On life-time-distributions of some one-dimensional diffusions and related exponential families

On life-time-distributions of some one-dimensional diusions and related exponential families

Uwe Kuchler

Humboldt-Universitat Berlin

Institut fur Mathematik, Unter den Linden 6 10099 Berlin

Abstract

Di usions^Xon the positive halfaxis with elastic killing boundary at zero and inaccessible bound- ary at¹are considered. The life-time is nite and ^h (^x) =^Exexp( ),ⁿ 0, is an -excessive function. The^{h( )}-transforms of^Xdene a family of di usions, such that the life-time distributions of them form an exponential family of distributions, the inverse local times at zero form an exponen- tial family of processes with independent increments and the spectral measures of them are connected simply by translation.

1 Introduction

1.1.

Let us given a ltered statistical space ( ^{F F}t (P )) with t 0 ² R, (^Ft) right- continuous and ^F=^W_t ^Ft.

We shall suppose 0².

Moreover, assume X = (Xt t 0) is a real valued process on ( ^F) with ^Ft = (Xs s t), t 0.

Assume that X has independent stationary increments under every P and that it holds P (X⁰= 0)1.

Let ( ² ) be its Levy-characteristics, occuring in the characteristic function E expiXt] = expt(i ^;2

2 ²+^Z

R e^iy;1^; iy 1 + y²

(dy))]

with t > 0, ²R, ².

1.2.

We say that ^P = (P ² ), together with X, forms an exponential familyof processes with independent stationary increments, if

(i) P (X_t ²dx) = expx^;()d]P⁰(X_t ²dx), t > 0, x ²R, ² for some function (). This property is equivalent with every of the following ones (see Kuchler, Kuchler (1981)):

(ii) P tP⁰tfor every t > 0 and

dP t

dP⁰_t = expX_t^;()t]

where P t denotes the restriction of P to^Ft,

(iii) For every t > 0 the "last observation" Xtis a sucient statistic for with respect to^Ft

(iv) ²⁰², d (y) = expy]d⁰(y), y²R, = ⁰+ ²⁰ +

Z

R

e ^y;1^;1 y1 + y²

⁰(dy) ²:

This equivalence does not hold for more general processes, e.g. Markov processes.

1.3. Example:

If X forms a Brownian motion with diusion constant equals to one and drift b(x) =^p2 tanh(^p2x) > 0 x²R

under P , then (iii) holds, but (i) and (ii) do not (see Kuchler (1982)).

1.4.

Thus, every of the properties (i)-(iii) (or slight changed variants of them) could be used to de ne exponential families of processes in the general case.

This was done by several authors, see e.g. Kuchler (1982), Kuchler, Soerensen (1989), Ycart (1989).

In Kuchler (1982) it was shown, that a family of conservative Markov processes under some conditions of regularity has the property (iii) if and only if for every ² there is a number () and a strictly positive function g on the state space with

(i⁰) P (Xt²dy^jX⁰ = x) = exp(^;()t)g (y)g (x)P⁰(Xt²dy^jX⁰= x):

Obviously it holds ^Z

g (y)P⁰(Xt²dy^jX⁰= x) = exp(()t)g (x) i.e., g is ()-regular for X under P⁰.

1.5.

Asmussen (1989) considered nonconservative Markov processes with nite life-time and obtains a family of Markov processes with a kind of property (iii), where the life-time distributions form an exponential family.

In this paper we shall present families of nonconservative diusions on 0 ¹) which have a property similar to (ii), the life-time distributions of which form an exponential family, and the inverse local times at zero form an exponential family of processes with independent stationary increments. Moreover, their spectral measures are simply connected by translation. This shows that also behind the processes with independent stationary increments there are connections between (i)-(iii), more precisely between dierent notions of exponential families.

Analogue results can also be obtained for gap- or quasi-diusions (see Kuchler (1986) for de nitions), but the formulations will be slightly more complicate.

2 Diusions, local times and spectral measures

Here we shall summarize some properties of one-dimensional diusions which are necessary in the sequel.

