The One-sided Barrier Problem for an IntegratedOrnstein-Uhlenbeck Process

By

C.H. HesseDepartment of StatisticsUniversity of CaliforniaBerkeley, CA 94720

Technical Report No. 250May 1990

(revised May 1991)

Department of StatisticsUniversity of CalifomiaBerkeley, California

The One-sided Barrier Problem for an IntegratedOrnstein-Uhlenbeck Process

By

C.H. HesseDepartment of StatisticsUniversity of CaliforniaBerkeley, CA 94720

THE ONE-SIDED BARRIER PROBLEM FOR AN INTEGRATEDORNSTEIN-UHLENBECK PROCESS

C.H. HesseDepartment of Statistics

University of California, BerkeleyBerkeley, CA 94720

ABSTRACTThe first-passage problem is considered for the process

t

X (t)= V (s) ds where V (s) is an Ornstein-Uhlenbeck process with an

additive constant term gt, friction parameter [3, and variance a2 startingfrom V (0) = v + g. Global explicit analytic approximations to the first-passage density are derived for the case of a one-sided fixed barrier at dis-tance x from X (0). By a series of simulations the quality of these approx-imations is studied for varies x, v, [, g,cY2. We also consider an applica-tion to stochastic models for particle sedimentation in fluids where thefirst-passage problem of an integrated Ornstein-Uhlenbeck process naturallyarises.

Key words: First-passage time, one-sided barrier, integrated Ornstein-Uhlenbeck process, local martingale, asymptotic expansion, perturbationmethods, particle sedimentation.

AMS(MOS) subject classification: Primary 60J65, 60J70 Secondary82A3 1.

1. INTRODUCrION AND OVERVIEW

Let B (t), t 2 0 be a Brownian motion process with variance parameter

1 starting at B (0) = V. Changing the time clock deterministically and res-

caling the state space, the process V (t) defined by

V (t) = ePt B ((a/272) (e213t - 1)) + , t > 0

is an Ornstein-Uhlenbeck process with variance parameter &2, friction

parameter ,B, and additive constant ,u starting from V (0) = v + p and for

this process

E V(t)) - ve' + p

var(V (t)) = (a2/2p)(1- e-22) (1.1)

COV(V(s),V(t)) = (a212f)(e2s- 1)e-P(t+s) for s 5 t

In this paper we consider the one-sided barrier problem for the integral of

the process V (t) to a fixed barrier located at distance x from the starting

position. An application where this first-passage problem of an integrated

Ornstein-Uhlenbeck problem arises naturally is given in the next section.

Specifically, the problem is this. Let

t

X(t) = JV(s)ds -x for x .00

and define

X= min{t:t.: 0, X (t) = 0)

where henceforth the minimum over the empty set is taken to be oo.

Clearly, t depends on x, v, p., 13 and aC2, so t = t(x ,v)p.,102) but we will

sometimes suppress this dependence in our notation. Also, let

g (t Px,v,j,a2) denote the density of X at t.

In Section 3 we derive the following approximations to the first-passage

density g:

gi(t)= [3x- (v + )t _ (3(x -ot)-vt)]4(x) (1.2a)

where 4 (x) is the density of the normal distribution with mean

= (v/f3)(l - e + gt and variance = (a2/2213)(2jt + 4et -e - 3),

and

2 (t ) = g 1 (t) ((I (X(:)) + ( (t))f14 (X())) (1.2b)

where 4 and (D are the standard normal density and distribution functions,

respectively, and

(t)= [3x - (v + )t - (t /4)(3(x -p)-vt)I(a t3'2 -a t52/8).

In particular, g 2 (t) is asymptotic to g (t) as (3x - (v + .) t) / 2t -+ o and

[3 -+ 0 in the sense that the ratio g2(0)/g (t) converges to 1 as

(3x - (v + )t)/2t - oo and 5 - 0. As will be seen, g2(t) performs

well as a global approximation to g over a wide range of values of the

parameters.

In Section 4 we present the results of a comparative simulation study to

demonstrate the quality of the derived approximations.

First-passage problems have been studied for many years. Only very

few exact analytic solutions for densities exist. Often the obtained results

provide Information about moments or the problem is solved to within the

Laplace transform of the first-passage density, see e.g. Abrahams [1] for a

review.

