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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 178, April 1973

EXIT PROPERTIES OF STOCHASTIC PROCESSES WITH

STATIONARY INDEPENDENT INCREMENTS

BY

P. W. MILLAR(l)

ABSTRACT. Let \X , t S 0| be a real stochastic process with station-

ary independent increments. For x >0, define the exit time T from the

interval (— oo; xi by T = inf\t >0: X > x\. A reasonably complete solu-X t Q

tion is given to the problem of deciding precisely when P \Xf = x\ > 0

and precisely when P \Xj- = x\ = 0. The solution is given in terms of

parameters appearing in the Levy formula for the characteristic function of

X . A few applications of this result are discussed.

1. Introduction. Let X = \X t > 0| be a real valued stochastic process

with stationary independent increments. Then E e = exp |/t/z(zz)|, where

(1.1) tb-iu) = iau - io2/2)u2 + j[e'ux - 1 - iux/il + x2)]vidx).

The measure v is called the Levy measure, and tp is called the exponent of

the process X. If o > 0, X is said to have a Gaussian component. If

/_. |x|iv((ix) < °o; then as is customary we will assume the exponent written in

the form

(1.2) djiu) = iau - io2/2)u2 + j[eiux - l]vidx).

It in (1.2), o=0, v\i- oo) 0)| = 0, a > 0 then the corresponding process has in-

creasing paths and is called a subordinator; the constant a is then called the

drift.

For x > 0, define T = inf 1/ > 0: X > x|. The basic problem of this paper

is to decide in terms of the exponent when P |X_ = x| > 0, and whenOr

P |X = x\ = 0. Thus in the latter case X jumps across the boundary on its

first exit from (- oo; x]. In the former case we will say for brevity that X has

continuous (upward) passages across the level x. Besides their intrinsic inte-

rest as descriptions of sample function behavior, results of this nature are often

Received by the editors June 5, 1972.

AMS(MOS) subject classifications (1970). Primary 60J30, 60G17; Secondary 60G10,60G40, 60J25, 60J40, 60J55.

Key words and phrases. Stochastic process, Markov process, stationary independent

increments, sample function behavior, local time, Levy measure, first passage time, first

passage distribution, subordinator.

( ) Supported by a National Science Foundation Postdoctoral Fellowship and partly

by NSF Grant GP-31091X1.Copyright C 1973, American Mathematical Society

459

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

460 P. W. MILLAR

tools in proving other properties of interest; see Theorem 1.1 below and the para-

graph at the end of this section.

The problem is obviously trivial unless the process hits points and has up-

ward jumps. Therefore we assume throughout that

(1.3) u\(0,co)\>0

and, if Hx = inf{f > 0: X( = x\,

(1.4) P°\H <~S>0, x>0.X '

Precise conditions under which (1.4) holds have been given by Kesten [ll]; see

also [6J. Moreover, in order to avoid discussing the uninteresting case where

the process has only a finite number of jumps in every finite time interval, we

further assume throughout that

(1.5) I/K-oo, oo)i = oo.

From assumptions (1.3) and (1.5) it follows that

(1.6) P°|XT _ = x < xT ! = P°\XT _ < x = xT \ = 0,X X X x

so that with probability one a given path either jumps across x strictly at time

T , or else hits x in a continuous manner. This result is well known; proofs

may be found in [ll], [13], [15]. It also turns out under the present hypotheses

(see Corollary 3.1) that either P°\XT = x\ > 0 for all x > 0 or P°JXT = x\ = 0' X ' X

for all x > 0. Finally, under the present hypotheses, Proposition 2.1 guarantees

that PÍX„ > x\ > 0 for all x > 0.' X

With these preliminaries aside we may now describe the main results of the

paper. It is convenient to distinguish the following possible conditions on the

exponent:

(a) ff2=0, j*1 \x\vidx)<°o, a' > 0,

(b) o2 = 0, f1 \x\v(dx)<oo, a' < 0,

(c) o2 ¿ 0,

(1.7)(d) a2 = 0, T0 \x\vidx) = °°, lxvidx)<°°,

J - l J o

(e) o2 = 0, J° \x\v(dx)< oo, J xvidx) = °o,

(f) a2 = 0, J° \x\v(dx) = j1 xvidx) = oo.

These are the only possibilities for which (1.4) can hold (see [ll]). In cases (b)

and (e) we show that P !Xr = x| = 0 for all x > 0, while in cases (a), (c), and

(d), P |X_ = x\ > 0 for all x > 0. In case (f) both possibilities can occur. An


EXIT PROPERTIES OF STOCHASTIC PROCESSES 461

analytic criterion in terms of ip is given in Theorem 3.2:

P°|XT =xi>0 for all x > 0 if and only ifX

P K(yV|(y, 1)1 dy < ~,(1.8)

where

Kiy) = lim f* [1 - cos uy]Re |(A - ipiu))' l \ du.Ain J -ooAlO

This criterion can be used to show that if ivj(- oo; - x)| = Oiv\ix, oo)|) as x 1 0,

then P |XT = x| = 0 for all x > 0 (see Theorem 3.6). In particular, the conclu-' x

sion of the preceding sentence holds for all symmetric processes and all stable

processes whose Levy measure is not concentrated on (- °°, 0). On the other

hand the criterion (1.8) can be used to show that (in case (f)) if v restricted to

(0, oo) is rather smaller than v restricted to (- oo, 0), then P \X = x| > 0 for' x

all x > 0 (see Theorem 3.4 for the precise statement).