For details the reader is refered to Ito, McKean (1974), Kac, Krein (1974), Kuchler (1986), Kuchler, Neumann (1991).

2.1.

Suppose X = ( ^F (^Ft) Xt Px) is a (regular) diusion on 0 L) with speed measure m() and scale s() in the sense of Ito, McKean (1974). Assume s(0) = m(0^;) = 0, s(L^;) = ¹ and let 0 be a killing, elastic killing or reecting boundary.

The in nitesimale generator A of X in L²(m) can be given by the restriction of the generalized second order dierential operator DmDsto

D_A:=^ff ²L²(m)^jD_mD_sf ²L²(m) aD_s^;f(0) = f(0)^g

where a²0 ¹] is xed. (The number a is connected with the killing rate of X at zero.) Because of s(L^;) =¹we have for a <¹that

Px( < X = 0) 1:

2.2. Remark:

Obviously, for every c > 0 the pair (cm c^;1s) together with the boundary condition acD^(c;1s)f(0) = f(0)

leads to the same process X. It should be noted here already, that certain quantities of X as the co- ecient in the boundary condition even mentioned but also transition densities, resolvent kernels, local times and spectral measures are uniquely determined only up to the factor c.

2.3.

For every complex and a²(0 ¹] let 'a and be the solution of

DmDs + = 0 (2.1)

satisfying the boundary conditions

'a(0 = 1 D^;_s'a(0 ) = 1a (0 ) = 0 D^;_s(0 ) = 1:

The functions ' := '¹ and are called the fundamental solution of (2.1). If < 0 then '_a( ) and ( ) are positive, increasing, and it holds

L

Z

0

'^;2(u )s(du) <¹: De ne

(x ) := '(x )^ZL

x '^;2(u )s(du) x²0 L) ²K (K denotes the set of complex numbers).

Then ( ) satis es (2.1) and for xed < 0 it is positive and decreasing. Moreover, we have for < 0 D^;_s(0 ) =^;1 (0 ) = lim_x

"L(x ) (x ) and because of S(L^;) =¹, D^;_s(L^; ) = 0.

The function G() de ned by

G() := (0 ) ²K^; := Kⁿ0 ¹) is called the characteristic functionof (m s) (more precisely, of m s^;1).

By assumption we have

G(0) := lim

"0

G() = s(L^;) =¹: (2.2)

Moreover, it holds

(x ) = G()'(x )^;(x ) x²0 L) ²K^;: (2.2a)

In the sequel we allow a to be zero and put '⁰:= .

2.4.

The Laplace transform of the rst hitting time of y _y:= inf^ft > 0^jX_t= y^g is given by

Exexp(y) = (y )=(x ) < 0 x y²0 L) (2.3)

with = 'a if xy and = if x y:

2.5.

TheWronskianW = W() is de ned by

W() := (x )D^;_s'a(x )^;'a(x )D^;_s(x ) ²K x²0 L):

It is independent of x, and we have

W() = ¹_aG() + 1 if a²(0 ¹] W() = G() if a = 0:

: (2.4)

For all functions f g such that DmDsf and DmDsg make sense Lagrange's identity holds:

x

Z

0

gDmDsf^;fDmDsg]dm = fDsg^;gDsf]^x;0+0: (2.5)

2.6

For every a²0 ¹] there exists the transition densityp(t x y):

Px(Xt²dy) = p(t x y)m(dy) t > 0 x y²0 L): (2.6) It satis es p(t x )²D_A and

@tp(t x y) = D@ ^mDsp(t x y) t > 0x y²0 L): (2.7) For theresolvent kernel

r^(a)(x y) :=

1

Z

0

e^tp(t x y)dt ²K^; x y²0 L) it holds

r^(a)(x y) = '^a(x^{^}y )(x^_y )

W() ²K^; x y²0 L): (2.8)

For every a ² 0 ¹] the spectral measure r = ra of DmDs^jD^A is de ned to be a measure on 0 ¹) (on (0 ¹) if a <¹) with

r^(a)(x y) =

1

Z

0

'a(x u)'a(y u)

u^; a(du) < 0x y²0 L): (2.9)

