A = Ar +Na t t t and the result follows. Let (Nt = Mt+ A) denote the unique Doob-Meyer decomposition of the CP (Nt) with respect to the family (< t). When this family (~ t) is free of t t t times of discontinuity the above Corollary 4. 3. 4 completely describes the discontinuities of the local martingale (Mt) and of the (- t) ICR (At): either (Mt) or (At) (but not both) have a discontinuity which is of size one and can only take place at a time of jump of (Nt). When the family (4z t) does have times of discontinuity the above statement is no longer necessarily true. Because it is likely for a (7 t) local martingale to have a jump at a time of

26 discontinuity for the family (7 t), the new following situations may now take place: (a) If T is a stopping time which is a time of discontinuity for the family ( t) and such that NT = 0 a. s. then it may happen that both AMT and AAT = - AM will be different from zero. In fact this can happen only if T has an accessible part which is not predictable. th (b) Let J'be the time of n jump of (Nt) which'is supposed to be accessible (but not predictable) and also a time of discontinuity for the family ( t). Then Theorems 4. 2. 2 and 4. 3. 2 (d) imply that both (At) and (Mt) have a discontinuity at J n The following example illustrates these two points. Example 4.3. 5: Take = {C) a12} with {}1)= pwhere O <p <1. Define the following CP (Nt): 0 t<l Nt(W1) 1 t>l Nt(2z) = 0 t>0 The family (Z ) is then given by ( {, 2) for t < 1 t l {ill {) t 2Q} fort> 1 and the unique time of jump of (Nt) by I 1 1 = )1 JY() = \ 00 0C = 02

27 This stopping time J is obviously accessible. Observe also that J is a time of discontinuity for the family (It): (see Definition 40-VII of [9]). Here (S = 1 - l/n) is an increasing sequence of stopping times, S < J for each n n n and the set {(): limS = J = {a()}' V n ={ (4,} n 1 S n n n Denote a (not necessarily unique) Doob-Meyer decomposition of (Nt) by (4. 12) N = M+A where (Mt) is a uniformly integrable martingale ((Nt) is bounded) and (A) an increasing (not necessarily natural) process. It is easy to see that the martingale (Mt) is given by 0 V, t < Mt() = fa =' t> 1 t Lb o =2, t> 1 where a and b are two constants such that (4. 13) _ -p) Then by (4. 12) we must have 0 V o, t<l At(o) = { l-a =1, t > 1 -b w=c2' t>l By the uniqueness theorem only one set of values a and b makes the increasing process (At) natural. These values are a = 1 - p and b = -p (this choice obviously satisfies (4. 13)),as in this case

28 At =,o) (t) is a deterministic hence natural process. Thus the ICR of (Nt) with respect to the family (7 t) is pI[1 )(t) and the martingale (Mt = Nt - pI )(t)) is given by r O o, t< 1 Mt = 0 1 - p =1 t 1 - -p = 2, t > 1 Therefore both the ICR pI[ 1 (t) and the above martingale have a discontinuity at the time of jump J of (Nt). This illustrates case (b). As stated above, this is a consequence of the fact that the accessible stopping time J is not predictable and is also a time of discontinuity for (7 t). Also if we define the stopping time T % 1 (1 =:)2 then 0 Wr=c1.T p a = 1 P c~:o02 even though NT = 0 for any w. This illustrates case (a). It is easy to check that T is a time of discontinuity for (9 t) which is accessible but not predictable.

29 5. 0 CONDITIONAL RATE. In the previous section we have seen that we can decompose uniquely any CP (Nt) adapted to a family ( t) into a sum of two CP's which are respectively regular and accessible with respect to this family ( t) (Theorem 4. 1.3). Regular CP's relatively to a family ( t) are precisely those which have a continuous ( t) ICR (Theorem 4. 2. 2). But a continuous ICR may not have absolutely continuous sample paths. For example, consider a CP of independent increments with a continuous, but not absolutely continuous mean. In the next theorem we give sufficient conditions under which the ICR (At) of a CP (Nt) with respect to a family ( S ) is absolutely continuous; in other words when does a random process (Xt) adapted to (/tr) exist such that we can express the ICR (A ) as t (5.1) A = S ds? t s 0 Under these conditions, we also have N -N (5.2) t = lim E ( t+h -t -t h- t and because of this relation we call the process (t) the "conditional rate" of the CP (Nt) with respect to the family t). Expression (5. 1) is then a justification for the terminology "Integrated Conditional Rate" (ICR) introduced in section 3.2, terminology used even though a conditional rate does not generally exist.

