On the Notion of Random Intensity

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ON THE NOTION OF RANDOM

INTENSITY

k1. Yashin

March 1985 WP-85-013

Working Papers a r e interi.m reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review, Views or opinions expressed h e r e i n do not necessarily r e p r e s e n t those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

O N THE NOTION O F RANDOM

INTENSITY

A.1. Yashin

DUCTI ION

The

problems of explaining t h e observed t r e n d s in mortality, morbidity a n d o t h e r kinds of individuals' transitions generated t h e numerous attempts of incorporating t h e covariates into t h e survival models. First models use t h e deterministic c o n s t a n t fac- t o r s as explanatory variables

[I,

2,3]. Gradually i t became clear t h a t t h e random a n d dynamic n a t u r e of t h e covariates should also be taken i n t o account [4,5,6]. This understanding h a s led t o t h e fact t h a t t h e notion of random intensity became widely used in t h e analysis of t h e asymptotic properties of t h e maximum likelihood a n d Cox- regression estimators [ 7 , 8 ] .

Having t h e clear intuitive sense t h e notion of random intensity can be introduced in different ways. The traditional way is to define t h e intensity in t e r m s of probability distributions of t h e failure time [9, 10. 111. Another way appeals t o t h e martingale theory [12] a n d defines t h e intensity in t e r m s of t h e predictable process, called

"compensator" [13, 141. For t h e deterministic r a t e s a n d simple cases of stochastic intensities t h e r e a r e already results t h a t establish a one-to-one correspondence between two definitions. The correspondence is r e a c h e d by t h e probabilistic represen- tation results for compensator [13,14]. Martingale theory guarantees t h e existence of t h e predictable compensator in more general cases. However t h e results on t h e proba- bilistic representation similar t o simple cases a r e still unknown. Meanwhile such representation is crucial, for instance, in t h e analysis of t h e relations between the duration of t h e life cycle of some unit and stochastically changing influential vari- ables. This paper shows t h e result of such representation for some particular case.

The generalization on t h e more general situations i s straightforward. The considera- tion will use some basic notions of a "general theory of processes" [12, 151.

-

Let T be t h e stopping time defined on some probability space ( R . H , H , P ) , where H

=

(Ht)t,o is t h e right-continuous nondecreasing family of o-algebras in R, such t h a t

H , =

H and Ho is completed by P-zero sets from

H.

If distribution of T is absolutely continuous t h e traditional definition of t h e intensity X ( t ) , related to t h e stopping time

T is a s follows

Martingale characterization defines the r a t e in terms of the process A ( t ) which is supposed to be H-predictable and such t h a t t h e process M ( t ) defined as

is an H-adapted martingale. It t u r n s out t h a t for process M ( t ) to be martingale, ~ ( t ) should have t h e form

where

A ( u )

is given by (1). The family H in this case is generated by t h e indicator pro- cess

4 = I ( t

2

T ) .

If t h e stopping time T is correlated with some random variable

z ( o ) ,

t h e n t h e traditional approach defines t h e intensity in terms of conditional pro- babilities

Martingale characterization shows t h a t the process

M z ( t )

is a martingale with respect t o t h e family of o-algebras

HZ.

generated in

R

by t h e indi- cator process

I ( t r T )

a n d random variable

z .

In the case of a discontinuous condi- tional distribution function for T, formula

( 2 )

should be corrected. The notion of cumulated intensity

h ( t , z )

is more appropriate in this case. The formula for it is

and martingale characterization is respectively [13,14]:

The situation is not so clear however if one has t h e random processes, say Yt, correlated with stopping time

T .

Assume for instance t h a t Yt simulates t h e changes of t h e physiological variables of some patient i n hospital a n d

T

is the time of death. It is clear t h a t in this case t h e process

3

should terminate a t time T, so observing Yt one can tell about t h e alive/dead s t a t e of the patient, t h a t is, can observe t h e death time.

I t

is clear also t h a t t h e s t a t e or t h e whole history of t h e physiological variable influences t h e chances of occurring death. The question is how can one specify t h e random intensity in terms of conditional probabilities in order to establish t h e correspondence between intuitive traditional and martingale definitions of t h e random intensities.

The idea t o use

h ( t

,

Y )

in t h e form

where a-algebra

q

is generated in

R

by t h e process Y,, up to time

t

fails because

q

contains t h e event

1 T I t

j. Taking

q-

instead of

HfV

seems t o improve t h e situation, however t h e event

fT

2

t 1

is measurable with respect to

q-.

So t o find t h e proper formula for random intensity one needs t o get t h e probabilis- t i c representation result for HY-predictable compensator.

RESULT

HO-TION

We will demonstrate t h e result and t h e ideas of proof on a simple particular case.

