A Note on Random Intensities and Conditional Survival Functions

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NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHORS

A

Note

On Random Intensities and Conditional

Survivd

F'unctians

A n a t o l i Y a s h i n

Eva

A r j a s

December 1986 WP-86-77

W o r k i n g P a p e r s are interim r e p o r t s on work of the International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily r e p r e s e n t those of t h e InsMtute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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One of t h e interesting directions of r e s e a r c h in IIASA's Population Program d e a l s with t h e methodological a s p e c t s of population heterogeneity dynamics. The c r u c i a l notion in t h i s analysis i s t h e s t o c h a s t i c intensity which i s widely used in t h e s t o c h a s t i c p r o c e s s e s models of human morbidity and mortality o r technical failure.

This p a p e r provides t h e probabilistic specification of t h i s notion which gives a n opportunity to use t h e r e s u l t s of modern g e n e r a l t h e o r y of p r o c e s s e s in analyz- ing f a c t o r s t h a t influence demographic c h a r a c t e r i s t i c s .

Anatoli Yashin Deputy Leader Population Program

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Acknowledgement

W e a r e grateful to Pentti Laara f o r a number of good comments.

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A Note on Random Intensities and Conditional Survival b c t i o n s

A n a t o l i Y a s h i n and E v a A r j a s

1.

Intmduction

Let

t =

( t t ) t a b e a random p r o c e s s and T a random time in some probability s p a c e . The intensity, o r h a z a r d r a t e , r e l a t e d t o t h e o c c u r r e n c e of T a n d given t h e observation of

ti = It,,

⁰⁵^s⁵^t

1 ,

i s often identified a s a limit of t h e form

X ( t , t ) = l i m - P ( t 1 < ~ ~ t + ~ l t k ; T a t ]

.

ArO A ( 1

Does such a definition mean t h a t t h e corresponding conditional s u r v i v a l function, when

t

i s o b s e r v e d , c a n b e obtained from t h e "exponential formulas"

Equality ( 2 ) i s often tacitly assumed in medical a n d epidemiological studies when dealing with survival analysis in t h e p r e s e n c e of o b s e r v a b l e influencing random fac- tors. I t t u r n s out t h a t t h i s formula does not always hold.

The exponential formula c a n b e viewed as t h e solution satisfying P ( T 2 0 )

=

1 of a corresponding differential equation. Thus, when t h e r e i s no conditioning, and assuming absolute continuity of t h e distribution function F ( t ) = P ( T 5 t ) , t 2 0 , t h e formulas

and

e x p r e s s a one-to-one c o r r e s p o n d e n c e between t h e h a z a r d rate X and t h e distribu-

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tion function F. This h a s a n obvious extension t o t h e case w h e r e conditioning i s on a fixed u-algebra, s a y , G o involving t h e conditional d i s t r i b u t i o n function p G O ( t )

=

P ( T d t ( G o ) a n d t h e c o r r e s p o n d i n g h a z a r d rate [1,2,3]. Why, then, i s i t t h a t formulas s u c h as (3a) a n d (3b) d o n o t n e c e s s a r i l y hold f o r meaningful h a z a r d rates when t h e conditioning i s "dynamically" o n time d e p e n d e n t random f a c t o r s ?

A f i r s t o b s e r v a t i o n i s t h a t knowledge of

<;

may d i r e c t l y t e l l w h e t h e r f T S t { holds o r not. In o t h e r words, P(T

s

t

1 [@

may b e e i t h e r 0 or 1 , w h e r e a s t y p i c a l

t

values of t h e function 1

-

^exp(-

^f

^{h ( s},<)cis) would b e s t r i c t l y between 0 a n d 1. A s a

0

c o n c r e t e example, o n e could think t h e s u r v i v a l of a n individual, assuming t h a t [ monitors t h e blood p r e s s u r e . A second a n d more formal problem with (2) i s t h a t t h e left-hand s i d e should b e defined f o r a l l sample points of t h e p r o b a b i l i s t i c s p a c e while A(t ,<) in (1) is only p a r t i a l l y defined (on f T 2 t

1).

