Integrated Brownian Motion, Conditioned to be Positive

INTEGRATED BROWNIAN MOTION, CONDITIONED TO BE POSITIVE

BY PIET GROENEBOOM, GEURT JONGBLOED AND JON A. WELLNER1

Delft University of Technology, Vrije Universiteit and University of Washington

We study the two-dimensional process of integrated Brownian motion and Brownian motion, where integrated Brownian motion is conditioned to be positive. The transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process. Explicit expressions for the transition density in terms of confluent hypergeomet-ric functions are derived, and it is shown how our results on the hitting time distributions imply previous results of Isozaki_{᎐Watanabe and} Gold-man. The conditioned process is characterized by a system of stochastic

Ž .

differential equations SDEs for which we prove an existence and unicity result. Some sample path properties are derived from the SDEs and it is shown that t_{¬ t}9r10 is a ‘‘critical curve’’ for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve t¬ t␣ is finite for␣ - 9r10 and infinite for␣ G 9r10.

Ž .

1. Introduction. Let U, V be the two-dimensional process of

inte-Ž . Ž .

grated Brownian motion IBM and Brownian motion BM , where U repre-sents IBM and V reprerepre-sents BM. This process is often called the Kolmogorov

w x diffusion since its study was apparently initiated by 7 .

Ž

It is well known and easily verified by computing expectations and .

covariances of the Gaussian process involved that the transition density of ŽU, V is given by. pt

Ž

x , y ; u, v

.

2

'

3 6 u

Ž

y x y ty

.

6 v

Ž

y y u y x y ty

. Ž

.

s ₂ exp

½

y ₃ q ₂ ␲ t t t 1.1

Ž

.

2 2 v

Ž

y y

.

y

5

; t w x Ž .

see 12 . Another way of writing this transition density that often is useful

Received March 1998; revised August 1998.

1

Supported in part by NSF Grant DMS-95-32039 and NIAID Grant 2R01 AI291968-04.

AMS 1991 subject classifications. 60J65, 60G40, 60J25.

Key words and phrases. Conditioning, confluent hypergeometric functions, hitting times,

integrated Brownian motion, Kolmogorov diffusion, stochastic differential equations.

is pt

Ž

x , y ; u, v

.

2

'

3 6 u

Ž

y x

.

6 u

Ž

y x v q y

. Ž

.

s ₂ exp

½

y ₃ q ₂ ␲ t t t 1.2

Ž

.

2 v2_{q vy q y}2

Ž

.

y

5

. t Ž .

We want to characterize the process U, V , where U is conditioned to be

Ž . Ž . Ž

positive and where U, V s 0, 0 at time zero U has slope zero at time .

zero . This process arises naturally in several contexts. Our motivation for studying this process originated in a study of the limiting behavior of the nonparametric maximum likelihood estimator of a convex density and non-w x parametric estimators of convex regression functions; see, for example, 6

w x w x

and 11 . Another motivation can be found in the work of 16 on the convex hull of integrated Brownian motion with a parabolic drift. In both cases, one encounters excursions of integrated Brownian motion above certain curves at which the integrated Brownian motion touches at the endpoints of the excursion. Using the Cameron᎐Martin formula, these excursions can be described by excursions of integrated Brownian motion above a line. These excursions, in turn, can be related to integrated Brownian motion, condi-tioned to be positive, in a way that is somewhat analogous to the relation

Ž . Ž .

between Bessel 3 bridges and the Bessel 3 process for ordinary one-dimen-w x

sional Brownian motion; see 5 .

We determine the structure of this process in Sections 2 and 4. It is shown

Ž .

that the transition density of the process U, V , where U is conditioned to be positive, is of the form

y1

h x , y

Ž

.

pt

Ž

x , y ; u, v h u, v ,

.

Ž

.

x) 0, y g ⺢,

Ž .

where p is the transition density of the process U, V , killed when U hits_t

Ž .

zero, and where h x, y is proportional to

1r4

_

₄

1.3 lim t P ␶ ) t ,

Ž

.

Ž x , y . 0

tª⬁

denoting by␶ the first time that U hits zero.0

 4

This motivates the study of the asymptotic behavior of P_{Ž x , y .}␶ ) t , as0

w x

tª ⬁. This has been studied by 4 , but we give a simple direct approach to this problem in Section 2, avoiding the use of Laplace transforms, Tauberian theorems and separation of cases. Our approach leads to an integral repre-sentation of h, valid for all values of the arguments. In fact, we obtain the asymptotic behavior of the density

'

1.4 P ␶ g dt, V ␶ t g ydz dt dz ,

Ž

.

Ž x , y .

½

0

Ž

0

.

5

Ž

.

as tª ⬁, showing that this joint density asymptotically behaves as the

'

Ž . Ž .

Ž .

tª ⬁. Moreover, we show that the function 1.3 has an explicit representa-Ž .

tion in terms of confluent hypergeometric functions; see part iii of Theorem Ž .

2.1 and part ii of Lemma 2.1.

We also study the behavior of the density

1.5 P



␶ g dt, V ␶ g ydz

4

dt dz ,

Ž

.

Ž x , y . 0

Ž

0

.

Ž

.

if t is fixed and zx0, showing that this density is of order z3r2_{, as zx0; see}

Ž .

part ii of Theorem 2.1. This result provides us with the transition density of

Ž . Ž Ž . Ž .. Ž Ž . Ž .. Ž .

an ‘‘excursion’’ of the process U, V , where U 0 , V 0 s U 1 , V 1 s 0, 0

Ž . Ž .w Ž .x

and U t ) 0, t g 0, 1 see 2.26 .

Ž .

In Section 3 we discuss how our results on the asymptotic behavior of 1.4

w x w x

can be specialized to yield previous results of 2 and 4 . The latter compari-son reveals at the same time a curious relation between the hypergeometric function F and gamma functions that was unknown to us and seems to be_{2 1} nonstandard. This comparison also reveals that Goldman’s result seems to be off by a factor 6.

Using the results of Section 2, we determine the marginal density of the conditioned process in Section 4. Next we show in Section 5 that the condi-tioned process can be characterized by a system of stochastic differential

Ž .

equations SDEs , and derive from the structure of these equations that U will not hit zero after time zero and will drift off to⬁, as t ª ⬁. The SDEs, together with the analytic properties of the function h, yield a very simple tool for proving these facts.

Finally, we deduce in Section 6 from Theorem 4.1 in Section 4 another

˜

sample path property of the process U, where we denote the conditioned

˜ ˜

9r10

Ž .

process by U, V . This is the property that the curve t¬ t is a ‘‘critical

˜

curve’’ for the process U in the sense that the expected amount of time the

˜

␣

process U spends below any curve t¬ t is finite for␣ - 9r10 and is infinite for ␣ G 9r10.

{ } w x

2. The asymptotic behavior of P_{( x, y )} ␶ )₀)))) t for large t. By 8 ,

Theoreme 1, page 388, we have, for x, z

_{´ `}

) 0,

'

PŽ x , y .

½

␶ g dt, V ␶0

Ž

0

.

t g ydz

5

Ž

dt dz

.

'

'

s z t p x, y; 0, yz t

½

t

Ž

.

⬁ t _{q q}

'

y

_H

_H

ptys

Ž

x , y ; 0, w P

.

Ž0 ,yz t_'.



␶ g ds, V ␶0

Ž

0

.

g dw

4

5

t ss0 ws0 2.1

Ž

.

'

s tz q x, y; 0, yz t

½

t

Ž

.

⬁ t _{q q} y

_H

_H

qtys

Ž

x , y ; 0, w P

.