The first-passage problem for a special integrated process has been stu-

died by Goldman [4]. There an integrated Brownian motion process is

considered under the restriction of negative starting velocities, i.e. 3 = 0,

p = 0 and v < 0. Among other items, Goldman obtains the following

exact expression, in our notation,

g (t ,x,v ,O,0, 1) =-[ 3/2ct3]112(3xt-1 - v)exp (-3 (x - vt)2/2t3) (1.3)2

00 t 00

- Jdb fJb PO,b (tl e ds,hl e dh)[p (t - s,O,v,x,b) -p (t - s,O,v,x,-b)o 00

where p is the transition density of (X (t), V (t)) starting from (a , b ), i.e. as

is easily derived

p (t,a,b,t,rn) = Prob(X(t) e dt,V(t) e drn)/d4drj=(31/2/ht2)exp(-6( - a - bt)2 t-3 + 6(ri- b)(t-a -bt)t-2- 2(1 - b)2 t-1)

and P Ob is the probability measure of the "1/2-winding time" t, and the

hitting place hI which was derived by McKean [13]:

PO,b (tl E dt,h1 e dh) (1.4)4bh It

- (3h/r2la2t2)exp(-2(b2- bh + h2)/t) J exp(-30/2)/i 12o12d0dt dh0

Wong [16] shows that the integral in (1.4) may be expressed in tenns of

theta functions and gives results concerning the different problem of deter-

mining the distribution of intervals between consecutive zero-crossings of

certain zero-mean stationary Gaussian processes.

All of these approaches lead to rather complicated integral expressions

which one does not seem to be able to evaluate analytically. These are

therefore of limited applicability. In this paper we aim by different

methods, to derive a useful accurate analytic approximation to the more

general quantity g (t ,x , v, ,,), i.e. for an Ornstein-Uhlenbeck process

and allowing also for both positive and negative starting velocities.

In the following section we describe an application where the passage

problem of an integrated Ornstein-Uhlenbeck process arises.

2. AN APPLICATION

In recent years the chemical engineering community has become

interested in stochastic models for particle sedimentation in viscous fluids.

By sedimentation, we mean the phenomena arising in a two-phase solids-

fluid system (i.e. a large number of rigid, identical, and usually small parti-

cles without charge is immersed in a quiescent viscous Newtonian fluid)

that evolves from some initial state under the influence of forces such as

gravity.

Theoretically, this is a problem in continuum mechanics. It is possible

to set up the equations of motion for all particles and fluid flows. How-

ever, since the number of particles involved is large this many-body prob-

lem becomes very high-dimensional and hence intractable, see e.g. Hess

and Klein [6].

Therefore, Pickard and Tory [14] introduced the basics of a stochastic

model to describe the phenomena arising during sedimentation. The sto-

chastic approach and the resulting model have since been refined and

revised, see Pickard and Tory [15] and Hesse [7,9]. The present author

became involved in this work in 1984 upon joining the Department of

Statistics at Harvard University to work with D. Pickard.

As we shall see, the model fitting procedure as developed in Hesse [9]

for the stochastic model is complicated and the use of existing engineering

data bases will lead to the first-passage time problem for integrated

Ornstein-Uhlenbeck processes stated above.

For the simplest case of vertical settling of N identical spherical, rigid

particles of mass m and radius a being immersed in a liquid which is con-

tained in a sedimentation vessel Q (thought of as a subset of IR2 for sim-

plicity) of infinite length the model heuristics will be briefly discussed now.

For N = 1, it is well known that the relevant equations of motion (for

position and translational velocity for the center of the particle) are

Langevin's equations:

dX(t) = V(t)dt (2.1)

dV (t) = Gdt - V (t) dt + F (t) dt

where X(t), V(t) denote position and velocity respectively; G is a force

(per unit mass) resulting from gravity, and F(t) represents those contribu-

tions of the force (per unit mass) exerted by the fluid on the particle

through molecular bombardment which are not already captured by the

linear (Stokes) friction term -f3V(t). F(t) is usually modelled as stochas-

tic and on intermediate time scales (i.e. larger than the order of magnitude

of molecular collisions and smaller than the systems charactenrstic damping

force) the following specifications are sufficiently precise approximations:

E (F(t)) = 0 (2.2)E (F (t1) F (t2)) = 2D 8 (t2 - t1)

Adding a normal distribution assumption, F (t) will be modelled as the for-

mal derivative of the Wiener process. Solving (2.1), (2.2) leads to an

Ornstein-Uhlenbeck process for V(t) (and hence an integrated Ornstein-

Uhlenbeck process for X (t)). The constant D may be derived from the

equipartition theorem which implies that the stationary velocity distribution

of the particle should be Maxwellian, so that D = m1IkTf (k =

Boltzmann's constant, T = temperature), see Lebowitz and Montroll [12].