It exists and is uniquely determined. Moreover, it holds (see Kuchler, Neumann (1991))

1 a + 1

G()

;1= r^(a)(0 0) =

1

Z

0

a(d)

u^; < 0 a²(0 ¹] (2.10)

and 1

G() =

1

Z

0 1

u^; 1 u^;

⁰(d): (2.11)

We have

G() = r⁽¹⁾ (0 0) =

1

Z ¹(d)

u^; < 0 (2.12)

and using G(0^;) =¹we get from (2.9)

1

Z

0

a(u)

u = a a²(0 ¹]: (2.13)

2.7

Fix a =¹and denote by l = l(t x) thelocal time of X de ned by

t

Z

0

f(Xs)ds =^ZL

0

f(x)l(t x)m(dx) f bounded, measurable.

The function l( x) is nondecreasing and l(t x) can be supposed to be continuous in (t x), t > 0 x²0 L).

For every x ² 0 L) let l^;1( x) be the right-continuous inverse of l( x). Then (l^;1(t x) t 0) is a stricly increasing process with independent stationary increments. It holds

E⁰e^l;1(t0)= exp

;

r⁽¹⁾ t(0 0)

= exp

;

G()t

t > 0 < 0: (2.14) Thus, the Levy-measure (du) of l^;1( 0) can be calculated from (2.11) and (2.14):

(du) =

1

Z

0

e^ul0(dl)du u > 0: (2.15)

In particular, we get

E⁰e^l;1(t0)= expt

1

Z

0

(e^u;1)(du)] t > 0 < 0: (2.16) Note that (" ¹)) gives the intensity of jumps of l^;1( 0) with size greater than ", i.e. the intensity of excursions of X from 0 of length greater than " in local time scale. Thus

Q_"(A) := (A^\(" ¹))

((" ¹)) A^2B

is the probability that a jump of l^;1( 0) has a size in A conditioned it is greater than ".

2.8. Examples:

Assume X is a diusion on 0 L) with in nitesimale generator ²(x)

2 d²

dx² + (x) ddx:

Then we can choose (assuming that all occuring integrals exists) dm(x) = 2²(x) exp

2^Zx

0

(y)²(y)dy

dx and ds(x) = exp

;2^Zx

0

(y)²(y)dy

dx:

a) Brownian motion on 0 ¹) with elastic killing at zero and with drift 0:

Put ²(x)1, (x)0 and x a²(0 ¹]. Then it holds G() = ( + (^2;2)¹²)^{;1 2}K^; a(du) = (2u^;2)¹²

(_a¹+ )²+ 2u^;2]1I ²²¹⁾(u)du u 0 (2.17) and, if <^;1_a, a has an additional mass of amount^;( +¹_a) at the point^;_a¹( +²¹_a). For a = 0 (killing boundary at zero) it holds

⁰(du) = (8u^;2)¹²

8 1I

2

8¹⁾(u)du:

Thus, it follows

(du) = e^82uu^;12 1

4^p2du u > 0:

b) Ornstein-Uhlenbeck-process on 0 ¹) with elastic killing boundary at zero:

Assume ²(x)1, (x) =^;x ( > 0 x 0).

Then we get

G() = 12^p ;(^;2)

;(^12;2) ²K^; where ; denotes the Gamma-function.

The spectral measure a(du) (a²0 ¹]) is concentrated in the points uk, k 0 with 0u⁰< u¹< ::: < un< :::

and lim_n

!1

un=¹, which are poles of (¹_a+_G¹⁽⁾)^;1. It holds a(^fuk^g) = Res

1 a + 1

G()

;1

⁼u^k: In particular we get

a = 0 : uk = (2k + 1) ⁰(^fuk^g) =

2(k + 1) k

^p

p2^2k a =¹: uk = 2k ¹(^fuk^g) = ^p

p2^;k

2k k

:

2.9.

Now we shall summarize some properties of the life-time distribution. We know, that for a <¹the life-time is nite and that X^;= 0.