Ž0 ,yz t_'.



␶ g ds, V ␶0

Ž

0

.

g dw ,

4

5

ss0 ws0

where␶q0 denotes the first time U passes zero after time zero and 2.2 q x , y ; u, v s p x, y; u, v y p x, y; u, yv .

Ž

.

t

Ž

.

t

Ž

.

t

Ž

.

w x Ž .

The function q was already an important tool in 2t who called it p* . In w x_{12 the joint density of␶ and V ␶}q _Ž q_{. under P is derived: for z) 0,}

0 0 Ž0,yz. PŽ0 ,yz.



␶q0 g ds, V

Ž

␶q0

.

g dw

4

3w 2 2 2 s ₂ exp

½

y

Ž

z y zw q w

.

5

'

s ␲ 2␲ s 2.3

Ž

.

= 4 zwrs_␰y1r2 _{exp y}3_{␰ d␰ dw ds.}

H

½

₂

5

0

It will be shown that the dominating asymptotic behavior, as tª ⬁, but also if zx0 and t is fixed, is coming from the double integral on the

Ž .

right-hand side of 2.1 . We first state a preliminary result, giving the integral representation of the crucial function h and also its relation to a function of one argument g that can be expressed in terms of standard confluent hypergeometric functions.

Ž . Ž .

LEMMA 2.1. Let the functions g: ⺢ ª 0, ⬁ and h: 0, ⬁ = ⺢ ª ⺢ be

defined by ⬁ ⬁ 3r2 h x , y

Ž

.

s

_H

_H

w qs

Ž

x , y ; 0,yw ds dw

.

ss0 ws0

'

⬁ ⬁ 2 3 3r2 2 3 2 2 2 s

_H

_H

w exp

_

y6 x s y 6 xys y 2 y q w s

_Ž

_.

₄

␲ ss0 ws0 2.4

Ž

.

=sinh 6 xws2_{q 2 yws ds dw}

Ž

.

and ⬁ ⬁ _3r2 2.5 g y s w q 1, y ; 0,yw ds dw s h 1, y

Ž

.

Ž

.

_H

_H

s

Ž

.

Ž

.

ss0 ws0

and write DD_{Ž x , y .} for the differential operator

⭸ 1 ⭸2 2.6 DD s y q .

Ž

.

Ž x , y . _{⭸ x 2 ⭸ y}2 Note that 2.7 h x , y s x1r6_{g yx}y1r3 _.

Ž

.

Ž

.

Ž

.

Then:

Ž .i The function h is harmonic for DD_{Ž x , y .} in the sense that DD_{Ž x , y .}h x, yŽ .s 0, and the function g is analytic on⺢ and satisfies the second-order differential equation

2 2 1

2.8 g⬙ y s y g⬘ y y yg y , yg ⺢.

Ž .ii The function g has the representation 1r6 2 1 4 2 3 2.9 g y s yU , , y , y) 0,

Ž

.

Ž

.

Ž .

9

Ž

6 3 9

.

1r6 2 1 1 4 2 3 2.10 g y s y yV , , y , y- 0,

Ž

.

Ž

.

Ž .

9 6

Ž

6 3 9

.

y1r6 2 1 1 2.11 g 0 s lim g y s ⌫ r⌫ ,

Ž

.

Ž .

Ž

.

Ž .

9

Ž .

3

Ž .

6 yª0

where U and V are the confluent hypergeometric functions, as defined on page w x

256 of 13 .

Ž . Ž .

PROOF. i The infinitesimal generator of the process U, V is given by

Ž .

the partial differential operator DDŽ x , y ., defined by 2.6 , and therefore, as

w x Ž .

noted in, for example, 10 , page 1302, the transition density p x, y; u, v oft

Ž . Ž .

the process U, V satisfies the backward Kolmogorov equation ⭸

2.12 DD p x , y ; u, v s p x , y ; u, v ,

Ž

.

Ž x , y . t

Ž

.

t

Ž

.

⭸ t implying that also

⭸ 2.13 DD q x , y ; u, v s q x , y ; u, v .

Ž

.

Ž x , y . t

Ž

.

⭸ t t

Ž

.

Hence we get, if x) 0, ⬁ ⬁ ⭸ 3r2 DDŽ x , y .h x , y

Ž

.

s

_H

w

_H

⭸ sqs

Ž

x , y ; 0,yw ds dw

.

ws0 ss0 ⬁ _3r2 s

H

w lim qs

Ž

x , y ; 0,yw dw

.

sª⬁ ws0 ⬁ _3r2 y

_H

w lim qs

Ž

x , y ; 0,yw dw

.

sx0 ws0 s 0. Ž . This implies, by 2.7 , ⭸2 _⭸ y1r2 y1r3 h x , y

Ž

.

s x g⬙ yx

Ž

.

s y2 y h x , y

Ž

.

2 _{⭸ x} ⭸ y 2.14

Ž

.

2 1

2 y7r6 y1r3 y5r6 y1r3

s y x g⬘ yx

_Ž

_.

y yx g yx

_Ž

_.

.

3 3

Ž . Ž

Evaluating this for xs 1, we get 2.8 . The analyticity of the function g on

. Ž . Ž .

⺢ follows from 2.5 together with the integral representation 2.4 of h. Ž .ii Let M be the Žstandardized confluent hypergeometric function. Ža

. Ž .

version of the so-called Kummer function , defined by 9.04 on page 255 of w x13 . A straightforward computation, using the fact that M and U satisfy the

Ž w x .

Ž .

solution of the differential equation 2.8 is of the form

1 4 2 3 1 4 2 3

2.15 y A⭈ M , , y q B ⭈ U , , y ,

Ž

.



Ž

6 3 9

.

Ž

6 3 9

.

4

for constants A and B. We are going to specify this function to our function g Ži.e., determine the constants A and B at. q⬁, since determining A and B at

Ž .

a finite point like 0 seems much harder in this case! In fact, by determining

Ž .

the behavior at ⬁ we will find a relation between special hypergeometric

w x w x

functions at zero, allowing us to compare the results in 2 with those in 4 Žshowing that there is in fact a discrepancy; see the end of Section 3 ..

2 3 _{Ž .} w x

Denoting 9y by z, we have, by 10.07 , 13 , page 257,

1 4 2 3 z y7r6 1 2.16 M , , y ; e z r⌫ , yª ⬁

Ž

.

Ž

6 3 9

.

Ž .

6 Ž . w x and, by 10.01 , 13 , page 256, 1 4 2 3 y1r6 2.17 U , , y ; z , yª ⬁.

Ž

.

Ž

6 3 9

.

On the other hand, using the change of variables sª ysr3 and w ª wy, we have g y

Ž

.

s h 1, y

Ž

.

'

⬁ ⬁ 2 3 3r2 3 2 2 2 s

_H

_H

w exp



y6s y 6 ys y 2 y s y 2w s

4

␲ ss0 ws0 =sinh 6ws2_{q 2 yws ds dw}

Ž

.

⬁ ⬁ 2 2 7r2 3r2 3 2 2 s y

_H

_H

w exp

½

y y s s q 3s q 3 q 3w

Ž

.

5

'

9 ␲ 3 ss0 ws0 2 3 =sinh

ž

y ws s

Ž

q 1 ds dw

.

/

3 7r2 _{⬁ ⬁} 2 y 2 2 3r2 3 2 3 ;

_H

_H

w exp

½

y y s 1 q w

Ž

.