If, more generally, one considers an ensemble of N macroscopic parti-

cles, then in addition to these forces, particles interact with each other

through fluid flows. Specifically, superposition of the velocity fields gen-

erated by individual moving particles will induce a force field which in

turn influences incremental particle movement. It is analytically complex

to incorporate these hydrodynamic interactions into the equations of motion

in phase space even if one resorts to low-order approximations, for exam-

ple by incorporating the Burgers-Oseen interaction tensor with pre-

averaging, see Hess and Klein [6].

In view of this complexity, the stochastic approach instead models the

N particle system with hydrodynamic interaction as a system of coupled

Ornstein-Uhlenbeck processes where the coupling is localized, short-range,

configuration-dependent and acts through the parameters of the Ornstein-

Uhlenbeck processes. Specifically, let tj = j At, j = 0,1,2,... and for

k = 1,9...,N

d Xk (t) = Vk (t)dt (2.3)

dVk (t) = j(c (Xk @),X,))d: -t(c (Xk (t ), k))V(t)dt +a(c (Xk(tj) ))dk(t)

describes the dynamics of the system in position-velocity phase space for

t e [ tj, tj1+) starting from [ (X 1 (tj), V1 (tj)), . , (XN (tj), VN (tj)) ]. Here

N

PXk = XA J) (2.4)i.k

with %ti) being the indicator function of the set

A (t,i )bexE IRn2x-Xi (the < ai. In addition, define

C (Xk (t)Ptk) = K (Xk (t)-y)Pk (y)dy (2.5)

for a kernel K IR2 -* JR with K(y)dy = 1 and K(y) = 0 outside of

some neighborhood of the origin. Heuristically, the reason for the

definitions (2.3), (2.4), (2.5) is the hydrodynamic fact that the state of the

entire system in phase space (this includes positions and velocities of parti-

cles, fluid flows and possibly internal pressures or external forces) deter-

mines incremental evolution of position-velocity trajectories. If an indivi-

dual particle has exact knowledge of the systems position in phase space it

"invokes" the laws of physics to compute its trajectory. However, for the

purpose of modelling, the state of the system in this sense is much too high

dimensional to be useful. The set-up (2.3), (2.4), (2.5) therefore focuses on

the main determinant of incremental particle motion, namely local particle

density, see e.g. Happel and Brenner [5]. The model interprets this in the

sense of (2.5), as a smoothed version of the N - 1 particle configuration

Pt . The local particle density profile at time tXk

C(t) = (C (X1 (t),PI1), * . ,C(XN (t),PxN))

then parametrizes the equations of motion through p (.), [B ( ), a ( ).

If particles are equipped with only partial information (as contained in

C (t)) concerning the state of the entire system they sample their incremen-

tal velocity transitions from Omstein-Uhlenbeck processes parametrized by

(9 (C (X1 (01),Pxl ),***, (C (XN (0)'PXIN ))'(2.6a)(5(C (X 1 (t),Pxt))**l (C (XN (t PXNI)) (2.6b)

((T (C (X I (t ), ptI) *I ,( C(N t) X ) (2.6c)

From physical considerations these functions are assumed to be continuous.

Hence, in summary, in the stochastic model particle velocities are

governed by a parametrized equicontinuous family of Ornstein-Uhlenbeck

processes which are coupled through local particle concentration via a

nested parametrization scheme. Since local particle concentration is itself

stochastic and evolves with the system the model structure is similar, in

general, to that of random systems in random media.

Due to this nested, two-stage parametrization scheme parameter estima-

tion and model fitting is difficult and no non-ad-hoc procedure was previ-

ously available. Our aim is to develop a satisfactory method that allows to

make use of the existing extensive industrial data bases containing transit

times of sedimenting particles. In other words, for particles starting at

t = 0 with velocity V (0), the time is observed that it takes the particle to

cross a barrier at distance x for the first time. Since velocity is an

Ornstein-Uhlenbeck process, this is simply the first-passage problem for an

integrated Ornstein-Uhlenbeck process. In view of the structure of the sto-

chastic model one will, for the purpose of estimating parameters, restnct

oneself to regions where particles remain under the influence of a single

Ornstein-Uhlenbeck process (at least approximately), i.e., particle trajec-

tories along which local particle concentration and hence Ornstein-

Uhlenbeck parameters remain essentially constant.

