CONDITIONAL PROBABILITY GENERATING FUNCTIONS OF COUNTING PROCESSES F. B. Dolivo and F. J. Beutler Computer, Information and Control Engineering Program The University of Michigan, Ann Arbor, Michigan 48104 September 1974 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 Conditional probability generating functions (CPGF) of counting processes (CP) are studied; these determine expressions for the probabilities of numbers of counts in an interval for special cases, and are in general required for applications to prediction, estimation and detection. Martingale theory-in particular the Meyer-Doob decomposition and the Doleans-Dade integral equation-leads to the desired CPGF in terms of the integrated conditional rate (i. e., natural increasing process) appearing in the Doob-Meyer decomposition. It is shown that the integrated conditional rate is nonrandom iff the CP has independent increments; the CP then generalizes the Poisson process in the sense that the mean of the CP need not be continuous. For a CP mean with discontinuities, the CPGF involves coefficients furnished by a specified infinite product. The other special case requires a conditional independence condition between the count and the rate. Here the CPGF is used to derive the probability of the number of counts in an interval. The resulting formula looks like the analogous expression for a Poisson process, but is actually a generalization in which the rate (known for a Poisson process) is replaced by the conditional expectation of the rate, given the past of the CP. 1

2 1. INTRODUCTION Recently martingale theory has been used to study Counting Processes (hereafter abbreviated CP). This has given rise to the notion of Integrated Conditional Rate (ICR)([5]). We examine here an efficient martingale technique to compute some conditional probability generating functions for CP's. First we obtain a general expression (involving the ICR) of this conditional probability generating function. For CP's with independent increments this expression gives actually an integral equation for this conditional probability generating function; furthermore, due to a result of DoleansDade [4], the unique solution of this equation, i. e. the conditional probability generating function, can be obtained. As a first consequence we derive a unique characterization of CP's with independent increments. Then we compute the probability of having n jumps in an interval (s, t]. This result is well known when the mean of the process is continuous, but our derivation extends to the general case where no conditional rate is assumed to exist. Finally we show how similar results can be obtained for CP's having a conditional rate satisfying some kind of conditional independence property. 2. PRELIMINARIES Let (Q2, 7, P) be a complete probability space. By (Xt) we denote a real valued stochastic process defined on R+ (the positive real line) and by a Counting Process (CP) we mean

3 Definition 2. 1: A CP is a stochastic process having sample paths zero at the time origin, and consisting of right-continuous step functions with positive jumps of size one. Let ( t) be a right-continuous increasing family of ar-subalgebras of 4 with o containing all the P negligible sets. Now if (Nt) is a CP adapted to 7t, with the sole assumption (i) the random variable Nt is a. s. finite for each t, then as a consequence of the Doob-Meyer decomposition for supermartingales we can associate to (Nt) a unique natural increasing process (At), dependent on the family ( t), which makes the process (Mt Nt - At) a square integrable ( t) local martingale [3]. When the mean mt = ENt of the CP is finite then the process (Mt) is actually a martingale. This decomposition (Nt = Mt + At) is intuitively a decomposition into the part (Mt) which is not predictable and (At) which can be perfectly predicted. This unique process (At) is called the Integrated Conditional t Rate (ICR) of (Nt) with respect to ( (t) ("the ( t) ICR of (Nt)") and has been studied in [5]. The terminology ICR is motivated by the fact that when t (Nt) satisfies some sufficiency conditionsits ICR takes on the form (f \ ds) ~~~t~~~~~~~0 s where (\t) is a nonnegative process called the conditional rate (with respect to ( )) satisfying \ = lim E[h (Nt+h N t) t] ([5], Section 2.5). t h t+h t t The existence of CP's which possess a bounded conditional rate with respect to the family of or-algebras generated by the process itself has been first shown in [1] and in [5]. Sufficiency conditions for the existence of a conditional rate have been given in [5].