5

sinh

ž

y sw ds dw,

/

'

3 3 ␲ 3 ss0 ws0

as yª ⬁. However, the last displayed expression equals

'

3 y ⬁ 1 1 3r2 w y dw

H

½

2 2

5

2␲ 0 1y w q w 1q w q w 5r2

'

3 y ⬁ w

'

s

_H

_{2 4} dws y . ␲ 0 1q w q w Ž . Ž . Ž .

Because of 2.16 and 2.17 , it now follows that, in the representation 2.15

Ž .

of g on 0,⬁ , the coefficient A has to be zero and hence that

1r6

2 1 4 2 3

Ž . Ž w x. for yg 0, ⬁ . On the other hand, we have using 13.5.8, page 508 of 1

1 ⌫

Ž .

y1r3 3 1 4 2 3 2 lim yU

Ž

6, , y3 9

.

s

Ž .

9 1 . ⌫ yx0

Ž .

6

The obvious candidate for the analytic continuation at zero is provided by the confluent hypergeometric function V, defined by

2.18 V a, c, z s ez_{U cy a, c, yz ,}

Ž

.

Ž

.

Ž

.

since V also satisfies the confluent hypergeometric equation and has the desired vanishing behavior at ⬁. In fact, it is immediate from 13.5.8, page

w x 508, 1 , that

1 1 4 2 3 7 1 4 2 3

⌫

Ž .

6 lim yU

Ž

6, , y3 9

.

s ⌫

Ž .

6 lim

Ž

yy V , , y ,

.

Ž

6 3 9

.

yx0 yx0

and, using the integral representations of U and V, it is easily verified that equality also holds at the level of the derivative. Hence we have a complete representation of the function g in terms of the confluent hypergeometric functions U and V. I

With this preliminary result in hand, we are prepared for the following theorem.

THEOREM 2.1. Let ␶ be the first time that U hits zero, if the process₀

ŽU, V. starts at x, y at time zero, where xŽ . ) 0. Then: Ž .i As tª ⬁ we have, for any z ) 0,

'

PŽ x , y .



␶ g dt, V ␶ r t g ydz0

Ž

0

.

4

Ž

dt dz

.

3r2 3r2

_

2

4

3⭈ 2 z exp y2 z ; _{3r2 5r4} h x , y .

Ž

.

␲ t 2.19

Ž

.

Ž .ii As zx0 through strictly positive values of z, we have, for any t ) 0, PŽ x , y .



␶ g dt, V ␶ g ydz0

Ž

0

.

4

Ž

dt dz

.

3r2

'

⬁ 4 3 z t _{3r2 y1r2} ;

H

H

w s p 0, w ; 0, 0s

Ž

.

'

2␲ ss0 ws0 2.20

Ž

.

=qtys

Ž

x , y ; 0,yw ds dw.

.

Žiii. 3⌫ 1r4 h x, y

Ž

.

Ž

.



4

2.21 P ␶ ) t ; as tª ⬁.

Ž

.

Ž x , y . 0 ₂3r4 3r2_{␲ t}1r4 Ž . Ž .

PROOF. i and ii The elementary but somewhat technical proofs of these

Žiii Integrating w.r.t. dz in 2.19 , we get. Ž . 3⌫ 1r4 h x, y

Ž

.

Ž

.



4

2.22 P ␶ g dt dt ; .

Ž

.

Ž x , y . 0 3r4 3r2 5r4 2 ␲ 4 t Ž .

From this we get 2.21 by integrating w.r.t. t. Note that the positivity of

 4

P_{Ž x , y .}␶ ) t for all x ) 0 and y g ⺢ implies₀

2.23 g y ) 0 for all y g ⺢. I

Ž

.

Ž

.

Ž .

We introduce the following notation for the result in part ii of Theorem 2.1. Let h be defined by h t , x , y

Ž

.

'

⬁ 4 3 t _{3r2 y1r2} s

_H

_H

w s p 0, w ; 0, 0s

Ž

.

'

2␲ ss0 ws0 2.24

Ž

.

=qtys

Ž

x , y ; 0,yw ds dw.

.

Ž .

Since q_tys x, y; u, v satisfies the backward Kolmogorov equation for the

Ž . Ž .

process U, V , for all x) 0, and lim_{sx 0}q x, y; 0, w_s s 0, if x ) 0, it follows that the function

w

t¬ h 1 y t, x, y ,

Ž

.

tg 0, 1 ,

.

w .

is ‘‘space᎐time harmonic’’ on 0, 1 in the sense that ⭸

q DDŽ x , y . h 1

Ž

y t, x, y s 0,

.

ž

⭸ t

/

Ž . Ž .

where DDŽ x , y . is defined by 2.6 . Since h 1y t, x, y has the interpretation

y3r2

2.25 lim z P



␶ g du, V ␶ g ydz

4

du dz ,

Ž

.

_zx0 Ž t , x , y . 0

Ž

0

.

Ž

.

us1

 Ž . 4

where P_{Ž t , x , y .}␶ g du, V ␶ g ydz denotes the probability that ␶ g du_{0 0 0}

Ž . Ž .

and V ␶ g ydz, if the value of the process is x, y at time t, the transition0

Ž . w x Ž .

density of the ‘‘bridge’’ of U, V on 0, 1 , starting at 0, 0 , where U is

Ž Ž . Ž .. Ž .

conditioned to be positive and where U 1 , V 1 s 0, 0 , is given by

y1

2.26 h 1y s, x, y p x , y ; u, v h 1y t, u, v ,

Ž

.

Ž

.

tys

Ž

.

Ž

.

Ž .

if 0- s - t - 1. Here p is the transition density of the process U, V , killed when U hits zero, and can be written for x, u) 0 as

pt

Ž

x , y ; u, v

.

s p x, y; u, vt

Ž

.

⬁ t y

_H

_H

ptys

Ž

0,yw; u, v P

.

Ž x , y .



␶ g ds, V ␶ g ydw ;0

Ž

0

.

4

ss0 ws0 w x Ž . Ž .

see 9 relation 3 , page 1054. In particular, since p x, y; u, v_t s

Ž . Ž . Ž . Ž w x Ž .

.

page 1054 , letting ux0, we get, for x, z ) 0, p x , y ; 0,

Ž

yz s p 0, z; x, yy

.

Ž

.

s p x, y; 0, yzt

Ž

.

⬁ t y

_H

_H

ptys

Ž

x , y ; 0, w

.

ss0 ws0 2.27

Ž

.

= PŽ0 , z .



␶ g ds, V ␶ g ydw0

Ž

0

.

4

s zy1_P

_

_{␶ g ds, V ␶ g ydz dt dz.}

_Ž

_.

₄

Ž x , y . 0 0 Ž . Ž .

The final step follows from 2.1 . Now 2.26 can be checked as follows. Due to the Markov property, the transition density of the bridge equals

P U t , V t



Ž

Ž .

Ž .

.

gdu dv U s , V s s x, y ,

Ž

Ž .

Ž .

.

Ž

.

U 1 , V 1

Ž .

Ž .

s 0, 0 ,

Ž

.

␶ ) 0 rdu dv

4

Ž

.

0 p_1yt

Ž

u, v ; 0, z

.

s p x, y; u, v limt

Ž

.

p x , y ; 0, z zx0 1ys

Ž

.

PŽ t , u , v .



␶ g dw, V ␶ g ydz0

Ž

0

.

4

s p x, y; u, v limt

Ž

.

, P



␶ g dw, V ␶ g ydz

4

zx0 Ž s , x , y . 0

Ž

0

.

_ws1 Ž . Ž .

which proves 2.26 according to 2.25 .

Ž . Ž .