2.10. Lemma:

Assume a²(0 ¹). Then the Laplace transformh^{( )}(x) of is given by h^{( )}(x) := Exexp( ) = (x )(0 )

a1

1a +_G^{1( )} 0 x²0 L): (2.18)

Proof:

We have

Px( > t) =^ZLp(t x y)m(dy)

and thus we obtain using (2.6):

Exe =

1

Z

0

e^tPx( ²dt) =^;

1

Z

0

e^td

dtP^x( > t)dt =

= ^;e^tPx( > t)^j10 +

1

Z

0

e^tPx( > t)dt

= 1 + ^ZL

0

r^(a)(x y)m (dy) < 0 x²0 L):

Therefore we get from (2.8), (2.5) and (2.4) W()Exe = (x )^Zx

0

'a(y )m(dy)+'a(x )^ZL

x (y )m(dy)+W()

=^;(x )^Zx

0

dD_s^;'a(y )^;'a(x )^ZL

x dD^;_s(y ) + W()

=^;(x )D^;_s'a(x )+(x )1a+'^a(x )D^;_s(x ) + W()

= 1a(x ) = (x ) (0 )

a1

1a+_G⁽¹⁾ W(): ²

2.11. Corollary:

(i) The life-time distribution underP⁰ is a mixed exponential distribution with P⁰( ²dt) = 1a

1

Z

0

ue^;uta(du)

u dt: (2.19)

(ii) The life-time distribution under Px is the convolution of the distributions of the rst hitting time ⁰ under Px and the life time underP⁰.

Proof:

Use (2.10) for (i). (ii) is obvious from (2.18). ²

2.12 Example:

Brownian motion on 0 ¹) with elastic killing at zero and drift 0: We have P⁰( ²dt) =

1 a

1

Z

2

2

e^;ut(2u^;2)¹² (_a¹²+²_a + 2u) du +1I^(;1;a¹

)() 1

a ^j + 1a^je^1a(+2a1)t

dt P_x(⁰²dt) = e^x x

p2x t^;

3

2 exp^h;1 2 x²

t + ²tⁱdt:

Thus, the life-time distribution under Px is the convolution of these two distributions.

3 The

^h()

-transformed diusion

3.1.

Consider the function h^{( )}() de ned above by

h^{( )}(x) := Exe 0 x²0 L)

and x an < 0.

Because of

e ^tExh^{( )}(Xt) = Ex^;Ex^;e ^(t+t)jFt 1I^f_>t^g

= Ex^;e 1I^f_>t^g Exe = h⁽⁾(x) x²0 L) and lim_t

#0

Exh^{( )}(Xt) = h^{( )}(x), x² 0 L), this function h⁽⁾() is -excessive (- -excessive in the termi- nology of Blumenthal-Getoor (1968)). In particular

P^{( )}(t x dy) := e ^tP(t x dy)h^{( )}(y)

h^{( )}(x) t > 0 x y²0 L) (3.1) forms the transition function of a new Markov process X^{( )}, the so-called h^{( )}-transformed process of X.

From (3.1) and (2.6) it follows

P^{( )}(t x dy) = e ^t p(t x y)

h^{( )}(x)h^{( )}(y) h⁽⁾(y)]²m(dy):

Thus X^{( )}is a diusion on 0 L) again. the choose of speed measure and scale is not unique as was noted in Remark 2.2 above. We put

dm^{( )}(x) = ^{( )}(x)dm(x) ds^{( )}(x) = ^{( );1}(x)ds(x) (3.2)

with ^{( )}(x) = c²()h⁽⁾²(x)

where c²() is a constant with respect to x but is allowed to depend on . It will be speci ed later.

m^{( )} (s^{( )}) is thespeed measure(thescale, respectively) we have chosen for X^{( )}. Obviously, it holds

s^{( )}(L^;) = c^;2ZL

0

h^();2(x)s(dx) =¹ (note that h^{( )}is decreasing and nonnegative if < 0).