30 Although there is great emphasis in the literature ([3], [1], [11], [12], [13], [14]) on CP's which admit a conditional rate, the problem of existence of these CP's has been treated only lately by Breimand ([1]) where a partial answer to this problem is given: the existence of CP's which possess a bounded random rate with respect to the family of r —algebras generated by the CP itself is demonstrated by the use of absolutely continuous changes of measures. This technique is discussed and extended in [5]. We now give sufficient conditions under which a CP with finite mean does possess a conditional rate: THEOREM 5. 0. 1: If for a CP (Nt) with finite men and adapted to a t family ( t) (i) for each t the following limit exists a. s. lim Q (t, h, a) = \ (t, w) m= 1 2,.. h m rn h —O where Q (t, h, o) P{Nt+h Nt t (ii) for almost all cthere exists h (o) > 0 such that the series (Q (t, wha) converges uniformly for h E (0, h ()] and hm M. m t which is bounded by a function a(t, ) such that J a(s,w)ds < oo 0 for each t. Then (a) The series F\ is convergent. Define the process m (\Xt = m). We have the relation: t mr m

31 N -N t+hE t ) a. s. for every t h —P0 (b) The ( t) ICR of (Nt) is given by t A = f k ds Proof;: By (i) and (ii) (5.3) lim(tlim (t,h, wC) =lim a (t, o) (, c) (5.3) lim (t~w) Xt((W) m h mn tM h->0 mrn m h-0O m where the first equality follows by the uniform convergence on (0, h (o)]. 0 Assumption (ii) also implies for almost all a and h < h (oo) C Qa (t,h,,) < a(t, )h () < oo m o m and this is enough to justify the equality D m(Q -Q m+) = Q m m+l m m m But m nm+l = {Nt+h t t} so that the above relation gives for h < h () 0 (5.4) E( -hNt t)= Q (t,h, ) m and by (5. 3) N -N (5.5):lim Qm(t,h,) = lim E (t+h t ) t h t h h-* O h —.O (b) The CP (Nt) is right-continuous? t

32 there exists a right-continuous modification for the submartingale (E(Nt+hl|t)) (see Definition 27-VII of [9]) and we denote by (Pht) a right-continuous modit+h t fication of the process (E( t+h t t We have seen above that lim ph = t a. s. By (ii) and (5.4) h-O0 0 ~ -n t h m~ m rn for h < h( ). Hence the integral t f p h ds 0h s is well defined for almost all o and by the dominated convergence theorem t t (5.6) lim j p\ds= J \ds a.s. h-'.O 0 0 Denote by (At) the (t/ ) ICR of (Nt) and define as usual the martingale (M, ~ N -t t (Mt Nt - At). Let c be any positive constant and define (5.7) P (Ac A It is easy to check that (P ) is a potential and by Theorem 29-VII of [91 we know that for each t t (Ca8 S 1 E(pC E pC 1 00 ~~(5.8 J ~h s s+h E s)ds h -0 tAC where convergence is in the weak sense in L. Now (A N - M ) tAC tAC tAC so that by (5. 7) P t [E(Acl,i) + M )] N t c ~ t tA C tAC

33 where (Mt ) is not only a ( ) but also a ( ) martingale. Hence tAC tAC t for s < t and if we choose c > t + h E(PC- P +h ) = E(Ns+h Nsl s) s s+h s s+h s' s Thus on the one hand by (5. 8) t: (L1, L ) f E(N - N- )Ws. A Joh Es+h s s h 0 t 0 and on the other by (5.6) t as t J0s +h s s h -- -0 s so that we must have a. s. for each t t A = f ds - t s 0 Our last result shows that the two conditional rates of a same CP (N) t but with respect to two families ( t) and (. t) such that 0t ot o t are related by a simple expression. THEOREM 5. 0. 2: Let (Nt) be a CP with finite mean. Denote its conditional rate with respect to the family (-? t) by (Xt) Let ( t) be another A family such that 2t c t c.t' Then the conditional rate (t) of (N) with respect to (t ) exists and is given by t t E(\tl t) Remark: Note that this result makes good intuitive sense, the conditional rate (t\) being the best mean square estimate of the conditional rate (Xt). t t

34 Proof: Part of this proof is a consequence of the innovation theorem ([1], Theorem 1. 1), i.e., the process (N - ds) is a ) martingale. t Now the process (J x ds) is increasing, continuous hence natural and conse0 quently is the (:t) ICR of (Nt) by the uniqueness of the Doob-Meyer decomposition.

