Working Paper MORTAL1TY.AND AGING I N A HETEROGENEOUS POPULATION: A STOCHASTIC PROCESS MODEL WITH OBSERVED AND UNOBSERVED VARIABLES A n a t o l i I. Y a s h i n IIASA, h e n b u r g , Austria K e n n e t h G. M a n t o n Center for Demographic S t u d i e s , Duke U n i v e r s i t y , Durham, North Capo l i n a J a m e s W. V a u p e l I n s t i t u t e o f Policy Sciences and PAbZic A f f a i r s , Duke U n i v e r s i t y , Durham, North C a m 2 i n a S e p t e m b e r 1983 WP-83-81 International Institute for Applied Systems Analysis A-2361 Laxenburg,Austria NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHORS MORTALITY. AND AGING IN A HETEROGENEOUS POPULATION: A STOCHASTIC PROCESS MODEL WITH OBSERVED AND UNOBSERVED VARIABLES A n a t o l i I. Y a s h i n IIASA, Luxenburg, Austria K e n n e t h G. Manton Center for Demographic Studies, Duke University, Durham, North CaroZina J a m e s W. V a u p e l I n s t i t u t e o f Policy Sciences and Pub Zic Affairs, Duke University, Durham, North Caro lina September 1983 WP-83-81 Working P a p e r s a r e i n t e r i m r e p o r t s on work o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems A n a l y s i s and have r e c e i v e d o n l y l i m i t e d review. Views or o p i n i o n s e x p r e s s e d h e r e i n do n o t n e c e s s a r i l y r e p r e s e n t t h o s e o f t h e I n s t i t u t e o r o f i t s N a t i o n a l Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a Low f e r t i l i ty 1 eve1 s in IIASA countries are creating aging populations whose demands f o r health care and income maintenance (social security) will increase to unprecedented levels, thereby calling forth policies that will seek to promote increased family care and worklife f l e x i b i l i t y . The new Population Program will examine current patterns of population aging and changing l i f e s t y l e s in IIASA countries, project the needs f o r health and income support that such patterns a r e 1 ikely to generate during the next several decades, and consider alternative family and employment policies that m i g h t reduce the social costs of meeting these needs. A central feature of the Population Program's research agenda i s the development of a theoretical model of human agiqg and mortality. This paper reports the results of some preliminary e f f o r t s along that l i n e . In i t , a Soviet mathematician, Dr. Yashin, collaborating with a demographer and a pol icy analyst from the United States, describes a mu1 tidimensional stochastic process model that generalizes earl i e r models of aging dynamics. The authors introduce the effects of non-Markovian behavior, unobservable variables, and measurement e r r o r , showing how additional information about s t a t e variables influences an observer's understanding of temporal changes in the physiological sys tem . Andrei Rogers Leader Population Program A number of m u l t i v a r i a t e s t o c h a s t i c p r o c e s s models have b e e n de- v e l o p e d t o r e p r e s e n t human p h y s i o l o g i c a l a g i n g and m o r t a l i t y . In t h i s p a p e r , we e x t e n d t h o s e e f f o r t s by c o n s i d e r i n g t h e e f f e c t s of unobs e r v e d s t a t e v a r i a b l e s on t h e a g e t r a j e c t o r y of p h y s i o l o g i c a l p a r a m e t e r s . T h i s i s a c c o m p l i s h e d by d e r i v i n g t h e Kolmogorov-Fokker-Planck equations f o r t h e d i s t r i b u t i o n of t h e s t a t e v a r i a b l e s c o n d i t i o n a l l y on t h e p r o c e s s of t h e o b s e r v e d s t a t e v a r i a b l e s . P r o o f s a r e g i v e n t h a t t h i s form of t h e p r o c e s s w i l l p r e s e r v e t h e G a u s s i a n p r o p e r t i e s of t h e d i s t r i b u t i o n . S t r a t e g i e s f o r e s t i m a t i n g t h e p a r a m e t e r s of t h e d i s t r i b u t i o n of t h e uno b s e r v e d v a r i a b l e a r e s u g g e s t e d b a s e d on an e x t e n s i o n of t h e t h e o r y of Kalman f i l t e r s t o i n c l u d e s y s t e m a t i c m o r t a l i t y s e l e c t i o n . Implications of i n d i v i d u a l d i f f e r e n c e s on t h e t r a j e c t o r i e s of t h e u n o b s e r v e d p r o c e s s f o r o b s e r v e d a g i n g changes a r e d i s c u s s e d a s w e l l a s t h e c o n s e q u e n c e s of s u c h modeling f o r d e a l i n g w i t h o t h e r t y p e s of p r o c e s s e s i n h e t e r o g e n e o u s populations. CONTENTS I. INTRODUCTION 11. ALTERNATIVE FORMULATIONS OF A MODEL OF AGING AND MORTALITY 111. ESTIMATING THE UNOBSERVED VARIABLE IV. APPLICATIONS V. DISCUSSION APPENDIX A: PROOF OF THE GENERALIZED KOLMOGOROV-FOKKERPLANCK EQUATION ACKNOWLEDGEMENTS REFERENCES I. INTRODUCTION Background A. T h e r e h a v e b e e n a number o f e f f o r t s t o d e v e l o p a t h e o r e t i c a l model f o r human a g i n g and m o r t a l i t y . an e a r l y such a t t e m p t . The l a w of m o r t a l i t y due t o Gompertz (1825) was H e r e , human m o r t a l i t y i s modeled a s a uni-dimen- s i o n a l f a i l u r e p r o c e s s b a s e d on a c o n s t a n t l o s s of v i t a l i t y . esting It is i n t e r - t h a t t h e "Gompertzian model" of human a g i n g dynamics c o n t i n u e s t o b e a p p l i e d e s p e c i a l l y f o r m o r t a l i t y a t advanced a g e s ( F r i e s , 1 9 8 0 ) . Such s i m p l e t ' E a i l u r e p r o c e s s " models of human a g i n g and m o r t a l i t y , although perhaps u s e f u l d e s c r i p t i v e t o o l s , a r e n o t t o t a l l y s a t i s f a c t o r y models of human a g i n g p r o c e s s e s f o r a number o f r e a s o n s . p l y t h a t human a g i n g p r o c e s s e s a r e u n i - d i m e n s i o n a l . F i r s t , t h e y im- It seems e x t r e m e l y u n l i k e l y t h a t t h e p h y s i o l o g i c a l dynamics of t h e g e n e t i c and e n v i r o n m e n t a l d e t e r m i n a n t s of human a g i n g c o u l d b e d e s c r i b e d by a u n i - d i m e n s i o n a l process. Second, c o n s i d e r a b l e e m p i r i c a l e v i d e n c e h a s a c c u m u l a t e d t o show t h a t human m o r t a l i t y p a t t e r n s a t l a t e r a g e s a r e n o t w e l l - d e s c r i b e d by t h e Gompertz f u n c t i o n ( e . g. , H o r i u c h i and C o a l e , 1 9 8 3 ; W i l k i n , 1982) . Third, we o f t e n h a v e a w i d e r a n g e of p h y s i o l o g i c a l c o v a r i a t e s a v a i l a b l e f o r anal y s i s from l o n g i t u d i n a l l y f o l l o w e d p o p u l a t i o n s . The s i m p l e model of Gom- p e r t z i a n a g i n g dynamics c a n n o t u s e i n f o r m a t i o n on t h o s e c o v a r i a t e s . In- d e e d , s u c h models do n o t e x p l i c i t l y d e s c r i b e t h e p h y s i o l o g i c a l mechanisms underlying t h e aging process. Thus, i t i s n e c e s s a r y t o d e v e l o p models which c a n s u c c e s s f u l l y u t i l i z e t h i s i n f o r m a t i o n . A number of models of human a g i n g and m o r t a l i t y h a v e b e e n d e v e l o p e d which do d e s c r i b e t h e p h y s i o l o g i c a l mechanisms u n d e r l y i n g a g i n g c h a n g e s . Several of t h e s e a r e r e p o r t e d i n C h a p t e r 7 of S t r e h l e r ( 1 9 7 7 ) . Perhaps o n e o f t h e most s u c c e s s f u l of t h e s e models was d u e t o S a c h e r and T r u c c o (1962). T h i s model d e s c r i b e s p h y s i o l o g i c a l a g i n g a s a p r o c e s s by which homeostasis w a s maintained i n a m u l t i - v a r i a t e state space. was d e s c r i b e d i n t h e model i n o n e of two ways. Mortality F i r s t , i f one assumed t h a t t h e s t a t e s p a c e was o f h i g h d i m e n s i o n a l i t y , m o r t a l f t y was d e s c r i b e d as a permanent l o s s of h o m e o s t a s i s due t o t h e e x c e e d a n c e of some p h y s i o l o g i c a l threshold. S i n c e s u c h a f o r m u l a t i o n would o n l y b e of t