Hazard Rates and Probability Distributions: Representation of Random Intensities

NOT FOR QUOTATION WITHOUT PERMISSION OF

THE

AUTHOR

HAZARD

RATES AND PROBABIUTY DISIIZIBUTIONS:

REFRESENTATION

OF

RANDOM INTENSITIES

AI. Yashin

March 1984 WP-84-21

Working m e r s are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSI'ITUTE FOR

APPLIED SYSTEMS

ANALYSIS 2361 Laxenburg, Austria

PREFACE

Recent a t t e m p t s to applysthe results of martingale theory in proba- bility theory have shown t h a t i t is first necessary to i n t e r p r e t this abstract mathematical theory in more conventional terms. One example of t h i s is t h e need t o obtain a representation of t h e dual predictable pro- jec tions (compensators) used in martingale theory in terms of probabil- ity distributions. However, up to now a representation of this type has been derived only for one special case.

In this paper, t h e author gives probabilistic representations of the dual predictable projection of integer-valued random measures t h a t correspond t o jumps in a semimartingale with respect t o the a-algebras generated by this process. The results a r e of practical importance because s u c h dual predictable projections a r e usually interpreted as ran- dom intensities or hazard rates related t o jumps in trajectories: applica- tions a r e found in such fields as mathematical demography and risk analysis.

ANDRZEJ WIERZBICKI

Chairman

System and Decision Sciences

HAZARD RATES AND PROBABILITY DISI?IIBUTIONS:

REPRESENTATION OF RANDOM INTENSITIES

International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria

The development of the martingale approach in the theory of random processes has made i t possible t o formalize and t h e n generalize many of t h e intuitive notions commonly used in applied fields. One of these is concerned with t h e concepts of h a z a r d and hazard rate.

The t e r m hazard r a t e is usually associated with t h e probability of occurrence of some unexpedted event or series of such events. This notion, which is popular i n risk analysis, corresponds to t h e idea of a compensator or d d predictable projection in martingale theory [1,2.3,4,5,6,7.8,9.10

1.

Many important results from this theory a r e formulated in t e r m s of compensators:

t h e s e include convergence of the parameter estimators and conditions for absolute continuity a n d singularity of the probabilistic measures [ 5 , 6 ] .

Probabilistic representation of t h e compensators provides a bridge between theory a n d applications. This paper is concerned with a generalization of Jacod's important r e s u l t [I] in this area.

2. BASIC NOTATION

AND

DEFINITIONS

bet ( Q , H . H , P ) be a probabilistic space, where H

= (H~)~,,,

is some non- decreasing r i g h t c o n t i n u o u s family of a-algebras, H

= H,,

and a-algebra Ho is completed by F z e r o s e t s from H.

A real-valued random process

q ,

t ² 0 , is said t o be H-adapted if for any u _r 0 random variable Y,, is &-measurable.

A non-negative random variable T is called the H-stopping time if t h e indi- c a t o r process

7 =

I ( T 6

t ) , t

r 0 , is H-adapted. We will use t h e notation

TAS

t o describe t h e stopping time T

=

min (1,s).

For any H-stopping time T there exists a o-algebra

HT

in f2, generated by events Afrom H such that for any t

r

0 we have A

n

t T S t E Ht.

The H-adapted process mt is called an H-marfingale if

E

] m t

I

^{S m} for any t r O a n d E ( m t ( H u ) = m u f o r a n y t S u r 0 .

A real-valued H-adapted process is a local H-martingale if there exists a sequence of H-stopping times (Tn)n,O such that lim Tn

=

^m and for any n

r

0

n +-

the processes m t A q . t > 0, a r e uniformly integrable martingales.

A

real-valued process Yt is K w e l l - m e m u ~ a b l e if t h e mapping ( o . t ) ^-B Yt is measurable with respect to t h e a-algebra W(H) in fl x (0,m) generated by all H- adapted, right-continuous processes.

A real-valued process Yt is H-predictable if the mapping ( o , t ) ^-B q ( o ) is measurable with respect to t h e a-algebra n(H) in

Q

x (0,m) generated by all H- adapted, left-continuous processes.

A stopping time T is said to be H-predictable if t h e process

5 =

I(T s t ) , t > ^0, is H-predictable.

The H-adapted process Yt, t

r

0, is an H-semimartingale if it may be represented in the form:

where

4

, t ² 0 , is a locally integrable variation process and Mt is an H-adapted local martingale.

We shall l e t ( E ~ B ( E ~ ) ) denote the measurable space such t h a t EA

=

EuA , where A is some auxiliary point, B(EA)

=

B(E)

y

[A{, E is Lusin space and B(E) is t h e Borelian u-algebra on E.

We will use the t e r m random meusure to describe t h e non-negative transi- tion measure t7(o;dt.&) from ( f l , ~ ) over (0,m) x

Ek

Let n ( ~ ) denote the o-algebra in

Q

x (0,-) x E defined by:

A random measure ⁷⁾ is called H-predictable if for each non-negative n ( ~ ) - measurable function X t h e process ( T X ) ~ (o) , t r 0, defined by

is H-predictable.

Hereafter we will omit the symbol o for simplicity.

We will also use t h e notation GvF to describe the a-algebra in Q generated by s e t s from a-algebras G and F.

3. J A W S -ATION RESULT

Jacod's formula for t h e random intensity function deals with t h e . c a s e in which environmental factors a r e random variables a n d consequently do not change over time. The general process whose intensity is of i n t e r e s t is a sequence of random times and random variables called a multivariate point process.

Some additional formal constructions will be useful in deriving t h e representation of t h e random intensity in this particular case.

3.1. Multivariate point processes

According t o [I], a multivariate point process is a sequence (Tn,&),,,, where the Tn a r e H-stopping times and the

Z,

a r e HTn-measurable random vari- ables with values in ( E ~ , B ( E ~ ) ) . Note t h a t

Z, =

A if and only if Tn

=

m , a n d t h a t t h e stopping times Tn have the following properties :

(ii) Tn

>

Tn , if Tn

< -

^,

(iii) Tn+]

=T,,

if Tn

= .

I t follows from these assumptions t h a t sequence (Tn)n,O has a unique accumulation point T,

=

lim Tn C

-.

We will assume t h a t

T , =

^m , To

=

0.

n*

A sequence of stopping times (Tn)n,O satisfying conditions (i)--(iii) is called a univariate point process or simple point process. Any arbitrary discrete-time random process is naturally also a multivariate point process.

A multivariate point process is uniquely c h a r a c t e r i z e d by t h e integer- valued random measure /I on (0.m) x E defined by t h e equality:

In the r e s t of t h i s paper we shall use p t o denote t h e integer-valued random measure generated by some multivariate point process (T,,%),,~.