Similarly, by i of Lemma 2.1, the function h defined by 2.4 is harmonic

Ž .

for the differential operator DD_{Ž x , y .}. Since h x, y is proportional to

1r4

_

₄

lim t P_{Ž x , y .} ␶ ) t ,0 tª⬁

Ž .

this function gives the transition density of the process U, V , where U is conditioned to be positive,

y1

h x , y

Ž

.

ptys

Ž

x , y ; u, v h u, v ,

.

Ž

.

for x, u) 0. This process is characterized by a system of stochastic differen-Ž

tial equations in Section 5, where it will be shown that U as first component .

of the conditioned process will drift off to⬁ and will never hit zero after time zero.

3. The results of Isozaki–Watanabe and Goldman. Set f r , a, A

Ž

.

' PŽ0 , 0.

Ž

U t

Ž .

- r q at for all 0 F t F A

.

and

g r , a,␴ ' P U t - r q at q␴ t2 _{for all 0F t F ⬁ .}

Ž

.

Ž0 , 0.

Ž

Ž .

.

w x

Sinai 16 showed that

f r , a, A

Ž

.

7 Ay1r4 as Aª ⬁ and

where f7 g means that frg lies between two positive and finite constants. w x

Isozaki and Watanabe 4 sharpen these results to 3.1 f r , a, A ; C r, a Ay1r4 as Aª ⬁

Ž

.

Ž

.

Ž

.

and to 3.2 g r , a,␴ ; D r, a ␴1r2 _{as ␴ ª 0,}

Ž

.

Ž

.

Ž

.

Ž . Ž .

where they give explicit formulas for C r, a and D r, a . They do this by

Ž 2 Ž ..

deriving an asymptotic expression for 1y E_{Ž x , y .}expya␴ ␶ y b␴ V ␶0 0 for

Ž . 2 Ž .

all aG 0, b G 0, and x, y g ⺢ with x F 0. The result 3.1 , where r ) 0

Ž .

and ag ⺢, is directly related to 2.21 . Indeed, due to the symmetry of Brownian motion started at 0, it is easily seen that



4

f r , a, A

Ž

.

s PŽ r , a. ␶ ) A .0

Ž . w x Ž .

Thus, 1.5 of 4 says that, changing the notation to agree with 2.21 ,

'

1 3 2

y1r4 1r6 y1r3

PŽ x , y .

Ž

␶ ) t ; t0

.

⌫ 3r4 x ␺ yx

Ž

.

,

Ž

.

'

␲ 2 2 ⌫ 1r6

'

Ž

.

where the function ␺ is defined by

1r6

¡

⬁ 9 y5r6 3 v y q v exp

Ž

yv dv,

.

y) 0,

H

ž

₂

/

0

~

␺ y s

Ž

.

_{2 y< <}3 _{⬁ 9} 1r6 ₂ y5r6 3 < < exp y

_H

v vq y exp

Ž

yv dv,

.

yF 0.

¢

_ž

9

_/

0

ž / ž

2 9

/

It is possible to express ␺ in terms of the hypergeometric functions U and V Žsee e.g., 1 , 13.2.5 , page 505 and 2.18 in Section 2 ,w x Ž . Ž . .

1r6 1 2 1 4 2 3

¡

⌫

_{Ž . Ž .}

_{6 9} yU

_Ž

₆, , y_{3 9}

_.

, y) 0,

~

␺ y s

Ž

.

_1r6 7 2 1 4 2 3

¢

y⌫

_{Ž . Ž .}

_{6 9} yV

_Ž

₆, , y_{3 9}

_.

, yF 0. Hence this corresponds to the results found in Section 2.

w x

Before specializing our results to that in Goldman 2 , we note that there is a factor 1r6 missing in his Proposition 2. Following the indicated steps

Ž . Ž .

between Goldman’s 3.1 and 3.2 , it becomes clear that the factor 3 in front

Ž . y1r3

of 3.2 should not be there: it disappears when the substitution ty s ª w is performed. Moreover, reducing the series of multiple integrals just before Proposition 2 to the expression involving the hypergeometric function F , a_{2 1} factor 1r2 is lost. Therefore, using our notation, Goldman’s Proposition 2 should actually read

PŽ x , 0.

Ž

␶ g dt dt0

.

x1r6 _{3⭈ 6}1r12 _{⌫ 5r4 ⌫ 7r4 ⌫ 5r12}

_Ž

_.

_Ž

_.

_Ž

_.

; _t5r4 _8␲2 _{⌫ 3r2} 2F 51

Ž

r12, 7r4; 3r2; 3r4

.

Ž

.

x1r6 ₃25r12_{⌫ 5r12}

Ž

.

s _t5r4 ₂65r12_{␲ ␲}

_'

2F 51

Ž

r12, 7r4; 3r2; 3r4

.

Ž . Ž .

for x) 0 as t ª ⬁. Substituting y s 0 in 2.22 and using 2.11 , our corre-sponding result reads,

x1r6 34r3⌫ 5r4 ⌫ 1r3

Ž

.

Ž

.

PŽ x , 0.

Ž

␶ g dt dt ;0

.

_t5r4 ₂11r12_{⌫ 1r6 ␲ ␲}

_'

.

Ž

.

Equality of these two asymptotic expressions leads to the following result, which we were unable to locate in the literature on special functions:

5 7 3 3 ⌫ 5r4 ⌫ 1r3 29r2

Ž

.

Ž

.

F , ; ; s .

2 1

ž

_{12 4 2 4}

/

_{⌫ 1r6 ⌫ 5r12 3}3r4

Ž

.

Ž

.

Numerical verification shows that both sides are equal to 2.0353 . . . . We finally show how the quantity

F 5

Ž

r12, 7r4; 3r2, 3r4

.

2 1

of Goldman’s Proposition 2 emerges from our integral representation in Ž2.19 , since this might not be immediately obvious. This follows by writing. the integral as a power series, using the power series for the sinh-function,

3r2

_

2 3 2

4

2 w exp y6 x s y 2 sw sinh 6 xws dw ds

Ž

.

H

_⺢2 q 2 nq1 ⬁

_Ž

_{6 x}

_.

_⬁ 2 nq5r2 2 s

_Ý

H

w exp

Ž

y2w

.

dw 2 nq 1 !

Ž

.

ws0 ns0 ⬁ _{3 nq1r4 2 3} =

_H

s exp

Ž

y6 x s ds

.

ss0 1r6

'

⬁ x ␲ ⌫ n q 7r4 ⌫ n q 5r12

Ž

.

Ž

.

n s₆5r12_{⭈ 2}11r4

_Ý

_{⌫ n q 3r2 n!}

Ž

3r4

.

Ž

.

ns0 1r6

_'

x ␲ ⌫ 5r12 ⌫ 7r4

Ž

.

Ž

.

s₆5r12_{⭈ 2}11r4 _{⌫ 3r2}

_Ž

_.

2F 51

Ž

r12, 7r4; 3r2, 3r4 .

.

4. The marginal distribution of the conditioned process. In Section

'

 4  Ž .

2 we analyzed the behavior of P_{Ž x , y .}␶ ) t and P0 Ž x , y .␶ g dt, V ␶ r t g0 0

4

ydz for large t when x ) 0. Now we extend those results to x s 0 and

˜ ˜

Ž . Ž .

obtain the marginal density of the process U, V started from 0, 0 ; recall

˜ ˜

Ž . Ž . Ž .

that U, V is the process U, V conditioned on U t G 0 for all t G 0. Our main result in this section is the following theorem.

˜ ˜

Ž . Ž .

THEOREM 4.1. The marginal density of U, V started at 0, 0 is given by

˜

˜

f u, vt

Ž

.

s PŽ0 , 0.