h e o r e t i c a l u s e , i t w a s a r g u e d t h a t m o r t a l i t y m i g h t a l s o b e modeled as a n a b s o r b i n g bound- ary. Such a b s o r b i n g boundary f o r m u l a t i o n s of m o r t a l i t y l e a d t o s e r i o u s d i f f i c u l t y i n e m p i r i c a l a p p l i c a t i o n s s i n c e : a . ) t h e y imply t h a t one must d e a l w i t h t r u n c a t e d d i s t r i b u t i o n f u n c t i o n s , and b .) t h e y r e p r e s e n t m o r t a l i t y a s a d e t e r m i n i s t i c f u n c t i o n of t h e s t a t e s p a c e v a r i a b l e s . To d e a l w i t h t h i s problem, Woodbury and Manton (1977) p r e s e n t e d a t h e o r y of human a g i n g and m o r t a l i t y composed of two p a r a l l e l p r o c e s s e s . The f i r s t i s a m u l t i - v a r i a t e s t o c h a s t i c p r o c e s s d e s c r i b i n g t h e change i n t h e d i s t r i b u t i o n f u n c tion for the s t a t e variables. The second i s a jump p r o c e s s w h i c h r e p r e - s e n t s m o r t a l i t y a s a p r o b a b i l i s t i c f u n c t i o n of a n i n d i v i d u a l ' s s t a t e s p a c e values. T h i s model h a s b e e n s u c c e s s f u l l y a p p l i e d t o b o t h e p i d e m i o l o g i c a l s t u d i e s of c h r o n i c d i s e a s e r i s k (Woodbury e t a l . , 1979) and t o l o n g i t u d i n a l s t u d i e s of normal a g i n g p r o c e s s e s (Woodbury a n d Nanton, 1983; Manton and Woodbury, 1983) . I n t h e Woodbury and Manton (1977) model, i t i s assumed t h a t a l l r e l e v a n t s t a t e v a r i a b l e s a r e observed. sumption i s o n l y a n a p p r o x i m a t i o n . C l e a r l y , i n p r a c t i c e such a n as- C o n s e q u e n t l y , i n t h i s p a p e r we ex- t e n d t h e Woodbury and Manton t h e o r y of human a g i n g and m o r t a l i t y t o i n - ' c l u d e e x p l i c i t c o n s i d e r a t i o n of t h e e f f e c t s of u n o b s e r v e d s t a t e v a r i a b l e s i n t h e process. A G e n e r a l i z a t i o n of Aging Dynamics To D e a l With Observed and B. Unobserved S t a t e V a r i a b l e s : The Problem I n Woodbury and Manton, a t h e o r y of human a g i n g i s b a s e d on a mathe m a t i c a l model of t h e change o v e r t i m e of a m u l t i v a r i a t e d i s t r i b u t i o n funct i o n t h a t d e s c r i b e s t h e l o c a t i o n of a p o p u l a t i o n i n a m u l t i d i m e n s i o n a l s p a c e of s t a t e v a r i a b l e s . A l t e r n a t i v e l y , t h e d i s t r i b u t i o n f u n c t i o n can b e in- t e r p r e t e d as d e s c r i b i n g t h e p d a b i l i t y c h a r a c t e r i s t i c s a t some a g e . t h a t a n i n d i v i d u a l h a s some set of The s t a t e s p a c e d o e s n o t i n c l u d e a l l f a c t o r s r e l e v a n t t o t h e t i m e p a t h and s u r v i v a l of a n i n d i v i d u a l . t o r s m a n i f e s t t h e m s e l v e s i n two ways. i n t h e s p a c e i s t o some e x t e n t random: The o m i t t e d f a c - F i r s t , t h e movement of a n i n d i v i d u a l a n i n d i v i d u a l ' s t i m e p a t h i s gov- e r n e d by a s e t of s t o c h a s t i c ( r a t h e r t h a n d e t e r m i n i s t i c ) d i f f e r e n t i a l equations. Second, a n i n d i v i d u a l ' s p o s i t i o n i n t h e s p a c e does n o t d e t e r - mine m o r t a l i t y , b u t m e r e l y t h e h a z a r d o r f o r c e of m o r t a l i t y . Woodbury and Manton d e s c r i b e t h e change i n t h e m u l t i v a r i a t e d i s t r i b u t i o n of t h e s t a t e v a r i a b l e s by a Kolmogorov-Fokker-Planck ( U P ) equation. I n t h e KFP e q u a t i o n , t h e y s p e c i f y f o u r t y p e s of p h y s i o l o g i c a l dynamics: d r i f t (i.e., s y s t e m a t i c change i n mean v a l u e s ) , r e g r e s s i o n ( i . e . , conver- g e n c e t o mean v a l u e s , d u e p e r h a p s t o h o m e o s t a t i c t e n d e n c i e s ) , d i f f u s i o n (i.e., d i v e r g e n c e due t o random i n f l u e n c e s ) , and m o r t a l i t y s e l e c t i o n ( i . e . , l o s s from t h e p o p u l a t i o n of f r a i l i n d i v i d u a l s ) . t h e y assume t h a t t h e p r o c e s s i s Markovian. To a p p l y t h e U P e q u a t i o n Some a s p e c t s of a n a g i n g pro- c e s s , however, may depend on a n i n d i v i d u a l ' s e n t i r e l i f e h i s t o r y . I n t h i s p a p e r , we g e n e r a l i z e Woodbury and Pianton's model t o d e a l w i t h non-Markovian ror. processes , unobs e r v a b l e v a r i a b l e s , and measurement e r - We p r e s e n t o u r r e s u l t s i n a way d e s i g n e d t o show how a d d i t i o n a l i n f o r - m a t i o n a b o u t t h e s t a t e v a r i a b l e s i n f l u e n c e s a n o b s e r v e r ' s u n d e r s t a n d i n g of t h e temporal change of t h e p h y s i o l o g i c a l s y s t e m . Our model assumes t h a t e a c h i n d i v i d u a l i s c h a r a c t e r i z e d by a s e t o f v a r i a b l e s t h a t change o v e r t i m e . Some of t h e s e v a r i a b l e s a r e measured; t h e r e s t a r e n o t o b s e r v e d o v e r t i m e , b u t as i n t h e Woodbury-Manton model, some i n f o r m a t i o n i s a v a i l a b l e a b o u t them. S p e c i f i c a l l y , we assume know- l e d g e of t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e u n o b s e r v e d v a r i a b l e s a t t h e i n i t i a l t i m e z e r o a s w e l l a s of t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s des c r i b i n g t h e i r random t i m e p a t h . The s t o c h a s t i c i t y i n t h e a g i n g p r o c e s s i s g e n e r a t e d by a Wiener ( i . e . , Brownian m o t i o n ) p r o c e s s , as w e l l as by t h e randomness i n t h e i n i t i a l v a l u e s of u n o b s e r v e d v a r i a b l e s . The f o r c e of m o r t a l i t y i s a f u n c t i o n of a n i n d i v i d u a l ' s p o s i t i o n i n t h e s t a t e s p a c e W e d e a l w i t h t h e o b s e r v e d v a r i a b l e s by d e v e l o p i n g a form of t h e KFP e q u a t i o n t h a t d e s c r i b e s t h e c h a n g e i n t h e d i s t r i b u t i o n of t h e u n o b s e r v e d v a r i a b l e s c o n d i t i o n a l b o t h on s u r v i v a l t o a g e t and on t h e t r a j e c t o r i e s of t h e o b s e r v e d v a r i a b l e s . W e t h e n show t h a t i f t h e f o r c e of m o r t a l i t y f o r a n i n d i v i d u a l i s a q u a d r a t i c f u n c t i o n of t h e unobserved v a r i a b l e s , i t i s p o s s i b l e t o estimate t h e means and v a r i a n c e s of t h e unobserved v a r i a b l e s over t i m e . The e q u a t i o n s u s e d a r e similar t o t h e Kalman f i l t e r e q u a t i o n s d e v e l o p e d by communication t h e o r i s t s t o estimate s i g n a l s . The e q u a t i o n s , however, g e n e r a l i z e t h e u s u a l Kalman f i l t e r e q u a t i o n s t o i n c l u d e m o r t a l i t y . The f o r c e of m o r t a l i t y a s a f u n c t i o n of a g e and o b s e r v e d l i f e h i s t o r y can be d i r e c t l y e s t i m a t e d . A s n o t e d above, however, e s t i m a t e s b a s e d d i r - e c t l y on t h e o b s e r v e d d a t a p e r t a i n o n l y t o t h e s u r v i v i n g p o p u l a t i o n and n o t t o t h e p o p u l a t i o n a s a whole o r t o any homogeneous s u b g r o u p w i t h i n i t . The s u r v i v i n g p o p u l a t i o n d i f f e r s from t h e e n t i r e p o p u l a t i o n b e c a u s e o f systematic mortality selection. S p e c i f i c a l l y , individuals a t high mortal- i t y r i s k on t h e u n o b s e r v e d v a r i a b l e s w i l l d i e o f f more r a p i d l y and t h u s w i l l be underrepresented i n the surviving population. Thus, t o r e t r i e v e t h e p a r a m e t e r s of t h e p r o c e s s f o r t h e who>e p o p u l a t i o n , o r f o r s e l e c t i n - d i v i d u a l s , o n e ' s model