The in nitesimal generator A^{( )}of X^{( )} in L²(m^{( )}) is given by

A^{( )}f = 1h^{( )}Afh^{( )}+ f (3.3)

with f ²D_A^{( )} if and only if fh^{( )2}D_A: Thus, every f ²D_A^{( )} satis es the boundary condition

D_S^{( )}f(0) = ^{( )}(0)Dsf(0) = ⁽⁾(0)Ds

fh^{( )} h^{( )}

(0) =

= ^{( )} f^{( )}Ds(fh^{( )})^;h^{( )}fDsh^{( )} h^{( )}]² (0) = c²()

h^{( )2}f1

a^;fh^{( )}Dsh^{( )}(0)

= ^{( )}(0)

1 a + 1

G()

f(0):

Now it is plausible to choose c() = h^{( );1}(0), ²(^;1 0].

With a :=1

a + 1 G()

;1 (3.4)

we obtain the boundary condition

a D_s^{( )}f(0) = f(0) f ²D_A^{( )}: (3.5)

3.2. Remark:

As we have seen, for every < 0 the transition function P^{( )}(t x dy) and therefore the processes X⁽⁾are well-de ned. But in the choose of (m^{( )} s^{( )}) there is some arbitrariness because of the factor c(). It turns out that the most formula below become more simple if we choose c() = h^{( )}(0)]^;1. In particular we get ⁽⁾(x) =^hh_h^{( )( )((0)x)i2}=_G^{2(2x( ))}.

3.3. Proposition:

The characteristic function G^{( )}()of (m^{( )} s^{( )})satises G⁽1⁾() = 1

G( + )^; 1

G() ²K^;: (3.6)

Proof:

We start with the computation of the fundamental solution '⁽⁾ and ^{( )}of (m^{( )} s^{( )}).

3.4. Lemma:

It holds

^{( )}(x ) = h^{( )}(0)

h^{( )}(x)(x + ) = (x + )

(x ) G() (3.7)

'^{( )}(x ) =

G()'(x + )^;(x + ) (x )

: (3.8)

Proof:

To prove (3.7) note that ^{( )} is the unique solution on (x ) = s^{( )}(x)^;Zx

0

(s^{( )}(x)^;s⁽⁾(u))(u )m^{( )}(dm) or, equivalently, "

h^{( )}(x) h^{( )}(0)

#

2

Ds(x ) = 1^;Zx

0

(u )

"

h^{( )}(u) h^{( )}(0)

#

2

m(du) with (0 ) = 0.

Using Lagrange's identity (2.5) it is easy to see that the right-hand side of (3.7) satis es this equation.

The equation (3.8) is proved similary. ² Applying (3.7) and (3.8) we obtain

G () = lim1 ^x"L'^{( )}(x )

^{( )}(x ) = 1

G( + )^; 1

G(): ²

3.5.

Now we are ready to calculate the other quantities connected with X^{( )}: the spectral measure _a^{(( ))} and the Levy-measure ^{( )}.

3.6. Proposition:

It holds

1

Z

0

d_a^{(( ))}(u) u^; =

1

Z

0

da(u) u^;( + ) =

1

Z

0

a(du + )

u^; (3.9)

i.e. _a^{(( ))}()chargesduasa()chargesdu +^fg.

Proof:

By (2.10), (3.4) and (3.6) we have

1

Z

0

d⁽⁽_a^{( ))} (u)

u^; = 1

a1 +_G¹

(⁾ = 1

a1+_G^{1( )}+_G^{(1+ ;}_G^{1( )} =

1

Z

0

da(u) u^;;: ²

Consequently, the Levy-measure _a() of the inverse local time l^;1(t 0) of X^{( )}at zero, can be calculated as ^{( )}(du) =^R e^;ul_a^{(( ))}(dl) = e^l(dy) and for the Laplacian of l^;1(t 0) we get

E^{0( )}expl^;1(t 0)] = exp^; t G ()

< 0 t > 0:

3.7. Remark:

If we do not x c() = h^{( )}(0)]^;1, we obtain G⁽1⁾() = c()h (0)

1

G( + )^; 1 G()

a = c()h (0)1

1 a + 1

G()

^{( )}(du) = e ^u(du)c()h^{( )}(0):

This shows that the quantities a , G () and ^{( )}() as well as m^{( )}and s^{( )}are de ned up to a constant c^{( )}for every process X^{( )}, but this constant can vary with . What we have shown is that there exists an appropriate choice of c() such that we get a simple connection between these quantities for dierent .What is independent of the choice of c() is e.g. for every " > 0

^{( )}(du)

^{( )}((" ¹)) = e ^u(du)

1

R

" e ^v(dv)=: Q⁽_"⁾(du) u > "

which express the conditional probability of the length of an excursion of X⁽⁾ from 0 given that this excursion is longer than ". Now the following Corollary is obvious.