In o r d e r t o s e t t l e t h e s e questions in t h e most convenient way w e switch o v e r to t h e c u r r e n t l y well-known a n d e x t r e m e l y flexible formalism involving counting p r o c e s s e s a n d t h e i r a s s o c i a t e d compensators [see e.g. J a c o d [2] or L i p t c e r a n d S h i r y a y e v [3].

2. The Reaulta

Let N

=

(Nt ) t M with Nt

=

1

b e t h e p r o c e s s which c o u n t s "one" at T. L e t G

=

(Gt)tzo b e t h e o b s e r v e d h i s t o r y o n which t h e assessment of t h e T-related ha- z a r d i s based, a n d define H

=

(Ht ) t by Ht

=

Gt V ufN,, s

s

t {. Clearly, if T i s a G-stopping time, w e h a v e H

=

G . Both G and H are assumed to s a t i s f y "the usual conditions" r e g a r d i n g right-continuity and completeness [4].

I t i s well known t h a t , u n d e r r e g u l a r i t y conditions, if G i s t h e <-generated histo- r y , h ( t ,<) of (1) s a t i s f i e s t h e r e q u i r e m e n t

The p r o c e s s ( h ( t , t ) l l ~ * ~ j ) t M is called t h e s t o c h a s t i c H-intensity corresponding to T. In f a c t , (4) i s t h e n used d i r e c t l y as t h e definition of s u c h a n intensity, instead of s t a r t i n g from a limit s u c h as (1).

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Let F

=

(Ft )t;rO be t h e process Ft

=

P ( T 5 t

I

Gt ). Clearly,

F

is t h e o r d i n a r y distribution function of T if G is trivial, while F

=

N if T i s a G-stopping time. In g e n e r a l F need not be monotone. I t is easily verified, however, t h a t F is a G- submartingale. We denote the G-compensator of

F

by A, i.e., A

=

(At)tM is t h e unique increasing G-predictable p r o c e s s , with A(0)

=

0 , s u c h t h a t t h e d i f f e r e n c e

F

- A i s a G-martingale (see, e.g. Jacod [Z] o r Liptcer and Shiryayev [3]). Let N and H be as above, and denote by A

=

( % ) t d the H-compensator of N . H e r e is the main r e s u l t of this p a p e r :

Theorem A h a s t h e r e p r e s e n t a t i o n

Proof. First observe t h a t this claim i s trivial if T i s a G-stopping time. In t h e g e n e r a l case where Gt c Ht

,

t

r

0 , i t i s enough t o p r o v e t h a t (i) A i s H-predictable, and (ii)

N-A

i s a n A-martingale.

t

W e start with (i). The integrand of At

'

^dsis left-continuous and

= J

⁰ ¹^-Fs-

H-adapted, t h e r e f o r e A-predictable, while A is G-predictable (by definition) and t h e r e f o r e a l s o A-predictable. The R-predictability of A follows.

In o r d e r t o p r o v e (ii), denote f i r s t m

=

N - A . I t is clear t h a t E

1

mt

1 <

^mf o r all t 2 0 . T h e r e f o r e i t remains t o show t h a t

holds f o r s

<

t

.

We have

The f i r s t t e r m on t h e right-hand side c a n b e written as

while t h e second term becomes

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T h e r e f o r e , (6) i s equal t o

However, h e r e t h e second t e r m vanishes, because, by t h e well-known p r o p e r t i e s of t h e compensator,

W e now show how t h i s t h e o r e m c a n b e used in o u r problem concerning t h e ex- ponential formula. F o r t h i s w e need t h e following two conditions:

(C1): F

=

(Ft )tH, i s absolutely continuous ; (C2): F i s of finite v a r i a t i o n

.

Under t h e s e conditions we have A

=

F , t h e theorem implies in a n obvious way t h e solution t o o u r problem. W e h a v e , when denoting d 4

=

h t d t , t h e following r e s u l t .

Corollary. Suppose

(a)

a n d (C2). Then, d e n o t i n g

the s t o c h a s t i c H - i n t e n s i t y c o r r e s p o n d i n g to T i s g i v e n b y A t

=

Yt llTat 1 , 2 2 0.