of t h e p r o c e s s must a d j u s t f o r s e l e c t i o n o n b o t h o b s e r v e d and u n o b s e r v e d s t a t e v a r i a b l e s . We show t h a t , g i v e n t h e e s ti- m a t e s of t h e means and v a r i a n c e s of t h e u n o b s e r v e d v a r i a b l e s , one c a n c a l c u l a t e t h e f o r c e of m o r t a l i t y f o r i n d i v i d u a l s a t a g e t w i t h i d e n t i c a l observed a s w e l l a s unobserved c h a r a c t e r i s t i c s . Thus, t h e i m p a c t on a g i n g and m o r t a l i t y of e a c h of t h e o b s e r v e d and u n o b s e r v e d v a r i a b l e s c a n be identified. C. Orientation Our p r e s e n t a t i o n i s o r g a n i z e d a s f o l l o w s : --We d e s c r i b e t h r e e d i f f e r e n t f o r m u l a t i o n s of a model of a g i n g and m o r t a l - i t y b a s e d on Noodbury and Maneon's s u g g e s t i o n s . The f i r s t f o r m u l a t i o n d e s c r i b e s t h e p r o c e s s f o r a s i n g l e unobserved v a r i a b l e u s i n g a simple vers i o n of t h e Woodbury-Manton model. The s e c o n d f o r m u l a t i o n shows how t h e b a s i c p r o c e s s i s m o d i f i e d t o i n c l u d e o b s e r v a t i o n s of t i m e of d e a t h . The t h i r d f o r m u l a t i o n i n t r o d u c e s a s e c o n d s t a t e v a r i a b l e which i s c o n t i n u o u s l y monitored over time. For t h e s e t h r e e c a s e s , we d e r i v e t h e e q u a t i o n s , b a s e d o n t h e KFP e q u a t i o n , t h a t g i v e t h e ( c o n d i t i o n a l ) d e n s i t y of t h e u n o b s e r v e d variable. We d i s c u s s huw t h e v a r i o u s i n c r e m e n t s i n i n f o r m a t i o n a f f e c t t h e d e s c r i p t i o n of t h e dynamics of t h e a g i n g and m o r t a l i t y p r o c e s s . In a fourth s e c t i o n of t h i s p a r t of t h e p a p e r , we s k e t c h two e x t e n s i o n s of t h e model: we a l l o w t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s t h a t d e s c r i b e t h e t r a j e c t o r i e s of t h e v a r i a b l e t o depend on t h e e n t i r e h i s t o r y of t h e o b s e r ved v a r i a b l e , and we i n d i c a t e how t h e model c a n b e g e n e r a l i z e d t o an a r b i t r a r y number of o b s e r v e d and u n o b s e r v e d v a r i a b l e s . --We t h e n b r i e f l y r e v i e w t h e r e s t r i c t i o n s and a s s u m p t i o n s s u g g e s t e d by Wood- b u r y and Manton t o e s t i m a t e t h e d i s t r i b u t i o n of t h e u n o b s e r v e d v a r i a b l e s . We make some a n a l o g o u s r e s t r i c t i o n s and a s s u m p t i o n s and p r o v e some r e s u l t s c o n c e r n i n g t h e G a u s s i a n form of t h e d i s t r i b u t i o n . By e x t e n d i n g t h e t h e o r y of Kalman f i l t e r s , we p r e s e n t e q u a t i o n s f o r t h e mean and v a r i a n c e of t h i s distribution. I n a d d i t i o n , we g i v e t h e e q u a t i o n f o r c a l c u l a t i n g t h e f o r c e of m o r t a l i t y of i n d i v i d u a l s a t t i m e t w i t h any s p e c i f i e d s e t of o b s e r v e d and unobserved c h a r a c t e r i s t i c s . --Next we d i s c u s s a p p l i c a t i o n s of t h e model t o e m p i r i c a l s t u d i e s of a g i n g and m o r t a l i t y p r o c e s s e s w i t h o b s e r v e d and unobserved v a r i a b l e s . --We c o n c l u d e w i t h a d i s c u s s i o n of how o u r model of human a g i n g and mort a l i t y r e l a t e s t o o t h e r a t t e m p t s t o s t u d y t h e g e n e r a l problem of d e t e r m i n i n g t h e e f f e c t s on a s t o c h a s t i c p r o c e s s of s y s t e m a t i c p o p u l a t i o n l o s s due t o s e l e c t i o n o r t r a n s i t i o n t o an a l t e r n a t e s t a t e . 11. ALTERNATIVE FORMULATIONS OF A MODEL OF A G I N G AND MORTALITY A. The B a s i c Model I n t h i s s e c t i o n we d e s c r i b e a model of a g i n g and m o r t a l i t y of t h e g e n e r a l t y p e s u g g e s t e d by Woodbury and Manton ( 1 9 7 7 ) . For e a s e of compari- s o n w i t h t h e a l t e r n a t i v e f o r m u l a t i o n s p r e s e n t e d below, we d e s c r i b e t h i s model i n t e r m s o f a s i n g l e p h y s i o l o g i c a l o r e n v i r o n m e n t a l v a r i a b l e Y ( t ) : g e n e r a l i z a t i o n t o an a r b i t r a r y number of v a r i a b l e s i s s t r a i g h t f o r w a r d . I n a d d i t i o n t o t h e p r o c e s s d e s c r i b i n g changes i n p h y s i o l o g i c a l s t a t e s we w i l l r e p r e s e n t t i m e of d e a t h by a n o n n e g a t i v e random v a r i a b l e T whose d i s t r i b u t i o n depends on t h e v a l u e of Y ( t ) . Y(t) Hence, i n a d d i t i o n t o t h e e v o l u t i o n of d e s c r i b e d by a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n , t h e model i n c l u d e s an a d d i t i o n a l random p r o c e s s t h a t i s d e s c r i b e d by a m o r t a l i t y i n d i c a t o r I(t). The t i m e p a t h of e a c h i n d i v i d u a l i s t h u s d e s c r i b e d by I ( t ) where I ( t ) = 1 i f T>t, otherwise I ( t ) = 0 , ( 1) and by Y ( t ) s a t i s f y i n g dY(t) = a ( t , Y ( t ) ) I ( t ) d t + b ( t , , Y ( t ) ) I ( t ) dW(t). (2) I n ( 2 ) , W i s a Wiener p r o c e s s t h a t i s i n d e p e n d e n t of t h e i n i t i a l v a l u e Y(O), which i s a random v a r i a b l e w i t h known d i s t r i b u t i o n . I t i s assumed t h a t t h e c o e f f i c i e n t s a and b a r e known, b u t t h a t no o b s e r v a t i o n s a r e a v a i l a b l e on Y ( t ) o r I ( t ) . Note t h a t when an i n d i v i d u a l d i e s , t h e e f f e c t of I ( t ) i n ( 2 ) i s t o make f u r t h e r change i n t h e c o e f f i c i e n t s a and b i r r e l e v a n t : is reasonable f o r physiological processes. v a r i a b l e s , I ( t ) can b e o m i t t e d from ( 2 ) : on t h e s u r v i v a l of a g i v e n i n d i v i d u a l . this I n t h e c a s e of e n v i r o n m e n t a l a i r t e m p e r a t u r e does n o t depend The c o n d i t i o n a l d i s t r i b u t i o n of T i s g i v e n by where p i s a bounded f u n c t i o n , assumed known, t h a t can b e i n t e r p r e t e d a s t h e f o r c e of m o r t a l i t y f o r i n d i v i d u a l s a t t i m e t w i t h c h a r a c t e r i s t i c Y ( t ) , and where Y t r e p r e s e n t s t h e e n t i r e h i s t o r y of Y from t i m e 0 t o t i m e t . 0 The d e n s i t y f u n c t i o n of Y ( t ) may b e w r i t t e n a s A s Woodbury and Manton n o t e , t h e change i n t h i s d e n s i t y f u n c t i o n o v e r t i m e i s governed by t h e Kolmogorov-Fokker-Planck equation: The t h r e e a d d i t i v e t e r m s i n t h i s e q u a t i o n r e f l e c t t h e d i f f e r e n t f o r c e s a f f e c t i n g t h e dynamics of change i n t h e d i s t r i b u t i o n of Y ( t ) . The f i r s t t e r m d e s c r i b e s t h e e f f e c t s u s u a l l y c a l l e d d r i f t and r e g r e s s i o n ; t h e s e c o n d t e r m , t h e e f f e c t s of d i f f u s i o n ; and t h e t h i r d t e r m , t h e e f f e c t s of m o r t a l i t y selection. B. The Model When Death Is Observed Suppose now t h a t i n d i v i d u a l s ' d e a t h s a r e o b s e r v e d , s o t h a t i t is known w h e t h e r T, t h e t i m e of d e a t h f o r a n i n d i v i d u a l , e x c e e d s t . D e f i n e t h e con- d i t i o n a l d e n s i t y of Y ( t ) by: Then i t f o l l o w s from t h e more g e n e r a l p r o o f o u t l i n e d i n Appendix A t h a t - u(t,y) f;(y) + a t ) ft(y) 9 where - ~ ( t = ) El.u(t,y) 1 T>tl. ( 8) T h i s g e n e r a l i z a t i o n of t h e KFP e q u a t i o n i s similar t o ( 5 ) e x c e p t f o r t h e a d d i t i o n a l f a c t o r g i v e n by ( 8 ) . T h i s f a c t o r , w h i c h may b e i n t e r p r e t e d as t h e o b s e r v e d f o r c e of m o r t a l i t y a t t i m e t , can b e c o