The following is a simple but amusing application of some of these ideas.

For x > 0, define T_x = inf 1/ > 0: Xf < - x\.

Theorem 1.1. Let X be a real process with stationary, independent incre-

ments such that

(i) P°|Wx < oo| > 0 for all x;

(ii) P°|Tx < oo| = P°|T_x < ooj = 1 for all x > 0;

(iii) P°|XT = x| = P°|X_ =-x|= 0 for all x > 0.' X ' —X

// x / 0, then starting from 0, X jumps across the point x infinitely often before

hitting it.

The proof, of course, is a simple application of the strong Markov property.

Symmetric processes that hit points and the stable processes of index 2 > a > 1

(with v\i- oot 0)| > 0, i/j(0, oo)j > 0) satisfy the hypotheses of this theorem.

The main problem of this paper is a special case of a much more general

problem that may be formulated as follows. Let X be a process with stationary

independent increments with values in R". Let ß be a closed set in R" which

is a positive distance from 0. Let T = inf iz: > 0: X £ B\. The problem is then

to determine when

(1.9) P°\XT £dB, TR <oc¡ = 0.z ß

In spite of its intrinsic interest in the description of sample function behavior,

and in spite of the fact that (1.9) appears as a hypothesis in at least one impor-

tant paper [lO], very little seems to be known regarding the solution of this

problem.

The organization of the paper is as follows. §2 contains the proof of the

rather simple cases (a) and (b), while §3 treats the more difficult cases (c), (d),


462 P. W. MILLAR

(e), (f). Notation and terminology belonging to the theory of Markov processes

follow that of [3]. As usual, the process X is assumed (without loss of general-

ity) to be a Hunt process. The notation Px is the measure for the process starting

at x; if x = 0, then the superscript will usually be omitted.

Acknowledgment. It is a pleasure to thank Professor Harry Kesten for a

great number of helpful conversations concerning the contents of this paper. In

particular, it was he who, at an early stage in my work, first made the key ob-

servation that the problem of this paper is equivalent to determining whether or

not the subordinator JV ! (see §3) has a positive drift, and he made available to

me Fristedt's as yet unpublished work [8] on [Y \. I also thank him for permission

to include in Proposition 3-4 a rather useful result of his on local time.

2. The simple processes: cases (a) and (b). This section treats the rela-

tively simple cases (a) and (b) of (1.7). The proofs below could in certain cases

be shortened by introducing considerations involving local time (as in §3); how-

ever, it was felt that appeal to such a sophisticated concept was a bit inelegant

in such simple situations as the ones discussed in this section.

The reader wishing to go directly to §3 need read only Proposition 2.5 of

this section. Recall that assumptions (1.3), (1.4), (1.5) are in effect throughout;

these assumptions will not always appear explicitly in the hypotheses.

Proposition 2.1. Let 8 > 0 and x > 0. Then P[X_ e (x, x + 8)\ > 0.' X

Proof. It is obviously enough to prove the result for 8 small. Suppose first

that v\(0, °°)î = °°. Let a > 0 be a point of increase for v such that a < x.

Since u\(0, oo)i = oo; there are then positive numbers a > a > • • • > a > 0

which are points of increase of v such that a, + • • • + a is within e of x +* In

a A A; here 0 < í < a,/12. Let /. be a small open interval about a. such that if

*■ e /,■ then x,+•••+ x is within 2( oí x + a,/A. Let v. be v restricted to• • 1 71 1 ;

/. and let x , • • > , x" be the independent compound Poisson processes corres-

ponding to fj, •••, vn ÍEeiuX'i = expi- tiff {u)\ where iff {u) = J; [l - eiux\vidx)).

Then we can write X = X + • • • + X" + Y where all the processes in this

representation are independent. Let A > 0 be chosen so small that

Písup0;Sís |yj < t? > 0. Then using the independence of X1, ..-, X" and Y

we have with positive probability: X , • • • , X" will jump exactly once in the

time interval [0, A], X will jump last, and V will remain less than e through-

out [0, A]. The result of this will put us within 3<r of x + ax/A, and moreover,

up until the time X jumps (in this scheme) we will not yet have exceeded x.

Thus T coincides here with the time of the jump of X , and x < X^. < iax) +

x. Since a may be chosen as small as desired, this completes the proof in

case v\(0, oo)j = 00.



If iv|(0, oo)j < tx>) then by (L5), v\i— oo) 0)| = °°, and we can proceed as follows.

Let a > 0 be a point of increase of v. Let n be a positive integer such that x +

a/A - na < 0, and let e > 0 be a small number. Using the argument above one can

with positive probability get within ( of x + a/4 - na before there are any positive

jumps at all and before the process rises above x. Having got here with positive

probability we can then experience (with positive probability) zz successive posi-

tive jumps of approximately size a before the rest of the process changes by more

than f. This will land us within 3f of x + a/A with positive probability; appro-

priate choice of ( completes the proof.

Proposition 2.2. Let hix) = P\XT > x\. Then hix) > 0 for every x > 0 andx

the map x —, hix) of (0, oo) to [0, l] is lower semicontinuous.

Proof. The positivity of h is immediate from Proposition 2.1. Let I\A\ be

the indicator function of the set A. By (1.6) lim _ l\co: XT > x > XT _| =

I\co: X > z > X _| on |cü: X > z > X _| , implying that lim inf hix) >' z ' z ' z 'z x—z

hiz), as desired.

Remark. Continuity of h is proved in §3.

Corollary 2.1. // 0 < a < b, then \hiz): a < z < b\ is bounded away from 0.