The parameter estimation procedure which we plan to present in a

future article is based on the approximations to the first-passage density

derived in the next section.

3. APPROXIMATIONS TO THE FIRST PASSAGE DENSITY

The derived approximations are based on approximate solutions of par-

abolic partial differential equations for the Laplace transform of the first-

passage density derived via a local martingale approach, and for the first

passage time density itself. Using methods of global analysis, such as the

method of dominant balance, and also perturbation methods, an asymptotic

expansion of the solution is obtained, i.e. an expression is obtained which

satisfies the partial differential equation (for the first-passage density) and

the initial-boundary conditions asymptotically as -* 0 and

(3x - (v + .)t)/2t -* oo.

First-passage problems for continuous Markov processes have first been

considered by Darling and Siegert [3] and have been solved to within

Laplace transform of the first passage density. But problems of the type

considered here are complicated by the fact that the integral of a Markov

process is in general no longer Markovian. However, the bivariate process

(X (t), V (t)), t > 0 is Markovian and this provides a starting point for

further analysis.

Unfortunately, it turns out that the boundary and initial conditions for

the bivariate process (X (t), V (t)) provided by the context do not guarantee

that the partial differential equation has a unique solution. We circumvent

this difficulty by generating, via a truncation technique, a related problem

with sufficient initial boundary information and with the same asymptotic

solution, hence obtaining an asymptotic expansion.

Consider the process {X (s),V (s)), s . 0 starting from (-x,v +j)

where x > 0. Assume that the plane X = 0 is an absorbing boundary. Let

p (xt, vt, t I x, v + p.) be the probability density associated with X (0) = -x,

V (0) = v + g, X (t) =x,, V (t) = vt and {X (s), V (s)} did not reach the

boundary in [ 0, t).

Also, let

YJxpWYt Ix' v +) pJ(xt, vt, rx, v +g)dvtdt

and

t(x,v,lpV ,,9 ) = min(:t .O,X(t)= 0

so that

g (t,x,v,gO,f,&2) -at P (0,t Ix,v + )

and g (t ,x, v, g, f, 2) is the density of t. We denote its Laplace transform

by

00

rs (x,v + I.t,f,a2) - jexp(-st')g (t',x,v,(,f,a2)dt' (3.1)0

but for notational convenience will usually ignore all arguments but the

first two. For t < t (x, v, l,p,, a2) one may consider the random density

g (t' , -X (t), V (t) _ p., p, 3, a2) and the random Laplace transform

]Fs (-X (t), V (t)) which will be of use in Theorem 1. We further introduce

the additional notation

alFs (Z1,Z2) = (aaZ1)IF (Z1,Z2)a2 Fs (Z1Z2) = (a/az2)s (Z1,Z2)a22Is (Z1,z) (aZaZ)175 (Z1,Z2)

Theorem 1: With the Laplace transform FS (*,*) of (3.1), define the

process {Z(t)=exp(-st)1rs(-X(t),V(t)); 0s t . t(x,v4l,PI,a2)) for

fixed positive s. Then the stochastic Ito differential d Z (t) of Z (t) is

given by

d Z(t) = exp(-st)a2Fs (-xt,v,)adW (t) (3.2)a2

+ exp(-st)[ a22rs (-x,,v)-V V(t)ai Fs(-x,,vt)

+ ,8(V(t) - ))a2Fs (-Xt,vt) - srs(-xt,vt)] dt

where W (t) is standard Brownian motion.

Proof: By Wto's formula or the following argument we have:

Az = Z(t+At)-Z(t)= (exp (-s (t + At))- exp (-st)) [ rS (-xt+& , vt+&)- r (-x , vt )J

+ (exp(-s (t + At)) - exp(-st))Jrs (-xt,vt)+ exp(-st)[Fs (-xt+&,vt+At) - rs (-x, vt)].