3.8. Corollary:

For every choice of c() and for every xed " with (" ¹) > 0 the probabilities

fQ⁽"⁾ 0^gform an exponential family of distributions.

4 Exponential families related to

^X()

Let X = ( ^F X_t ^F_t P_x) be a diusion on 0 L) as above. De ne h⁽⁾(x) := Exe x²0 L) 0

and consider the family (X^{( )} 0) of diusions corresponding to the transition functions P^{( )}(t x dy) = e ^tP(t x dy)h^{( )}(y)

h^{( )}(x):

For abbreviations of the notations we shall put a(A) = 0 for every Borel-set A(^;1 0).

4.1. Theorem:

For the family(X^{( )} 0) of diusions the following properties hold:

(i) For everyt > 0,(t^{^} Xt^{^})is a su cient statistic for with respect to^Ft, for every Px. (ii) Assume 0y < x < L. Then the rst hitting time distributions

;P_x^{( )}(y ²dt) 0 form an exponential family of distributions:

P_x^{( )}(y²dt) = exp(t)P^x(y²dt)

1

R

0

exp(s)Px(y²ds): (4.1)

(iii) For everyx²0 L)the life-time distributions

(P_x^{( )}( ²dt) 0) form an exponential family of distributions:

P_x^{( )}( ²dt) = exp(t)P^x( ²dt) h^{( )}(x) : If x = 0then we have from (2.10) and (2.18)

P^{0( )}( ²dt) = 1a

1

Z

0

e^;ta(d +^fg)

dt t > 0 0 (4.2)

i.e., has a mixed exponential distribution underP⁰⁽⁾ too, and the mixing (probability) measure is a1 a(d +^fg)

0

with _a¹ =¹_a +_G^{1( )}.

(iv) The inverse local times (l^;1(t 0) t 0) of X⁽⁾ at zero form an exponential family of processes with independent stationary increments, and their Levy-measures are given by

(du) = e ^u(du) = e ^u

1

Z

0

e^;us0(ds): (4.3)

(l^;1(t 0)),t 0 is killed with constant killing ratea . (v) The spectral measure a of(m s ) is a shift ofa:

_a⁽⁾(u) = a(u + ) u 0: (4.4)

Proof:

(i): Because of the de nition of h^{( )}() we have dPx^{( )}

dPx (!) = exp( (!))h^{( )}(x) x²0 L) (4.5) and for the restrictions P_xt⁽⁾and Pxt of Px⁽⁾and Px, respectively, to^Ftit holds

dP_xt^{( )}

dPxt(!) = exp( ^{^}t)]h^{( )}(Xt^{^})

h^{( )}(x) (4.6)

where in this formula is assumed that

h⁽⁾(X) = 1 Px^;a.s. for all x²0 L):

For the proof the reader is refered to Asmussen (1989).

(ii): If y < x then by (2.3) it follows

E_x^{( )}exp_y] = ^{( )}(y )

^{( )}(x ) < 0 0: (4.7)

From (2.2a), (3.7) and (3.8) it follows, that this expression equals (x + )

(y + )

(x ) (y ):

Applying (2.3) once again we obtain that

E_x^{( )}expy] =

1

R

0

e^te ^tPx(y²dt)

1

R

0

e ^tPx(y²dt) : Now (ii) is obvious.