The next lemma essentially says that if there are continuous crossings of a

level x, then the closer to x one starts, the more likely is one to cross x con-

tinuously. The definitive version of this lemma is in Corollary 3.2.

Proposition 2.3. Consider the following statements:

(a) P|XT = x| > 0 for some x > 0.

(b) lim inXinf P|XT >y| = 0.y 1 Z y

Then (a) implies (b).

Proof. Abbreviate T = T ,T =inf|/>0;X>x- (1/«)|, and set A =_ x n t ' n

\T . < 7| = |XT < x|, A = II A . Note that A 3/1 .. If (a) is assumed then77 1 n 77 77 77+1

PÍA) > 0 since |X_ = x| C A . ThenO = P(X.r_ < x; A] = lim P|X_._ < x; /4 |v ' x n X n—oo T n

= j¿™„^fAnP Tn\XT-<xUP>-Vímn-~$A P T"\XT-<*\dP. Hence

P "|X_._ < x| —» 0 in probability on A as n—> oo. A subsequence therefore

converges to 0 a.s. (P ) on A, so that for some sequence iy !, y î x,

lim ^^ Pyr>\XT_ < x| = 0. Statement (b) now follows from spatial homogeneity.

Proposition 2.3 permits an easy proof of the stable case. In case X is sym-

metric stable, this was proved by S. Watanabe by a different method [22].

Corollary 2.2. Let X be a stable process of index a, 1 < a < 2, with

v|(0, oo)i > 0. Then P\X > x| = 1 for every x > 0.


464 P. W. MILLAR

Proof. By the scaling property, for any r > 0 the process \r X(rí)¡ has

the same distribution as the process {X(/)(. Then, since the property of jumping

across a given level at the first passage time is independent of contractions of

the time scale, we have

P\XÍt) exceeds x for the first time by jumping!

= Pjr Xirt) exceeds x for first time by jumping!

= P\XÍrt) exceeds r~ x for first time by jumping!

= P[XÍt) exceeds r~ x for first time by jumping!.

Choosing r = x , we see from this calculation and Proposition 2.1 that P\Xr > x!x

= hix) is positive and constant for x > 0. Hence (b) in Proposition 2.3 cannot

hold, so P\X = x\ = 0. Since PjT < °°\ = 1 in the present case, P[X > x\1 x x ' X

= 1 as desired.

We can now settle case (b) of (1.7).

Theorem 2.1. Suppose v\i— °°, °°)i = °°, ^{(0, °°)\ > 0, f . |x|f(ú?x) < °°, and

a' < 0. Then P[X = x] = 0 for every x > 0.' X

Proof. Under the present hypotheses, lim ¡0 X Jt = a a. s., so that X is

negative in an initial random time interval. (This fact is well known; see, for

example, Shtatland [2l].) Consequently there is a number b > 0 such that if

T_h = infir > 0: X( < - b\ and TQ = infí/ > 0: X( > 0| then P\T_y < TQ \ > 0.

Let [F , t >0\ be the usual sigma fields associated with the Markov process

¡X i (see [3D; and let I\A \ denote the indicator function of a set A. Choose

n > 0 so large that P\T_b < TQ, XT_ e [-n, - b]\ > 0. Then

P\XT > xi> P\T_b < Tx, XT >x!X x

>P\T_b<T0,XT e [- n, - b], XT >x!— b x

FT

= ££ ~hI\T_b <T0, XT e[-n,-b\\l\XT > x\— b x

A ~

= £/!T , <Tn, XT e[-n,-b]\P ~b\X > x\.~h ° T-b X

It follows from Proposition 2.2 that the function of y given by y—» P~y{X_. > x!

= P !X_ > x + y\ is bounded away from 0 for y restricted to the intervalX +y

[b, n\. Moreover if x is restricted to [0, l] then the same proposition allows us

to choose this positive lower bound independently of x. Letting B be this posi-

tive lower bound, we see that for x £ (0, l],

PSXr >xi> BP\T_h < T0, XT e[-n, - b]\>0.x - b



It follows that statement (b) of Proposition 2.3 cannot hold, and this proves the

theorem.

The next proposition can be strengthened (see Corollary 3.4) but it is enough

for presept purposes.

Proposition 2.4. Let X have Levy measure v. Let ia, b) be an open inter-

val containing 0, and let X be the truncated process obtained from X by re-

moving all the jumps having size in Í— °°, a] U [b, oo) iso that EexplzzzX | =

expida), with ibxiu) = iffiu) - f fc) [eiux - l]vidx)). Let Tlx =

inf 1/ > 0: X1 > x\ and let 8 > 0. // P\Xl x = x| > 8 for all sufficiently small x,

then P\X = x\ > 8/2 for all small x. T*X

Proof. This is true for case (b) by Theorem 2.1. In all other cases, 0 is

regular for (0, oo) and so lim T = 0 = lim Jn T a.s. (see Rogozin [20]; 0

regular for (0, oo) means P |Tn > 0| = 0). However, using the Ito construction

of X and X1 , one sees that the paths of X agree with those of X in an initial

time interval and so for all sufficiently small x (how small depends on <u) the

paths of X and X will agree up to time T .

The following formula for P|X_ > xj will be used several times. As usual,

IAix) = 1 if x e A, = 0 if x 4 A.

Proposition 2.5.

(2.1) P\XT >xi = EÍ°° f x I. ,ÍX)dsvídy).Tx J y = 0 J s=0 (x-y.x) S y

Proof. Let fíu, v) be a nonnegative Borel function from R x R to R that

vanishes on the diagonal. According to the theory of Levy systems (see S.