3.2. Daal predictable projections of intege~valued random masures

We shall d e h e

H/'

as a a-algebra in

Q

generated by the multivariate point process or, equivalently, by t h e integer-valued random measure p u p t o t i m e t:

a n d l e t

Ho

be some fixed a-algebra in Q. Denote by H# t h e non-decreasing fam- ily of a-algebras

where

a r e a-algebras in Q g e n e r a t e d by t h e union of and

Hf.

_t 1 0. The family is known to be right-continuous [2

1.

According t o [I], t h e r e is one a n d only one (up t o a modification on a P- null set) H#-predictable random m e a s u r e vo on ( 0 , ~ ) x E such t h a t for each non-negative n(H#)-measurable function X we have

Measure vo is called the dual *-predictable projection of I t t u r n s o u t t h a t one can choose a version of vo which P a . s . satisfles the inequality:

We shall u s e t h e following equivalent formulation of the above r e s u l t in this paper:

The random measure vo is characterized by (2) and

(i) t h e process v0((0,t],

. ) ' I

_t

a

0 . is Hg-predictable for any

r

^{E B ( E )}

HAZARD RATES AND PROBABILITY DISI'RlBUTIONS:

REPRESENTATION OF RANDOM INTENSITIES

International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria

1. INTRODUCTION

The development of the martingale approach in the theory of random processes has made i t possible t o formalize and t h e n generalize many of t h e intuitive notions commonly used in applied fields. One of these is concerned with t h e concepts of h a z a r d and hazard r a t e .

The t e r m h a z w d r a t e is usually associated with the probability of occurrence of some unexpedted event or series of such events. This notion, which is popular in r i s k analysis, corresponds to the idea of a compensator or d d predictable projection in martingale theory [I, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10

1.

Many important results from this theory a r e formulated in t e r m s of compensators:

t h e s e include convergence of t h e parameter estimators and conditions for absolute continuity a n d singularity of the probabilistic measures [5,6].

Probabilistic representation of the compensators provides a bridge between theory a n d applications. This paper is concerned with a generalization of Jacod's important r e s u l t [I] in t h i s area.

2. BASIC NOTATION

AND

DEFlMTIONS

bet (O,H,&P) be a probabilistic space, where H = is some non- decreasing right-continuous family of a-algebras, H

=

H,, and a-algebra Ho is completed by P-zero sets from H.

A real-valued random process

5 ,

t ² 0 , is said to be H-adapted if for any u t 0 random variable

Y,

is %-measurable.

A non-negative random variable T is called the H-stopping t i m e if t h e indi- c a t o r process

5 = 1(T s

t ) , t ^{2 0 ,} is H-zdapted. We will use the notation TAS to describe the stopping t i m e T

=

min (T,S).

For any H-stopping t i m e T there exists a u-algebra HT in R, generated by events A ^from H s u c h t h a t for any t r 0 we have A r\ [ T 5 t j E Ht

.

The H-adapted process m t is called an H-martingale if E

lmt I

^{S m for any}

t

r

0 and E(mt

I

^H,) = m u for any t

r

u 2 0.

A real-valued H-adapted process is a local H-martingale if t h e r e exists a sequence of H-stopping times (Tn)nso such t h a t lim Tn

=

^m and for any n

r

0

n ^+-

the processes m t ~ l f , , t 2 0, a r e uniformly integrable martingales.

A

real-valued process Yt is H-well-measu~able if t h e mapping ( a t ) -r

5

^is

measurable with respect t o t h e o-algebra w(H) in R ^{x (0,-)} generated by all H- adapted, right-continuous processes.

A real-valued process Yt is H-predictable if t h e mapping ( o , t ) -, &(w) is measurable with respect to t h e o-algebra n(H) in h2 x (0,m) generated by all H- adapted, left-continuous processes.

A stopping time T is said to be Hpredictable if t h e process

Yt

=

I(T 5

t ) ,

t 2 0, i s H-predictable.

The H-adapted process

Yf

, t r 0, is an H-semimartingale if i t may be represented in t h e form:

where

4 ,

t ^{2 0 ,} is a locally integrable variation process and Mt is a n H-adapted local martingale.

We shall l e t (EA, B(EA)) denote t h e measurable space such t h a t EA

=

EVA,

where A is some auxiliary point, B(EA)

=

B(E)

u

^[A], E is Lusin space and B(E) is t h e Borelian a-algebra on E.

We will use t h e t e r m random measure to describe t h e non-negative transi- tion measure rl(w;dt,dz) from (R,H) over ( 0 , ~ ) x EA.

Let n ( ~ ) denote t h e o-algebra in

n

^{x (0,m) x} E defined by:

A random measure 17 is called H-predictable if for each non-negative n ( ~ ) - measurable function X t h e process

(9at

^{(w). t r 0,} defined by

is H-predictable.

Hereafter we will omit the symbol GI for simplicity.

We will also use t h e notation GvF to describe t h e a-algebra in R ^generated

by sets from a-algebras G and F.

3. JACOD'S REPRESEXTATION RESULT

Jacod's formula for t h e random intensity function deals with t h e . c a s e in which environmental factors are random variables and consequently do not change over time. The general process whose intensity is of interest is a sequence of random times and random variables called a multivariate point process.

Some additional formal constructions will be useful in deriving t h e representation of t h e random intensity in this particular case.

3.1. Multivariate point processes

According to [I], a multivariate point process i s a sequence (Tn.G)n,O, where t h e T, a r e H-stopping times and the 2& a r e Hm-measurable random vari- ables with values in (EA,B(EA)). Note t h a t

Z,

= A if and only if

Tn

= ^m, and t h a t t h e stopping times T, have the following properties :

(ii) Tn

>

^{T, , if}

^Tn <

= , (iii) Tn = Tn, if Tn = =

I t follows from t h e s e assumptions t h a t sequence (Tn)n,o has a unique accumulation point T, = lim Tn

<

^{m .} We will assume t h a t T,

=

, To = 0.

n-

A sequence of stopping times (T,)n,O satisfying conditions ( i ) - ( i i i ) is called a univariate point process or simple point process. Any arbitrary discrete-time random process is naturally also a multivariate point process.

A multivariate point process is uniquely characterized by t h e integer- valued random measure p on ( 0 , ~ ) x E defined by t h e equality:

In t h e r e s t of t h i s paper we shall use 1 t o denote t h e integer-valued r a n d o m m e a s u r e g e n e r a t e d by s o m e multivariate point process (T,.%),,~.