4 We assume here that modern martingale theory ([3], [7]) is known. Recall that a semimartingale (X ) is a process which can be written as a sum (Xt = X + L +At) where X is a -measurable, (Lt) is a ( ) local martingale,and (At) is a right-continuous process adapted to ( t) with A = 0 a. s. and having sample paths of bounded variation on every finite interval (see [3]). A result basic to this study and due to Doleans-Dade [4] is the following: the stochastic integral equation t Z= 1 + Z dX t 0 s- s with (Xt) a semimartingale has a unique solution, which is a semimartingale given byt t t 2 t St s s<t where the product in the right hand side converges a. s. for each t. Here we define (<>t) as the unique natural increasing process associated to the continuous part of the local martingale (Lt); (<Xc>) is identically zero when (Xt) is a semimartingale with sample paths of bounded variation on every finite interval (see [3]). 3. A FORMULA FOR THE CONDITIONAL PROBABILITY GENERATING FUNCTION Let (Nt) be a CP with Nt a. s. finite for each t and adapted to a family ( t ). Its conditional probability generating function 4(z, t, s) is defined for When f tis a right-continuous function with left-hand limits Aft denotes the jump f - f. tt

5 t >s by: (NN-N) nn p (3. 1) (z, t, s) = E[z t I ] E z P{Nt N nl n where z is a complex number with zl < 1. Denote by (At) the ( t) ICR t t of (Nt). Recall that the process (Mt N -At) is a ( ) local martingale Lemma 3. 1: The conditional probability generating function is given by t (N -N ) V- S (3.2) q$(z,t,s) = 1 + (z-1)E ( z N sdA Iji 8 Proof: The CP (Nt) is a right-continuous step process with ANt being either zero or one so that for t > s N N N N N t s A v D,v vz -z = Az = ( V - z ) S<v< t s<v < t AN N = (z - 1)z s<v< t N (z - 1) z VAN s<v < t t N = (z- 1) z dN s where L is the sum over the discontinuities of (Nt) in (s, t]. Using the t expression (Nt Mt + At) in the above gives N N t s z = - t N t N (3.3) + ('z- 1) z dM + z dA s s

6 Let (T ) be a sequence of stopping times reducing the local martingale (M) n t i. e. the stopped process (Mt ^ T ) is a uniformly integrable martingale n for each n (see [3]). The sample paths of (Mt ^ T ) are of bounded variation n on every finite interval and I z < 1 so by Proposition 2 of [3] the process t N t N (J v z- dM T ) is a martingale. In particular E(f z dM AT s n -s n = 0. Now t N t N f z v dM T = f Z {v < T} dM v AT v- n v s n s and it follows from the bounded convergence theorem that t N E(f z V dM l) = 0. s -N Substituting the above relation in (3. 3) and multiplying both sides by z we get the desired result (3. 2). l Formula (3. 2) can be generalized to the case where the jumps of the process (Nt) are of random size. This formula would then contain, in place of the term (z-1), a random quantity which is a function of the random size jumps ANt and z. This additional randomness makes this formula difficult to manipulate and perhaps of less value. Accordingly, we shall limit our future considerations to CP's. 4. APPLICATION TO PROCESSES OF INDEPENDENT INCREMENTS Suppose now that (Nt) is a CP of independent increments with finite mean mt and consider the family of -algebras ( (N t)) mean m and consider the family of cr-algebras (,I'j ='a- (N, 0 < u < t)) generated by the process itself up to and at time t. We will show that CP's of independent increments are uniquely distinguished by the fact that their

7 (1t) ICR is deterministic and given by the mean mt. Also, the probability generating function 4(z, t, s) and the probability P{N - N n} will be comt S puted. The method used to devise these formulas is appealing as it does not require the mean m to be continuous, and hence generalizes currently known formulas [8]. Theorem 4. 1: Let (N ) be a CP with finite mean mt for each t. Denote its t t (Nt) ICR by (At) Then (a) (Nt) is a CP of independent increments if and only if the ICR (At) is deterministic. (b) If the ICR (At) is deterministic then At = m (c) The probability generating function of a CP with independent increments is given by (4. 1) (z, t, s) = exp[(z-l)(mt-ms)]' I [l+(z-l)m v]exp[(1-z)in ] s <v<t (d) Denote by ti the (at most countable) times of jump of mt on the interval (s, t]. Define (4. 2) t m -m- Am 5 - t- Ms - V s <v<t (1) If the number of jumps of mr in (s, t] is infinite then the product II [ + (z-l)Am ] is uniformly convergent in the i 1 region I z < 1 to an analytic function and we denote by ak the coefficients of the Taylor expansion of the above infinite product.

8 (2) If the number j of jumps of mt in (s, t] is finite the coefficients ak of the Taylor expansion can be computed by formulas (4. 9) to (4. 11) below. In particular, if mt is continuous (j=O) then a = 1, ak = 0, k > 1 and t 0~ t 6 = m-m. s t s (3) We have n ak t n-k (4. 3) PN -N=n) = exp{-6t} ( (6 )k) s s S k=O s Remark: Observe that when m is continuous we get the well known formulas 4(z,ts) = exp[(z-l)(mt-m)] and L S (4.4) P{N-N = n = (m-m )n exp[-(-m )] for the Poisson counting process with variable rate. Proof: (a) ( => ) It is easy to show that the process (N - mt) is a (/t) martingale.t Furthermore the increasing process mt is natural because it is deterministic, so (Nt) has the unique Doob-Meyer decomposition (Nt = (Nt-mt) + mt)and by definition mt is the (/ t) ICR of (Nt). (<= ) From formula (3. 2) t (N -N) J(z,t,s) 1 + (z-l) E( Zv- dA \ The process (At) is deterministic and by Fubini's Theorem we can obtain t (N -N ) t N -N E( z V SdA ) = f E(z v- )dA s s t Here is the completion of the o —algebra generated by {N, 0 < t}.

9 so that (z,, t, s) satisfies the following integral equation: t (4.5) q(z,t,s) = 1 + (z-1)f 4(z,v-,s)dA v s By the Doleans-Dade result, Theorem 1 of [4], the above equation has the unique solution (4.6) (z, t, s) = exp[(z-l)(At-A )] 7 [1+(z-l)AA ]exp[(l-z)AA ] s<v< t The right hand side of this relation is a deterministic function and it follows that (Nt) is a process with independent increments. A (b) By definition of the (t) ICR (At) the process (Mt N -At) is a local martingale. This process is in fact a martingale since mt = ENt is finite for each t (see [5], Theorem 2. 3. 1) and the result follows from EM = 0 = EN - At. L Lt t. Part (c) is a restatement of (4. 6) where we have used the fact that At =mt(d) For a process of independent increments we clearly have P{N -N = n } = P{Nt - N = n. Now define t = m -m - nt. s t s 1 Formula (4. 1) can be rewritten with the above relation in the form (4. 7) 4(z,t, s) = exp{-6t}exp{z6t} II [1 + (z-l). s s.i t. i 1 We examine now the infinite product (4.8) II [1 + (z-l)Am ] i i

10 Observe that: (a) for each n the partial product n f (z) = n [1 + (z-l)mt ] n. t. i=l 1 is analytic in the complex plane and (b) the series Z |(z-l)Am I i ti is uniformly convergent in the region zl < 1. This last point follows from the Weierstrass test: (z-l)Amt I < 2Amt i 1 and because the mean m is finite for each t the series _m Am < < m < oo t. t i 1 is convergent. Conditions (a) and (b) above imply that the infinite product (4. 8) converges uniformly to a function f(z) which is analytic in the region j z < 1 (see [6], Corollary to Theorem 8.6.3; or [2], Theorem 5.4.8). Hence we can get a Taylor series expansion for f(z) = II [1 + (z-l)Amt] i i1 in the region zl < 1. f(z) = F az = I [1 + (z-l)Amt J i it This power series can also be differentiated term by term in the region I z < 1. It is then easy to compute from (4. 7)