Watanabe [23]; a fairly simple proof for the present situation may be found in [14]):

Z AX... X.) - fs__Q fyeR fiX, Xs + y)vidy)ds

is a martingale of mean 0, assuming expectations exist. Define / iu, v) = 1 if

u < x — n~ < x < x + n~ < v, f iu, v) = 0 otherwise, and fiu, v) = 1 if u < x

< v, fiu, v) = 0 otherwise. Let M > 0. By the optional sampling theorem,

rT AME I'/(*,_. X,) = eJ* flofnÍXs,Xs + y)vídy)ds.

s<T A/M- x

Let zz J oo and then M 1 oo. Using the monotone convergence theorem one obtains

T

E Z fiXs_,Xs) = EJ0Xrof(X,Xs + yMdy)s<T

ds

from which (2.1) follows.

Let us now take care of case (a).


466 P. W. MILLAR

Theorem 2.2. Assume v[í— °°, °o)¡ = <*>, v{(0, <x)\ > 0, / |x|i/(¿zx) < °°, and

a' > 0. Then P[XT = x\ > 0 for all x > 0.

Proof. The process X satisfying the hypotheses above may be written X =

a t + X - X where X and X are independent subordinators. We first prove

that P[Xj. = x\ > «r > 0 for all sufficiently small x. According to Proposition 2.4

we may assume that there are no jumps bigger then 1 in magnitude (i.e., v[[x, °°)!

= i/{(— °°, - x]i = 0 for all x > 1) so that X , X have finite expectations, and

by the same proposition we may assume that EX = ct, where 0 < c < a , by

truncating v on (- °°, 0) even further if necessary. Then according to Proposi-

tion 2.5:

P\XT >x!= f1 E ( x I, AX )dsvidy).'„ J v=0 J s = 0 (x-v,x) s '

But■ T

E f x I. ,ÍX )ds<EÍT -T )J o (x—y,x) s — x x—y

< ET if x > y

rT - yE\,X l(x-y.xiXs)ds<ETx if »<y<l.

so that

(2.2) p\XT >x\<(xETyvidy) + ETx\îxvidx).

Let S = inf {/ > 0: a t - X > x\. Since X has nonnegative increasing paths,

S > T . Let n be a positive integer. Since the process a t - X has no up-

ward jumps

Eia'Sx Atz -X'¿ a„!£x,

so a'ES A 7Z < x + £Xl' A = x + c£(5 A 77) (where the equality comes from the

optional sampling theorem). Hence (zï - c)£5^ < x, implying that ETx

<x/ia -c). Therefore by (2.2)

(2.3) P|XT >x\<K(Xyvidy) + Kxjxvidx) ÍK = ia - c)~x).

Since / . |x|iy(iix) < o°, the right side of (2.3) goes to 0 as x 1 0, so that

P[XT = x\ > 0 for all sufficiently small x, say x < 8. To complete the proof,X

we verify that P[XT = x\ > 0 for all x > 0. Let x > 8 and let R =

inf [t > 0: X( > x - (S/2)i. By Proposition 2.1, P¡XR e ix - (8/2), x - (8/4))! >

0. The theorem for general x now follows by the strong Markov property and the

fact that if holds for all x < 8.



We close this section by indicating an alternative approach to case (a). This

approach, however, depends on local time. A perusal of Kesten's Lemma 2.4

[11, p. 16] and his calculation on p. 21 of [ll] reveals that P\XT = x| > 0 for' x

x in a set of positive measure. Appeal to Corollary 3-1 in the next section finishes

the proof.

3. The cases (c), (d), (e), (f). Let X = jX t > 0| be a real-process with

stationary independent increments having 0 regular for (0, oo). Let X* =

sup X . Let Q = \Q , í > 0| be the local time at 0 for the Markov process

¡X*— XI (that Q exists follows from the assumption that 0 is regular for (0, oo)).

Let

(3.1) r; = inf|s>0: Qs > t\

be the right continuous inverse of Q. Fristedt has shown [8] that the process

y = {Yt,i>0) defined by

(3.2) Y, = X

is a subordinator. Since 0 is regular for (0, oo); (see [20]) T = inf 1/ > 0: X > x|

= inf 1/ > 0: X > x| a.s. and since Y has strictly increasing paths,

Sx s inf Í/ > 0: Y t> x\ = inf \t > 0: Y( > x] a.s.

Proposition 3.1. Let X be a process with stationary independent increments

having 0 regular for (0, oo). Then for each x > 0

(3.3) Ys =XT a.s.X X

Proof. From the definitions it follows that

(3.4) Yt< x if and only if r f < T,

From (3.4) and the fact that T is strictly increasing (up to its terminal time) and

has the continuous inverse Q , we see that

(3.5) S = inf {/> 0: Y > x] = inf h > 0: r > T ! = O 0T a.s.X t — t — X ^ X

Moreover, if / is a point of right increase for 0 (cd),

(3.6) ro0(W = ¡.

Since P\T is a point of right increase of Q I = P\T < oo| by V.3.5 of [3] (es-

sentially), it follows from (3.6) that Ys = Xf _q7. = X a.s.

Corollary 3.1. Let X satisfy the hypotheses of Proposition 3.1 and let

x > 0. Then PJX = x| > 0 if and only if the subordinator Y has positive drift.z *■


468 P. W. MILLAR

Moreover, either P[X = x\ > 0 for all x > 0 or P[X_ = x! = O' for all x > 0.' X ' X

Proof. Immediate from (3-3) and Theorem lc of Kesten's monograph [ll].