Retaining terms of order At only, we obtain

AZ -s exp(-st)Jrs (-xt,v,)At+ exp(-st)[-a,rs (-xt,vt)V(t)A + a2rs (-xt,vt)AV(t)

+ -.222lrs (-xt,vt)(AV(t))2I2

Since V (t) = Vo(t) + i, where Vo(t) is an Ornstein-Uhlenbeck process

starting from v, we have that

A Vo (t) = -Vo (t) At + (AW (t)

and hence

AV(t) = -,B(V(t) - ) At + (AW (t)

(AV(t))2 -T2(A W(t))2

where AW (t) is an increment of standard Brownian motion and (AW (t ))2

is therefore an infinitesimal of order At. Terms such as At AW(t) and

higher order terms were again rejected. Hence

AZ = exp(-St)a2rs(-Xt,Vt)aAW(t)

+ exp(-st) [ 2 a22rs (-x,,v,)- V(t)airs (-x,,vt)

+ O(V(t) - s)a2rs (-Xt,vt) - Srs(-xt,vt)]Att.

Replacing initesimal increments by differentials the result follows.

Theorem 2: The process (Z (t),O s t . 'r(x,v4,j3,a2)) as defined in

Theorem 1 is a local martingale relative to the family of a-algebras gen-

erated by {Z (s ), s < t).

Proof: The simple proof uses stopping time arguments of which the

details are omitted.

It is now immediate from the integral representation

£

Z(t) = rs (x,v + ) + fexp(-st) 2r5adW() (3.3)0

t 2

+ Jexp(-s) [ 2-22rs -V(C)a,rs + (V( )-)a2rs - s rs ]dt0

corresponding to (3.2), that the local martingale property of Z(t) requires

the third summand of (3.3) to vanish identically for all t. This in turn

implies

a2 a2Fs (X ,v +[L) ars(x v+i') ars (x ,v +g)2 ~~~(v+g) .v_Srs*"V+g

(3.4)

The absorbing boundary condition is

rs (O,V + p, 2) = 1 for all s > 0 and v +g > 0. (3.5)

The first-passage density g satisfies

2 av2 -(v + ) ax -v av-

at 0 (3.6)

and

g (t,O, v, 3,p, c) = 8 (t) (3.7)

g (O,X,V1,v, ,2) = 8(x)

where 8 is Dirac's delta function. Also

lim g (t ,X ,V gR3 a2) = lim g (t,x ,v ,ijP3,a2) =limg (t,X ,v ,j.t3, &) = 0.x-)c v4 t4

The system (3.4), (3.5) does not determine rP uniquely because an mni-

tial condition is missing and cannot be easily obtained from the problem

context. To get a handle on (3.4) and (3.5) and to generate a closely

related problem with sufficient initial and boundary information, define for

v > v + ,u the process

X (t)ifV(s) < v forall s < tX;(t) = (~X(t0) for all t > to = min{t: V(t) = v

The bivariate process (Xv (t), V(t), t . 0) has the same sample patis as

(X (t), V (t)) except when the velocity process hits v, then the first coordi-

nate is stopped dead at X (to). Also, let x; be the first hitting time of

Xv (t) on 0, i.e.

x; = min(t: t 0,X; (t) =O}

and let Is denote the corresponding Laplace transform. Then rs also

satisfies (3.4) and (3.5) and in addition

rs (x, v, ,, a2) =O for alls and x > 0. (3.8)

Both xv and X are stopping times with respect to the family of a-algebras

generated by (X (t), V(t)) and xv 2 X for every sample path, with strict ine-

quality whenever the process V (t) hits v. Since

lim T. = a.s.V --40O

we also have

lim rs (x,v + 1,43,2) - I' (X,V +v C

A possible approach to (3.4), (3.5) and (3.8) is the use of perturbation

methods to transform the problem into a WKB-theory problem for the

Laplace transform (which due to the linearity of the equation is amenable

to boundary-layer theory). One then aims to approximate the system by a

sequence of equations valid in outer and inner regions and in the boundary

regions and to combine the solutions by asymptotic matching techniques

into a globally valid approximation, see Bender and Orszag [2]. Since we

were unable to carry through this program in a fashion resulting in

mathematically simple global approximations, we decided on a different

strategy.

A Special Case: First observe that for p = ( = 0 and a2 = 1, the system

greatly simplifies. Then with

IF= (v) = exp (-cx)1I's (x,v,O, l)dx0

(3.4), (3.5), (3.8) transforms into

2 a2f -v (c rsc - rs (0,v,0, 1)) - s rsc= 0

with

rJc(v) = 0 forall s >Oandc >0.

The substitution z = av + b with a = 21/3 c 1/3 and b = 21/3 c-2/3 s gen-

erates the one-dimensional nonhomogeneous Airy equation

-zrsc (z) = -C1 z +2113c513s (3.9)

with fsc (z;) = 0 where z; = 2 + 213c-23s.