3.2. Dual predictable projections of integer-valued random measures

We shall define

Hf

a s a a-algebra in R g e n e r a t e d by t h e multivariate point process o r , equivalently, by t h e in teger-valued random m e a s u r e p u p t o t i m e

t

:

H/ =

u ~ ~ ( ( o . u ] , r ) , u s

t , r

^{E B(E)J}

a n d l e t

go

be some fixed a-algebra i n R. Denote by

%

t h e non-decreasing fam- ily of a-alg e b r a s

where

a r e a-algebras in R g e n e r a t e d by t h e union of

hTo

and

HP,

f 2 0. The family i s known t o b e right-continuous [2

1.

According t o [I], t h e r e is one a n d only one (up t o a modification on a P - null s e t ) %-predictable r a n d o m m e a s u r e vo on (0,m) x E s u c h t h a t f o r e a c h non-negative PI(%)-measurable function X we have

Measure vo i s called t h e dual w - p r e d i c t a b l e projection of 1t t u r n s o u t t h a t o n e c a n choose a version of vo which P-a.s. satisfies t h e inequality:

We shall u s e t h e following equivalent formulation of t h e above r e s u l t in t h i s paper:

The r a n d o m m e a s u r e vo i s c h a r a c t e r i z e d by (2) a n d

(i) t h e process vo((O,t], r ) ,

t

^{1 0 ,} is Hg-predictable for any

r

^{E B ( E )}

(ii) the process (p((0,t],

r)

^{- v0((0,t],}

r)),

^{t 2} 0, is an %-adapted local mar- tingale.

Dual predictable projections of integer-valued random measures may be interpreted as generalized cumulative random intensity functions.

Remark Notice here t h a t a-algebra

Ho

is not necessarily formed by events independent of H t . For instance,

go

could be the a-algebra in R corresponding t o the past history of some random process u p to time 0. We shall consider t h e a-algebra

Ho

generated by a Wiener process up t o time m.

3.3. Probabilistic representation of random intensity functions

In some senses, Jacod's representation of dual H#-predictable projections serves as a bridge between the abstract theory of random processes with jumps currently developing in the framework of the martingale approach, and t h e wide range of applications based largely on t h e knowledge of probabilistic distri- bution f u ~ c t i o n s .

In order to express Jacod's result, we have to define regular versions of t h e

HbfTn -conditional probabilities of events [T,,l

s

uj

n f & , l ~ rj,

^{2 u j,}

where u ² 0 , E B (E) and n = 1,2,

... .

This may be done using equalities

I t should be emphasized once again t h a t in t h i s part of t h e paper we a r e consid- ering t h e case in which additional information about events and variables influencing t h e multivariate point process perceived by the statistician (observer) does not change over time.

The following theorem was proved in [I].

Theorem 1. 7he following is a representation of a dual %-predictable projec- tian vo of r n e m r e p:

Corollary. Notice t h a t if t h e u-algebra

%

does not provide t h e observer with any information about t h e events, and if all he has a t time t is information

about events from t h e history

Hf,

the hazard r a t e coincides with t h e dual

'- I$

predictable projection v of the measure p and the formula for v is a simple corollary of equation (3):

In t h e case of a pure point process (a sequence of random times T,), t h e formula for t h e dual WL-predictable compsnsator A(t ) becomes:

where the a-algebras

f l ,

^{t r 0,} a r e generated by t h e values of t h e point pro- cess (T,),,~ or, equivalently, by t h e values of the counting process

Nu,

u r 0, u p to time t .

Equations (3') and (3") produce h o w n results when applied to well-studied processes. Thus, for a Poisson process with deterministic local intensity func- tion X(t), t h e dual predictable projection A(t) defined by equation (3") coin- cides with t h e cumulative hazard rate and is given by t h e equality

For a 6nite-state. continuous-time, Markov process

Ct

, t 2 0, with s t a t e s 11,2

...., ^Nj

and intensities )4,(t), i , j

=

(1,2

,...,

N), t s 0 , equation (3') gives t h e dual predictable projection in t h e form:

Now assume t h a t t h e observer h a s some additional information about t h e intensity function, for instance, t h a t h e knows the value of some random vari- able Z which influences t h e frequency of t h e jumps. This means t h a t he is deal- ing with the history H g t a s determined by the equality ( I ) , where u-algebra

&

coincides with t h e u-algebra U(Z) generated by random variable Z in R

.

^{In this}

case t h e observer needs t o use t h e dual w - p r e d i c t a b l e projection of p as a ran- dom intensity function. Thus, in t h e c a s e of a double stochastic Poisson pro- cess

Nt

,

t

2 0 , with random intensity function ZX(t)

.

^{t 2 0} (where Z is some

positive r a n d o m variable), it is necessary t o use eqn. (3) which gives t h e follow- ing dual %-predictable projection of Nt , t 2 0:

Note t h a t if two observers have different information about t h e processes occurring i n s o m e r e a l s y s t e m ( f o r example, if one of t h e m knows t h e value of t h e variable Z a n d t h e o t h e r does not), t h e y will u s e different h a z a r d r a t e s t o e s t i m a t e t h e probability of change. In t h e c a s e of continuously distributed jump-times in a double stochastic Poisson process, t h e relation between t h e two i n t e n s i t i e s (derived from a comparison between eqns. (3") a n d (3')) m a y be r e p r e s e n t e d a s follows:

where h ( t ) i s t h e ~ ~ - ~ r e d i c t a b l e local h a z a r d r a t e . This i s r e l a t e d t o t h e dual

~ ~ - ~ r e d i c t a b l e projection x(t ) of

Nt

by t h e equality:

We s h a l l now prove t h e relation between x ( t ) a n d h ( t ) . Consider t h e integral ~ , ( t ) defined by t h e equality:

which i s t a k e n from t h e right-hand side of eqn. (3"). Let cpp(u) be t h e density f u n c t i o n of conditional distribution P(Tp+l

s

u

I HF

^). Note t h a t t h e following

P

equality holds:

where f ( z ) i s t h e density distribution function of random variable Z. Using t h i s equality a n d noting t h a t for Tp l u

we have

Since

we can derive t h e formula for the

A ( t )

From eqn. (3").

Remark. Equation (4) shows t h a t to calculate intensity functions A ( t ) or h ( t ) it is first necessary t o calculate the HF-conditional mathematical expectation.