Define hix) = P[XT > x\, gix) = P[X = xlx Tx

Corollary 3.2. Let X be any real process with stationary independent incre-

ments satisfying either (1.5) or o > 0. Then g and h are continuous functions

on (0, <x>). If gix) > 0 for some x > 0, then lira ,Q gix) = 1.

Proof. If the process does not hit points then g is trivially continuous. The

case where X hits points and 0 is not regular for (0, °o) ¡s treated in Theorem

2.1, which shows that g is identically 0 in this case. Finally, if 0 is regular

for (0, °°) then gix) = P[XT = x\ = P[YÇ = x!; in this case g is either identi-' X JX

cally 0 or positive, continuous, and with the announced limit at 0 by Proposition

6, p. 120 of Kesten's monograph [ll]. Hence in any case, g is continuous.

Since gix) + h(x) = P[T < °°i, one proves h continuous by proving x—>PjT < °<=i

continuous. If x j, z, then T I T so P[T < °°\ is right continuous and decreases

as x increases. If S > 0, then

P[Tx < <*>!> P\TS < oo\P\Tx_s < oo!;

so if 0 is regular for (0, oo)( then limgj0 P¡Tg < °°| = 1, implying in this case that

P[T < ooj ¡s also left continuous. In the only remaining cases, X_ _ < x < Xx ' X ' X

on [T < °°i, which can be shown to imply left continuity here as well.

Remark. From Theorem 1 of [10], one may deduce that h is excessive for

the subprocess obtained by killing X upon first leaving (- °°, x). This observa-

tion provides an alternative route to proving some of the regularity properties of h.

The next two corollaries give refinements of Propositions 2.3 and 2.4. In

each case the proof is nearly immediate.

Corollary 3.3. The following statements are equivalent:

(a) P[ XT = x( > 0 for some (hence all) x > 0;

(b) limyl0XPÍXT >y! = 0.

Corollary 3.4. Let X be obtained from X by truncation as in Proposition

2.4. Then

P\Xr = x\ > 0 for some (hence all) x > 0 if and only ifX

P[XX . = x\ > 0 for some (hence all) x > 0.T

X

This corollary states that the manner in which a process passes over a given

level is independent of the large jumps of the process. Hence, for solving the



main problem of this paper we may assume (without loss of generality and when-

ever convenient) that P\T < ooj = 1 for all x > 0 and that, for example, the pro-

cess X has no jumps larger in magnitude than (say) 1.

Assume now for the rest of this section that (1.3), (1.4), (1.5) hold, and that

one of the cases (c)—(f) is present. Under these hypotheses it is known (Breta-

gnolle [6]) that 0 is regular for itself so that for each real number y there is a

continuous additive functional Ly = \Ly, t > 0| called the local time at y. This

additive functional is unique only up to constant multiples; we may choose Ly

fot each y in such a way, however, as to be jointly measurable in it, y, cA) and

to satisfy for all Borel sets B simultaneously:

(3.7) ¡BL>dy-foIBiXs)ds.

This result is due to Blumenthal and Getoor ([2]; see also [9]).

The following proposition gives a necessary and sufficient condition that

P|XT = x| > 0 for all x > 0, assuming X has local times.' X

Proposition 3.2. Let X = jX , t > 0| be a real process with stationary in-

dependent increments having local times Ly as described above. Then

P\XT = x| > 0 for all x > 0 if and only if flQE°L°T v[y]dy < oo, where v[y\ =

v\íy, oo)¡.

Proof. Without loss of generality, assume v has no mass outside the inter-

val (- 1, 1). Then by (3-7) and Proposition 2.5:

T

P|XT >xj = E° f°° f X I. AX )dsvidy)Tr J y-0 Js=0 (x-y.x) s s

(3.8)

= E°r r L» duvidy) = E° r r L-i/Uyw«J y=0 J x-y ' x J u = -oo J y-x—u ' x

= E°/X0o LT ^x-«]rfa = E° f~ Lx~yV[y]dy

= P Shit x - y before exceeding x| E L_ v[y]dy

since v[y] = 0 if y > 1. Let / = X - X_Using the same argument as in

Proposition 2.5 (with fiu, v) = v - u if u < x < v, fiu, v)= 0 otherwise) we find

(3-9) E°hx - £° ;;=0 û *'(*.,.* w*"<*>;

and if l4yj = /°° uvidu) = f uvidu), (3.9) reduces under the present hypotheses to

(3.10) E°/T = J1 P°|hit x-y before exceeding x|E°L^. V[y]dy.


470 P. W. MILLAR

Also under the present hypotheses, lim ,QT =0 and 0 < Jr < 1; hence,

lim(3.11) lim £°/r =0

by the dominated convergence theorem. Note that 0 < E L_ <EL <°Oj0<i y ^

y < 1; 0 < V[y], for 0 < y < a, some a > 0; and jQ V[y] dy < oo. For each x,

let / (•) be the function / (y) = P[hit x - y before exceeding x\. It then follows

from (3.11) that the family of functions / converges to 0 in measure (for Lebesgue

measure on [0, a]) as x | 0.