Ž

U t

Ž .

g du, V t g dv du dv

Ž .

.

4.1

Ž

.

29r4 1r6 y1r3 s 2 u g vu

Ž

.

h t , u,

Ž

yv ,

.

Ž .

3r2 1r2

REMARK. Writing us ut and vs vt and using the change of

vari-1r2

ables sª st and w ª wt in the definition of h, we get

29r4 y2 1r6 y1r3

4.2 2 t u g vu h 1, u,yv ,

Ž

.

Ž

.

Ž

.

˜

y3r2

_˜

y1r2

Ž Ž . Ž . .

showing that the joint density of U t t , V t t does not depend on t, which also follows from consideration of Brownian scaling.

PROOF OFTHEOREM 4.1. First note that



4

p 0, z ; u, v Pt

Ž

.

Ž u , v . ␶ ) s y t0

4.3 f u, v s lim f u, v s lim lim .

Ž

.

t

Ž

.

t , z

Ž

.

_q

P



␶ ) s

4

sª⬁

zx0 zx0 Ž0 , z . 0

Ž . Ž . Ž .

Here ft, z is the density at time t of the process U, V started at 0, z ,

Ž . Ž .

where U is conditioned to be positive on 0,⬁ . By Theorem 2.1 iii , it follows that 12⌫ 5r4 u

_Ž

_.

1r6_{g vu}

_Ž

y1r3

_.

1r4

_

₄

lim s P_{Ž u , v .} ␶ ) s y t s_{0 3r4} .

'

sª⬁ 2 ␲ ␲ Ž .

Moreover, using 2.3 , we get for all z) 0,

3⌫ 5r4

Ž

.

1r4 q

_'

lim s PŽ0 , z .



␶ ) s s0

4

_␲3r2 11r2₂ z . sª⬁ Therefore, 29r4 1r6 y1r3 y1r2 4.4 f u, v s 2 u g vu z p 0, z ; u, v

Ž

.

t , z

Ž

.

Ž

.

t

Ž

.

Ž . and we get, using Theorem 2.2 ii ,

p 0, z ; u, vt

Ž

.

PŽ u ,yv.



␶ g dt, V ␶ g ydz0

Ž

0

.

4

Ž

dt dz

.

lim _1r2 s lim _3r2 z z zx0 zx0 4.5

Ž

.

s h t, u, yv ,

Ž

.

Ž . Ž . Ž . Ž .

where h is as defined in 2.24 . Combining 4.3 , 4.4 and 4.5 , we get the

Ž .

expression given in 4.1 . I

5. Stochastic differential equations and sample path properties.

We now study the system of SDEs,

5.1 dU t s V t dt, dV t s c U t , V t dtq dW t ,

Ž

.

Ž .

Ž .

Ž .

Ž

Ž .

Ž .

.

Ž .

where the function c is defined by ⭸ y1 5.2 c x , y s h x, y h x , y , x) 0, y g ⺢,

Ž

.

Ž

.

Ž

.

Ž

.

⭸ y Ž .

and h is defined by 2.4 . Several difficulties arise in analyzing this system. 1. The system clearly does not define a two-dimensional diffusion, since the

Ž . Ž .

2. The function x, y ¬ c x, y is not uniformly Lipschitz, nor is this func-tion bounded.

Ž . Ž .

3. The growth of the function x, y ¬ c x, y is faster than linear, as y ª y⬁.

Ž .

Note that, for all x) 0, the function y ¬ h x, y , y g ⺢, is one-to-one, since ⭸ 5.3 h x , y ) 0, yg ⺢, x ) 0.

Ž

.

Ž

.

⭸ y Ž . Ž . Ž .

Also note that the function x, y ª c x, y is positive since h x, y and Ž⭸r⭸ y h x, y are both positive for all x ) 0, y g ⺢, as is easily seen from. Ž . Ž2.23 , 2.9 and 2.10 , using the explicit representation of g in terms of the. Ž . Ž . confluent hypergeometric functions. We also have

5.4 lim h x , y s 0 for all y - 0 and lim h x , y s 0.

Ž

.

Ž

.

Ž

.

xx0 xx0, y ­0

Ž . Ž . Ž Ž . Ž ..

Since U t can only hit zero for values V t F 0, we can define h U t , V t s Ž .

0, if U t s 0.

In spite of the difficulties mentioned above, we have the following existence

Ž .

and unicity result for the system 5.1 , showing that the system actually characterizes our conditioned process.

Ž .

THEOREM 5.1. The system of SDEs 5.1 has a unique strong solution

˜ ˜

˜

˜

ŽU, V , for any starting point U 0 , V 0. Ž Ž . Ž ..s x, y , with x ) 0. Furthermore,Ž .

˜ ˜

Ž . Ž .

let the function h be defined by 2.4 and suppose that the process U, V

Ž . Ž .

solves 5.1 for a starting value x, y , at time zero, with x) 0. Then:

˜ ˜

Ž .i The transition density p of the process U, V is given by

˜

t Ž .

y1

5.5 p x , y ; u, v s h x, y p x , y ; u, v h u, v ,

Ž

.

˜

t

Ž

.

Ž

.

t

Ž

.

Ž

.

˜ ˜

Ž . Ž .

that is, U, V is distributed as the process U, V , for U away from zero. Ž .ii The process

˜

˜

t¬ 1rh U t , V t ,

Ž

Ž .

Ž .

.

tG 0

˜ ˜

Ž .

is a local martingale w.r.t. the natural filtration, induced by U, V .

˜

Žiii With probability 1, U never hits 0,.

˜

PŽ x , y .



U t

Ž .

) 0 for all t ) 0 s 1.

4

˜

Živ The process U is transient, that is,.

˜

PŽ x , y .

½

lim U t

Ž .

s ⬁ s 1.

5

tª⬁

Ž .

PROOF. We prove the existence of a unique strong solution to 5.1 by a

localization argument. Let, for N) 0, the function c be defined byN

Then cN is globally Lipschitz. Hence it follows from Theorem 3.1, page 164, w x

Chapter IV of 3 that the system

dU t

Ž .

s V t dt,

Ž .

dV t

Ž .

s c U t , V tN

Ž

Ž .

Ž .

.

dtq dW t

Ž .

Ž . Ž .

has a unique strong solution U , VN N for each N) 0. Moreover, U , V is aN N

 Ž .

solution of the original system up to time TNs inf t ) 0: U t - 1rN orN

˜ ˜

Ž . 4 Ž .

VN t - yN . Pasting these solutions yields a solution U, V to the system Ž5.1 up to time T. s sup T . Below we show that T s ⬁.N N

˜

˜

Ž Ž . Ž ..

Ito’s formula shows that the process t

ˆ

¬ 1rh U t , V t is a nonnegative

˜

˜

Ž Ž . Ž .. local martingale and hence a supermartingale. Hence t¬ 1rh U t , V t

w x

satisfies Doob’s supermartingale theorem; see 15 , Theorem 49.1 and Corol-w x

lary 49.2, page 147. By the Fatou lemma, 14.3, 14 , page 22, we then get that

˜

˜

5.6 lim 1rh U t , V t

Ž

.

Ž

Ž .

Ž .

.

tªT

Ž . Ž .

exists almost surely and is finite, for any starting point x, y g 0, ⬁ = ⺢ of

˜ ˜

˜

Ž . Ž . Ž

the process U, V . By 5.4 this implies that U does not hit zero up to and .

including time T.

˜ ˜

Ž . Ž .