The method of dominant balance (for an account of which, see Bender

and Orszag [2]) is then used on (3.9). The basic strategy is to first peel off

the leading asymptotic behavior then, after having removed this, to deter-

mine the leading behavior of the remainder, remove this, and so on.

This leads to

r (z) C-1 _ Z 1/3 C-513 i (3n)! Z-3n-1SC ~~~~~n=O 3" n!

as z ±±oo where by the notation

00

y (x) - a + an x"kn as x ->+oon=O

(i.e. "y (x) is asymptotic to the power series a + : an x ) is meant

that

Nlim (y(x) - a - I anx-kn)fxk = 0

x-4±0 n=O

for every N. A power series may be asymptotic to a function without

being convergent. In fact,

c-l - z1/3 c-5/3S (3n)! Z-3n-1n=O 3n n!

formally satisfies (3.9) exactly but that sum does not converge. It is well

known that many problems in perturbation analysis and the theory of dom-

inant balance lead to such divergent series. These series are still useful;

under certain conditions formal solutions (e.g. divergent series) of

differential equations are asymptotic expansions of actual solutions and

optimally truncated divergent series may provide accurate approximations

to exact solutions.

Now, after transforming and formally inverting the asymptotic expan-

sion of the double Laplace transform rSC one obtains an approximation (a

first order approximation) to the density g (t), namely

g 1(t,x ,v ,0,0,1)= 1 [3/2rt3 ]1/2 (3xt-- v)exp(-3(x -vt)2/2t3). (3.10)2

It is easy to see that

g(t,x,v,O,O,'l) = - -P (Y > x)

if Y N (Vt, t3 / 3) and this is in fact the distribution of the displacement

t

process V (s) ds when ,u=13=O and (a2 = 1, so that g1 constitutes the0

approximation derived from

P ( < t) Z P (X(t) > ). (3.11)

For higher order correction terms we again utilize the method of dominant

balance setting, in view of (3.10),

g 2 (t, x, v, 0, 0, 1) = G (x, v, t) (3/2nt3)1/2 exp (-3 (x - vt)2 / 2t3)

and G (x,v, t) satisfies

- vaGav- -axaG 3(vt-x) aGat t2 av

With

a2G _° as (3x - vt)/2t -caV2one gets

3x - vtG t +2t

t3'2 3x-vt2(3x_-Vt)2 )( t/

where 0, as before, is the standard normal density.

Continuing in this fashion, i.e. with

G (x,v,t) = 3 vt t3/2+ | Vtt)2 + Gl(x,v,t)j

1-2 (3x -v

one arrives (after tedious computations) at the full asymptotic expansion

G (x,v,t)- 3X2-vt + [ (-I)'+'1 - 3 5- (2i - 1) I3x+/2] ( 3vz )

This asymptotic series is reminiscent of Mill's ratio M (z) = -

1 - )( (z)

(see Kendall et al. [1 1]) and indeed one may simplify

3x -vt +2t

t1/22

( 3x - vt )M ( 3x - vt )2t V t)M(

Hence the following approximation to the first passage density is

obtained

1 a2G2 av2

p( 3x - vtt3/2

G (x,v,t) -

3x-vt4¢( 3/2

g2(X,V,O,O,1) =3x-vt )+(t(I/2n ( 3x-vtW( 3x-vt))(3x vt)](3/2rCt3)1/2exp( -3 (x - vt)2

The General Cases: Without the above restrictions on the parameters (i.e.

without , = 0, Fi = 0, a2 = 1) using (3.11) gives

gl(t, ,D 2)[3x(v+ )t x 13(3 (x - gt) - vt)]¢(x) (3.12)2t 8

where 4 (x) is the density of the normal distribution with mean =

(vI/f)(l - e-5') + gt and variance = (a2/2J33)(2pt + 4eH3 - e - 3).

In the factor multiplying ¢ (t) we have only retained the term linear in P.

Note that g1 satisfies (3.6) and (3.7) asymptotically as

(2t-1 [ 3x - (v + )t] -oo and 5 0.