Now assume that fQ E LT v[y] dy < oo. Then since [fx\ converges in mea-

sure as x 1 0, it follows from (3.8) that lim ,0 PjX^. > x! = 0. An application

of Corollary 3.3 completes the proof in this case. (One may also complete this

part of the proof by using the argument at the end of Theorem 2.2, and thus avoid

the use of the subordinator Y). Conversely, assume that P|X_ = x) > 0. ThenX

gix) = P[Xj. = x! is continuous, positive, and lim ,. gix) = 1, so that gix) > S

> 0 for all x e (0, A, say. Also, for y > 0, P (hit x - y before exceeding x\ >

P°[X =x-y! = g(x-y). But then by (3.8)' x —y

l>PÍXr >e\>8 f£0E°L°T v[y\dy

which implies that fQELT v[y] dy < «=•

Proposition 3.2 gives a quick solution to case (d).

Theorem 3.1. Suppose that X satisfies (1.3) and that JQxvidx) < <*>,

f°, \x\vidx) = °°. Then P[X = x\ > 0 for all x > 0.1 ' X

Notice that the proof works whether or not there is a Gaussian component.

Proof. JqEL^ v[y]dy < ELt f¿ v[y]dy < °°, since the hypothesis

jQxvidx) < °° implies fxQv[y]dy < <*>.

Next, define

(3.12) H = inf{/>0: X =*!.'X t

Obviously H > T . The next proposition gives an improvement of Proposition

3.2; the improvement lies in the fact that ELj, happens to be more easily ana-n *

lysed than EL" , as we will see below.' X

Proposition 3.3. Let X have local times as described at the beginning of

this section. Then PiXT = x! > 0 for all x > 0 if and only if A EL?, v[y]dy' X u "y

< oo.

Proof. If J EL v[ y]dy < oo, then since EL >ELT ; it is evident that

fçEL® v[y]dy < oo. Proposition 3.2 then guarantees that P|X_ = x! > 0 for ally x

x > o.



Conversely, suppose that P|XT = x| > 0 for all x > 0. Since L increases

"only" when X visits 0 (see [3, V.3] for the precise result) and because of the

strong Markov property, we have

EL?, = EL<L +P°{T <H and X hits 0 between T and H |EL°Hy Ty y y y y Hy

<EL% + P0|T <H \EL°U .— I y y Hy y

But if P|Xr = xj > 0 for every x > 0, then by Corollary 3.2, lim lQ P\T < H \

= 0 (Iß = T }=» |X_ =xj). Hence there exists S> 0 such that P\T < H \<V¡,X X l x y y —

0 < y < <5, so that

(3.13) EL°H <2EL°T , 0<y<8.v y

Also, if P|Xr = x] > 0 for ail x > 0, then flQEL^ v[y]dy < oo, so

fQEL v[y]dy < oo from (3.13), as desired.

The next proposition, which is due to Kesten, is the first step in reducing

the criterion of Proposition 3.3 to a tractable analytic expression. A different

proof of this proposition appears in [12]; the present simpler proof is also due to

Kesten.

Proposition 3.4 (Kesten). Let X be a process with stationary independent

increments having a local time at 0. Let zzx(x) = Ex f°° e~ d L°. Then

(3.14) ELi = lim uXÍ0)[l - Ee~XHyEe~XH-y].y Ai.0

Remark. Observe that according to this proposition EL = EL® , a fact"y n _y

that will be useful later.

Proof. Let À > 0. Then using the strong Markov property:

uHo) = e° re-xtdL°J o t

.H.-AH= E0 f y e-*'dL° + E°e yEy (°° e~*<dL°

Jo t Jo t

E° j"y e-XtdL°t + E°e~XHyEye~XH°E0 j

E° \Hy e-*'dL° + E0e~XHyFA'e~X"°uK0).J n t

-AH -AH,

0 t

Hence,


472 P. W. MILLAR

£° f y e-^dL°^uH0)[l-E°e yEye °]J o t

. « —An ^ -An

= zAo)[l - £0e yE°e -y].

Since £° f"ye~Xt dL° T E0L° as A 1 0, the proof is complete.o t y

Using Proposition 3.4 we next obtain an analytic necessary and sufficient

condition for P\XT = x! > 0 for all x > 0.' X

Theorem 3.2. Let X have local times as described earlier in this section.

Then P[XT = x\ > 0 for all x > 0 if and only if fXv[y]KÍy)dy < oo, where Ay] =

'lAiy, l)\ and

Kiy) = lim f °° [1 - cos uy] Re-iU - if,iu))~ X\du.

Proof. From Proposition 3.3, it is enough to show that

(3.15) EL°H ^7T-xKÍy) as y 10.y

Since 0 is regular, Theorem 2 of Kesten's monograph [ll, p. 7] guarantees that

(3.16) f°° Re (A - if/iu))- X du < oo for all A > 0.J -oo

Moreover, according to Bretagnolle [6] there is for each A > 0 a bounded contin-

uous density ux(x) such that

(3.17) £ f°° e-XíIAÍXt)dt = f uxix)dx for every Borel set A

and also

— XH(3.18) uHx) = uXiO)Ee~ x.