(iii): From (4.5) it follows

E⁽_x⁾e= E^x

;e^{(+ )}

h^{( )}(x) = h^{( +)}(x)

h^{( )}(x) x²0 L) < 0 0:

Thus, (Px^{( )}( ²dt) 0) forms an exponential family:

P_x^{( )}( ²dt) = e^tPx( ²dt)

h⁽⁾(x) x²0 L) t > 0: (4.8)

For x = 0 we get from (2.18)

P^{0( )}( ²dt) = aexp(t)

1 a + 1

G()

P⁰( ²dt) = exp(t^;ln aa )P⁰( ²dt):

Using Corollary 2.11 (i) and _a¹ = _a¹ +_G^{1( )} we get (4.2). Because of (4.8) and the fact, that the life- time distribution under Px^{( )}is the convolution of the hitting time distribution of ⁰ under Px^{( )}and the life-times distribution under P⁰⁽⁾ (see Corollary 2.11 (ii)), we obtain now that

(P_x⁽⁾( ²dt) 0) forms an exponential family of distributions also.

(iv) It suces to show that for every t > 0 the distributions of l^;1(t 0) under P^{0( )}form an exponential family. To do this consider

E^{0( )}e^l;1(t0)= e^;Gt() = exp

;t

1

G( + )^; 1 G()

= e^;

t

G(+ )

e^{;G( )t} : (v) Was already proved in Chapter 3. ²

4.2. Remark:

It is possible to derive a spectral representation for the distribution of ⁰ under Px in terms of the spectral measure ⁰, see Kuchler, Salminen (1989).

The result of Theorem 4.1 (iv) does not remain valid if X can be killed at more than one place. To illustrate it, let us assume additionally that s(L^;) + m(L^;) <¹, i.e. L is a regular boundary in Fellers terminology. Moreover, let h be the constant occuring in the boundary condition at L:

hD⁺_sf(L) + f(L) = 0:

Then we have

4.3. Proposition:

h^{( )}(x) = E_xexp ] = 1hr (x L)+ 1

ar (x 0) = (x )(0 )

a1

1a +_G^{1( )} + '^a(x ) '_a(L )

h1

h1 +_H¹⁽⁾ 0 x²(0 L):

HereGand H denote the characteristic function of(m s)together with the boundary constant hand of (m(L^;)^;m(L^;) s(L^;)^;s(L)) together with the boundary constant a, respectively.

The proof is very similar to those of Lemma 2.10 and omitted.

Now h^{( )}is of more complicate form and one can not expect that (l^;1(t 0)

t > 0) form an exponential family as above. But from the general theory (Asmussen (1989)) it follows, that the distributions of the life-time under P_x still form an exponential family.

References

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Blumenthal, R.M. Getoor, R.K. (1968): Markov process and Potential theory, Academic Press, New York, London.

Ito, K. McKean, H.P. (1974): Diusion processes and their Sample Paths, 2nd Printing, Springer, Berlin

Kac, I.S. Krein, M.G. (1974): On the spectral functions of the string, Amer. Math. Soc. Transl., 103(2), 19-102.

Knight, F.B. (1981): Characterization of the Levy measures of inverse local times of gap diusion, Progress in Prob. Statist. 1, Birkhauser, Boston, Mass.

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Statistics, 13(1), 57-69.

Kuchler, U. (1986): On sojourn times, excursions and spectral measures connected with quasidiusions, J. Math. Kyoto Univ. 26-3, 403-421.

Kuchler, I. Kuchler, U. (1981): An analytical treatment of exponential families of stochastic processes with independent Stat. increments, Math. Nachr., 103, 21-30.

Kuchler, U. Neumann, K. (1991): An extension of Krein's inverse spectral theorem to strings with nonreecting left boundaries, Lecture Notes in Math., 1485 (Seminaire des Probabilites XXV), 354-373

Kuchler, U. Salminen, P. (1989): On spectral measures of strings and excursions of quasi-diusions, Lecture Notes of Mathematics 1372, Springer, Berlin, New York, Heidelberg, 490-502.

Kuchler, U. Soerensen, M. (1989): Exponential families of Stochastic Processes: A Unifying Semi- martingale Approach, Int. Stat. Rev. 57(2), 123-144.

Ycart, B. (1989): Markov processes and exponential families on a nite set, Stat. Probab. Lett. 8(4), 371-376.