11 P{Nt-N = n = n d( tn) dz 1 t d(n): exp(-6t) (exp z6t s alz s dzn s n an-k = exp (6t) (n-k (6t)n s k=0 (n k) Now if the mean mt has only a finite number of jumps j > 1 in the interval (s, t then the coefficients ak are such that a az = n1 [+ (z-l)Amt ]; ~1=1 Q=1 1 they can then be computed by (4.9) a = S (1 - nt ) i=l 1 For 0 <k <j k i ak 1 k ( I Am )[ I (1-Am )] all permutations q=l I q=k+l Q (4. 10) q q=l,.,} of {1,........j} k (4. 11) ak = Am for k=j, 1 and finally for k > j, ak = 0. If j = 0 (continuous case) then II [ + (z-l)Am ] = 1 so that a = 1 and ak =0, i 1 k,,and result (4. 3) reduces to (4.4) (6 = mt - m in this case). I1 -- S t S'

12 If we define a non-homogeneous Poisson process (N t) as being a CP Nt of independent increments with a characteristic function E z given by t exp{(z-1) f \ ds} where X is a nonnegative function called the rate, then s t we have: Corollary 4. 2: A CP (Nt) of independent increments with finite mean mt for each t is a non-homnogeneous Poisson process if and only if the mean m is absolutely continuous. The rate Xt is then given by the Radon-Nikodym dm derivative. dt Proof: By Theorem 4. 1 it is easy to see that N t E z = exp{(z-1) \ ds} 0 if and only if mt = f \ds. 0 5. APPLICATION TO COUNTING PROCESSES WITH A CONDITIONAL RATE Assume now that (Nt) is a CP with finite mean for each t and for which a conditional rate (X ) with respect to a family ( bt) exists and satisfies the condition N N (5. 1) vEz vl1 ) = E(z v- )E(I'1 ) for all v > s. This condition will be discussed later on. From (3. 2) we get t (N -N ) (5.2) 4(z,t,s) = 1 + (z-l)E ( z v- x dvl ) 5 ~

13 t Now the CP has a finite mean so that E f X ds is finite which implies 0 t (N -N) E lz I Xdv < oo for Izl <1 v s Then by Fubini's Theorem t (N -N ) t (N -N) E[f z x dvls ] = f E[z x x ]dv v s v s s s Hence by the above relations (5. 1) and (5. 2) one has t A s (5.3) L(z,t,s) = 1 + (z-l)J (z,v,s)X dv v s where X = E(k i ) i.e., (s ) is the minimum mean square error v s v prediction of (k ) based on past information up to and at time s. As before, this equation has a unique solution which is a semimartingale (Theorem 1, [4]) t As (5.4) t(z,t,s) = exp{(z-l) X dv} and (5. 5) {Nt-NJ = nl } - dv) exp{-n Xdv} s s It is interesting to note that both these formulas generalize the corresponding expressions for Poisson processes directly, with the best estimate X of X v v replacing the latter, which is deterministic for a Poisson process. All this is vr a alin s true only if condition (5. 1) is satisfied. This condition which can be rewritten (by adding and subtracting terms) as

14 (N — N ) (N -N) (5.6) E[z ( - ) = E[z ]E[( ]V is difficult to interpret. But in the particular case where s = 0 and 5 = {% 1, } (this is the case for = 1it)the above condition (5. 1) becomes N N v- V(5.7) E(z v ) = E(z )E(\ ) v>0 V V - and is satisfied if for each t the two random variables Nt and t are independent. This seems a reasonable assumption if we suppose the value of N does not influence the rate at time t. Then under this condition (5. 7) relation (5. 5) gives t t P tN =} = n (J (EX)dv)n exp{- (EX )dv) 0 0

15 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. J. Depree and C. C. Oehring, Elements of Complex Analysis, Addison-Wesley, Reading, Massachusetts, 1969. 3. C. Doleans-Dade and P. A. Meyer, Integrales stochastiques par rapport aux martingales locale, Seminaires de Probabilites IV, Lecture Notes in Mathematics No. 24, Springer-Verlag, Berlin, 1970, pp. 77-107. 4. C. Doleans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, Z. Wahrscheinlichkeitstheorie verw. Geb., 16(1970), pp. 181-194. 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. E. Hille, Analytic Function Theory, Blaisdell, Waltham, Massachusetts, 1963. 7. P. A. Meyer, Probability and Potentials, Blaisdell, Waltham, Massachusetts, 1966. 8. E. Parzen, Stochastic Processes, Holden Day, San Francisco, California, 1962.

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