This density #*(•) is, of course, the same object that appears in Proposition 3.4

(consult [9] for more detail on «*). Using the inversion formula for characteristic

functions (see [7, Theorem 6.2.1]), (3.16) and (3.17) we find

jy_ uXiz)dz=f™ e-X'PS|X(|<y!^

(3-19) = 77- ' f°°^ iT X sin zzy Re (A - dj(u))~ x du

= n~ \ dz \ cos zzzRe (A - if/iu))~ du,JO J -00

implying that

(3.20) zzX(y) + uxi-y) = ir'1 f" cost/yReU-rM«))-'1^,J — OO



(3.21) uXÍ0) = Í2n)-1 f~ Re (A-.¿(a))"1 ¿a.J -oo

(The calculation (3A9) is in [9] and [12].) Next

X H X H — X H

aA(0)[l-E«?~ yEe~ " A = aMO) - uHy)Ee~ ~y

-AH

= uHO) - uXiy) + Ee y[uXÍ0) - uH- y)]

< 2aX(0) - uHy) - uH- y)

= tr~ ' (°°^ [l - cos ay]Re (A - ifiiu))' l du,

using the fact that ax(0) > ax(x) by (3.18). On the other hand, zzx(x) < a*(0) and

H —* 0 in probability as y —• 0 (Bretagnolle [6]). Hence for all sufficiently

small y (how small depends on e and not on À £ (0, l)):

AH — A H

aX(0)[l - Ee~ ' yEe ' ~y] > (l - í)zt_ ! f°° (l - cos uy) Re (A _ ibiu))~ l duJ -oo

for 0 < e < 1, so that

EL° ~ lim 77" 1 f°° (1 - cos ay) Re (A- Mu))'1 du"y AlO J-~

as desired.

In certain cases of common occurrence, one can evaluate Kiy) more conve-

iently by taking the limit under the integral.

Corollary 3.5. // lim inf \\ Re \-i¡iíu)\ > 8 > 0, then

EL°H ^7r-1K(y) = 77-! P"^ [1 -cos uy]Re\-ibiu)\\X - ibiu)\~2 du.

This hypothesis is satisfied if the measure corresponding to the characteristic

function exp i/r(a) is not purely singular relative to Lebesgue measure.

Proof.

JToo (1 " cos"y)Re(A - 0("))"' du =f (1 - coszvyÍRei-^HA-^zz)!-2^

+ A|i i il-cos uy)\X + y/iu)\~2 du

+ A f. . (1 -COS7Vy)|A + l/zU)r2¿ZZ.I tz|>l

By monotone convergence,

f ^ (1 - cos zvyVRe |-t/z(a)i | A + djiu)\~2 du

-. f°° il -cos uy) Re \-ifÁu)\\djiu)\~2 du <77EL(l < oo.l ""O "y


474 P- W. MILLAR

as A j 0; while, since |<A(zz)| > const zz2 for |«| < 1,

A f , il -eos uy)\k + tfÁu)\-2 du -^0.J | u¡^ 1

Finally, (this is the only place where the hypothesis is used):

A f. , (1 -cos«y)|tA(zz) + A|-2¿«<A f. . (1 - cos uy)\ifjíu)\-2 duJ |"|2l -* |u| 21

<AS_1 f, . il - cos uy) Re \-if,iu)}\iffiu)\-2 duJ | U | 2 1

<nk8-1EL°H — 0,y

using the first part of the proof. This completes the proof since the implication

of the second sentence of the corollary is well known. I do not know whether the

hypothesis of the corollary can be removed.

We will now apply the criterion of Theorem 3-2 to several cases of particular

interest. Let us first consider the case when X has a Gaussian component.

Theorem 3.3. Let X be a process with stationary independent increments

having a2 > 0. T¿erz PJX_ = x! > 0 for all x > 0.' x

Proof. Using Corollary 3.5 we will verify that if o2 > 0, then EL <y

const y for all sufficiently small y. The conclusion will then follow from Theo-

rem 3.2, since

I £L„ v[y\dy < const I yv[y\dy = const I y vidy) < oo.

However, from (3.15) and an elementary inequality

EL° <tt-1 f"" (1 - cos uy)[We[-djiu)S\-x du"y J -OO

Kn-1 j^^il - cos uy)[ia2/2)u2]-1 du

-1 -2 f°° z \ — 2= y4z7~ o~ I (1 — cosz)z~ dz,

as desired. We have used in this calculation the fact that the real part of the ex-

ponent of a process is negative.

In order to state the next result it is convenient to have some further notation.

If v is the Levy measure of the process X, let v+ be v restricted to (0, oo) and

v_ be v restricted to (- oo, 0). Assuming that X has no Gaussian component,

the next intuitively appealing result says essentially that if v is rather larger

than v+, then X will have continuous passages upward (i.e. P¡X_ = xi > 0 for

all x > 0). In order to give a precise meaning to this we introduce the indices



ß, ß" oí Blumenthal and Getoor [l]: let X be a process with no Gaussian part

having exponent t/y and Levy measure v. Define

(3.22) ß(x)= inf|a>0: lim |y |~ aRe<-t>(y)l = ol -I |y|-oo )

(3.23) /3"(X) = sup/a>0: lim |y raRe|-^(y)| = 4-I |y|~« )

For a given process X, let X be the process with Le'vy measure v+, and let

i/z+ be the exponent of X :

xb¿u) = J"^ [e'ux - 1 - iux/il + x2)]t7 + Ux)

= [°°[e>"*-\ -iux/il +x2)]vidx).J 0

Let X" = X - X and let \fi_ be the exponent of X". Then we have the following

theorem.

Theorem 3.4. Assume ß"iX~)>l, and ß"ÍX~) > ßixA. Then P\XT = x|

> 0 for all x > 0.

Remarks. Examples of such processes may be constructed as follows. Let

X be a stable process with index a and Levy measure concentrated on (0, oo),

and X a stable process, independent of X , with index a and Levy measure

on (- oo, 0). If a > 1 and a > a , then X + X is an example of the pro-

cesses described in Theorem 3.4. According to Theorem 3.5 these processes

will not have continuous passages in the negative direction.