Although t h e proof i s obvious from t h e Theorem, some comments o n t h i s r e s u l t should b e helpful. F i r s t l y , (7) i s c l e a r l y equivalent t o

(assuming t h a t P ( T

>

0

1

^Go)

=

1 ) . The c r u c i a l point h e r e i s not t h e equivalence of (7) a n d (8). b u t t h e fact t h a t Y

=

(Yt)tM, being multiplied by l l ~ , ~ ) , i s t h e H- intensity f o r T.

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Secondly, ( C l ) is c l e a r l y n e c e s s a r y f o r (7) t o b e a meaningful definition, and f o r (8) to hold. However, (C2) may need a comment. H e r e i s a simple sufficient con- dition f o r (C2):

(C2'): For all t r 0

,

P ( T 5 t

I

G t )

=

P ( T St

1

G,) a s .

The r e a s o n is t h a t under (C2') F becomes monotone. (C2') postulates t h e conditional independence between [ T St

I

and G,, given by G t . Using t h e terminology of Pitman and Speed (1973). one c a n s a y t h a t T satisfying (C2') i s a randomized G-stopping time.

Notice t h a t o u r conditions f o r (8) are actually quite subtle: If T i s a G-stopping time, (C2') i s c l e a r l y met; however, ( C l ) cannot t h e n hold. In a s e n s e , t h e r e f o r e , w e must think of G , or of

t ,

as information exogenous to t h e a c t u a l counting p r o c e s s N.

3. Conclusion

Mathematical models based on counting p r o c e s s e s and martingales have proved extremely useful in many applied fields, such as biostatistics, reliability t h e o r y , and risk analysis. The m o s t important asset of t h i s a p p r o a c h i s i t s flexibility combined with t h e powerful methods of t h e s t o c h a s t i c calculus. A s this study shows, however, one should b e v e r y cautious when assuming t h a t well-known formulas, such as t h e exponential formula h e r e , automatically have formally similar extensions.

Lastly, a word a b o u t extending o u r r e s u l t s to m o r e g e n e r a l point processes.

Above, w e only considered "the single point process" Nt

=

^lITStt 2 0. If t h e r e are m o r e points, s a y at (O<)T1

<

^T2

< -

, w e could easily switch into t h e counting p r o c e s s

$ =

^j,t ²0. T h e r e f o r e N i s t h e sum of "single point processes",

i P I

and t h e corresponding H-compensator i s automatically a sum of p r o c e s s e s like (5).

e a c h corresponding to s o m e p a r t i c u l a r point Ti. A similar extension of t h e Theorem holds f o r marked point p r o c e s s e s .

On t h e o t h e r hand, t h e Corollary does not s e e m to generalize in a useful manner. The formal generalization of t h e exponential formula would b e

however, t h e left-hand side d o e s not a p p e a r t o have interesting i n t e r p r e t a t i o n s .

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REFERENCES

[I] Chou, C.S. a n d P.A. Meyer (1974) S u r la R e p r e s e n t a t i o n d e s Martingales comme I n t e g r a l e s S t o c h a s t i q u e s d a n s les P r o c e s s u s P o n c t u e l s . Lecture Notes in Mathematics, 381. Berlin: Springer-Verlag

.

[Z] J a c o d , J. (1975) Multivariate Point P r o c e s s e s : P r e d i c t a b l e P r o j e c t i o n , radon- Nicodim Derivatives, R e p r e s e n t a t i o n of Martingales. Z e i t s c h r i , fir Wahrscheinlichkeitstheotie u n d Verwandte Gebieten 31 :235-253.

[3] L i p t c e r , R.S. a n d A.N. S h i r y a y e v (1978) S t a t i s t i c s of Random Processes, Vol. II, Applications. Heidelberg: Springer-Verlag.

[4] D e l l a c h e r i e , C. a n d P . A . Meyer (1982) Probabilities a n d Potential. N o r t h Hol- land.

[5] Pitman, J.W. a n d T.P. S p e e d (1973) A Note o n Random Times. Stochastic Processes AppL. 1:369-374.