This problem can be overcome by using a n approach based on filtering of t h e jumping processes (see, f0.r instance, [ l l , 121). In the simplest life-cycle models, which are characterized only by stopping time T (time of death) and a r e widely used in reliability and demographic analysis, t h e random intensity describes differences in susceptibility to death or failure [13]. Notice t h a t in t h e case of life-cycle processes, equation (3") gives t h e following relation between the H-adapted compensator A(t) and t h e local intensity function X(t):

Recall that. from t h e definition of t h e compensator, t h e process

is an H-adapted martingale.

4. GENERAL F'OKMULA

MIR

REPRESENTATION OF FtANDOM INTENSITY

In spite of t h e fact t h a t dual predictable projections exist for a wide class of families of a-algebras, probabilistic representations a r e known only for a- algebras with s t r u c t u r e (3). However, in practical situations u-algebras often have a more general s t r u c t u r e . In particular, new information may be gen- erated not only by the multivariant point process but also by some additional process q t . In this case the u-algebras describing t h e observation history have t h e form Ht

=

H t v v H t , where a-algebras H t are generated by some process 77:

which is observed simultaneously with t h e multivariate point process ( ) Detailed probabilistic characterization of dual predictable

projections is often useful when applying t h e results of the general theory of random processes in practice.

In this section we will give t h e probabilistic representation of t h e dual predictable projection of integer-valued random m e a s u r e s corresponding to t h e jumps of t h e semimartingale with r e s p e c t t o t h e family of o-algebras g e n e r a t e d by this process.

4.1. Generation of jumps

Let r a n d o m process

4 ,

t ² 0, defined on probability space (R,H,H,P) be H- adapted, take values in F@ a n d have right-continuous, left-limited sampling paths. Denote by

IF

t h e family of o-algebras

a n d a s s u m e t h a t

H;;

is completed by t h e s e t s from

Hf

= Hf, with a P-probability of zero. Assume also t h a t process

4

m a y be r e p r e s e n t e d a s follows:

where

4

^{a n d}

^B,

a r e F - a d a p t e d m a t r i c e s of appropriate dimensions. m a t r i x

B,

i s non-singular for any u 2 0,

E = ~ ^-

^{101, and w,} is a n H-adapted, k - dimensional Wiener process independent of

Xo.

Note t h a t , in general, t h e dimension of

4

may be g r e a t e r t h a n t h a t of t h e jumping changes. This c a n m e a n , for instance, t h a t we a r e also considering situations with two-component processes of which one is pure jumping a n d t h e o t h e r is continuous.

I t follows from (3) t h a t t h e m e a s u r e p is I F - a d a p t e d a n d t h a t

Let f l denote t h e dual IF-predictable projection of integer-valued random m e a s u r e y Our m a i n aim i s t o derive t h e probabilistic r e p r e s e n t a t i o n of meas- u r e #.

4.2. The form of the hazard rate

The main r e s u l t of this subsection is formulated in t e r m s of auxiliary processes defined by t h e equalities:

where A , B, ^{w ,} p have b e e n defined previously. Introducing t h e m e a s u r e s pn defined by t h e equalities

eqn. ( 8 ) m a y be rewritten a s follows:

Notice t h a t m e a s u r e & m a y be considered a s a m e a s u r e of t h e jumps of t h e process

IS,

,

t

r 0.

Introduce a-algebras I$' a n d H? s u c h that:

a n d define t h e r e g u l a r versions of t h e conditional probabilities of e v e n t s

using t h e equalities

The next assertion is t h e m a i n r e s u l t of t h i s paper.

Theorem 2. Assume that coefficients A and

B

aTe such thut a strong s o l u t i o n o j equation (3) e z i s t s and is u n i q u e . Then w e have the jollowing ~ e p r e s e n t a t i o n o j the dud ) f Z - p e d i c t a b l e projection 9 o j i d e g e ~ - v d U @ d TtZndom measuTe p:

This theorem is proved in t h e Appendix.

Sometimes i t is m o r e convenient t o use a n o t h e r form of representation for 17, transforming t h e conditional probabilities on t h e right-hand side of (7). For t h i s purpose we introduce t h e function F (u,

I?),

making use of the equality

where P(2&+, E I ' ( H t + I - ) i s t h e regular version of t h e Hc+l--conditional pro- bability of event € I'j. The dual p - p r e d i c t a b l e projection v2 of m e a s u r e p may t h e n be r e p r e s e n t e d i n t e r m s of this function a s follows:

5.1. Conditional Gaussian property

Let p r o c e s s ~ ( t ) . t 2 0 , satisfy t h e linear stochastic differential equation

where Yo is a Gaussian r a n d o m variable with m e a n m,, a n d variance yo, w ( t ) is an H-adapted Wiener process, H = (Ht)t20 i s some non-decreasing, right- continuous family of o-algebras, a n d Ho is completed by F z e r o s e t s from

H =

H,. Denote by I$' t h e family of a-algebras in R g e n e r a t e d by t h e values of t h e random process Y(u), i.e.,

Assume t h a t process ~ ( t ) determines t h e random r a t e of occurrence of some unexpected e v e n t c h a r a c t e r i z e d by t h e random time T. t h r o u g h t h e equality:

Notice t h a t process ~ ( u )

=

? ( u ) , u 2 0 , may be interpreted as the frailty of an individual changing stochastically over time. Using t h e terminology of martingale theory one can say that the process

is an W-predictable compensator of the life-cycle process

4 =

1(T

<

f ) ,

t r

0.

This means t h a t t h e process Mt

=

I ( T

< t ) -

~ ( t ) ,

t

r 0 i s an @-adapted mar- tingale. Associating t h e stopping time T with t h e t i m e of death, we m a y describe t h e process y 2 ( f ) , t r 0, a s t h e age-specific mortality r a t e for an indi- vidual with history

flo

⁼ f ~ ( u ) j , 0 I u I

t .

Letting x ( t ) ,

t

² 0, denote t h e observed age-specific mortality r a t e we h a v e i ( t ) = h ( t ) a t ) , t 1 0 , w h e r e Z ( t ) = ~ t ~ ~ ( t ) I ~ > t j [13].

In order to calculate t h e observed mortality r a t e x ( t ) , t 2 0 , i t is neces- s a r y first t o calculate t h e second moment of t h e conditional distribution of t h e Y ( U ) given t h e e v e n t I T r 0

j .

I t t u r n s out that this m o m e n t may be calculated quite easily using t h e result of the Following theorem.

Theorem 3. A s s u m e t h a t p r o c s s s ~ ( t ) a n d s t o p p i n g t i m e T are r e l a t e d t h ~ o u g h eqns. ( 8 ) a n d ( 9 ) . T h e n t h e conditional d i s t r i b u t i o n of

~ ( t

) g i v e n IT r t j is Gam- sian. 2he p a r a m e t e ~ s of this d i s t r i b u t i o n , i.e . , t h e m e a n mt a n d t h e v a r i a n c e yt , are g i v e n b y t h e following e q u a t i o n s :

7hs f o w n u t a f o r x(t ^{) is} t h e n h ( t )

=

h ( l ) (mt2

+

^{y t )}

.