Proof. From Corollary 3-5, EL°H Ktr'1 /^ (1 - cos ay)(Re |- !/z_(a)|)_ ldu.

Since for |a| < 1, Re|-i//_(a)| > const u ,

n~ Il i , (1 - cosay)(Ret-i/f (zz)i)"1 < const y2.J |up 1 "■

Choose ß subject to ß" ÍX~) > ß > 1, ß> z3(X+). Then for some constant A,

Re tfj_iu) > Au", \u\ > 1, implying that

n~ Jlt/Ul ^ ~ cos "yXRe {—v>_(ii)})-1 < const J, , (l - cos uy)u~ß du

z3-l= const y

Hence ELH < const y^"1. Let y be chosen to satisfy ß > y > ßiX+). By The-

orem 2.1 of [l], if y > 0, then v+[y] = v[y] < const y_r. /JeE° v[y]dy <

const ¡\yP~Xy~y dy < oo and the result follows from Theorem 3.2.


476 P. W. MILLAR

The remaining results of this section are perhaps most easily deduced by

using a recent result of Fristedt [8]. Recall the definitions of the subordinators

[t , t > 0! and ¡V , t > 0! from (3.1) and (3.2). Fristedt has then shown that

(3.24) F expl- A^ - A^l = expi- tifjiky k2)\

where, if F is the distribution of X

(3.25) 4,ikx, k2) = exp/j^_o i"1 dtf~_Q [e'Xl' - e~ ̂ '^F((dx)\ .

It then follows that [Y{\ will have positive drift if and only if

lim ArVd, A,) = c>0;

that is, \Y \ has positive drift if and only if

(3.26) f™ rXdt f™[e-' -e-'-Xx]F tidx)-log k — c2 > - oo

as A —> oo.

Recall the definition T_ = inf [t > 0: X < - x!, x > 0. The next result says

that the only processes that have continuous passages both upwards and down-

wards are those processes having a Gaussian component.

Theorem 3.5. Suppose X is a process with stationary independent increments

for which PJXX = *! > 0 and P\XT = - x\> 0 for all x > 0. Then X must have' X ' —X

a Gaussian component.

The converse to this theorem was established in Theorem 3.3-

Proof. Let Y = ¡Y^i be the subordinator defined in (3-2) and let Y' = \Y'\

be the analogous subordinator for the process ¡- X !. By Corollary 3.1, Y and

Y both must have positive drift. From the criterion (3.26) both (i) and (ii) hold

as A —> oo;

(i) fîe-'r1'* /~[1 - e~Xx]FAdx) - log A ̂ c^ > - oo,

(ii) f™e-'rX dt p_J\ - e~X\x\]Ftidx) - log A ̂ c4 > - oo,

so that

(3.27) re-T'dtT [l-e-Aix]F Ux)-log(A2) -O-oo.JO J -oo Í

However, by the first part of the proof of Theorem 2.1 of [17],

f [l-e-*\*\]F(dx) = n-1 r [1-«-'**>>] (1+y2)-1^.J _oo t J -oo

Moreover, if z is a complex number, Re z > 0, then

(3.28) Re J~ e-WHl - e~tz)dt = log \z + l|.


EXIT PROPERTIES OF STOCHASTIC PROCESS 477

Hence, from (3.27) and (3.28),

(3.29) limzr-1 f" log ||1 -iA(Ay)|A-21(1 +y2)-ldy = c>-oo.J -oo

By a well-known result (possibly first noticed by Bochner [4, Theorem 3.4.2]),

lim . ^Ji/KAy^A-2 = 0 if and only if there is no Gaussian part. Hence (3.29) can

hold only if there is a Gaussian part, and this is what we wanted to prove.

We can now easily dispose of case (e) of (1.7).


having o2 = 0, /* xvidx) = oo, j°_ j |x|iy(a?x) < oo. Then P\XT = x\ = 0 for all

x > 0.

Proof. By Theorem 3.1, P\X~ = - x| > 0 for all x > 0. Since X has no' —x

Gaussian part, Theorem 3.4 implies that P{XT = x| = 0 for all x > 0.' x

We conclude this section with the following useful criterion.


having no Gaussian component but admitting a local time at 0. Assume that

v\i- oo, - x)| = OiiAix, oo)|) as x [0. Then P\Xt = x| = 0 for all x > 0.* X

Proof. Let v+[y] = lAíy, oo)¡, v_[y] = iyj(- oo, - y)j for y > 0. According to

Theorem 3.5 and Proposition 3-3 at least one of the following must occur:

(i) J¿ EL°HyV+[y]dy = oo,

(ii) fl0EL°Hy v_\y]dy = oo.

By Proposition 3.4, ELH = EE„ so that (ii) is the same as

(ii)' /> ELlvvJyV7- »■ VSince at least one of (i), (ii) must hold and since by hypothesis v+ is bigger

than v_, clearly (i) must hold. Hence by Proposition 3.3, F|X_ = x| = 0 for

all x > 0.

The following special case is worth pointing out.

Corollary 3.5. Suppose X has no Gaussian component. If v\i— oo, — x)| =

OivKx, oo)!) and if v\(x, oo)| = OÍiAÍ- <*>, - x)|) as x { 0, then P\XT = x| =

P|X_ =-x}= 0 for all x > 0.' —x

In particular any symmetric process and any stable process that does not

have its Levy measure concentrated on a half-axis will have continuous passages

in neither the upward nor downward direction.

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DEPARTMENT OF STATISTICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA

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