n s i d e r e d a c o r r e c t i o n t e r m a r i s i n g from t h e a d d i t i o n a l i n f o r m a t i o n known a b o u t w h e t h e r a n i n d i v i d u a l is a l i v e . C. The Model When D e a t h And A V a r i a b l e A r e Observed Now s u p p o s e t h a t t h e r e i s a n a d d i t i o n a l p h y s i o l o g i c a l o r e n v i r o n m e n t a l v a r i a b l e X(.t) t h a t i s o b s e r v e d o v e r t i m e . I n p a r t i c u l a r , suppose t h a t i n a d d i t i o n t o ( 1 ) t h e f o l l o w i n g two e q u a t i o n s d e s c r i b e t h e t i m e p a t h of an individual : and where W 1 and W 2 a r e Wiener p r o c e s s e s i n d e p e n d e n t of each o t h e r a n d of t h e i n i t i a l values X(0) and Y ( 0 ) . D e f i n e t h e c o n d i t i o n a l d e n s i t y of Y ( t ) by where X: t. r e p r e s e n t s t h e e n t i r e h i s t o r y o f t h e p r o c e s s X from t i m e 0 t o t i m e Then a s i n d i c a t e d i n Appendix A , where Note t h e s i m i l a r i t y of (11) t o ( 5 ) and ( 7 ) . The a d d i t i o n a l , f i n a l t e r m i n ( 1 1 ) d e s c r i b e s t h e e f f e c t of o b s e r v i n g X ( t ) . D. F u r t h e r E x t e n s i o n s Of The Model The p r o c e s s e s c o n s i d e r e d up u n t i l now have b e e n Markovian p r o c e s s e s : t h e coefficients i n the stochastic d i f f e r e n t i a l equations ( 2 ) , (9), depend o n l y on t h e c u r r e n t v a l u e s o f t h e v a r i a b l e s . and (10) T h a t i s , i t i s assumed t h a t t h e c u r r e n t v a l u e s on t h e i n d i v i d u a l ' s p h y s i o l o g i c a l v a r i a b l e s a r e r e a s o n a b l e a p p r o x i m a t i o n s o f t h e i n d i v i d u a l s ' p h y s i o l o g i c a l " s t a t e " a n d , conseq u e n t l y , w i l l d e s c r i b e t h e f u t u r e changes of t h a t s t a t e e x c e p t f o r s t o c h a s t i c innovations. When X ( t ) i s o b s e r v e d , i t i s p o s s i b l e t o g e n e r a l i z e t h e p r o c e s s t o depend on t h e e n t i r e h i s t o r y of X t 0' This i m p l i e s t h a t t h e p r i o r physio- l o g i c a l c h a r a c t e r i s t i c s o f t h e i n d i v i d u a l , and p o s s i b l y t h e t r a j e c t o r y of change of t h o s e p h y s i o l o g i c a l c h a r a c t e r i s t i c s , must b e i n c l u d e d i n t h e d e f i n i t i o n of p h y s i o l o g i c a l s t a t e . For example, h a v i n g e l e v a t e d b l o o d p r e s s u r e a t t h e c u r r e n t t i m e may n o t b e s u f f i c i e n t t o d e s c r i b e t h e s t a t e of t h e i n d i v i d ual with respect t o mortality r i s k s . Risk may b e more dependent upon accumulated damage ( p e r h a p s r e p r e s e n t e d by- t h e e l e v a t i a n s f p r e s s u r e o v e r a l o n g p e r i o d of t i m e ) o r upon e x t r e m e v a l u e s ( e . g . , p r e s s u r e t h r e s h o l d was e x c e e d e d ) . X(t) i n (9) , (10) , t h e number o f t i m e s a b l o o d Such p r o c e s s e s may b e modeled by r e p l a c i n g (11) and ( 1 2 ) b y xi. A s k e t c h of t h e p r o o f i s g i v e n i n Appendix A. Each of t h e t h r e e f o r m u l a t i o n s p r e s e n t e d above can b e r e a d i l y e x t e n d e d t o t h e g e n e r a l c a s e o f any number of s t a t e v a r i a b l e s . This extension r e q u i r e s t h e s u b s t i t u t i o n of t h e a p p r o p r i a t e m a t r i c e s . 111. ESTIMATING THE UNOBSERVED VARIABLE Woodbury and Manton (1977) s u g g e s t some a s s u m p t i o n s and r e s t r i c t i o n s f o r e s t i m a t i n g t h e p a r a m e t e r s of t h e o b s e r v e d p r o c e s s . Some of t h e s e w i l l b e u s e f u l f o r e s t i m a t i n g c h a r a c t e r i s t i c s of t h e u n o b s e r v e d v a r i a b l e s . I n t h e f o l l o w i n g we a p p l y t h e i r g e n e r a l t i m e s e r i e s a p p r o a c h t o t h e v a r i o u s f o r m u l a t i o n s d e s c r i b e d above.. A. The B a s i c Model C o n s i d e r t h e f i r s t f o r m u l a t i o n of t h e model, p r e s e n t e d above i n s e c t i o n I I A , i n which n e i t h e r d e a t h n o r t h e s t a t e v a r i a b l e a r e o b s e r v e d . This c a s e i s p r i m a r i l y of t h e o r e t i c a l i n t e r e s t a l t h o u g h i f enough p a r a m e t e r e s t i mates a r e a v a i l a b l e from a u x i l i a r y d a t a , t h e e q u a t i o n s below w i l l d e f i n e t h e e v o l u t i o n of t h e d i s t r i b u t i o n of t h e unobserved v a r i a b l e s . Assume t h a t t h e o b s e r v e d v a r i a b l e f o l l o w s a G a u s s i a n d i s t r i b u t i o n a t t i m e 0. Further- more, r e s t r i c t t h e s t o c h a s t i c e q u a t i o n i n ( 2 ) a s f o l l o w s : dY(t) = a t ) 0 + al(t) Y(t)dt + b(t)dWl(t). (13) I t i s o b v i o u s t h a t t h e d i s t r i b u t i o n of Y ( t ) i s G a u s s i a n a t any t i m e t . The mean, m ( t ) , and v a r i a n c e , y ( t ) , of t h i s d i s t r i b u t i o n a r e g i v e n by: B. The Model When Only Death Is Observed Now c o n s i d e r t h e second f o r m u l a t i o n p r e s e n t e d above. Assume t h a t t h e unobserved v a r i a b l e f o l l o w s a Gaussian d i s t r i b u t i o n a t t i m e 0 and t h a t t h e f o r c e of m o r t a l i t y i s a q u a d r a t i c f u n c t i o n of t h i s v a r i a b l e : F u r t h e r m o r e , r e s t r i c t t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n i n ( 2 ) as f o l l o w s : dY(t) = I ( t ) . [ao(t) + al(t) Y(t)dt + b(t)dwl(t)]. (17) It f o l l o w s t h a t t h e d i s t r i b u t i o n of Y ( t ) c o n d i t i o n a l on I ( t ) = 1 o r T>t ( i n o t h e r w o r d s , among t h e s u r v i v i n g p o p u l a t i o n ) i s G a u s s i a n a t any t i m e t: proof of t h i s i s a s p e c i a l c a s e of t h e more g e n e r a l proof s k e t c h e d i n Appendix A; a s p e c i f i c proof may b e found i n Y a s h i n ( 1 9 8 3 ) . The mean, m ( t ) , and v a r i a n c e , y ( t ) , of t h i s d i s t r i b u t i o n a r e g i v e n by: and Note t h e a d d i t i o n a l terms i n (18) and ( 1 9 ) compared w i t h ( 1 4 ) and ( 1 5 ) . The o b s e r v e d f o r c e of m o r t a l i t y i s g i v e n by t h e f o l l o w i n g f o r m u l a : t ) = t 2 (20) + m ( t ) p l ( t ) + (m ( t ) + y ( t ) ) u , ( t ) . & I f r e s t r i c t i o n s a r e p l a c e d on t h e p ' s i n t h i s formula--e.g., s o t h a t they are c o n s t a n t o r f o l l o w c e r t a i n s p e c i f i e d f u n c t i o n a l forms--then i t may b e p o s s i b l e t o e s t i m a t e t h e i r v a l u e s g i v e n t h e o b s e r v e d v a l u e s of i.Another approach i s t o r e s t r i c t ( 1 6 ) t o : 2 u ( t ,Y(t>)= Y ( t ) ' ~ ( t ) . (21) T h i s c o n s t r a i n t i s a n a l o g o u s t o t h e f o r m u l a t i o n i n Vaupel e t a l . ( 1 9 7 9 ) . Y' 7 corresponds t o t h e v a r i a b l e c a l l e d " f r a i l t y " . The f o r m u l a i n ( 2 0 ) reduces t o s o t h a t t h e t i m e p a t h of p ( t ) can b e c a l c u l a t e d from t h e o b s e r v a t i o n s of i ( t ) and t h e e s t i m a t e s o f m ( t ) and y ( t ) C. . The Model When Death And X ( t ) Are Observed Suppose now t h a t X ( t ) i s o b s e r v e d . Assume t h a t t h e d i s t r i b u t i o n of t h e unobserved Y(0) c o n d i t i o n a l on t h e o b s e r v e d X(0) i s G a u s s i a n and t h a t t h e f o r c e of m o r t a l i t y i s a q u a d r a t i c f u n c t i o n of Y ( t ) : IrL a d a i t i o n , r e s t r i c t t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s as f o l l o w s : dy(-t) = [ a ( t , x t ) 0 0 + al(t,x;) Y(t)]dt + t bl(t,Xo)dWl(t) t + b2(t,~0)dW2(t) (24) and t d ~ ( t ) = [ A ~ ( ~ , x : ) + A ~ ( ~ , X ;~) ( t I )d t + ~ ( t , x ~ ) d W ? ( t ) Note t h a t ( 2 4 ) a n d ( 2 5 ) a r e more g e n e r a l t h a n ( 9 ) and ( 1 0 ) . c o e f f i c i e n t s may depend on t h e e n t i r e h i s t o r y of e x t e n s i o n t o t h e non-Markovian case. x0t .' (25) First, the t h i s represents the Second, t h e f i r s t e q u a t i o n now de- pends on b o t h Wiener p r o c e s s e s ( i . e . , W 1 This is a s t r a i g h t - and W2). f o r w a r d g e n e r a l i z a t i o n t h a t may b e u s e f u l i n e s t i m a t i o n . A s o u t l i n e d i n Appendix B , i t f o l l o w s t h a t t h e d i s t r i b u t i o n of Y ( t ) c o n d i t i o n a l on X ( t ) and T>t i s G a u s s i a n . F u r t h e r m o r e , t h e mean and v a r i a n c e of t h i s c o n d i t i o n a l d i s t r i b u t i o n a r e g i v e n b y : t t d m ( t ) = [ a ( t , ~ : ) + a l ( t , X o ) m ( t ) - y ( t ) p1(t,X0) 0 t t t b 2 ( t ,Xo) B ( t ,XO) + Al(tyXO) ~ ( t ) + [ 2 I B (t,~:) [dX(t) and - t (A ( t , X O ) 0 t + ~~(t,X~)m(t))dtl. - t y ( t ) m ( t ) lJ2(t,Xo)1dt (2 6 ) These two e q u a t i o n s a r e s i m i l a r t o t h e p r e v i o u s e x p r e s s i o n s f o r t h e mean and v a r i a n c e i n (18) and (19) e x c e p t f o r t h e f i n a l t e r m s ( a n d t e r m s a r i s i n g from t h e i n c l u s i o n of W i n (24)). 