Since, by the second equation of the system 5.1 , any solution U, V of the

˜

Ž . Ž . Ž .

system 5.1 has to satisfy V t G W t , for all t G 0 for which the solution is

˜Ž .

Ž .

defined, we cannot have V t s y⬁ ‘‘explosion to y⬁’’ at a finite time t.

˜

Ž . Ž .

Also, by 5.6 , U cannot hit zero see above . So the only way in which

˜Ž .

explosion could occur is when V t s ⬁ at a finite time t. However, this possibility is actually excluded by the growth condition on the function

˜

Žx, y.¬ c x, y , as y ª ⬁, using that minŽ . _{tgw0, M x}U tŽ .) 0 for each time

w x

interval 0, M . Thus Ts ⬁, implying that we have a unique strong solution

Ž . Ž . Ž .

to the system 5.1 . We now also have proved ii and iv .

˜

˜

Ž Ž . Ž .. Now note that the infinitesimal generator of the process U t , V t is given for any test function ␸ by

y1 D

DŽ x , y .␸ x, y s h x, y

Ž

.

Ž

.

DDŽ x , y . h x , y

Ž

.

␸ x, y .

Ž

.

Ž .

This corresponds to the transition density 5.5 . Since 1rh is harmonic for

Ž . Ž .

D

DŽ x , y . we now have i . In fact, ii follows from the harmonicity of 1rh for DDŽ x , y ., as was seen above by applying Ito’s formula.

ˆ

Ž .

Part iii would follow from

˜

˜

5.7 lim E 1rh U t , V t s 0

Ž

.



Ž

Ž .

Ž .

.

4

tª⬁

˜

˜

Ž Ž . Ž ..

and the fact that 1rh U t , V t has, almost surely, a finite limit, as t ª ⬁, since almost sure convergence to a finite limit implies convergence in

proba-˜

˜

Ž . Ž Ž . Ž ..

bility to the same limit, and since 5.7 implies that 1rh U t , V t converges

Ž .

to zero in probability. However, by 5.5 for the transition density of the

˜ ˜

Ž .

process U, V , it follows that

⬁ ⬁ y1

˜

˜

EŽ x , y .1rh U t , V t s h x, y

Ž

Ž .

Ž .

.

Ž

.

H

H

pt

Ž

x , y ; u, v du dv

.

us0 vsy⬁ y1



4

s h x, y

Ž

.

PŽ x , y . ␶ ) t ª 0 as t ª ⬁,0

Ž .

for any starting point x, y such that x) 0. So we get

˜

˜

5.8 lim E 1rh U t , V t s 0,

Ž

.

Ž x , y .

_

Ž

Ž .

Ž .

.

₄

tª⬁ and hence

˜

˜

5.9 lim 1rh U t , V t s 0,

Ž

.

Ž

Ž .

Ž .

.

tª⬁ Ž .

with probability 1. Now 5.9 implies

˜

˜

5.10 lim h U t , V t s ⬁

Ž

.

Ž

Ž .

Ž .

.

tª⬁

˜

Ž .

˜

Ž .

˜

with probability 1. If V t tends to⬁, then also U t tends to ⬁, since U is the

˜

˜Ž .

Ž .

integral of V. On the other hand, if V t does not tend to infinity, 5.10 can

˜Ž .

Ž . Ž

only happen if U t tends to infinity, using 5.3 the monotonicity of h in the

˜

. Ž .

second argument . So we obtain in all cases that U t tends to infinity with probability 1. I

6. A critical curve. Our investigation was originally motivated by the

˜

Ž

question whether the expected amount of time that the process U i.e., .

integrated Brownian motion, conditioned to be positive spends below any line of positive slope is finite. The following result answers this question negatively.

THEOREM 6.1. Suppose that k) 0 and 0 -␣ - 3r2 and let the constant

c⬘ ) 0 be given by 35r4 _⬁ 9⭈ 2 6.1 c⬘ s g v g yv dv.

Ž

.

H

Ž .

Ž

.

'

5␲ ␲ vsy⬁ Then

˜

␣ 5r3 5␣ r3y5r2 6.2 P U t - kt ; c⬘k t as tª ⬁.

Ž

.

Ž0 , 0.

Ž

Ž .

.

˜

_{Ž . Ž .} ␣

Hence, if T denotes the amount of time U t spends below the curve u t_␣ s kt , - ⬁, if␣ - 9r10,

6.3 E T

Ž

.

Ž0 , 0. ␣

½

s ⬁, if␣ G 9r10.

Ž . Ž . Ž . Ž .

PROOF. Let ␣ g 0, 3r2 . Using 4.1 , 4.2 and 2.24 and denoting the

constant 229r4 _{by c, we get for any k) 0,}

˜

␣ P U t



Ž .

- kt

4

⬁ ␣y3r2 kt 1r6 y1r3 s c

H

H

u g vu

Ž

.

h 1, u,

Ž

yv dv du

.

us0 vsy⬁

'

⬁ ⬁

4 c 3 kt␣y3r2 1 1r6 3r2 y1r3 y1r2

s

H

H

H

H

u w g vu

Ž

. Ž

1y s

.

'

2␲ us0 vsy⬁ ss0 ws0

By the change of variables vª vu1r3, sª su2r3 and wª wu1r3, we get

˜

␣

P U t



Ž .

- kt

4

'

⬁ ⬁

4 c 3 _kt␣y3r2 _uy2r3 _y1r2

2r3 3r2 2r3 s

H

H

H

H

u w g v

Ž . Ž

1y su

.

'

2␲ us0 vsy⬁ ss0 ws0 = p1ysu2r3

Ž

0, wu1r3; 0, 0 q 1,

.

s

Ž

yv; 0, yw ds dw dv du

.

'

'

⬁ ⬁ ⬁ 4 c 3 3 _kt␣y3r2 2r3 3r2 ;

_H

_H

_H

_H

u w g v

Ž .

'

2␲ ␲ us0 vsy⬁ ss0 ws0 6.4

Ž

.

=q 1, yv; 0, yw ds dw dv dus

Ž

.

⬁ 36 c 5r3 Ž␣y3r2.Ž5r3. s k t

_H

g v

Ž .

'

5␲ 2␲ vsy⬁ ⬁ ⬁ _3r2 =

_H

_H

w q 1,s

Ž

yv; 0, yw ds dw dv,

.

ss0 ws0 Ž . as tª ⬁, yielding 6.2 .

˜

_{Ž .} ␣

The amount of time T that U spends below u t_␣ s kt can be written as T_␣s H₀⬁1_{U Ž t .- k t 4˜} ␣ dt. It now follows that the expected amount of time spent

below the curve ys kt␣ is

⬁ _␣

˜

EŽ0 , 0.T␣s

_H

PŽ0 , 0.

_Ž

U t

Ž .

- kt

_.

dt.

0

Ž . Ž . Ž .

By 6.2 this is finite when 5r3 ␣ y 5r2 - y1, and infinite when 5r3 ␣ y 5r2 G y1. Hence we get the conclusion that the expected amount of time spent below the curve ys kt␣ is finite when ␣ - 9r10, and infinite when ␣ G 9r10. I

APPENDIX

( ) Ž .

Proof of Theorem 2.1 i . For the first term of 2.1 we get, if z) 0,

'

tzq x , y ; 0,t

Ž

yz t

.

2 2

'

2 z 3 6 x 6 xy 2 y 6 xz 2 yz 2 s exp

½

y ₃ y ₂ y y 2 z sinh

5

_ž

_3r2 q _1r2

_/

␲ t t t t t t 2 2

_'

2 z exp

Ž

y2 z

.