The generalization on a more general situation i s straightforward.

Assume t h a t stopping time

~ ( w )

and t h e Wiener process

wt ( w )

a r e defined on some probability space ( R , H , P ) and are independent for any

t

r 0. Define

& =

Wt

- / ( T ~ t ) w ~ .

Let

I P = (q)t90, H = (Hyl)tro

where

q = ^utY,.u t j

The following s t a t e m e n t is t r u e .

Theorem

W - p r e d i c t a b l e c o m p e n s a t o r A Y ( t )

of

t h e process I ( T

S

t ) have the fol- lowing r e ~ e s e n t ation

The proof of t h i s theorem will be followed by some auxilliary lemmas.

Let

H;I =

ol&, u

s t 1,

where

4 = I ( T

5

t ).

I t is clear t h a t

= q

^v

H;I.

Denote

HI = HZ, = HT.

The n e x t assertion underlines t h e important property of t h e stopping times with respect t o

W .

Lemma 1.

Fbr a n y W stopping t i m e

7

one c a n find t h e r a n d o m v m i u b l e s

S

a n d

S'

s u c h t h t i n d i c a t o r I ( S

r

s )

is

m e a s u r a b l e f o r a n y s

r 0 ;

i n d i c a t a I ( S '

r s ) is

HT m e n s u r a b l e f o r a n y s

r 0

a n d :

Let u s first prove t h e first equality. From t h e definition of i t follows t h a t

G- n

^{t s zz}

^{T j = t s TI}

^;

t h e r e f o r e

( V V

^H&)

n1.5 T I = q n ^{t s s}

^TI

o r by virtue of coincidence of t h e s e t s @'

n ^{[ t s TI}

^{a n d}

(q

^{v H f )}

n ^{[ t}

^ZZ

^{T ]}

^{and con-}

tinuity of t h e flow of 8-algebras W :

H'f- n t s s T ] = g ' n 1 ~ 1 T j .

Taking t h e s e t [ r

<

^s

I

^{from II,YY} one c a n find t h e s e t

D,

from s u c h t h a t

- 6 -

~ T < s ] n

1s

s T ] = D s n

1s

s T ] .

Note now t h a t for T

<

s

lo, ^{U D , I} n

^{I S}

^s T I = D ~ n

^1s

^{T ]}

In f a c t

Define t h e random variable S by t h e relations

Where ^T a n d s a r e r a t i o n a l n u m b e r s we have

which completes t h e proof of t h e first equality. The s e c o n d equality c a n b e proved i n a similar way.

The n e x t lemma deals with t h e r e p r e s e n t a t i o n of IF"-adapted s q u a r e integrable martingales a s s t o c h a s t i c i n t e g r a l s with r e s p e c t t o Wiener p r o c e s s

Wt.

kmma 2. Let

h? = ( M t )t,O

be a W - u d a p t e d s q u u r e i n t e g r a b l e m a r t i n g a l e . Then f h e r e .is an Hw-adapted p r o c e s s f ( s , w ) , s r 0 _{s u c h}

that

f o r m y f

a n d

The proof of this lemma may be found in [13].

The next s t a t e m e n t will play an important role in t h e characterization of

W -

predictable processes.

lemma

3. Any W - w e l l measurable process is W - p r e d i c t a b l e .

Proof. Let T be t h e a r b i t r a r y stopping time with respect t o

H"'.

Denote by A(t) W - p r e d i c t a b l e compensator of t h e process I ( T

s

t ) . According t o t h e definition of

compensator t h e process Mt

=

I ( T 6 t )

-

A(t) is t h e bounded martingale with r e s p e c t t o

W .

According to Lemma 2 t h e r e exists H"'-adapted function f ( s , o ) s u c h t h a t

By virtue of the continuity of t h e stochastic integral's trajectories, t h e right-hand side of t h i s equality is W - p r e d i c t a b l e . This f a c t yields t h e IT"-predictability of t h e indica- t o r I ( T s t ) for a n y t t h a t in t u r n yields t h e H"'-predictability of t h e stopping time 7.

The predictability property for any stopping time with r e s p e c t t o

W

yields t h e coin- cidence of t h e well-measurable a n d predictable a-algebras.

lemma

4. Hv-adapted process

Z =

(q)t,o is P - p r e d i c f a b l e if and only if there is an W - m e a s u r a b l e process X

=

(4)t,o a n d Hy-well m e a s u m b l e process X'

=

^(x;'),,~

s u c h that

h o o f . Necessity. Let Zt

=

I ( t S T) where T is an a r b i t r a r y stopping time with r e s p e c t t o W . According t o t h e r e s u l t of Lemma 1 we have:

where S is some random variable, such t h a t indicator

I ( t

S S ) is @'-measurable for any

t

2 0. ,Setting

Yt = I ( t

s S ) we get t h e necessity of t h e condition (6) for t h e pro- cess Zt

= I ( t s

^{7 ) .}

Denote by

L

t h e s e t of bounded *-adapted functions which satisfy relation (6). I t is easy t o see t h a t

L

is closed with respect t o monotonic convergence t o bounded func- tions and contains t h e s e t of indicators

I ( t

1 T ) , where T is arbitrary stopping times with respect t o W . This s e t of indicators is t h e algebra; i t contains t h e 1, and gen- e r a t e s t h e @-predictable a-algebra in

R

x

R,.