This theorem m a y be proved in a similar way as t h e conditional Gaussian property for processes governed by stochastic differential equations of the diffusion type (see [14]).

6. APPENDIX PROOF OF THEOKEM 2

The proof uses r e p r e s e n t a t i o n ( 3 ) for vo. I t t u r n s out t h a t if t h e u-algebra

Ho

in eqn. (1) is of a p a r t i c u l a r form (which will be specified l a t e r ) t h e n t h e H t % ^- conditional probabilities of events ITp+l S u

jnI25,

^E

rj

^{a n d} fTp+l ² u r i l l P-a.s. coincide with t h e *-conditional probabilities of t h e s e e v e n t s on t h e integration intervals i n (7). Representation of m e a s u r e vo t h r o u g h

@-

conditional probabilities m a k e s it e a s i e r t o prove i t s Hz-predictability property.

It i s t h e n easy t o c h e c k t h a t t h e process

is an Hz-adapted martingale for a n y

I'

E B(E). The fact t h a t

IF

is unique shows t h a t IF a n d vo coincide P-a.s. Representation (7) i s derived from (3) t h r o u g h substitution of t h e conditional probabilities. Several auxiliary r e s u l t s will be useful i n t h e proof of Theorem 2: t h e s e are derived in t h e following subsections.

8.1. Auxiliary a-algebras

Introduce t h e auxiliary right-continuous families of o-algebras

W , W , Wlp,

c, _w8p

^{a n d}

p,

^where

where

H

is some o-algebra in R and o-algebras a n d

Hf

a r e completed by P-

z e r o s e t s from a-algebras

I?"'

^{and H p ,} respectively.

Recall also t h a t family

I F

i s defined a s follows:

6.2. Existence of

HZ

-predictable projections

We shall now establish t h a t dual Hz-predictable projections exist and a r e unique.

Lemma 1. m e dual H z - p r e d i c t a b l e p r o j e c t i o n s

vZ

of the i n t e g e r - v a l u e d r a n d o m m e m e p e z i s t and are u n i q u e .

Proof. Note that the sets [0] E IT,,

.T, E

belong to and have measure

less than or equal to 1. This means t h a t measure

M p

is a-finite on

((n

( O . t ] E),

W).

From [I] , this implies t h a t t h e lemma is true.

6.3. Characterization of

P

IF-stopping times For any t ² 0 l e t

The next assertion is a generalization of Lemma 3.2 in Jacod's paper [I].

Lemrna2. Let T be the P I P - s t o p p i n g t i m e . For a n y n r 0 t h e r e e z i s t s a r a n d o m variable

3 '

s u c h t h d i n d i c a t o r I(SL r u ) is <,-measurable f o r any u 2 0 and the Jollowing equality h o l d s :

Proof. It follows f r o m t h e definition of t h e u-algebra t h a t the following fami- lies of sets coincide:

Ht- n

^ITn

<

^u

s

Tn+,] = H t

n

^tTn

<

u

s

Tn+,j

and consequently t h e families of s e t s

( W H P - )

n

^{tTn < U <Tn+,j} = =vHffn

n

iTn

<

^u s T n + l j

also coincide. Take t h e s e t tT

<

^uj from

q v q - ,

and find t h e s e t

D ,

from W H Y n such t h a t

Note t h a t for r

<

u we now have

[(Dr n l T n

<

r j ) U(D, n I T n

< uj)ln!

^T,

^<

^{u I Tn+,j = D,}

IT, <

u I T,+,j . Define SL by t h e equalities

where t h e r are rational numbers. We then obtain

I T < u I n ! T n < u s T n + l j =

IF

< u I n I T , < U <Tn+li

thus completing t h e proof of Lemma 2.

8.4. Representation of martingales

The following result plays a fundamental role in the analysis of t h e predic- tability property.

Lemma 9. Let

&

be a right-continuous, l e f t - l i m i t e d , square-integrable, p - m a r t i n g a l e process. Then an *-adapted process f ( u . w ) , u

>

^0, e x i s t s s u c h t h a t

and

The proof of this lemma is similar to that of Theorem 5.5 in 1141. The following well-known result i s important in t h e proof of some auxiliary assertions.

kmma ^4. Let

L

be s o m e v e c t o r space of bounded real f u n c t i o n s defined o n R.

Assume t h a t

it

contains t h e c o n s t a n t 1, is closed wifh respect t o u n i f o r m con- vergence, and is a h

that

for a n y u n i f o r m l y bounded increasing s e q u e n c e of n o n - n e g a t i v e f u n c t i o n s

I,,

^{n 2 0 , f , E} L, the f u n c t i o n

I =

lirn also belongs

n +-

t o

L Let

Q be a s u b s e t of

L

w h i c h is closed with respect to multiplication. l b n

the space I, contains a l l bounded f u n c t i o n s , measured with r e s p e c t to the u- algebra H generated by the e l e m e n t s of Q .

R e m a r k This r e s u l t is known as t h e monotonic class t h e o r e m , a n d is proved in [15]

.

The t h e o r e m is also t r u e if

(a) L i s closed with r e s p e c t t o monotonic and uniform convergence and (b) Q i s t h e algebra a n d 1 E Q

or

(a') L is a s e t of functions closed with respect t o monotonic convergence t o t h e bounded function and

(b') Q is a vector space closed with respe'ct to operation A ( m a x i m u m of two functions) a n d 1 E Q.

8.6. Predictability of H,W-weU-measurable processes

I t t u r n s out t h a t H,w-well-measurable processes have t h e following r e m a r k - able property:

~emrna 5. Let be an a r b i f r a r y H,w-well-measurable process. m e n process F I f T n

<

t j .is Kw-predictable.

Proof. Let T be a n a r b i t r a r y H,w-stopping t i m e , a n d denote by h ( t ) t h e dual Qw- predictable projection of non-decreasing process I ( T S t ) . From t h e definition of h ( t ) t h e process

&

= I(T ^S t )

-

~ ( t ) is a n s w - m a r t i n g a l e .

Now consider t h e process v, = W ~ n + u

-

w

G'

a n d define

Sv

= ( q , t ) t s O , w h e r e X t t = u f v , , u ~ t j v ~ ~ ,

.