2 These f i n a l t e r m s can b e viewed a s c o r r e c t i o n s introduced because i n f o r m a t i o n i s a v a i l a b l e about xt0 ' The t e r m s w i l l l o o k f a m i l i a r t o s t u d e n t s of c o n t i n u o u s - t i m e Kalman f i l t e r s . I n d e e d , one way of i n t e r p r e t i n g (26) and (27) i s t h a t t h e y g e n e r a l i z e t h e u s u a l Kalman f i l t e r e q u a t i o n s t o i n c l u d e t h e f o r c e of m o r t a l i t y . The o b s e r v e d f o r c e of m o r t a l i t y can b e r e l a t e d t o t h e o b s e r v e d v a r i a b l e s and t h e d i s t r i b u t i o n of t h e u n o b s e r v a b l e v a r i a b l e s by D. D i s c r e t e Time O b s e r v a t i o n s I n most e m p i r i c a l s t u d i e s , t h e o b s e r v e d v a r i a b l e s a r e n o t m o n i t o r e d c o n t i n u o u s l y b u t a r e o b s e r v e d from t i m e t o t i m e . T h i s s e c t i o n d e s c r i b e s how t h e f o r m u l a s d e v e l o p e d above may b e a p p l i e d t o t h e c a s e o f d i s c r e t e time observations. Assume t h a t t h e unobserved p r o c e s s i s governed by t h e s t o c h a s - t i c d i f f e r e n t i a l equation dY(t) = ( a o ( t , X ) + al(t,X) Y(t)) dt + b(t,X)dW t' where t h e p r o c e s s X i s now t h e s e q u e n c e of ( t ,X ) , n>O. n n a s e q u e n c e of o b s e r v a t i o n t i m e s t ments X 1' X2' ..., X n . 1' t 2'"" That i s , t h e r e i s t n , and a s e q u e n c e of measure- The X s e q u e n c e can be d e s c r i b e d by t h e g e n e r a t i n g n procedure: where A(t,X) and D ( t , X ) ( a s w e l l a s a o ( t , X ) , a l ( t , X ) , b ( t , X ) ) a r e known f u n c t i o n s o f t and t h e e n t i r e h i s t o r y o f t h e p r o c e s s X up t o b u t n o t i n c l u d i n g t i m e t and where En i s a sequence of G a u s s i a n - d i s t r i b u t e d w i t h mean 0 and v a r i a n c e 1. random v a r i a b l e s From a s t r a i g h t f o r w a r d m a n i p u l a t i o n o f ( 3 0 ) , we can b e generated see t h a t t h e t i m e series of t h e u n o b s e r v e d v a r i a b l e s , Y (T,) from t h e o b s e r v e d t i m e s e r i e s i n X and t h e a s s u m p t i o n of t h e G a u s s i a n d i f f u s i o n p r o c e s s , w i t h a p p r o p r i a t e n o r m a l i z a t i o n , f o r Y(T,). L i k e w i s e , (30) il- l u s t r a t e s how t h e u n o b s e r v e d v a r i a b l e s a f f e c t t h e o b s e r v e d p r o c e s s . As before, w e assume t h a t t h e f o r c e of m o r t a l i t y may be r e p r e s e n t e d by v ( t , X , Y ( t ) ) = "0 ( t , 3 + Y(t) ill(t,X) where t h e p ( t , X ) a r e n o n n e g a t i v e , 1 + 2 Y ( t ) u -7 ( t .X) , (31) m e a s u r a b l e f u n c t i o n s of t and t h e e n z i r e h i s t o r y of X up t o b u t n o t i n c l u d i n g t i m e t . By g e n e r a l i z i n g t h e method o f p r o o f used i n Y a s h i n (1980) i t can b e shown t h a t t h e c o n d i t i o n a l d i s t r i b u t i o n of Y ( t ) g i v e n I ( t ) = 1 ( i . e . , T > t ) and X i s G a u s s i a n . The mean and v a r i a n c e of t h i s d i s t r i b u t i o n a r e : and These e q u a t i o n s may b e viewed a s g e n e r a l i z a t i o n s of b o t h c o n t i n u o u s t i m e and d i s c r e t e t i m e Kalman f i l t e r a l g o r i t h m s . IV. APPLICATIONS A. General Observations To u s e t h e model e m p i r i c a l l y , i t i s n e c e s s a r y t o p r o d u c e e s t i m a t e s of t h e v a l u e s of t h e c o e f f i c i e n t s i n t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s (25) and e i t h e r ( 2 4 ) o r ( 2 9 ) . Although d i s c u s s i o n of t h e d e t a i l s of s t a t i s - t i c a l e s t i m a t k o n ' l i s beyond t h e s c o p e of t h i s p a p e r , we n o t e t h a t i f o b s e r v a t i o n s a r e a v a i l a b l e on a p o p u l a t i o n of i n d i v i d u a l s a c r o s s t i m e and o v e r a g e , t h e n t h e c o e f f i c i e n t s of t h e s e e q u a t i o n s a r e e s t i m a b l e g i v e n t h e appropriate identifying constraints. For example, by s p e c i f y i n g t h a t i n e q u a t i o n ( 2 9 ) c e r t a i n c o e f f i c i e n t s can v a r y by a g e , b u t n o t t i m e ( i . e . , t h e con- s t r a i n t of no c o h o r t e f f e c t s o p e r a t i n g t h r o u g h X ) , w e c a n e s t i m a t e c e r t a i n c o e f f i c i e n t s f o r ( 2 4 ) i f c o h o r t e f f e c t s do emerge. Alternately, previous t h e o r e t i c a l and e m p i r i c a l r e s e a r c h may s u g g e s t v a l u e s o r f u n c t i o n a l forms f o r t h e coefficients t h a t w i l l f a c i l i t a t e estimation. have been In particular, there a number of l o n g i t u d i n a l s t u d i e s of a g i n g p r o c e s s e s ( e . g . , the f i r s t and s e c o n d Duke L o n g i t u d i n a l s t u d i e s of n o r m a t i v e a g i n g ) which can p r o v i d e e s t i m a t e s of t h e a g e r a t e of d e c l i n e of a b r o a d r a n g e of p h y s i o l o g i c a l parameters. These e s t i m a t e s c o u l d b e employed d i r e c t l y i n t h e e q u a t i o n s . Given t h e c o e f f i c i e n t s , ( 2 6 ) and ( 2 7 ) o r ( 3 2 ) and (33) p e r m i t e s t i m a t i o n o f t h e mean and v a r i a n c e of t h e c o n d i t i o n a l d i s t r i b u t i o n of t h e unobserved v a r i a b l e . - E q u a t i o n (28) can t h e n b e used a s t h e b a s i s f o r e s t i m a t i n g t h e f o r c e of m o r t a l i t y f o r an i n d i v i d u a l w i t h any s p e c i f i e d c h a r a c t e r i s t i c s and a t any a g e . A s n o t e d e a r l i e r , t h i s e s t i m a t i o n might r e q u i r e s p e c i f y i n g c e r t a i n f u n c t i o n a l forms f o r p O , p l , and p7. A l t e r n a t i v e l y , i t might b e * assumed t h a t y and p a r e e q u a l t o z e r o , i n which c a s e t h e v a l u e s of p 0 1 2 o v e r t i m e can be i m m e d i a t e l y c a l c u l a t e d from t h e o b s e r v a t i o n s of B. over time. Unobserved R i s k F a c t o r s The model may b e u s e f u l i n a v a r i e t y of a p p l i c a t i o n s where d a t a a r e a v a i l a b l e o v e r t i m e c o n c e r n i n g some v a r i a b l e s , b u t t h e r e i s r e a s o n t o bel i e v e t h a t o t h e r s i g n i f i c a n t v a r i a b l e s a r e unobserved. I n some c a s e s enough t h e o r e t i c a l o r e m p i r i c a l knowledge may b e a v a i l a b l e a b o u t t h e s e unobs e r v e d v a r i a b l e s s o t h a t t h e i n i t i a l p r o b a b i l i t y d i s t r i b u t i o n s and s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s can b e s p e c i f i e d w i t h some c o n f i d e n c e . In s u c h c a s e s e s t i m a t i o n o f t h e e v o l u t i o n of t h e u n o b s e r v e d v a r i a b l e s may b e of c o n s i d e r a b l e i n t e r e s t . I n o t h e r c a s e s , i t may b e s u s p e c t e d t h a t some unmeas.ured f a c t o r s u c h a s " f r a i l t y " i s an i m p o r t a n t s o u r c e of h e t e r o g e n e i t y i n the population. Such a v a r i a b l e may h a v e t o b e i n t r o d u c e d by i m - p o s i n g c o n s t r a i n t s i n t h e model. For i n s t a n c e , Vaupel e t a l . (1979) assume t h a t an i n d i v i d u a l ' s f r a i l t y i s c o n s t a n t o v e r a g e a n d t h a t t h e d i s t r i b u t i o n of f r a i l t y among i n d i v i d u a l s f o l l o w s some s i m p l e d i s t r i b u t i o n a l form. interest: I n some s t u d i e s t h e u n o b s e r v e d v a r i a b l e may n o t b e of much i t may b e viewed a s a n u i s a n c e i m p o r t a n t o n l y b e c a u s e i t ob- s c u r e s t h e a c t u a l r e l a t i o n s h i p s among t h e v a r i a b l e s of d i r e c t i n t e r e s t . A s a s p e c i f i c example of t h i s k i n d of a p p l i c a t i o n , c o n s i d e r a l o n g i t u d i n a l a n a l y s i s o f c h r o n i c i l l n e s s b a s e d on t h e k i n d o f i n f o r m a t i o n c o l l e c t e d , s a y , i n t h e Framingham s t u d y . "anton e t a l . ( 1 9 7 9 ) and Woodbury e t a l . ( 1 9 7 9 , 1 9 8 1 ) p r e s e n t a n a l y s e s o f t h i s s o r t , b a s e d on t h e i n s i g h t s of t h e Woodbury-Manton model. I n t h e i r a n a l y s e s , t h e change i n c o r o n a r y h e a r t d i s e a s e r i s k f a c t o r s i n t h e s t u d y p o p u l a t i o n was modeled a s a n a u t o regressive process adjusted f o r t h e e f f e c t s of systematic mortality selection. I t seems l i k e l y t h e p o p u l a t i o n was s u b j e c t t o r i s k f a c t o r s n o t f u l l y r e p r e s e n t e d by t h e a v a i l a b l e m e a s u r e m e n t s , i . e . , b l o o d p r e s s u r e , serum c h o l e s t e r o l , u r i c a c i d , e t c . s y s t o l i c and d i a s t o l i c The s t o c h a s t i c d i f - f e r e n t i a l e q u a t i o n s p r e s e n t e d h e r e , and t h e Kalman f i l t e r e q u a t i o n s gene r a l i z e d t o r e p r e s e n t t h e e f f e c t s of m o r t a l i t y s e l e c t i o n o f f e r a r a n g e o f s t r a t e g i e s f o r a . ) e s t i m a t i n g t h e i m p a c t of unobserved r i s k f a c t o r s , and b . ) i d e n t i f y i n g t h e " t r u e " e f f e c t s o f o b s e r v e d r i s k v a r i a b l e s . P a r t i a l l y Overlapging S t u d i e s C. Sometimes l o n g i t u d i n a l d a t a a r e a v a i l a b l e from s e v e r a l r e l a t e d s t u d i e s s u c h t h a t some v a r i a b l e s a r e o b s e r v e d i n a l l s t u d i e s , b u t o t h e r v a r i a b l e s a r e o b s e r v e d i n o n l y some s t u d i e s . Having a s e t of s u c h s t u d i e s can g r e a t l y f a c i l i t a t e t h e e s t i m a t i o n o f t h e model p a r a m e t e r s . For i n s t a n c e , t h e Wood- bury-Manton model h a s s e r v e d a s t h e b a s i s f o r a n a l y s e s of c o r o n a r y h e a r t d i s e a s e r i s k s n o t o n l y i n t h e Framingham s t u d y p o p u l a t i o n , b u t a l s o i n t h e popul a t i o n s o b s e r v e d i n t h e Duke L o n g i t u d i n a l S t u d y of Aging (Manton and Woodbury, 1 9 8 3 ) , and of a Kaunas, L i t h u a n i a s t u d y . P a r t i a l l y o v e r l a p p i n g s e t s of ob- s e r v e d v a r i a b l e s were a v a i l a b l e f o r t h e s e t h r e e a n a l y s e s . The Duke s t u d y d i f f e r e d from t h e Framingham s t u d y i n t h a t u r i c a c i d c o n c e n t r a t i o n s were n o t o b s e r v e d , b u t s c o r e s were t a k e n on t h e Wechsler A d u l t I n t e l l i g e n c e S c a l e . I n t h e Kaunas s t u d y , i n t e l l i g e n c e t e s t d a t a were n o t a v a i l a b l e , b u t u n l i k e t h e o t h e r d a t a s e t s , o b s e r v a t i o n s were a v a i l a b l e of smoking b e h a v i o r and of an i n d e x of body mass. To compare and s y n t h e s i z e s u c h i m p e r f e c t l y c o o r d i n a t e d d a t a s e t s , i t may b e u s e f u l t o employ a model t h a t i n c l u d e s a l l of t h e v a r i a b l e s o b s e r v e d i n any of t h e s t u d i e s . The model c o u l d t h e n b e a p p l i e d t o t h e d i f f e r e n t s t u d i e s by s p e c i f y i n g which v a r i a b l e s w e r e o b s e r v e d and which were n o t observed. The e f f e c t s o f a l l of t h e v a r i a b l e s a c r o s s a l l of t h e s t u d i e s c o u l d t h e n be compared. F u r t h e r m o r e , p r o c e s s p a r a m e t e r s e s t i m a t e d f o r an "ob- s e r v a b l e " i n o n e s t u d y c o u l d b e a p p l i e d t o a n o t h e r s t u d y where t h a t v a r i a b l e was "unobserved". D. Measurement E r r o r s and I n d i r e c t Measurements Most v a r i a b l e s can o n l y b e measured w i t h some e r r o r : n o i s e can b e s e v e r e . sometimes t h e I n o t h e r c a s e s , a v a r i a b l e of prime i n t e r e s t can n o t b e o b s e n e d d i r e c t l y , b u t a c o r r e l a t e d v a r i a b l e c a n b e m o n i t o r e d and u s e d a s an i n d e x . For L n s t a n c e , t h e e l a s t i c i t y of b l o o d v e s s e l s may b e i m p o r t a n t i n c o r o n a r y h e a r t d i s e a s e p r o c e s s e s , b u t o b s e r v a t i o n s may o n l y b e a v a i l a b l e on blood p r e s s u r e . I n d e e d , most of t h e measurements a v a i l a b l e i n s t u d i e s o f a g i n g p r o c e s s e s may o n l y i n d i r e c t l y r e f l e c t t h e u n d e r l y i n g p h y s i o l o g i c a l s t a t e variables. A s n o t e d a b o v e , t h e f o r m u l a s p r e s e n t e d f o r e s t i m a t i n g t h e mean and v a r i a n c e of t h e u n o b s e r v e d v a r i a b l e s can b e i n t e r p r e t e d a s e x t e n s i o n s of t h e Kalman f i l t e r e q u a t i o n s d e v e l o p e d t o d e t e c t s i g n a l s i n n o i s y measurements. Thus, t h e Kalman f i l t e r t y p e e q u a t i o n p r e s e n t e d h e r e c a n b e u s e f u l i n i d e n t i f y i n g t h h t r u e v a r i a b l e s of t h e p r o c e s s , i n t h e f a c e o f measurement e r r o r o r i n d i r e c t a s s e s s m e n t , from s t u d i e s w i t h m u l t i p l e measurements t a k e n over time E. . Assumptions E f f o r t s t o a p p l y t h e model w i l l , of c o u r s e , b e dependent on t h e r e a s o n a b l e n e s s of model a s s u m p t i o n s f o r a s p e c i f i c a p p l i c a t i o n . I n t h i s sec- t i o n , we d i s c u s s a s s u m p t i o n s and some s t r a t e g i e s f o r e x t e n d i n g t h e i r a p p l i cability to certain situations. 1. Gaussian D i s t r i b u t i o n The d i s t r i b u t i o n of t h e u n o b s e r v e d v a r i a b l e s c o n d i t i o n a l on t h e obs e r v e d v a r i a b l e s a t t i m e z e r o i s assumed t o b e G a u s s i a n . Furthermore, t h e model i m p l i e s t h a t this c o n d i t i o n a l d i s t r i b u t i o n among s u r v i v o r s w i l l b e G a u s s i a n a t any t i m e t . For some v a r i a b l e s t h i s may n o t b e t r u e , b u t a t r a n s f o r m of a v a r i a b l e may b e more o r l e s s G a u s s i a n d i s t r i b u t e d . For example, Manton and Woodbury (1983) u s e a s t h e i r v a r i a b l e s t h e l o g a r i t h m s o f p u l s e p r e s s u r e , d i a s t o l i c b l o o d p r e s s u r e , and serum c h o l e s t e r o l 12~21. C o n s i d e r a t i o n of t h e r e a s o n a b l e n e s s of t h i s a s s u m p t i o n must b e b a s e d on a v a i l a b l e t h e o r e t i c a l i n s i g h t a b o u t t h e dynamics of t h e u n o b s e r v e d v a r i a b l e ( s e e Manton and S t a l l a r d , 1 9 8 1 ) . 2. Q u a d r a t i c Hazard The f o r c e o f m o r t a l i t y i s assumed t o b e a q u a d r a t i c f u n c t i o n of t h e unobserved v a r i a b l e s . This assumption is c l o s e l y t i e d t o t h e Gaussian a s s u m p t i o n , a s t h e f o l l o w i n g example i l l u s t r a t e s . Let u ( t , Y ) b e t h e f o r c e of m o r t a l i t y a t t i m e t f o r an i n d i v i d u a l w i t h unobserved c h a r a c t e r i s t i c Y. Suppose where p ( t ) might b e i n t e r p r e t e d a s t h e f o r c e of m o r t a l i t y f o r some s t a n d a r d i n d i v i d u a l f o r whom Y e q u a l s one. Now c o n s i d e r a n a l t e r n a t i v e f o r m u l a t i o n : where z i s a c h a r a c t e r i s t i c t h a t e q u a l s Y 2 . T h i s f o r m u l a t i o n i s t h e one u s e d i n t h e " f r a i l t y " model proposed by Vaupel e t a l . (1979) and a p p l i e d i n s t u d i e s by Manton e t a l . (1981) and H o r i u c h i and Coale ( 1 9 8 3 ) . Finally, c o n s i d e r t h e f o r m u l a t i o n where ~ ( t , x >= u ( t ) e x , (36) where x i s a c h a r a c t e r i s t i c t h a t e q u a l s t h e l o g a r i t h m of Y 2 . T h i s approach h a s b e e n a d o p t e d i n a v a r i e t y o f s t u d i e s , i n c l u d i n g Heckman and S i n g e r ( 1 9 8 2 ) . Given t h e a p p r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n s , a l l t h r e e f o r m u l a t i o n s can b e made e q u i v a l e n t . For i n s t a n c e , t h e f i r s t f o r m u l a t i o n w i t h Y f o l l o w i n g a G a u s s i a n d i s t r i b u t i o n w i t h mean z e r o and v a r i a n c e one i s e q u i v a l e n t t o t h e second f o r m u l a t i o n w i t h z f o l l o w i n g a Gamma d i s t r i b u t i o n w i t h s c a l e p a r a m e t e r one and s h a p e p a r a m e t e r 0.5. I n some r e s p e c t s t h e s e c o n d f o r m u l a t i o n , i n v o l v i n g z , i s t h e most t r a n s p a r e n t s i n c e z c a n b e i n t e r p r e t e d a s m e a s u r i n g t h e r e l a t i v e r i s k of m o r t a l i t y f o r an i n d i v i d u a l compared t o some " s t a n d a r d " i n d i v i d u a l . Since Y does n o t have t o b e a s i n g l e v a r i a b l e , b u t can b e a v e c t o r of v a r i a b l e s , i t i s p o s s i b l e t o c o n s i d e r z d e f i n e d by where a is a matrix. I n t h i s c a s e , z w i l l h a v e a d i s t r i b u t i o n known a s a q u a d r a t i c form o f t h e G a u s s i a n d i s t r i b u t i o n . f l e x i b l e and c a n t a k e on a v a r i e t y of s h a p e s . Such q u a d r a t i c forms a r e v e r y Thus, t h e assumption t h a t e a c h v a r i a b l e i n t h e u n o b s e r v a b l e s e t of v a r i a b l e s Y i s G a u s s i a n d i s t r i b u t e d c a n b e r e a d i l y g e n e r a l i z e d t o t h e c a s e where t h e u n o b s e r v e d v a r i a b l e s c a n , i n e f f e c t , f o l l o w a q u a d r a t i c form o f t h e G a u s s i a n d i s t r i b u t i o n . Biologically t h e q u a d r a t i c f o r m of t h e h a z a r d i s r e a s o n a b l e f o r p h y s i o l o g i c a l p a r a m e t e r s subject t o homeostatic forces. That i s , v a r i a b l e s t h a t a r e e s s e n t i a l t o p h y s i o l o g i c a l f u n c t i o n i n g s h o u l d have a v i a b l e i n t e r i o r r a n g e and nonv i a b l e e x t e r n a l r a n g e s where h o m e o s t a s i s b r e a k s down. 3. D i f f e r e n t i a l Processes B o t h t h e o b s e r v e d and u n o b s e r v e d v a r i a b l e s i n o u r model a r e assumed t o b e c o n t i n u o u s and g o v e r n e d by a d i f f e r e n t i a l p r o c e s s . s t u d i e s t h i s may b e s a t i s f a c t o r y . I n a v a r i e t y of I n some i n s t a n c e s , however, c a t e g o r i c a l v a r i a b l e s t h a t a r e e i t h e r c o n s t a n t o v e r t i m e o r t h a t f o l l o w some jumping p r o c e s s may b e i m p o r t a n t . Constant c a t e g o r i c a l v a r i a b l e s , l i k e s e x , r a c e , o r n a t i o n a l o r i g i n , can b e h a n d l e d by s t r a t i f y i n g t h e d a t a . Discrete- s t a t e v a r i a b l e s t h a t jump from one s t a t e t o a n o t h e r p o s e a much more d i f f i c u l t problem. Examples of s u c h v a r i a b l e s t h a t may b e r e l e v a n t t o s t u d i e s of a g i n g and m o r t a l i t y i n c l u d e m a r i t a l s t a t u s , t y p e o f employment, p l a c e of r e s i d e n c e , and s u c h f a c t o r s a s w h e t h e r an i n d i v i d u a l i s h o s p i t a l i z e d o r i n a n u r s i n g home, h a s had a s t r o k e o r a h e a r t a t t a c k , h a s q u i t smoking, and s o on. I t i s p o s s i b l e t o e x t e n d t h e models p r e s e n t e d h e r e t o t h e more g e n e r a l c a s e where some of t h e o b s e r v e d o r unobserved v a r i a b l e s f o l l o w a jumping p r o c e s s a s opposed t o a d i f f e r e n t i a l p r o c e s s . V. DISCUSSION I n b o t h e m p i r i c a l and t h e o r e t i c a l s t u d i e s o f human a g i n g and m o r t a l i t y , t h e need f o r m o d e l i n g i n d i v i d u a l d i f f e r e n c e s i n a g i n g p r o c e s s e s h a s been repeatedly demonstrated ( e . g . , S t r e h l e r , 1 9 7 7 ; Economos, 1 9 8 2 ; Manton and Woodbury, 1983). U n f o r t u n a t e l y , t h e r e a r e many i n s t a n c e s where t h o s e d i f f e r e n c e s a r e due t o u n o b s e r v e d v a r i a b l e s . I n d e e d , t h e n a t u r e of t h e s o u r c e s of t h e s e d i f f e r e n c e s , such a s d i f f e r e n c e s i n t h e age-related l o s s of f u n c t i o n a l " v i t a l i t y " o r t h e impact on l o n g e v i t y of g e n e t i c f a c t o r s , s u g g e s t t h a t d i f f i c u l t i e s i n measurement and c o n c e p t u a l i z a t i o n w i l l d i c t a t e t h a t s u c h i n d i v i d u a l p r o p e r t i e s w i l l remain a t l e a s t p a r t i a l l y h i d d e n f o r a l o n g t i m e . N o n e t h e l e s s , s u c c e s s f u l l y c o p i n g w i t h t h e e f f e c t s on a g i n g p r o c e s s e s of such l a t e n t h e t e r o g e n e i t y w i l l b e a n e c e s s a r y component of a d e q u a t e models of human a g i n g and m o r t a l i t y . f o r t h e n e c e s s i t y of j o i n i n g individual aging ratesl'with For example, Economos (1982) h a s a r g u e d "Simm's i d e a of s t a t i s t i c a l l y d i s t r i b u t e d Gompertz's c o n c e p t of " a c c e l e r a t e d d e c l i n e of v i t a l i t y l ' i n o r d e r t o r e l a t e t h e o b s e r v e d p a t t e r n of r a t e s of a g i n g w i t h t h e o b s e r v e d p a t t e r n of t h e r a t e s o f d y i n g . I n d e e d , t h e l o g i c by which t h e s e c o n c e p t s a r e r e l a t e d i s t h a t of a d i f f u s i o n p r o c e s s where t e m p o r a r y s o j o u r n s above a t h r e s h o l d v a l u e c a u s e the r a t e of i n c r e a s e i n m o r t a l i t y r a t e s t o b e more r a p i d t h a n t h e r a t e of d e c l i n e of p h y s i o l o g i c a l v i t a l i t y . The model we h a v e p r e s e n t e d p r o v i d e s a f l e x i b l e s t r a t e g y f o r a s s e s s i n g t h e i m p a c t of s u c h h e t e r o g e n e i t y on human a g i n g and m o r t a l i t y p r o c e s s e s . I n p a r t i c u l a r , i t g e n e r a l i z e s t h e n o t i o n of t h e e f f e c t s of h e t e r o g e n e i t y from t h a t of a f i x e d d i s t r i b u t i o n t o t h e e f f e c t s of an unobserved p r o c e s s . Thus, i t c a n l e a d t o a n e m p i r i c a l s t r a t e g y f o r a s s e s s i n g b o t h f u n c t i o n change and m o r t a l i t y which i s r i c h enough t o r e p r e s e n t t h e c o m p l e x i t y of c u r r e n t c o n c e p t u a l models of human a g i n g and m o r t a l i t y . We p r e s e n t e d o u r model a s a development of t h e Woodbury-Manton model of a g i n g and m o r t a l i t y p u b l i s h e d by t h i s j o u r n a l . Our model c a n a l s o be viewed a s h a v i n g r o o t s i n a n a l y s e s done by numerous r e s e a r c h e r s i n a v a r i e t y of d i s c i p l i n e s . t i s t i c s ( e . g. , O f t e n a n a l y s t s working i n t h e v a r i o u s f i e l d s of s t a Lundberg, 19401, l a b o r economics and McCarthy , 1955) , s o c i o l o g y ( e . g. l i a b i l i t y engineering (e.g., ( e . g. , , ( e . g. , Blumen, Kogan S i n g e r and S p i l e r m a n , 1974) , r e - Harris and S i n g p u r w a l l a , 1 9 6 8 ) , demography Sheps and Menken, 1923) , and h e a l t h p o l i c y a n a l y s t s ( e . g. , Shepard and Z e c k h a u s e r , 1 9 7 7 ) , were o n l y p a r t i a l l y aware o f t h e m u t u a l r e l e v a n c e of t h e i r m e t h o d o l o g i c a l r e s e a r c h . The t h r u s t of much of t h i s d i v e r s e body of r e s e a r c h i s how t o cope w i t h t h e e f f e c t s of p o p u l a t i o n h e t e r o g e n e i t y on t h e p a r a m e t e r s of t h e p r o c e s s of i n t e r e s t . The most common c o n c e p t u a l i z a t i o n of t h e problem i s t h a t t h e r e i s some unobserved v a r i a b l e t h a t i n f l u e n c e s t h e l i k e l i h o o d t h a t an i n d i v i d u a l w i l l " d i e " a t some p a r t i c u l a r t i m e . Sometimes t h i s v a r i - a b l e i s of d i r e c t i n t e r e s t ; i n o t h e r c a s e s , i t i s e s s e n t i a l l y a n u i s a n c e . When i t i s of d i r e c t i n t e r e s t , methods t o e s t i m a t e p a r a m e t e r s of i t s d i s t r i b u t i o n , may b e i m p o r t a n t . But w h e t h e r i t i s of i n t e r e s t o r j u s t a n u i s a n c e , o n e must b e c o n c e r n e d w i t h i t s e f f e c t s i n o r d e r t o u n c o v e r t h e u n d e r l y i n g r e l a t i o n s h i p between t h e f o r c e of " m o r t a l i t y " and t h e v a r i a b l e s of i n t e r e s t . APPENDIX A. Proof of t h e G e n e r a l i z e d Kolmogorov-Fokker-Planck Equation C o n s i d e r t h e random p r o c e s s (YxX) d e f i n e d on p r o b a b i l i t y s p a c e ( R , H , P ) by t h e r e l a t i o n s : + d ~ ( t )= ( a ( t , ~ ( t ,)x i ) ~ ( t ) d t b ( t , ~ ( t ,)x i ) l ( t ) d w l ( t ) , Y ( o ) ( ~ 1 ) and d ~ ( t )= ( ~ ( t , ~ ( t ) , x i~) ( t ) d t+ ~ ( t , ~ ( t ) , x g tI )( L ) ~ w ~ ( ~ ) , x ( o ) (A21 where W ( t ) and W ( t ) a r e i n d e p e n d e n t Wiener p r o c e s s e s t h a t a r e a l s o inde1 2 pendent of t h e i n i t i a l c o n d i t i o n s Y(0) and X(0). C o e f f i c i e n t s a , A , and b a r e m e a s u r a b l e f u n c t i o n s of t , Y(.t), and t h e e n t i r e h i s t o r y of t h e p r o c e s s X from t i m e 0 t o t i m e t . B i s a p o s i t i v e , m e a s u r a b l e f u n c t i o n of t and t h e e n t i r e h i s t o r y of t h e p r o c e s s X. I ( t ) i s a two s t a t e (1,O) c o n t i n u o u s t i m e process w i t h I(O)=l, such t h a t t h e t r a n s i t i o n i n t e n s i t y f u n c t i o n ' I , y ( t , Y ( t ) , X , I ( t ) ) = u ( t , Y ( t ) ,XI I ( t ) ' I , (-43) t where y ( t , Y ( t ) , X ) i s a m e a s u r a b l e f u n c t i o n of t , ~ ( t ) ,and t h e e n t i r e h i s 0 t o r y of t h e p r o c e s s X up t o t i m e t . The proof of t h e g e n e r a l i z e d Kolmogorov-Fokker-Planck equation f o r t h e d e n s i t y of t h e unobserved v a r i a b l e c o n d i t i o n a l on I ( t ) = l and X t i s b a s e d on 0 t h e f o r m u l a f o r t h e c o n d i t i o n a l m a t h e m a t i c a l e x p e c t a t i o n of an a r b i t r a r y , bounded, doubly d i f f e r e n t i a b l e f u n c t i o n F ( Y ( t ) ) . T h i s f o r m u l a may b e de- r i v e d a s a consequence of t h e g e n e r a l e s t i m a t i o n approach b a s e d on semimarting a l e t h e o r y ( J a c o d , 1979; Bremand, 1 9 8 1 ) , as w e l l a s t h e methods of f i l t r a t i o n of random p r o c e s s e s w i t h jumping components (Yashin, 1969) and t h e a n a l a g o u s methods g i v e n i n L i p t z e r and S h i r j a e v (19 77). Using Bayes' f o r m u l a , one can w r i t e ~ ( f ( ~ ( t ) ) l ~ =( t1, ) x i ) = E ' ( F ( Y ( ~ ) .~ t ) ) where @ ( t ) i s t h e l i k e l i h o o d r a t i o g i v e n by Here we s k e t c h t h e p r o o f . where i s t h e Wiener p r o c e s s w i t h r e s p e c t t o t h e f a m i l y of 0 - a l g e b r a s g e n e r a t e d by t h e p r o c e s s X and where and The symbol E' means t h e o p e r a t i o n of m a t h e m a t i c a l e x p e c t a t i o n w i t h r e s p e c t t o t h e m a r g i n a l p r o b a b i l i t y measure c o n c e n t r a t e d on t h e component W 1 of t h e Wiener p r o c e s s . Using I t o ' s d i f f e r e n t i a l r u l e ( L i p t s e r and S h i r j a e v , 1 9 7 7 ) , one can r e a d i l y t r a n s f o r m ( A 5 ) i n t o t h- e d i f f e r e n t i a l r e l a t i o n s h i p A ( t , Y ( t ) ,x;) - ~ (,x;)t d N t ) = O(t) dW(t) ~ (,x:)t (-49) I n o r d e r t o c a l c u l a t e ( A 4 ) , r e p r e s e n t t h e p r o d u c t of F ( Y ( t ) ) and $ ( t ) by using I t o ' s d i f f e r e n t i a l rule. F ( Y ( ~ ) )9 ( t ) = F ( Y ( o ) ) $ ( o ) +i This y i e l d s t F ' ( Y ( ~ ) )e c u ) a ( u , Y ( u ) ,x0) du where F' and F" a r e t h e f i r s t and second o r d e r d e r i v a t i v e s o f F w i t h r e s p e c t Taking t h e m a t h e m a t i c a l e x p e c t a t i o n E' of b o t h s i d e s of (AlO), we g e t t E(F(Y(~)~ ) ( t = ) 1 , ~ =~ E) ( F ( Y ( O ) I(O) = 1 , ~ +~ it )0 E ' ( F ' ( Y ( U ) ) a ( u , Y ( u ) , x ) @ ( u ) ) ~ u I I By a g a i n u s i n g Bayes' f o r m u l a one c a n show Using t h e a r b i t r a r y doubly d i f f e r e n t i a b l e f u n c t i o n F(Y) s u c h t h a t and r e w r i t i n g (A12) i n t e r m s of t h e i n t e g r a l w i t h r e s p e c t t o t h e c o n d i t i o n a l density one can f i n a l l y g e t t h e c o n d i t i o n a l Kolmogorov-Eokker-Planck equation given i n t h e main t e x t . B. Proof t h a t t h e C o n d i t i o n a l D i s t r i b u t i o n i s Gaussian I n o r d e r t o prove t h a t t h e c o n d i t i o n a l d e n s i t y f (y) i s Gaussian, t some a d d i t i o n a l a s s u m p t i o n s a r e needed. A , and p h a v e t h e f o l l o w i n g forms: We assume t h a t t h e c o e f f i c i e n t s a , of t i m e and of t h e e n t i r e p a s t of t h e p r o c e s s X from t i m e 0 up t o t i m e t . We assume a l s o t h a t t h e i n i t i a l c o n d i t i o n Y(0) i s Gaussian d i s t r i b u t e d cont d i t i o n a l on I ( O ) = l and Xo, F ( Y ( ~ )=) e i a Y(t) (B2 ) Denote by For t h i s s p e c i a l c a s e (A12) may b e w r i t t e n a s where X' and X" d e n o t e t h e f i r s t and second d e r i v a t i v e s w i t h r e s p e c t t o a and t m(t) = E(Y(t) I I ( t ) = l , X O ) Denote by m 0 and y o t h e mean and v a r i a n c e of t h e c o n d i t i o n a l d i s t r i b u t i o n of Then t h e f u n c t i o n 'Y Yo. Y 0 = exP {iam 0 0 can be w r i t t e n a s - 4 a2 yo} Given t h i s p a r t i c u l a r form and t h e e q u a t i o n f o r Y form: t' we s e e k Y i n t h e s i m i l a r t where m t and y t s a t i s f y t h e f o l l o w i n g s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s dm(t) = c l ( t ) + dt dl(t) di(t) 08) + d2(t) di(t) dy(t) = C2(t) d t The c o e f f i c i e n t s i n (B8) c a n b e found from ( B l ) and (B7) . Using t h e e q u a l i t i e s y' = Y -4 y a 2 , y, W = - + Y a4 and comparing t h e s t o c h a s t i c d i f f e r e n t i a l of $t r e p r e s e n t e d i n t e r m s of m t and yt w i t h t h e r i g h t hand s i d e of (B5), we have I t remains t o b e shown t h a t t h e e q u a t i o n f o r y t h a s a unique s o l u t i o n . Proof of t h i s f o l l o w s e a s i l y from t h e approach s u g g e s t e d by L i p t z e r and S h i r j aev (19 77). 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