The necessity of the condition (6) of t h e lemma follows as a result of montone classes theorem [15]. In t h e same way the necessity of condition

( 7 )

can be proved

Sufficiency. Note t h a t for any arbitrary P - a d a p t e d and left continuous process Lt t h e process

I ( t s T)Lt

is @-adapted and left continuous; therefore

where

P ( P )

a n d

P ( @ )

denote El""-predictable and W-predictable a-algebras, respec- tively. According t o Lemma 3 t h e process

Yt

from t h e relation (6) is W-predictable, a n d process

X

from

( 7 )

is Hw-predictable. Consequently t h e processes

4 I ( t

^I

T )

and

&I(T

⁵ f ) a r e @-predictable.

Define t h e process

A ( t )

by t h e equality

The following statement concerns t h e measurability property of

A ( t )

which is impor- t a n t for predictable specification of t h e compensator.

Lemma 5. ?he process h(t ) is

W

-predictable.

Proof. For h(t ) one can write

The first addendum on t h e right-hand side is an W-predictable process in accordance with Lemma 4. The second is an W-adapted, left-continuous process and consequently i t is also P - p r e d i c t a b l e .

The next lemma demonstrates t h e last effort i n proving t h e main result.

Lemma 6. 7he process

=

I(T s t )

-

^{A(t ) is}

W

-martingale.

Proof. Consider t h e process

It i s clear t h a t

E 1

^{~ ()}

1

^t

<

=. Let u s prove t h a t

E ( M ( ~ )

1 =

M(S)

.

We have

The first t e r m can be transformed i n t o

In order to transform t h e second addendum l e t us assume for simplicity t h a t t h e conditional distribution of T is absolutely continuous a n d consequently A(t) can be r e p r e s e n t e d i n t h e form

t

where h ( w , u ) is FP-adapted process

we have

Taking into account formula ( 9 ) for X ( W . U ) , t h e last addendum can be transformed into t h e form

Subtracting ( 1 0 ) from (8) a n d taking i n t o account ( l l ) , one can s e e t h a t

The uniqueness of W-predictable compensator yields t h e coincidence of A ( t ) a n d A Y ( t ) t h a t completes t h e proof.

Remark. It t u r n s out t h a t even if t h e o-algebra has t h e more general struc- t u r e t h a n in t h e example above, for instance,

=

@ v

p,

t h e r e s u l t of t h e proba- bilistic representation of t h e P - p r e d i c t a b l e compensator of t h e process

X(t )

=

I ( T

s

t ) given by formula (5) is t r u e .

Example. The relevance of t h e results such as formula (5) becomes evident from the following example. Assume t h a t Wiener process

Wt

and stopping time T a r e inter- related and t h e random intensity of occurrence

T

is h ( t ) ~ f 2 . Formula (5) gives immediately the form of t h e conditional survival function

In survival analysis the stopping time

T

is associated with the death or failure time and the research is often focused on the properties of t h e survival function S ( t )

=

P ( T

> t

) [8,11,10,7]. The straightforward way of its calculation is the averag- ing of the conditional survival function S ( t (

w;).

It t u r n s out t h a t (see for instance

161)

For the Wiener process t h e conditional mathematical expectation on t h e right-hand side of this formula can be easily calculated ( t h e condition

W o =

0 is used there)

where y(t ) is t h e solution of t h e differential equation

+ ( t )

=

1 - 2 h ( t ) ? ( t ) , y(O)=O

.

When A(t) i s constantly equal, say, t o

- 1

the straightforward calculations lead to the 2

formula

which coincides with t h e r e s u l t based on the Cameron a n d Martin formula 1131.

CONCLUSION

Formula (5) c a n be generalized in more complex c a s e s including t h e sequence of stopping times and semimartingale as an influential stochastic process. I t can be use- ful in t h e field of survival analysis, reliability theory and risk analysis. I t shows which particular conditional distribution functions should be used i n specification of t h e random intensities. The specification of the influential process and t h e measurement schemas provide t h e particular forms for t h e distributions and t h e intensities. Some examples, r e l a t e d to t h e biomedical and demographical applications a r e discussed in [17, 18, 161.

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