35 REFERENCES 1. P. M. Bremaud, "A Martingale Approach to Point Processes," Memorandum No. ERL-M345, Electronic Research Laboratory, University of California, Berkeley, August 1972. 2. F. J. Beutler and 0. A. Z. Leneman, "The Theory of Stationary Point Processes, "Acta Mathematica, vol. 116, 1966. 3. J. R. Clark, "Estimation for Poisson Processes with Application in Optical Communication, "Ph. D. Thesis, M. I. T., September 1971. 4. C. Doleans-Dade and P. A. Meyer, "Integrales stochastiques par rapport aux martingales locale, "Seminaires de Probabilites IV, Lecture Notes in Mathematics No. 124, pp. 77-107, SpringerVerlag, Berlin, 1970. 5. F. B. Dolivo, "Counting Processes and Integrated Conditional Rates: A Martingale Approach with Application to Detection, " Ph. D. Thesis, The University of Michigan, June 1974. 6. K. Ito and S. Watanabe, "Transformation of Markov Processes by Multiplicative Functionals, "Ann. Inst. Fourier, Grenoble, Vol. 15, No. 1, pp. 13-30, 1965.

36 7. H. Kunita and S. Watanabe, "On Square Integrable Martingales," Nagoya Math. J., Vol. 30, pp. 209-245, 1967, 8. P. A. Meyer, "A Decomposition Theorem for Supermartingales, " Illinois J. of Math., t. 6, pp. 193-205, 1962. 9. P. A. Meyer, "Probability and Potential, "Blaisdell, Waltham, Mass., 1966. 10. P. A. Meyer, "Un lemme de theorie des martingales, " Seminaire de Probabilites III, Lecture Notes in Mathematics No. 88, pp. 143144, Springer-Verlag, Berlin, 1969. 11. I. Rubin, "Regular Point Processes and their Detection, "IEEE Trans. on Information Theory, Vol. IT-18, No. 5, pp. 547-557, September 1972. 12. D. L. Snyder, "Filtering and Detection for Doubly Stochastic Poisson Processes, "IEEE Trans. on Information Theory, Vol. IT-18, No, 1, pp. 91-102, January 1972. 13. D. L. Snyder, "Smoothing for Doubly Stochastic Poisson Processes," IEEE Trans. on Information Theory, Vol. IT-18, No. 5, pp. 558-562, September 1972.

37 14. D. L. Snyder, "Information Processing for Observed Jump Processes, "Information and Control, Vol. 22, No. 1, pp. 69-78, 1973. 15. A. Segall, "A Martingale Approach to Modelling, Estimation and Detection of Jump Processes, "Technical Report No. 7050-21, Center for Systems Research, Stanford University, August 1973.

REFERENCES 1. P. M. Bremaud, A martingale approach to point processes, Memorandum No. ERL-M345, Electronic Research Laboratory, University of California, Berkeley, California, August 1972. 2. F. J. Beutler and 0. A. Z. Leneman, The theory of stationary point processes, Acta Mathematica 116(1966), pp. 159-197. 3. J. R. Clark, Estimation for Poisson Processes with Application in Optical Communication, Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, September 1971. 4. C. Doleans-Dade and P. A. Meyer, Integrales stochastiques par rapport aux martingales locale, Seminaires de Probabilites IV, Lecture Notes in Mathematics No. 124, Springer-Verlag, Berlin, 1970, pp. 77-107. 5. F. B. Dolivo, Counting Processes and Integrated Conditional Rates: A Martingale Approach with Application to Detection, Ph. D. Thesis, The University of Michigan, Ann Arbor, Michigan, June 1974. 6. K. Ito and S. Watanabe, Transformation of Markov processes by multiplicative functionals, Ann. Inst. Fourier, Grenoble, 15:1 (1965), pp. 13-30. 7. H. Kunita and S. Watanable, On square integrable martingales, Nagoya Math. Journal, 30(1967), pp. 209-245.

8. P. A. Meyer, A decomposition theorem for supermartingales, Illinois Journal of Math., 6(1962), pp. 193-205. 9. P. A. Meyer, Probability and Potentials, Blaisdell, Waltham, Massachusetts, 1966. 10. P. A. Meyer, Un lemme de theorie des martingales, Seminaire de Probabilites III, Lecture Notes in Mathematics No. 88, SpringerVerlag, Berlin, 1969, pp. 143-144. 11. I. Rubin, Regular point processes and their detection, IEEE Transactions on Information Theory, IT-18:5, September 1972, pp. 547557. 12. D. L. Snyder, Filtering and detection for doubly stochastic Poisson processes, IEEE Transactions on Information Theory, IT-18:1, January 1972, pp. 91-102. 13. D. L. Snyder, Smoothing for doubly stochastic Poisson processes, IEEE Transactions on Information Theory, IT-18:5, September 1972, pp. 558-562. 14. D. L. Snyder, Information processing for observed jump processes, Information and Control, 22:1 (1973), pp. 69-78. 15. A. Segall, A martingale approach to modelling, estimation and detection of jump processes, Technical Report No. 7050-21, Center for Systems Research, Stanford University, Stanford, California, August 1973.

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