RECURSIVE INTEGRAL EQUATIONS FOR THE DETECTION OF COUNTING PROCESSESt 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. A recursive stochastic integral equation for the detection of Counting Processes is derived from a previously known formula [5] of the likelihood ratio. This is done quite simply by using a result due to Doleans-Dade [4] on the solution of stochastic integral equations. 1. INTRODUCTION. Recently modern martingale theory has been used to describe Counting Processes (hereafter abbreviated CP) in a way specially appropriate to the problems of detection and filtering. This has given rise to the notion of Integrated Condtional Rate (ICR) [5], which generalizes the notion of random rate. Expressions for likelihood ratios (involving ICR's) for the detection of CP's have been obtained in [5] using a three-step technique introduced by Kailath [9] and Duncan ([6], [7]) in their works on detection of a stochastic signal in white noise. The three steps are the Likelihood Ratio Representation Theorem ([2], [5], [6]), the Girsanov Theorem ([5], [8], [13]) and the Innovation Theorem ([2], [5], [9]). By this method likelihood ratios for a large class of CP's can be found. These expansions represent a generalization of the formulas given in [1] and [12] in the context of Poisson processes and [2] in the context of CP's which admit a conditional rate. The purpose of this paper is not to present a proof of the likelihood ratio formula (for that see [5]) but to derive from this formula stochastic integral equations by which the likelihood ratio can be computed recursively. This can be done quite simply using auresult due to Doleans-Dade [4] on the solution of stochastic integrals equations involving semimartingales. These recursive equations are most useful in applications as they give a way of 1

2 implementing the computation of the likelihood ratio continuously in time. 2. PRELIMINARIES. Let (Q,, P) be a complete probability space. By (Xt) we denote a real valued stochastic process defined on t+, the positive real line and by a Counting Process (CP) we mean Definition 2. 1: A CP is a stochastic process having sample paths which are zero at the tire origin and consisting of right-continuous step functions with positive jumps of size one. th The time of n jump J of a CP (Nt) is the stopping time defined by n t inf {t: N n} J = 1 n oo if the above set is empty. Let (t) be a right-continuous increasing family of or-subalgebras of i with o containing all the P negligible sets, and suppose (Nt) is a CP, adapted to 7t with the sole assumption that ENt is finite for each t. Then, as a consequence of the Doob-Meyer decomposition for supermartingales we can associate to (N) a unique natural increasing process (At), dependent on the family (t), which makes the process (Mt Nt - At) a martingale (see [11]). This decomposition (N = M + 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 Rate (ICR) of (Nt) with respect to (4t) ("the (kt) ICR of (Nt)") and has been studied in [5]. The terminology ICR is motivated by the fact that when (Nt) satisfies some

3 t sufficiency conditions its ICR takes on the form (J k ds) where (Xt) is a 0 nonnegative process called the conditional rate (with respect to ( st)) satisfying \ lim E[h ~(Nt+h-Nt)I t] ([5], Section 2.5). The existence of - t+h t t CP's possessing a bounded conditional rate with respect to the family of ar-algebras generated by the process itself has been first shown in [2] and in [5]. Sufficiency conditions for the existence of a conditional rate have been given in [5]. By a change of time we can show similar results (i. e., existence (see [5], Corollary 3. 1. 3) and sufficiency conditions) for (i t) ICR's t t of the form (f Xdm ) where (\ ) is a locally bounded predictable process s s t 0 and mt. a deterministic increasing right-continuous function with m = 0. Denote by / (it) the class of all locally bounded predictable (with respect to (it)) processes (see [3], p. 98). For example, processes adapted to (it) t and having left-continuous sample paths belong to -" (t). t Remark 2. 2: Let the ICR (A ) be of the form (f X dm ) and denote ~t 0 s s by A the union of all intervals of IR+ on which the function m is constant. Observe that the ICR (At) is not affected by a change of values of ( t) for t E A and we may well have t = oo for t e A. To avoid problems due to this indeterminacy we adopt the following convention: for t E A we set t equal to unity. We assume here that modern martingale theory ([11], [3]) is known. Recall that a semimartingale (X ) is a process which can be written as a sum (Xt = X + + At) where X0 is ~ -measurable, (Lt) is a (it) local martingale and (At) is a right-continuous process adapted to ( t)

4 having sample paths of bounded variation on every finite interval and with A =0 a. s. (see [3]). A result basic to this study and due to Dole'ans-Dade [4] is the following: the stochastic integral equation t Z = 1+ Z dX Zt s- s where (Xt) is a semimartingale has a unique solution, which is a semimartingale given by Z exp(Xt -<XC>t) H (1 + AX )exp(-AX ) s<t where the product in the right hand side converges a. s. for each t. Here we define (<X >) as the unique natural increasing process (see [3]) associated to the continuous part of the local martingale (Lt); (<X >t) is identically zero when (Xt) is a semimartingale with sample paths of bounded variation on every finite interval (see [3]). 3. THE DETECTION PROBLEM. Let P and P be two measures 0 1 carried on ( i, ). Suppose that (N ) is a CP defined on (0, ) and denote by /t the minimal cr-algebra generated by (Nt) up to and at time t. The notation E. () for i=0, 1 is intended for the expectation operator with respect to the measure P.. 1 Definition 3. 1: For a (/t ) stopping time R (possibly infinite) de-R note by P for i=0, I the restriction of the measure P. to the cr-algebra i 1 When ft is a right-continuous function with left-hand limits Lft denotes the jump ft — ft -

5 We have the inclusion R c. so that if P0 << ~P then PR -R -R and the Radon-Nikodym derivative dP /dP R is well defined. We examine now the meaning of this Radon-Nikodym derivative. In the case where the stopping time R is equal to a constant a then TR =' a = r(N, 0 <u <a) R a U -a -a so that dP /dP is the likelihood ratio for testing the two hypotheses H. for 0 1 1 i=O, 1: P1 is the probability measure on (2, 7), by observations on the CP (Nt) for t < a. The detection scheme then consists in selecting H0 or H1. according as dP /dP~ is above or below a given threshold. Now in the case where R is a stopping time which is not a constant we know that c/R7 ^ ~(N 0 < u) (this follows from the fact that N is (7^) R\. U AR* ~ uAR R measurable by Theorem 49-IV of [11]) but the reverse inclusion is not -R - R necessarily true. For this reason dP /dP is not the likelihood ratio for our detection problem when the time of observation is the stochastic inter-R -R val [0, R], as one could have conjectured. But one can interpret dP /dP 0 1 as a likelihood ratio if we assume that the information accessible to the observer is 7R and not simply a(N 0 < u). For i=0, 1 with the measR uR' -'R ure P. carried on (Q, 6 ) suppose that the CP (Nt) has the process (f k dm ) 1 t s for ( t) ICR, where (;t) is a family of cr-algebras with D At' t t t (t) / ( t) is a positive process, and m is an increasing deterministic function with m = 0. It is known that we can make a change of measure under which (Nt) is a CP of independent increments with mean mt = ENt under the new measure P (Theorem 2. 6. 1 of [5]). Using this fact and the three-step technique tP << P means that the measure P0 is absolutely continuous with respect to P while P0 P indicates that the two measures are equivalent.

6 of Duncan and Kailath (see Introduction) the likelihood ratio for detecting CP's has been obtained according to THEOREM 3. 2 (Theorem 3. 4. 4 of [5]): For i=0, 1 let (Nt) be, under the measure P., the CP described above. Assume 1 (a) P0 << P and P - P and define for i=0, 1 the (P, 7't) martingale Li1 -00 t Lt E( co Ant); (b) For i=0, 1, the stopping times Ti are such that there exists ini i i creasing sequences of stopping times(T ) for which T = lim T a. s. and n n E(lnL ) < oo for eachn. Let T = T AT T' n t (c) For i=O, 1 E. \ dms < oo. Then (1) dPtAT J tAT 0 n A 1 A0 01) ii exp[f ( 1_- )dms] dPt J <tAT 0 1 n — where I n L A[ where t = E.(kt / ) for i=0, 1 and J is the time o n jump of (N By t i t L n t convention the product II( ) = 1 for J > t T. Remark 3. 3: (a) The stopping time T which is the first time after which the martingale (L ) can behave badly may take the value + oo. It is in fact desirable for T to be as large as possible. (b) By our convention (Remark 2. 2) condition (c) above insures that the process (\i) is well defined. t

7 4. RECURSIVE INTEGRAL EQUATIONS FOR LIKELIHOOD RATIOS We show here that the likelihood ratio (1) of our detection problem can be obtained as the unique solution of a stochastic integral equation. This stochastic integral equation can be mechanized by a feedback scheme tantamount to a recursive filter, as shown in Figure 1. THEOREM 4. 1: The likelihood ratio dPtT /dPtA of Theorem 3. 0 1 is the unique solution of the following stochastic integral equation: t (2) Z = + Z dX 0 ^ s-SAT where (3) X - dN t A s s s Proof: By assumption (Xt), i=0, 1, is positive a. s. finite for all t (by condition (c) of Theorem 3.2 and Remark 2. 2). The process (Nt) has a finite number of jumps in any finite interval so that the process tAT (J [( 0/A )-l]dN ) has sample paths of bounded variation on any finite 0 tAT interval; and so does the process (j ( -\ )dm ) by assumption (c) of 0 s s Theorem 3. 2. Hence (X ) is a semimartingale with sample paths of bounded variation on any finite interval so that (<XC> - 0) (see the tAT remark, on p. 90, following proposition 3 of [3]). Then by Theorem 1 of [4] the unique solution of (2) is given by (4) Z = exp(X T) II (1 + X ^)exp(-AX A ) A e sAT. SAT

8 Now AX = ((/l)- )AN T and hence the product in (4) becomes SAT S s SAT II ()= II [1+fjs 1 s jN. T]exP i LA~-T[A -l]sATI s ~t s<t ~sAr _8~ 1 A AT L1 d A <ntATK jS<tATj dtAT 5~0 z -=n nexpF dN J <tAT i 0 L Vn.JJ it Substituting the above relation and expression (3) in (4) gives the desired result (compare with (1)) Zt = n exp S (Si)dm = -t J E<tAT L dP nObserve that if under the measure P1 the CP (Nt) is a process of independent increments with mean mt then P - P1, x 1= 1 and Eq. (3) becomes (5) Xt = f (i'- l)d(N - m ) The process (M t N t - m) is a (P, 7Lt) martingale. Hence (5) shows that the process (Xt^T) is a local martingale. In turn, (2) then implies that the process (Zt) is a local martingale. In this case we in fact have Zt = Ei[(dPo/dP )| 77t T] i. e. the likelihood function is a uniformly integrable martingale. In applications, Eqs. (2) and (3) give a way of implementing the computation of the likelihood ratio continuously in time. They represent recursive

9 equations if one also obtains the best estimates (Xt) in a recursive manner. The block diagram of this implementation is given in Figure 1.

10 Nt IM MINIMUMUM MEAN MEAN SQUARE SQUARE ESTIMATOR ESTIMATOR 0I' At xt f\ \/.' LIKELIHOOD FUNCTION: ftl O)dm -= dPtT/d-Pt T S 00 0 X" ^ins- o/ T + ~~~t, DIVIDER Recursive Scheme for Obtaining the Likelihood Function Zt. Figure 1

11 REFERENCES 1. I. Bar David, Communication under the Poisson regime, IEEE Transactions on Information Theory, IT-15, January 1969, pp. 31-37. 2. P. M. Bremaud, A martingale approach to point processes, Memorandum No. ERL-M345, Electronic Research Laboratory, University of California, Berkeley, California, August 1972. 3. 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. 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. T. E. Duncan, On the absolute continuity of measures, Ann. Math. Stat., 41 (1970), pp. 30-38. 7. T. E. Duncan, Likelihood functions for stochastic signals in white noise, Information and Control, 16 (1970), pp. 303-310.

12 8. I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory of Probability and Its Applications, V:3 (1960), pp. 285-301. 9. T. Kailath, A further note on a general likelihood formula for random signals in a Gaussian noise, IEEE Transactions on Information Theory, IT-16, July 1970, pp. 393-396. 10. H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. Journal, 30 (1967), pp. 209-245. 11. P. A. Meyer, Probability and Potentials,Blaisdell, Waltham, Massachusetts, 1966. 12. B. Reiffen and H. Sherman, An optimum demodulator for Poisson processes: photon source detectors, Proceedings of the IEEE, 51, October 1963, pp. 1316-1320. 13. J. H. Van Shuppen and E. Wong, Transformation of local martingales under a change of law, Electronic Research Laboratory, Memorandum No. ERL-M385, University of California, Berkeley, California, May 1973.

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