Observe t h a t

GBt

⁼

x,Tn+t

a n d consequently t h a t family %v coincides with family

knnW = (qaT

^+t)taO. I t i s n o t difficult t o c h e c k t h a t v, is a Wiener

n

process with r e s p e c t t o

sv

a n d t h a t

qn = qn+t

i s a n s n l w - m a r t i n g a l e process.

From Lemma 3 we have t h e following representation of

q:

or, i n t e r m s of

q,

Taking Tn +u = t we get

f

1

The right-hand side of this equality is an %w-predictable process. Remembering t h e definition of Zt , we deduce t h a t t h e process I ( T I t ) I ( T , < t ) is

Kw-

predictable.

The result of the lemma may then be derived from the monotonic class theorem [15].

6.6. Characterization of

HW

mp-predictable processes

The following assertion describes the structure of Wlp-predictable processes.

kmma 6. An Hwap-adaptedprocess is Wlp-predictable if and only i f , f o r any n

r

0, there ezists an ~w-welZ-measurable process such that

Proof

Necessity. Consider the process = I ( t I T), where T is an arbitrary Hwlp- stopping time. It follows from Lemma 1 t h a t

which leads to equality (0) with

=

I ( t I

F).

That these conditions a r e neces- sary may be proved from t h e monotonic class theorem.

SiLflciency. Observe t h a t for an arbitrary KW-adapted process

5 ,

t h e process ItT, s t j Yt is P"p-adapted. This is because

and any arbitrary set from ( p v ~ f - ) n t ~ ,

<

f j is VvH/'-measurable. Left- continuous KW-adapted processes Yt generate left-continuous Hwmp-adapted processes IITn S t j

&.

This means t h a t the following inclusion is true:

where ~ ( I E , ~ ) and n ( p l p ) are o-algebras for

w-

and p r p - p r e d i c t a b l e s e t s respectively, and ]

~]I JT,,T,+

is t h e stochastic interval corresponding to the stopping times

Tn

and

Tn+l.

The inclusion ( k 2 ) yields:

From Lemma 5, t h e process < t ) is Rw-predictable. Inclusion (A.3) shows t h a t t h e process (which according to equality ( A I ) coin-

n

cides with process

q )

is W1p-predictable. This completes the proof.

6.7. A property of conditional distributions

Let H, G, F be u-algebras in R. Assume t h a t they a r e complete with respect t o measure P and such t h a t G s H , F

sH.

The next s t a t e m e n t will t h e n be use- ful i n analyzing t h e form of the dual predictable projection.

hmma 7 . Let B ^E H , P(B)

>

⁰ be such that t h e f a m i l i e s of s e t s F

n

B a n d G

n

^B c o i n c i d e P-a.s. m e n f o r m y H - m e a s u r a b l e i n t e g r a b l e r a n d o m v a r i a b l e q the f o l l o w i n g e q u a l i t y h o l d s :

Proof. For a n y A ^E

H

define t h e measure P ~ ( A ) as follows:

Let denote t h e mathematical expectation with respect to

$.

The families of s e t s G n B and F n B form o-algebras of t h e subsets of set B t h a t , generally speaking, a r e not a-algebras in R. Since these families a r e complete with respect t o m e a s u r e pB, they coincide 9 - a . s . with the u-algebras GvB and FvB, respectively.

It follows f r o m t h e conditions of t h e l e m m a t h a t for any A E H we have

or, equivalently,

and t h u s the following equality holds 9 - a . s . for any bounded random variable 7:

E~ ( 7

1

GvB) = E (q ) FVB)

.

This may be rewritten in t h e form

I(B) ~ ~ ( 7 1 GVB) = I(B) E ~ ( ~ ( F v B ) , P-a.s.

or

I(B) ~ ( 7 1 GvB) = I ( B ) E ( ~ ~ F v B ) , P-a-s. , t h u s completing t h e proof.

6.8. Some properties of conditional mathematical expectations

The next assertion will be useful in proving the predictable characteriza- tion of some random measures.

Lemma 8. Let A E H c + l . T h e n the following equalities are true f o r a n y

t >

0 :

Proof. Since B, is non-singular for any u r 0, t h e process wt may be represented as follows:

where

This shows t h a t t h e process wt is HI-adapted and leads to t h e inclusion:

m H & s H,I .

^{(A 5 )}

Consider now the bounded random variables

XI,

X2, X3 which a r e measur- able with respect t o a-algebras

q , ^H#

_n ^and

^H;"-,

respectively. Note t h a t X3 does not depend on events from Ht and consequently since

s

H t .

Define d =

E(X~X~X~I(T,

S t )

1(A)).

Using t h e

P v H Y

-measurability of the product

X1X2X3

this can be rewritten a s

Observe now t h a t the product

X 1 ~ Z I ( ~ n + l

(--

t)I(A)

is HfiU-measurable and conse- quently H;z-measurable. Using t h e fact t h a t

X3

is independent of the events of 0-algebra

Hf

we obtain

Since

I(Tn+, s t) = I(Tn+l

^I

t) I(Tn

<

t )

^equation (A.6) may be rewritten as fol- lows:

Noting t h a t events from

(H;Wvh7fn

)

filTn <

t { also belong to Hf and since

X3

is independent of

H;Z

we get

Thus

Using the monotonic class theorem we prove t h e f i s t p a r t of the lemma.

In a similar way it is possible t o prove t h e equalities:

which yield

6.9. Predictability analysis of the uo

The following assertion is a n i m p o r t a n t s t e p towards t h e proof of t h e m a i n r e s u l t .

Lemma 9. F a r a n y

r

^{E B(E)} the process vo((O,t].I'). t 2 0, isW 1 p - p r e d i c t a b l e . Proof. I t follows from Lemmas 5 a n d 6 t h a t t h e dual How8p-predictable projec- tion of integer-valued random m e a s u r e p ( d u , d z ) may be r e p r e s e n t e d as follows:

Observe t h a t t h e function on t h e right-hand side of this equality immedi- ately following t h e indicator I(T,

<

t ^I Tn+l) is W H t - m e a s u r a b l e , with right-continuous, left-limited sampling paths. This means t h a t t h e function is H,w-well-measurable. From Lemma 3 t h e p r o c e s s

is IE,w-predictable, a n d consequently (from Lemma 4) t h e process vo((O,t], i7)tro is lP"p-predictable for any

r

^E B(E). This completes t h e proof.

6.10. Measure v as the dual F o p - p r e d i c t a b l e projection of p

The next two l e m m a s give t h e probabilistic form of t h e dual Wlp- predictable projection of p.

Lemma 10. FOT a n y F ^E B (E) the p r o c e s s

is an W t p - a d a p t e d local martingale.