By a strategy similar to the one employed in the special case set

g (t,x,v,,f3,a2) - G (x,v,t)4)(x)

where now G satisfies

a2 D2 6 __ G _~

2 av2 -(v+p) _ _

and hence, using terms linear in 03 only,

a2 '(v2+d )JadaG +3(x-(v+g)t) +(P/4t)(3(x-pt)-vt) 0-y ~~~~-~~--(v+ii) L ~~~~~~(3.13)

and from (3.12)

G(t,x,v) _3x -(v +g)t (3(x- t) - vt)

asymptotically as (2t)-' [3x - (v + p)t I -4 oo and 1 -+ 0.

After repeated application of the method of dominant balance one

arrives at

G - cl + [c2 - clM (c11c2)14(c11c2)where M (y) is Mill's ratio and c1, c2 are given by

Cl =3x- (v + ) -t (3 (x- t)-vt)2t 8

and

[-t1/2 t3/2YC2 l2 16

In summary as 1 - 0 and (2t)-1 [ 3x - (v + p)t] - 00

g2(t,X,V,943,a2) = [C1 + [C2 - C1M (C1/c2)B0(c1/c2)J¢(x) (3.14)

is asymptotic to the first-passage density g (t). After some further manipu-

lations, this may also be represented as

g2(t,X,v,9,13T,a2) - g1(t,X,v,p.,,a2)(I(X(t)) +((t))_l¢(X(t)))with

(t) = [3x - (v + )t - (P3t /4)(3(x - gt) - vt)]/(at312 - a3t5/22/8).

The quality of the approximations g 1 and g2 has been checked in an

extensive series of simulations. The results of these simulations are

reported in Section 4.

4. SIMULATION RESULTS

First-passage time simulations of the position process X (t) were basedon the elementary relation

t+A

X(t+ At) = X(t)+ JV(s)ds X(t)+(V(t+At)+V(t))At/2t

approximating the integral by the trapezoidal rule of quadrature. Physically

this approximation corresponds to the assumption of constant acceleration

between times t and t + At and implies that position is a quadratic spline.

Therefore, approximate realizations of the position process X (t) can be

obtained from V (kAt), k = 0,1,2,... and a quadratic interpolation scheme

may be employed to approximate boundary crossing times within intervals

[(k - 1)At?kAt].

As in Pickard and Tory [15], if

X ((k - l)At) =y < 0, V ((k - 1)At) = u, V (kAt) =v

then for 0 < T < At:

X ((k - l)At + TI) = y + uri + (v-u)rn2/2At.

Zero-crossing occurs at ((k - 1) At + n1o) for

= (-u + (u2 - 2y (v - u)/A)112At)/(v -u)

if either

(u +v)At/2 2 -y

or if

u2At-2y(v -u) . 0 with u >0>v.

At time t = 0, we started 100,000 realizations of the bivariate process

(X (k A t), V (k A t)) from X (0) = -x, V (0) = v + g. The time increments

were set at At = 1O-1. For a2 = 1 and for various choices of x ,v , 3, [

empirical first passage-time probabilities were recorded for

t = 0,1,2, ... , 20. These were compared with the corresponding proba-

bilities derived from the approximations g 1 and g2.

The results of the simulations are reported in Tables 1-4. These con-

tain the empirical (based on 100,000 realizations) first-passage time distri-

butions evaluated at t = 0,1,2, . , 20 (for various combinations of f3, x,

v and g). Also given is

g

P ( < t) Jg2(s)ds0

(which was obtained by numerical integration using routines from IMSL)

and the first order approximation based on g 1 (t), i.e.

P ( < t) z 1 - ((x - mean)ISD)

where

mean = (v / e)(1-e) + ptSD2 = (a2/2p3)(2pt + 4e-Pt - e-2It - 3)

Both the tables and the plots (Figures 1-4) show that g2 (t) is a significant

improvement over the first order approximation and constitutes a very

accurate global approximation to the first-passage density. The parameter

choices represented in Figures 1-4 are only a subset of the extensive series

of simulations we conducted. The emerging pattern is that the accuracy of

the approximation increases with increasing x, decreasing v and p, and

decreasing 1 and decreases with increasing t. This is consistent with the

results of Section 3. Also observe that probabilities computed from g 1 (t)

underestimate (and quite significantly so for large t) corresponding true

first-passage probabilities while those computed from g2(t) tend to only

slightly overestimate these. Figures 5,6 demonstrate the effect of 3 on the

approximate first-passage density. Over the range of ,B from 0 to .15 the

maximum of the first-passage density decreases with increasing ,B. The

order of the curves in the tail of the distribution is reversed compared to

the order at the maximum.

ACKNOWLEDGMENTS

This work was partially supported by a COR faculty research grant from

the University of California at Berkeley and a National Science Foundation

grant under contract DMS-90-01710. I thank the Editors and two

anonymous referees for valuable suggestions. Our simulations were per-

formed in the Statistical Laboratory at Berkeley. I am grateful to S. Rein

for assistance with these simulations.

t

t g1(s)ds0

1234567891011121314151617181920

0.0000.0320.0910.1400.1780.2080.2320.2510.2670.2810.2920.3030.3120.3200.3270.3340.3400.3450.3500.355

Table 1t

g2(s)ds0

0.0000.0330.0930.1470.1910.2280.2590.2860.3100.3320.3510.3690.3850.4000.4140.4270.4390.4510.4620.473

First and second order approximations of P (t < t) and the empirical distri-

bution forKt = 0, B= 0.05, x = 1, v = -1.

empmcal

0.0000.0320.0930.1450.1880.2230.2530.2790.3000.3200.3380.3540.3680.3820.3940.4050.4150.4250.4340.443

Table 2t t

t f81 (s) ds Jg2(s)ds empirical0 0

1 0.000 0.000 0.0002 0.011 0.011 0.0123 0.045 0.046 0.0454 0.081 0.084 0.0845 0.113 0.119 0.1206 0.139 0.151 0.1517 0.160 0.178 0.1788 0.178 0.202 0.2029 0.193 0.224 0.22310 0.206 0.243 0.24211 0.217 0.261 0.25812 0.226 0.277 0.27513 0.234 0.292 0.28914 0.241 0.306 0.30215 0.248 0.318 0.31416 0.253 0.330 0.32517 0.258 0.340 0.33618 0.263 0.350 0.34719 0.267 0.359 0.35620 0.270 0.368 0.366

First and second order approximations of P (t < t) and the empirical distri-

bution for i= -0.3, 3= 0.15, x = 1, v = -1.

Table 3t t

t g1 (s) ds Jg2(S)ds empirical0 0

1 0.000 0.000 0.0002 0.037 0.037 0.0373 0.168 0.169 0.1704 0.269 0.273 0.2735 0.332 0.343 0.3426 0.372 0.393 0.3907 0.399 0.431 0.4268 0.418 0.461 0.4549 0.432 0.487 0.47710 0.442 0.508 0.49711 0.450 0.527 0.51312 0.457 0.544 0.52813 0.462 0.560 0.54114 0.466 0.574 0.55215 0.470 0.587 0.56216 0.473 0.599 0.57217 0.475 0.610 0.58118 0.478 0.620 0.58919 0.479 0.630 0.59720 0.481 0.640 0.605

First and second order approximations of P (t < t) and the empirical distri-

bution for .t = 0, f = 0.05, x = 3, v = 0.1.

t

t fg1 (s)ds0

1234567891011121314151617181920

0.0000.0090.0750.1480.2010.2390.2660.2860.3010.3120.3210.3270.3330.3370.3410.3440.3460.3480.3490.350

Table 4t

Jg2(s)ds0

0.0000.0100.0750.1490.2070.2510.2870.3160.3410.3630.3810.3980.4140.4280.4400.4520.4630.4730.4820.491

First and second order approximations of P (t < t) and the empirical distri-

bution for pt=-0.3, ,B=0.15,x =3,v =0.1.

empmcal

0.0000.0090.0760.1510.2090.2540.2890.3180.3430.3640.3820.3990.4120.4250.4370.4480.4580.4680.4770.485

Fig. 1: g1 (t), g2 (t) and empirical histogram for i = 0, r = 0.05, x = 1,

v =-1 andt e [0,20].

Fig. 2: g 1 (t), g2 (t) and empirical histogram for A = -0.3, f3 = 0.15,

x = 1, v = -1 and t E [0,20].

Fig. 3: g1(t), g2(t) and empincal histogram for x= 0, [ = 0.05, x = 3,

v = 0.1 and t E [0,20].

Fig. 4: g1 (t), g2 (t) and empirical histogram for i = -0.3, I = 0.15,

x = 3, v = 0.1 and t e [0,20].

Fig. 5: Second order approximations of the density g for

, = 0.01,0.05,0.10,0.15 with x = 3, v = .1, 1t = -0.3.

Fig. 6: Second order approximations of the density g for

p = 0.01,0.05,0.10,0.15 with x = 3, v = .1, p = 0.

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