3 6 x ; _3r2

_ž

q 2 y ,

_/

tª ⬁. t ␲ t

For the second term we get

⬁ t ytz

_H

_H

qtys

Ž

x , y ; 0, w P

.

Ž0 , z t'.



␶ g ds, V ␶ g dw0

Ž

0

.

4

ss0 ws0

'

⬁ 6 zt 3 t _{y2 y2} A.1 s ws ty s

Ž

.

₂

_H

_H

Ž

.

'

␲ 2␲ ss0 ws0

6 x2 _{6 xy 2 y}2_{q w}2

Ž

.

= exp y

½

3 y 2 y

5

ty s ty s ty s

Ž

.

Ž

.

2

_'

2 6 xw 2 yw 2 z t

_Ž

y zw t q w

_.

= sinh

ž

₂ q

/

exp

½

y

5

ty s s ty s

Ž

.

3

'

4 zw trs y1r2 =

_H

␰ exp

½

y ␰ d␰ ds dw.

5

2 0

'

By the change of variables wª w t and s ª st, we get

'

⬁ 6 z 3 1 _{y2 y2} ws

Ž

1y s

.

H

H

2

_'

t␲ 2␲ ss0 ws0 6 x2 _{6 xy 2 y}2 _{2 w}2 =exp y

½

_t3 _{1y s} 3 y _t2 _{1y s} 2 y _{t 1}

_Ž

y s

_.

y ₁y s

5

Ž

.

Ž

.

6 xw 2 yw 2 z2_{y zw q w}2

Ž

.

=sinh ₂ q _1r2 exp

½

y

5

3r2

ž

Ž

1y s t

.

Ž

1y s t

.

/

s 3 4 zwrs _y1r2 =

_H

␰ exp

½

y ␰ d␰ ds dw.

5

2 0

As will become clear in the sequel, the dominating behavior of this multiple integral, as tª ⬁, will come from a region of integration for s in a

neighbor-w x

hood of 1. We therefore first consider the region of integration sg 1r2, 1 . We define u 3 y1r2 A.2 ␺ u s ␰ exp y ␰ d␰ ,

Ž

.

Ž

.

_H



2

4

0

'

Ž .

Using the change of variables sª 1 y 1r st and w ª wr t , and using the

Ž .

notation A.2 , we can write the integral over this region as

'

⬁ ⬁ 6 z 3 _y2 w 1

Ž

y 1r st

Ž

.

.

H

H

2

_'

t␲ 2␲ ss2rt ws0

= exp y6 x2_s3_{y 6 xys}2_{y 2 y}2_{sy 2w}2_{s sinh 6 xws}2_{q 2 yws}



4

Ž

.

=␺ 4 zwty1r2_{r 1 y 1r st}

_Ž

_Ž

_.

_.

Ž

.

2 z

_Ž

2_{y zwt}y1r2_{q w}2_ty1

_.

= exp y

½

5

ds dw 1y 1r st

Ž

.

A.3

Ž

.

3r2 2

_'

⬁ ⬁ 24 z exp

Ž

; 2 z

.

3 _3r2 ; _{5r4 2}

_H

_H

w

'

t ␲ 2␲ ss0 ws0 = exp y6 x2_s3_{y 6 xys}2_{y 2 y}2_{sy 2w}2_s



4

= sinh 6 xws2_{q 2 yws ds dw, tª ⬁.}

Ž

.

The asymptotic equivalence of the last step can be proved in the following

w .

way. Restricting the region of integration for s to ␧, ⬁ for a fixed ␧ ) 0, we get, using Lebesgue’s dominated convergence theorem,

'

⬁ ⬁ 6 z 3 _y2 w 1

Ž

y 1r st

Ž

.

.

H

H

2

_'

t␲ 2␲ ss␧ ws0

=exp y6 x2_s3_{y 6 xys}2_{y 2 y}2_{sy 2w}2_{s sinh 6 xws}2_{q 2 yws}



4

Ž

.

2 z2_{y zwt}y1r2_{q w}2_ty1

Ž

.

y1r2 =␺ 4 zwt

Ž

r 1 y 1r st

Ž

Ž

.

.

.

exp

½

y

5

ds dw 1y 1r st

Ž

.

'

⬁ ⬁ 24 3 _3r2 2 3 2 2 2 ; _{5r4 2}

_H

_H

Ž

zw

.

exp



y6 x s y 6 xys y 2 y s y 2w s

4

'

t ␲ 2␲ ss␧ ws0 2



2

4

=sinh 6 xws q 2 yws exp y2 z ds dw,

_Ž

_.

tª ⬁.

w x

For the region sg 2rt,␧ , we get

'

␧ ⬁ 6 z 3 _y2 w 1

Ž

y 1r st

Ž

.

.

H

H

2

_'

t␲ 2␲ ss2rt ws0 2 3 2 2 2 2

=exp y6 x s y 6 xys y 2 y s y 2w s sinh 6 xws q 2 yws



4

Ž

.

2 z2_{y zwt}y1r2_{q w}2_ty1

Ž

.

y1r2 =␺ 4 zwt

Ž

r 1 y 1r st

Ž

Ž

.

.

.

exp

½

y

5

ds dw 1y 1r st

Ž

.

2 3r2 y2 z

_'

_{␧ ⬁} 24 z e 3 3r2 2 3 2 2 2

F _{5r4 2}

_H

_H

w exp

_

y6 x s y6 xys y2 y sy2w s

₄

'

t ␲ 2␲ ss0 ws0 2 = sinh 6 xws q 2 yws ds dw

Ž

.

3r2 2

_'

␧ ⬁ 24 z exp

Ž

yz

.

3 _3r2 F _{5r4 2}

_H

_H

w

'

t ␲ 2␲ ss0 ws0 =exp y6 x2_s3_{y 6 xys}2_{y 2 y}2_{sy 2w}2_s



4

< < 2 =ws 6 xsq2 y cosh 6 xws q2 yws ds dw

Ž

.

due to the inequality

< <

A.4 sinh u F u cosh u .

Ž

.

Ž

.

Ž

.

Now note that, by the change of variables wª wsy1r2, the last displayed integral is less than

␧ ⬁ _{5r2 y3r4 2 3 2 2 2} w s exp

_

y6 x s y 6 xys y 2 y s y 2w

₄

H

H

ss0 ws0 < < 3r2 1r2 = 6 xs q 2 y cosh 6 xws

_Ž

q 2 yws

_.

ds dw s OO

Ž

␧ ,

.

␧ x0. Ž .

w x

We next consider the region of integration sg 0, 1r2 . The corresponding

'

integral can be written by making the change of variables wª w s and next sª z2_{rs, as}

'

⬁ ⬁ 6 z 3 _y2 y1 2 ws

_Ž

1y z rs

_.

H

H

2

'

2 t␲ 2␲ ss2 z ws0 6 x2 6 xy 2 y2 2 w2 = exp y

½

_{3 2} 3 y _{2 2} 2 y _{t 1y z rs}2 y _{1y z rs}2

5

Ž

.

t

_Ž

1y z rs

_.

t

_Ž

1y z rs

_.

6 xw 2 yw 1r2 2 = sinh

ž

_Ž

_{1y z rs t}2

_.

2 3r2 q

_Ž

1y z rs t2

_.

1r2

/

exp



y2 s y ws

Ž

q w

.

4

3 4w s' _y1r2 =

_H

␰ exp

½

y ␰ d␰ ds dw.

5

2 0 Ž .

Using inequalities A.4 and

4w s' 3 y1 y1r2 y3r4 1r2 A.5 s ␰ exp y ␰ d␰ F 4s w ,

Ž

.

_H



2

4

0 Ž y3r2.

it is easily seen that this term is OO t , as tª ⬁. Concluding, we get, as tª ⬁,

'

PŽ u , v .

½

␶ g dt, V ␶0

Ž

0

.

t g ydz

5

Ž

dt dz

.

3r2 2

_'



4

24 z exp y2 z 3 ; _t5r4 2_{␲ 2␲}

_'

A.6

Ž

.

⬁ ⬁ _{3r2 2 3 2 2 2} =

_H

_H

w exp



y6 x s y 6 xys y 2 y s y 2w s

4

ss0 ws0 = sinh 6 xws2_{q 2 yws ds dw,}

Ž

.

Ž .

which yields the result 2.19 . I Ž .

PROOF OF THEOREM 2.1 ii . The proof of this property proceeds along

Ž .

similar lines. For the first term of 2.1 we get zqt

Ž

x , y ; 0,yz

.

2 2 2

'

2 z 3 6 x 6 xy 2 y 2 z 6 xz 2 yz s ₂ exp

½

y ₃ y ₂ y y

5

sinh

_ž

₂ q

_/

t t t ␲ t t t t 2

_'

2 2 2 z 3 6 x 6 x 6 xy 2 y ; _{␲ t}3

ž

_t q 2 y exp y

/

½

_t3 y _t2 y _t

5

s OO z2 _{, zx0.}

Ž

.

For the second term we get ⬁ t _{q q} y z

_H

_H

qtys

Ž

x , y ; 0, w P

.

Ž0 ,yz.

_

␶ g ds, V ␶0

Ž

0

.

g dw

₄

ss0 ws0

'

⬁ 6 z 3 t _{y2 y2} s ₂

_H

_H

ws

Ž

ty s

.

'

␲ 2␲ ss0 ws0 6 x2 _{6 xy 2 y}2_{q w}2

Ž

.

=exp y

½

3 y 2 y _{ty s}

5

ty s ty s

Ž

.

Ž

.

A.7

Ž

.

6 xw 2 yw 2 z2_{y zw q w}2

Ž

.

=sinh ₂ q exp

½

y

5

ž

_Ž

_{ty s}

_.

ty s

/

s 3 4 zwrs _y1r2 =

_H

␰ exp

½

y ␰ d␰ ds dw.

5

2 0 w x Ž .

We first consider the region of integration sg ␧, t , for ␧ g 0, t . Using Lebesgue’s dominated convergence theorem, we get that this integral is asymptotically equivalent to 3r2

_'

_⬁ 24 z 3 t _{3r2 y5r2 y2} w s

Ž

ty s

.

H

H

2

_'

␲ 2␲ ss␧ ws0 6 x2 _{6 xy 2 y}2 _{2 w}2 = exp y

½

3 y 2 y _{ty s}y _{ty s}

5

ty s ty s

Ž

.

Ž

.

6 xw 2 yw 2 w2 = sinh

ž

₂ q

/

exp

½

y

5

ds dw, ty s s ty s

Ž

.

w x

as zx0. We next consider the region of integration s g 0,␧ . The

corre-'

sponding integral can be written, by making the change of variables wª w s and next sª z2rs, as

'

⬁ ⬁ 6 z 3 _y2 y1 2 ws

_Ž

ty z rs

_.

H

H

2

_'

₂ ␲ 2␲ ssz r␧ ws0 6 x2 _{6 xy 2 y}2 _{2 w}2_z2_rs =exp y

½

₂ 3 y ₂ 2 y _{ty z rs}2 y _{ty z rs}2

5

ty z rs ty z rs

Ž

.

Ž

.

'

'

6 xwzr s 2 ywzr s 1r2 2 =sinh

ž

₂ 2 q 2

/

exp



y2 s y ws

Ž

q w

.

4

ty z rs ty z rs

Ž

.

3 4w s' _y1r2 =

_H

␰ exp

½

y ␰ d␰ ds dw.

5

2 0 Ž . Ž .

Again, using the inequalities A.4 and A.5 , it is easily seen that this term is Ž 1r4 3r2.

O

REFERENCES

w x1 ABRAMOWITZ, M. and STEGUN, I. A. 1964 . Handbook of Mathematical Functions. NationalŽ . Bureau of Standards 55, Washington, DC.

w x2 GOLDMAN, M. 1971 . On the first passage of the integrated Wiener process. Ann. Math.Ž .

Statist. 42 2150᎐2155.

w x3 IKEDA, N. and WATANABE, S. 1981 . Stochastic Differential Equations and Diffusion Pro-Ž .

cesses. North-Holland, Amsterdam.

w x4 ISOZAKI, Y. and WATANABE, S. 1994 . An asymptotic formula for the Kolmogorov diffusionŽ . and a refinement of Sinai’s estimates for the integral of Brownian motion. Proc.

Japan Acad. Ser. A 70 271᎐276.

w x5 ITO, K. and MCKEAN, H. P., JR. 1974 . Diffusion Processes and Their Sample Paths.Ž . Springer, Berlin.

w x6 JONGBLOED, G. 1995 . Three statistical inverse problems. Ph.D. dissertation, Delft Univ.Ž . w x7 KOLMOGOROV, A. N. 1934 . Zuffallige Bewegungen. Ann. Math. II 35 116Ž . ¨ ᎐117.

w x8 LACHAL, A. 1991 . Sur le premier instant de passage de l’integrale du mouvement brown-Ž . ´

ien. Ann. Inst. H. Poincare 27 385᎐405.´

w x9 LACHAL, A. 1992 . Sur les excursions de l’integrale du mouvement brownien. C. R. Acad.Ž . ´

Sci. Paris Ser. I 314 1053´ ᎐1056.

w x10 LACHAL, A. 1996 . Sur la distribution de certaines fonctionnelles de l’integrale du mouve-Ž . ´

ment Brownien avec derives parabolique et cubique. Comm. Pure Appl. Math. 49´

1299᎐1338.

w x11 MAMMEN, E. 1991 . Nonparametric regression under qualitative smoothness assumptions.Ž .

Ann. Statist. 19 741᎐759.

w x12 MCKEAN, H. P., JR. 1963 . A winding problem for a resonator driven by a white noise. J.Ž .

Math. Kyoto Univ. 2 227᎐235.

w x13 OLVER, F. W. J. 1974 . Asymptotics and Special Functions. Academic Press, New York.Ž . w x14 ROGERS, L. C. G. and WILLIAMS, D. 1987 . Diffusions, Markov Processes and Martingales 2.Ž .

Wiley, New York.

w x15 ROGERS, L. C. G. and WILLIAMS, D. 1994 . Diffusions, Markov Processes and Martingales 1,Ž . 2nd ed. Wiley, New York.

w x16 SINAI, Y. G. 1992 . Statistics of shocks in solutions of inviscid Burgers equation. Comm.Ž .

Math. Phys. 148 601᎐621.

P. GROENEBOOM G. JONGBLOED

DEPARTMENT OFMATHEMATICS DEPARTMENT OFMATHEMATICS

DELFTUNIVERSITY OFTECHNOLOGY VRIJEUNIVERSITEIT

2628 CD DELFT DEBOELELAAN1081A

THENETHERLANDS 1081 HV AMSTERDAM

E-MAIL: p.groeneboom@twi.tudelft.nl THENETHERLANDS

E-MAIL: guert@cs.vu.nl J. A. WELLNER DEPARTMENT OFSTATISTICS UNIVERSITY OFWASHINGTON P.O. BOX354322 SEATTLE, WASHINGTON98195-4322 E-MAIL: jaw@stat.washington.edu