Proof. From t h e definition of t h e vo.

: Y

is a n qwlp-adapted local m a r t i n g a l e for a n y

r

^E B ( E ) . Introduce t h e process

I t is e a s y t o s e e t h a t

4

is an H""P-adapted local martingale. However, i t follows f r o m Lemma 6 t h a t t h e process

qr

i s Wap-adapted and consequently coincides with

&r,

t h u s proving t h e lemma.

The following assertion provides a probabilistic c h a r a c t e r i z a t i o n of t h e dual w ~ p - p r e d i c t a b l e projection of m e a s u r e p.

Lemma 11. The ^{d u d} H?lp-predictable projection of i n t e g e r - v a l u e d r a n d o m m e a s u r e p coincides with the process vo.

This may be proved using Lemmas 6 a n d 7 and t h e uniqueness of t h e dual HWIJ'-predictable projection of p.

6.1 1. Probabilistic form of the dual

HZ

-predictable projection of p

The fact t h a t eqn. (4) has a s t r o n g solution for

4

yields t h e inclusion

which, together with ( l l ) , shows t h a t a-algebras

~ v H ( L

a n d

Hr

coincide. This in t u r n m e a n s t h a t t h e classes of P o p - a n d Hz-predictable processes coincide, a n d consequently t h a t vo i s I f - p r e d i c t a b l e .

The introduction of a non-singularity condition for B,, u ZL 0, m e a n s t h a t for any n

r

0 we have:

where

It follows f r o m t h e s e equalities t h a t process zut is HI"-adapted a n d consequently t h a t

Note also t h a t equations (6) have a strong, unique solution for % , t , n 1 0. This f a c t yields t h e inverse inclusion:

a n d consequently

From the definition of H? we have

~ ? n ( ~ ~ < t l = ~ c n ( ~ ~ c t j = H & ( T n < t j

a n d thus

Substituting t h e T v H f -conditional probabilities in eqn. (3) by _n

HP-

conditional probabilities we obtain:

From Lemma 7 a n d t h e coincidence of t h e a-algebras II;"vHP and H t for a n y t > 0, we deduce t h a t process

is an Hz-adapted local martingale. The uniqueness of the dual W-predictable projection means t h a t m e a s u r e s vo and

vZ

coincide, t h u s completing t h e proof of Theorem 2.

References

1. J. Jacod, "Multivariate Point Processes: Predictable Projection, Radon-Nicodim Derivatives, Representation of Martingales." Zeitschrift fiir

Wah7~~cheinlichkeitstheory und V e m . Gebiete 31, pp.235-253 ( 1 9 7 5 ) .

2. J. Jacod, C a l c d e s Stochastique e t Probleme & Marfingales. Lecture Notes in B a t h o n a t i c s , Vol. 714, Springer-Verlag, Heidelberg (1 979).

3. P. Bremaud, Point Processes and Queues, Springer-Verlag, New York, Heidelberg and Berlin (1980).

4. C. Dellacherie. Capacities e t Processus Stachcrstipes, Springer-Verlag, Ber- lin and New York (1972).

5. Yu.M. Kabanov, R.S. Liptzer, and AN. Shiryaev, "Absolute Continuity and Singularity of Locally Absolutely Continuous Probability Distributions,"

Math. S b o r n i k USSR (in R u s s i a n ) 35(5), pp.63 1-680 (1979).

6.

AI.

Yashin, "Convergence of Bayesian Estimations in Adaptive Control Schemas," P r o c e e d i n g s of t h e Workshop o n Adaptive Control, October 27-29,

1982, Florence, Italy, pp.51-75 (1982).

7. R. Rebolledo, "Central Limit Theorem for Local Martingales," Z e i t s c h r i f t f u r Wahrscheinlichkeitstheory u n d Verw. Ge b i e t e 51, pp. 269-286 (1980).

0. R.S. Liptzer and A.N. Shiryaev, "Functional Central Limit Theorem for Sem- imartingales," Probability m e o r y a n d Applications ( i n ^Russian) 25, pp.667- 688 (1980).

9. M.H.A. Davis, "Detection Of Signals With Point Process Observations."

D e p a r t m e n t o f C o m p u t i n g a n d Control, Publ. 73/8, Imperial College. Lon- d o n (1973).

10. C.S. Chou and P.A Meyer, gLL7 l a R e p r e s e n t a t i o n d e s Martingales C o m m e h t e g r a l e s S t o c h a s t i q u e s d u n s l e s P r o c e s s u s P o u n c t u e l s , L e c t u r e Notes in M a t h e m a t i c s , Vol. 465 , Springer-Verlag, Berlin ( 1975).

11.

AI.

Yashin, "Filtering of Jumping Processes," A u t o m a t i c a n d R e m o t e Con- trol 5, pp.52-58 (1970).

12. D.L. Snyder, R a n d o m Point P r o c e s s e s , John Wiley and Sons. New York (1975).

13.

J.W.

Vaupel and A.I. Yashin, 7he D e v i a n t D y n a m i c s o f Death ⁱⁿ Heterogene- o u s Bpuldions, R R - 8 3 1 , International Institute for Applied Systems Analysis, Laxenburg, Austria (1982).

14. R.S. Liptzer a n d A.N. Shiryaev, S t a t i s t i c s of R a n d o m Processes, Springer- Verlag, Berlin and New York

.

15. P . k Meyer, P r o b a b i l i t y a n d P o t e n t i a l s , Waltham, Blaisdell (1966).

(ii) the process h ( ( 0 , t

1, I?)

- v0((0,t

1, r)),

^{t 2} 0, is an w - a d a p t e d local mar- tingale.

Dual predictable projections of integer-valued random measures may be interpreted as generalized cumulative random intensity functions.

Remark Notice here that u-algebra

Ho

is not necessarily formed by events independent of H t . For instance,

Ho

could be the u-algebra in

Q

corresponding to the past history of some random process up to time 0. We shall consider the a-algebra

Ho

generated by a Wiener process up to time =.

3.3. Probabilidic representation of random intensity hrnctions

In some senses, Jacod's representation of dual w-predictable projections serves as a bridge between the abstract theory of random processes with jumps currently developing in t h e framework of the martingale approach, a n d t h e wide range of applications based largely on t h e knowledge of probabilistic distri- bution fuoctions.

In order to express Jacod's result, we have to define regular versions of t h e H6fG -conditional probabilities of events _l u {

y \ l K + l ~ r{,

ITn+l ² u

1.

where u 1 0 ,

r

^{E B (E) and n = 1,2,}

... .

