the one-sided barrier integrated ornstein-uhlenbeck process
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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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