This may be done using equalities

I t should be emphasized once again t h a t in this part of the paper we are consid- ering t h e case in which additional information about events and variables influencing t h e multivariate point process perceived by the statistician (observer) does not change over time.

The

following theorem was proved in [I].

Theorem 1. lrtrr IoUowing isa representation

03

a dual H&pre&table projec- tion vo o J m s m u e

p:

Corollary. Notice t h a t if the u-algebra

H,,

does not provide the observer with any information about the events, and if all he has at time t is information

about events from t h e history

HfCI,

the hazard rate coincides with the dual

w-

predictable projection v of the measure p and the formula for v is a simple corollary of equation (3):

In the case of a pure point process (a sequence of random times T,), the formula for the dual W-predictable compsnsator ~ ( t ) becomes:

where the a-algebras kItN, t r 0. are generated by t h e values of t h e point pro- cess (T,)n,o or, equivalently, by the valoes of the counting process

%,

^u r 0 , up to time t .

Equations (3') and (3") produce known results when applied to well-studied processes. Thus, for a Poisson process with deterministic local intensity Func- tion A(t), the dual predictable projection A(t) defined by equation (3") coin- cides with the cumulative hazard rate and is given by the equality

For a finite-state, continuous-time, Markov process

Ct

, t r 0, with states I1,2

.... ^,N]

and intensities h j ( t ) , i , j

=

(1.2

.... ^,N),

t r 0 , equation (3') gives the dual predictable projection in the form:

Now assume that the observer has some additional information about t h e intensity function, for instance, t h a t he knows the value of some random vari- able Zwhich influences the frequency of the jumps. This means t h a t he is deal- ing with the history H g t as determined by the equality (I), where a-algebra

&

coincides with t h e o-algebra a(@ generated by random variable

Z

in

Q .

In this care the observer needs to use the dual H,f-predictable projection of p a s a ran- dom intensity function. Thus, in the case of a double stochastic Poisson pro- cess

\

,

t

& 0 , with random intensity function ZA(t) , t

r D

(where

Z

is some

positive random variable), it is necessary to use eqn. (3) which gives the follow- ing dual %-predictable projection of

Nt . ^t

^{> 0:}

Note t h a t if two observers have different information about the processes occurring in some real system (for example, if one of t h e m knows the value of t h e variable Z and t h e o t h e r does not), they will use different hazard r a t e s to estimate t h e probability of change. In t h e case of continuously distributed j u m p t i m e s in a double stochastic Poisson process, the relation between the two intensities (derived from a comparison between eqns. (3") and (3')) may be represented as follows:

where h ( t ) is the H'-predictable local hazard rate. This is related t o t h e dual

~ ' - ~ r e d i c t a b l e projection x ( t ) of

Nt

by t h e equality:

We shall now prove the relation between X(t) and A(t). Consider t h e integral ~ , ( t ) defined by t h e equality:

which is taken from t h e right-hand side of eqn. (3"). Let p P ( u ) be the density function of conditional distribution P(Tp+I s u

I H!

). Note t h a t the following

P

equality holds:

where j ( z ) is t h e density distribution function of random variable 2. _Using t h i s equality and noting that for

Tp <

u

we have

Since

we can derive the formula for the K ( t ) from eqn. (3").

Remark. Equation (4) shows that to calculate intensity functions x ( t ) or x(t) it is &st necessary to calculate the H'-conditional mathematical expectation.

This problem can be overcome by using an approach based on filtering of t h e jumping processes (see, f0.r instance, [ll, 121). In the simplest life-cycle models, which a r e characterized only by stopping time T (time of death) and a r e widely used in reliability and demographic analysis, t h e random intensity describes diflerences in susceptibility to death or failure [13]. Notice that in the case of life-cycle processes, equation (3") gives t h e following relation between the H-adapted compensator A ( t ) and the local intensity function A(t ):

Recall that, f r o m the definition of the compensator, t h e process

is an H-adapted martingale.

4. GENERAL FORMULA

POR

R E P ~ A T I O N OF RANDOM

In spite of t h e fact t h a t dual predictable projections exist for a wide class of families of 0-algebras, probabilistic representations a r e known only for o- algebras with s t r u c t u r e (1). However, in practical situations u-algebras often have a more general structure. In particular, new information may be gen- erated not only by t h e multivariant point process but also by some additional process In this case the 0-algebras describing t h e observation history have t h e form Ht

=

H t v v H f , where a-algebras h!f are _generated by some process qt which is observed simultaneously with t h e multivariate point process

(L, Z,,)nlo.

Detailed probabilistic characterization of dual predictable

projections is often useful when applying the results of the general theory of random processes in practice.

In this section we will give the probabilistic representation of the dual predictable projection of integer-valued random measures corresponding to the jumps of t h e semimartingale with respect to the family of o-algebras generated by this process.

4.1. Generation of jumps

Let random process

4 ,

^{t 2} 0, defined on probability space (R,H,H,P) be H- adapted, take values in $ and have right-continuous, left-limited sampling paths. Denote by

IF

the family of u-algebras

and assume that is completed by the sets from

HI =

H1, with a P-probability of zero. Assume also t h a t process

4

may be represented a s follows:

where

4

and

B,,

a r e HI-adapted matrices of appropriate dimensions, matrix B, is non-singular for any u 1 0 , E

=

$

-

^IOj. and w,, is an H-adapted. k - dimensional Wiener process independent of Xo.

Note that, in general, the dimension of

4

may be greater t h a n t h a t of the jumping changes. This can mean, for instance, t h a t we a r e also considering situations with two-component processes of which one is pure jumping a n d t h e other is continuous.

I t follom from (3) t h a t t h e measure p is *-adapted and t h a t

Let

9

denote t h e dual *-predictable projection of integer-valued random measure p. Our main aim is t o derive t h e probabilistic representation of meas- ure

vf .

4.2. The form of t h e hazard rate

The main result of this subsection is formulated in terms of auxiliary processes defined by the equalities:

where A , B, w , p have been defined previously. Introducing the measures defined by the equalities

eqn. (8) m a y be rewritten a s follows:

Notice that measure p,, may be considered as a measure of the jumps of t h e process

&

,

t

r 0.

Introduce c-algebras

$

^and

^{H ?}

such that:

and define the regular versions of t h e conditional probabilities of events

using the equalities

The next assertion is t h e main result of this paper.

Theorem

2. Assume

that

c o e w n t s A a n d B m e such t h a t a s t r o n g s o l u t i o n oJ e q u a t i o n (3) % l i s t s and is unique. m e n we have the following r e p r e s e n t a t i o n oJ thr dud

IF

-pta&tabla p r o j e c t i o n vf oJ integer-valued random m e a s w e p: