On the Evaluation of First-Passage-Time Probability Densities via Non-Singular Integral Equations

Abstract

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.'s for Wiener and Ornstein-Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

Full text

On the Evaluation of First-Passage-Time Probability Densities via Non-Singular Integral
Equations
Author(s): V. Giorno, A. G. Nobile, L. M. Ricciardi, S. Sato
Source:
Advances in Applied Probability,
Vol. 21, No. 1 (Mar., 1989), pp. 20-36
Published by: Applied Probability Trust
Stable URL: http://www.jstor.org/stable/1427196
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Adv. Appl. Prob. 21, 20-36 (1989)
Printed
in N. Ireland
? Applied Probability
Trust
1989
ON THE EVALUATION OF FIRST-PASSAGE-TIME
PROBABILITY DENSITIES VIA NON-SINGULAR
INTEGRAL EQUATIONS
V. GIORNO,
*
University of Salerno
A. G. NOBILE,
*
University of Salerno
L. M. RICCIARDI,
**University of Naples
S. SATO, ***Osaka
University
Abstract
The algorithm
given by Buonocore
et al. [1] to evaluate
first-passage-time
p.d.f.'s
for Wiener
and Ornstein-Uhlenbeck
processes
through
a time-dependent
boundary
is extended to a wide class of time-homogeneous
one-dimensional
diffusion
proc-
esses. Several examples are thoroughly
discussed
along with some computational
results.
DIFFUSION PROCESSES; VARYING BOUNDARIES; RESTRICTED PROCESSES
1. Introduction
In a previous
paper (Buonocore et al. [1]) a new second-kind Volterra integral
equation was proposed to determine first-passage-time
p.d.f.'s through time-
dependent boundaries
for time-homogeneous
one-dimensional diffusion processes
with natural
boundaries.
This equation
was seen to be particularly manageable
for
computational
purposes
in the cases of Wiener
and Ornstein-Uhlenbeck
processes.
Indeed, for these the kernel of the integral
equation
can be made continuous,
thus
overcoming
some crucial difficulties
arising
from the singular
nature
of the integral
equations
for the first-passage-time p.d.f. Such continuity
was made possible by a
suitable
choice of two arbitrary
time-dependent
functions that appear
in the kernel
of the integral
equations.
Here, in view of the relevance of first-passage-time
problems
in numerous
applied
fields (see, for instance, Buonocore et al. [1] and references
therein), we intend to
provide
an extension
of the results
of Buonocore et al. [1] in two directions: on the
one hand, we shall prove that the method for regularizing
the kernel of the integral
equation
is valid not only for the Wiener
and the Ornstein-Uhlenbeck
processes
but
can also be used for a more general
class of diffusion
processes
whose free transition
Received
16 September
1987;
revision received
20 January
1988.
* Postal
address:
Dipartimento
di Informatica e Applicazioni,
University
of Salerno,
84100
Salerno,
Italy.
**
Postal
address:
Dipartimento
di Matematica
e Applicazioni,
Universith
di Napoli, Via Mezzocan-
none 8, 80134
Napoli, Italy.
***
Postal address:
Department
of Biophysical
Engineering, Faculty
of Engineering
Science, Osaka
University,
Toyonaka
(Osaka), Japan.
20

Evaluation
offirst-passage-time
probability
densities 21
p.d.f.'s are known; on the other hand, we shall see that the method can also be
employed to make the kernel continuous
if the diffusion
process is restricted
by a
reflecting
boundary
and the transition
p.d.f. in the
presence
of such
boundary
is known.
In the remaining part of this section we shall briefly review some background
results
from Buonocore et al. [1] and then see how they can be adapted to more
general instances. Some straightforward
intuitive considerations are then made in
Section 2 to indicate why the method of Buonocore et al. [1] can be extended
beyond the cases of Wiener and Ornstein-Uhlenbeck
processes. Rigorous proofs
will instead
be given in Sections 3 and 4 by making
heavy use of some fundamental
results
due to Fortet [4]. A few examples
are discussed in Sections 5 and 6.
Let S(t) and k(t) be continuous functions in [to, +oo), toE R, and let {X(t);
to
5 t _ T < +oo} be a time-homogeneous one-dimensional diffusion process with drift
A
1(x)
and infinitesimal variance
A2(x).
Let us denote
the
diffusion
interval
by
I = (rl, r2),
with rl and r2
natural boundaries. Then [1]:
g[S(t), tl xo, to]
=
(1.1) -2W[S(t), t xo,,
to]
+ 2f drg[S(r), r xo, to][S(t), tI
lS(r), r], xo<S(to)
to
2W[S(t),
t xo,
to]
- 2 drg[S(r),
r Ixo,
t
So][),S(
t
I
(Sr),
,], xo>
S(to)
where g[S(t), t I
xo, to] is the first-passage-time
p.d.f. of X(t) through S(t) condi-
tional upon X(to)
= xo and
(1.2) P[S(t), t I
z, -] = d F[S(t), t I
z, -] + k(t)f[S(t), t I
z, -].
(Throughout
this paper we shall take the function r(t) in Buonocore et al. [1] as
identically
zero. Indeed, the case r(t)# 0 does not play any role in what follows.)
Here
(1.3) F(x, t
lz, -) = P{X(t)5x I
X(-r)= z}
and
(1.4) f (x,
t z,
I ) - F(x,
tIz, r)
ax
denote the transition distribution
function
and the transition
p.d.f. of X(t). We have
the following
result.
Lemma
1.1. Let S(t) e Cl[to, T]. Then:
P[S(t), tI z, r] = (S'(t) - A1[S(t)] + JA'[S(t)] + k(t)}f[S(t), t I
z, r]
(1.5) a
+ ? A2[S(t)] -f(x, tI z, )
ax x=S(t)

22 V. GIORNO,
A. G. NOBILE, L. M. RICCIARDI
AND S. SATO
d d
where
we have set S'(t) = S(t) and
A2[S(t)] = A2(x)
dt dx x =S(t)
Proof. From (1.2) we have
d rs(t)
TY[S(t),
t I
z, -] = -
f
dxf(x, t I
z, r)
+ k(t)f[S(t), t I
z, -]
(1.6) = [S'(t) + k(t)]f[S(t), t I
z, -r]
+ dx- f(x, t I
z, -r).
The integral
on the right-hand
side can be calculated
by recalling
that
f satisfies
the
Fokker-Planck
equation
af
_ 1 a2
(1.7) af a [Al(x)f]
+ [A2(x)f]
(t 7x 2
aX2
and that no probability
mass flows at the natural
boundary
rl.
We recall that the arbitrariness of k(t) has been used in Buonocore et al. [1] to
make the kernel
of (1.1) continuous at r = t.
Let us now denote by r an arbitrarily
fixed state with rl
- r < r2
and assume
that r2
is a natural
boundary.
Further,
let us denote by f(r)(x, t z, r), (x ? r), the transition
p.d.f. of X(t) in the presence
of a zero-flux
condition
at r, by F(r)(x,
t I
z, -), (x - r),
the distribution
function and by g(r)[S(t),
tI xo, to] the first-passage-time
p.d.f. of
X(t) through S(t) in the presence of the zero-flux
condition at r. Let us define the
function
(1.8) p(r)[S(t), t I
z, -] =d F'r)[S(t), t I
z, r] + k(t)f(r)[S(t), t I
z, r].
We have the following
result.
Lemma 1.2. Let S(t) e Cl[to, T]. Then:
TP(r)[S(t),
t I
z, ] = {S'(t) - A1[S(t)]
+ JA'[S(t)] + k(t)}f(r)[S(t), t I z, T]
(1.9) + 10W a
+ 1A2[S(t)] f(r)(x, tI z, 0)
ax x=S(t)
Proof. It goes as for Lemma 1.1, the difference being that now a zero-flux
condition
at r must hold.
It is convenient
to consider
separately
the cases X(to)
x- o
> S(to) and xo
< S(to).
Lemma
1.3. Let xo
> S(to). Consider the following
two possibilities:
(a) rl is natural
(hence r > rl). Then:
(1.10) g'(r)[S(t),
t xo, to]- g[S(t), t l
xo, to], xo > S(to)
(so that g(r) can be obtained as solution of the second of Equations (1.1)).
(b) rl is regular or entrance, so that r _ rl. Then g(r)[S(t), t xo, to] is the solution

Evaluation
of first-passage-time
probability
densities 23
of the following equation:
g(r)[S(t), t I
xo, to]
= 2W(r)[S(t), t I
xo, to]
- 2 drg('[S(r),
Ixo,
toP(r)[S(t),
t S(r),T, xo
>
S(to)
where W(r)
is obtained from (1.2) by substituting
F and f with the corresponding
functions obtained
in the presence
of a zero-flux
condition
at r.
Lemma
1.4. Let xo
< S(to). Then g(r')
is the solution of the following
equation:
g(r)[s(t), t I
xo, to]
= -2T(r)[S(t), t xo, tol
(1.12)+ 2 drrg(r)[S(r), ,
irtolt(r)[s(t), tI S(r), r], x < S(to)
where the notation
used has the same meaning
as in (b).
The proofs of Lemmas
1.3 and 1.4 goes along similar
lines as in Buonocore
et al.
[1]. Note that Equations
(1.11) and (1.12) exhibit a weakly singular
kernel at r = t.
However, this kernel may be made continuous at t after a suitable choice of the
arbitrary
function
k(t) appearing
in (1.8) is made and once the boundary
S(t) has
been specified. The algorithm of Buonocore et al. [1] can then be applied to
evaluate numerically
the function
g(r)[S(t), t Ixo, to]. Some examples
will be briefly
discussed
in Sections
5 and 6.
2. Determination of the function k(t)
Here we shall outline some straightforward
intuitive
considerations
indicating
that
a determination
of the function k(t) in (1.2) can be achieved for processes other
than Wiener and Ornstein-Uhlenbeck.
Let Al(x) and A2(x) denote the drift and infinitesimal variance of a one-
dimensional time-homogeneous
diffusion process X(t) defined over the interval
I= (rl, r2), with rl and r2 natural boundaries. We may then think of X(t) as
generated
by the stochastic
differential
equation
(2.1) dX(t) = AI(x) dt + \FA2(x)
dB(t),
where B(t) denotes normalized
Brownian motion. To obtain the transition
p.d.f.
fx(x, t I
xo, to) of X(t) conditional
upon X(to) = xo, we argue as follows. We denote
by x' an arbitrarily
fixed point such that r1
< x' < r2 and set
x dO
(2.2) y h(x)=V 2()
so that Y(t)=h[X(t)] is also a diffusion process. We shall further assume that
transformation (2.2) maps I onto (-o, +oo). The process Y(t) satisfies (Ito's

24 V. GIORNO, A. G. NOBILE, L. M. RICCIARDI AND S. SATO
Lemma)
(2.3) dY(t) =
[A(x))
A (x) dt + V dB(t).
A2(x) 4 x=h-'(y)
The increment AY of Y(t) in [-, t] conditional on Y(r) = w h(z) is then
(2.4) AY(t) Y(t) - w A (z) - A
z=h-(w)
Since AB(t) is a zero-mean
stationary
normal
process
with variance
t - r, from (2.4)
we see that, for small t - -, Y(t) has the p.d.f.
1
fy(y, tI w, -)= 2
1
(2.5) S1 y
1 -w Al(z) - A'(z)/4
ep 2(t - -) V2(
_
Jz=h-'(w)
Expressing
y and
w in terms
of x and
z via (2.2), from
(2.5)
we obtain
(2.6) fx(x, tIZ,0
z() =
) fy[h(x), tl h(z), z]
A2(x)
or, for small t- r,
1
fx(x, t z, ) = r(t- 'r)A2(x)
(2.7)
Xexp{-2(t - r)A2(z) z - (A)
Let us now assume
that
S(t) is a smooth function.
Then,
making
use of Lemma 1.1
and
of the approximation
af(x, t I
z,
,) x - z - [Al(z) - A'(z)/4](t - r)
ax (t - r)A2(z)
(2.8) A'(x)
xf(x, tI z,)- f(x, tl z,
t)
2A2(x)
we obtain
),t A[S(t)] 1 S(t)- z A2[S(t)]
W[S(t),
tI z, r]
-S'(t) - A[S(t)]
+ 2 t - A2(z)
(2.9)
+ A(z) A[S(t)] +k(t)f[S(t), t I
z, T].

Evaluation
of first-passage-time
probability
densities 25
Hence
uA'[S(t)] 1 S(t)- S(r) AE[S(t)] ([S(t), t I
S(r), C]
- S'(t) -
AI[S(t)] +A~[S(t)] 1 S(t) - Sr) A2[S(t)]
(2.10) 1 fA[r Ai[S(r)] A2[S(t)
+24A AS() A2[S(r)] + it)f[S(t), t S(r), 4.
From
(2.10) we thus see that if
1 Ar[S(t)]
(2.11) k(t) = Al[S(t)] - S'(t)- AS(t)4
2 4
then
(2.12) lim [S(t), t I
S(i), r] = 0.
rTt
Indeed,
f has a singularity
(t - -r)- whereas the expression
in curly
brackets
on the
right-hand
side of (2.10) goes to 0 of order t - r due to the assumed
smoothness of
S(t) if k(t) is taken as (2.11). Hence, with the choice (2.11) for k(t) the kernel 'P of
Equations
(1.1) is continuous
at r = t. Note that (2.11) includes
the results derived
in Buonocore
et al. [1] for the Wiener and the Ornstein-Uhlenbeck
process.
In Sections 3 and 4 under the assumption
that S(t) C2[t0,
T], we shall give a
rigorous proof of (2.11) in the form of a necessary
and sufficient condition
not only
for the case when both endpoints of I are natural boundaries but also when a
zero-flux
condition
holds at one endpoint
of L
3. Two natural
boundaries
In this section use of some well-known
results
by Feller [2] and Fortet [4] will be
made to provide
a rigorous
extension
of the main conclusion drawn in Buonocore et
al. [1]. We shall assume throughout that X(t) is a diffusion process defined in
I= (rl, r2) with ri (i = 1, 2) natural boundaries. Let Al(x) and A2(x) be the
infinitesimal
moments of X(t). We shall assume that transformation
(2.2) maps I
onto (-o, +o). Then, if
(3.1) Bi(y)i 2A(x[A(x) -
=h(y)
and B'(y) exist and are bounded in (-oo, +oo) and if furthermore
B'(y) satisfies a
suitable Lipschitz condition (Fortet [4]), the transition p.d.f. of X(t), with
X(to)
= xo, can be expanded
as (Feller [2])
(3.2) f (x, t xo, to)= H,n,(x,
t xo, to)
n---=O

26 V. GIORNO, A. G. NOBILE, L. M. RICCIARDI
AND S. SATO
where
1 1 x dO 2
(3.3) Ho(x,
t l
xo, to)
= V2;r(t- to)A2(x)
expI 2(t- to)
-
o
VVA()
H,,(x,
t Ixo0, to)
= dr dz Ai(z) z)
(3.4) to r
az
Ho(z, r jXo, to)- H,_•(x, t z, z) (n = 1, 2,...).
As proved
by Fortet [4], the following inequalities
hold:
(3.5) IHn(x,
t I xo, to) a,(t - to)(n-2/2, n = 1,
2, ...
and
(3.6) H(x, t I
xo, to)
-
fn(t - to)(n-2)/2, n = 1, 2,
where ar,,
and
fi, are the terms of convergent sequences, both independent
of x, t, xo
and to. From (3.2)-(3.4) one can prove that
f(x, tI xo, to)
= Ho(x,
t ixo, to) 1 + Q1(x,
t IXo,
to) + 2to)t - to
(3.7) + Q3(x,
tI x, to)
[f
•A
]2 +Q4(x, t I
Xo, to)Vt
- to od
+ Qs(x, tI xo, to)(t - to)J + Q6(x, t Xo, to)I t- to
where Qi(x, t I
xo, to), (i = 1, 2, .- -. , 6) are bounded functions.
In particular,
Q1
and
Q2
are
given
by
(38)1 (x. exp (_- 2/2) - 1 dA2[p(?)]
(3.8) Ql(x, tj Xo, to) • dpf d A AI[4() 4
V/A2[4(f)] -4
dpV(A) J'
Q2(x,t3I
Xo,
to)
= - dp d2
9 exp
(_-2/2)
(3.9) x fA[( )]1 dA(0)]
a[)1 4 d)
[(J)
where
(3.10) (5) = h-1{1/2(t - to)p(1 - p) + h(xo) + p[h(x) - h(xo)]}.

Evaluation
of first-passage-time
probability
densities 27
Lemma
3.1. Let S(t) E C[to, T]. Then
limW[S(t), tI S(r), r]
(3.11)
Tt f[S(t), t S(r), r]
iff
1{ C AJ[S(t)] S()
(3.12) k(t) = {Ai[S(t)] - t- S'(t)
Proof. From (3.2)-(3.10) we have
lim
{f[S(t), t I
S(r), r]}l1
-'f [, t I
S(), r]
,rTt ax x=S(t)
A'[S(t)] S'(t) QI[S(t), t l S(r), r]
(3.13) + lim
2A2[S(t)] A2[S(t)] rTt VA2[S(t)]
_ _ 3
A2[S () -S'(t) + A[S(t) A[S(t) .
A2[S(t)] 4
Making
use of Lemma
1.1, from (3.13) we obtain
u[s(t),
t S(r), •] s'(t) AI[S(t)] A'[S(t)]
(3.14) lim + k(t)
+2
rTt f[S(t), t S(T), r] 2 2 8
which
vanishes
if and only if (3.12) holds.
Theorem
3.1. Let S(t) E C2[to,
T]. Then
(3.15) lim
WI[S(t),
tI
S(r), r] = 0
rTt
iff (3.12) holds.
Proof. Let k(t) be given by (3.12). From (1.5) we then have
W[S(t), I, r] St) A [S(t)] +-AI[S(t)] f[S(t), t Iz, r]
2 2 8
(3.16) A2[S(t)] t I
2 axf Ix=S(t)
Recalling
that Ho[S(t),
t I
S(r), r] exhibits a singularity
of the type (t - r)-1 as
tends to t and using (3.8) and (3.9), due to the boundness
of function
Qi, we obtain
1
i
1 {[S'(t) Al[S(t)]+ A[S(t)]
lim W[S(t), t S(r), r] (limt AS(
xtt -2rA2[S(t)] t (t r) 2 2 4
(3.17) x (t - /) A2S(t)] tiS)
A2[S(t)] Q1[S(t), t S(t), (]-
+v'A2[S(t)] (t- )

28 V. GIORNO,
A. G. NOBILE,
L. M. RICCIARDI
AND
S. SATO
The
right-hand
side of (3.17)
is an indeterminate form.
After a twofold
application
of l'Hospital's
rule and
recalling
that
S(t)
E
C2[t0,
T], we finally
obtain
(3.15).
This
proves
the sufficient
part
of the theorem. To prove
the necessity,
let us assume that
(3.15)
holds,
i.e. that
Y[S(t),
tIS(r),
r]
(3.18) lim I[S(t), tI
S(r), r] limf[S(t), tI
S(r),
'] 0.
rTt rTt f[s(t),
tI
S(t),
r]
Since
f[S(t), t I
S(r), r] has a singularity
of the type (t - r)-I as r grows to t, the
vanishing
of P[S(t), t I
S(r), r] as nrt implies
the vanishing
of the ratio in (3.18) as
rTt.
This, in turn, by Lemma
3.1 implies
that k(t) is given by (3.12).
4. One natural
and one reflecting
or entrance
boundary
In this section we extend Lemma 3.1 and Theorem 3.1 to the case when the
diffusion
process is constrained to the interval
JI= (r, r2), ri? rl, such that r2 is a
natural
boundary
while r is either entrance or regular
with a zero-flux
condition.
With some obvious
changes,
the results of this section can be proved
to hold in the
case when J= (rl, r), r 5 r2. In what follows, we shall assume that the
transformation
(4.1) y = • h(x)
maps
J onto (0, +oo).
Furthermore,
we assume
that
(4.2) C(y) - A(x) A 4 x=h'(y)
and C'(x) exist and are bounded in (0, +oo) and that C'(y) satisfies a suitable
Lipschitz
condition
(Fortet [4]).
Let us set
Do(x, t I xo, to) = exp l-1 [X dO ]21
D(x, tX2o, to) r(t - to)A2 ex p- 2(t - to) oA2(0)
1 X
dO
d
2
(4f3)2(t - to)
- If fVA
(432
r-2(t
) L
VA2(z-0)LrVA(0)J
D,,(x, t xo, to)= dr dz Ai(z)- A(z)]
(4.4)r
x Do(z, I
xo,
to) zD
O_(x, tI
zo,
ro) (n=
1,2,-..).
If
(4.5) lim exp2 • dz ) Do(x, txo, o)=
o A2(z)J aD0,

Evaluation
offirst-passage-time
probability
densities 29
then the transition
p.d.f. f(r)(x, t I
xo, to) of X(t) conditional on X(to) = xo and in the
presence
of a zero-flux condition
at r can be expanded
as
(4.6) f(r)(x, t l
xo, to)= > D,(x, t Xo, to),
n=O
where
the functions
Dn(x,
t I
xo, to) and - D,(x, t I
xo, to), n >_-
1, are bounded
as in
&xo
(3.5) and (3.6). This can be proved by showing that the right-hand
side of (4.6)
satisfies
the Kolmogorov
backward
equation,
the initial
delta-condition
and, because
of (4.5), also the zero-flux
condition
(4.7)xolimexp {2fdzA f')(x, t Ixo, t) = 0.
xot) A2 XOr), Ot=
One can further
prove that
f(r)(x, t xo,
to)
= Ho(x,
t
x o, to) 1 + Q t(x,
t I
Xo,
to) •dO
+ Q*(x,
t I
Xo,
to)t - to
+ Q3*(x,
t
I
Xo,
to)
(4.8) X[f dO ]2 xo) dO
x fo VA )+ Q (x,
tI
Xo,
to)Vt-
to o
VA
+ Q5*(x,
t I
x, to)(t - to) + Q(xt, t to)
t0,
where Q*(x, t xo, to), (i = 1, 2, - -, 6) are bounded functions. In particular,
we
have:
1
A4() { A1[
Ia[41(0)]}
Q*(x,,
t x0, to)
= dpf A1[1()] - 4
(4.9) 1 4 4
X E(p,,(x, t Ixo,
to)F(p, )(x, t I
x,
to),
Q~f(x, t xo, to) = - dp d?
P A2[1
(4.10) x
A,[(?)]- A2[4,()]
x E(p
?)(x,
t IXo,
to)F(p, )(x,
t lxo,
to)
where
we have set
Ep,(xt I
xo, to)=1+ exp {-2 t o
P 0
(4.11a) xx
2]
Vo dIOO o d }
x
exp
_(t
- to)P
or?

30 V. GIORNO,
A. G. NOBILE, L. M. RICCIARDI
AND S. SATO
-
t
(o -0) _rd
FSg(x,
txo,
to)
=exp(-52/2)-exp
t
-
to
do
2
@
x exp - (t-1(t - to)(1 - ) 2Vd
(4.11b) 1xo do x do rx do
x
V(t - to)p(1 - p) + + )+ p
x (1 - p4)
x (- - (t- to)p(1
- P)]
(4.11c) 1p(j)
= hi'1{• 2(t
- to)p(1
- p) + hi(xo)
+ p[hl(x)
- hl(xo)]}
(4.11d) l= v(t - to)p(1 - p) r 2(0- xo 2(0)
Lemma
4.1. Let S(t) E C1[to,
T]. Then
(4.12)[S(t),
tl S(r), z]
(4.12) lim = 0
,tr
f()[S(t),
t S(t), -]
iff
1{ Aa'[S(t)]
(4.13) k(t) = -2 AA[S(t)] - 4 - S'(t) .
2 4
Proof. Similar
to that of Lemma
3.1. Indeed, we have
lim {f(')[S(t), tI S(T), r]})-1 f(r)[x,
tI S(), r]
(4.14) T =S(t)
A'[S(t)] S'(t) Q
?[S(t), t S(r), c]
+ lim
2A2[S(t)] A2[S(t)] rT A
Since
1
---4A[S(t)]1
lim Q*[S(t), t I
S(r), T] =- ArA tIAI[S(t)]
(4.15) rTt 2 [S(t)]
1o
+= 2 1 A [S(t)]
x dpf d exp - =~/-} A[S(t)] - A
,S(t)
substituting (4.15) in (4.14) and making
use of Lemma 1.2 the proof of Lemma
4.1
follows.
Theorem
4.1. Let S(t) E C2[t0,
T]. Then
(4.16) lim WII(')[S(t),
tI S(t), r] = 0
iff (4.13) holds.

Evaluation
of first-passage-time
probability
densities 31
Proof. Recalling (1.9) and Lemma 4.1, the necessary and sufficient parts of
Theorem 4.1 can be proved as in Theorem 3.1. However, we remark that the
evaluation
of ~Pr)[S(t), t I
S(r), r] as r tends to t requires
a cumbersome
proof that
for brevity
will be omitted.
5. Examples
(a) The
lognormal
process. Let
(5.1) Al(x) = mx, A2(x) = 2 (x 0),
with m and 2 > 0 arbitrary
real numbers. The diffusion
process having moments
(5.1) is defined over the diffusion interval (0, +o), with the endpoints natural
boundaries. Transformation (2.2) thus maps the diffusion interval onto (-oo, +oo),
while condition (3.1) and the other regularity
conditions of Section 3 are satisfied.
Then, from (3.12) and (5.1) we have
1 /
(5.2)
k(t)
=1[(m
-
m
S(t)
-
S'(t)
.
2
2
Since
(5.3) f(x, t z, r)=n exp - r22- r) 1
xu /23r(t - r) 12&(t - r)
from (1.5) and (5.2) we obtain
(5.4) W[S(t),
t I
z, r] = - S'(t)- tIn [St(t), t z, r].
2
t
-'r
z
Since the right-hand
side of (5.4) vanishes identically for z = S(r) and S(t)=
exp (at + b), where a, b E R from (1.1) we have
eat+b
(5.5) g(eat+b, t I
xo, to) = Iln
xo - ato - bl f (eat+b, t I
xo, to) (xo / eat+b).
t - to
(b) The
hyperbolic
process. Let
A - B exp (-2px/Ir2)
(5.6) Al(x) A + B exp (-2x/)' A2(x) A2 = 2,
with A, B real numbers such that AB > 0. We incidentally
note that the Wiener
process is a special case of the process having moments (5.6). Now the diffusion
interval
is the entire real line, with the points at infinity
natural
boundaries.
Clearly,
transformation
(2.2) maps the diffusion
interval onto itself. From (3.12) and (5.6)
we obtain
i a -B exp [-21iS(t)/u2]
(5.7) kt= g2t A + B exp [-2,iS(t)/a2] - JJ.

32 V. GIORNO, A. G. NOBILE, L. M. RICCIARDI
AND S. SATO
Since
(5.8) f(x,tz,) t- A + B exp (-2xz/o2) Cexp
1I- 2 (t
_ r)
(rV2xr(t- r) A + B exp (-2pz/or2) 2or2(t
--
)
_'
from
(1.5)
and
(5.7)
we obtain
1 S(t) - z f[S(t), i
z' T].
(5.9) W[S(t),
t I
z, r]
= s'(t)- S(t)- [S(t) t z, r].
(A detailed
study
of the process (5.8) will be presented
elsewhere.)
We note that the
right-hand
side of (5.9) is identically
0 for z = S(r) and S(t) = at + b. From (1.1) it
then follows that
1xo
- at0 - b
(5.10) g(at + b, t xo, to)
= f(at + b, t I
xo, to) (xo
* ato
+ b).
t - to
(c) Wiener
process with a reflecting boundary. Consider a Wiener process with
infinitesimal
moments
(5.11) A1
=p , A2 = &2
defined in [r, +oo), with r a reflecting boundary.
Since condition
(4.3) and the other
regularity
conditions of Section 4 are satisfied and since (4.5) is also valid, from
(4.13) we have
(5.12) k(t) = ?[p - S'(t)].
Furthermore,
since
1
[
[
x-z
-1i(t-
-)2
f'(r)(x,
ti Z, r) exp
2_(t-_r)
OV2x (tJ- r) I f 2u2(t - r)
(5.13) +exp( 2p(zr) }exp [x + z - 2r - p(t - r)]2
I 222(t - r)
Sexp
2(x - r) x- +z - 2r
+(t - )
from (1.9) we obtain
2 1 S(t)- z
t(r)[S(t), tI z, r]
=2S'(t)-St-
x fexp [S(t) - z - P(t - r)1]2
2o2(t - r) +2aoV2r(t- r)
(5.14) x [S'(t)
S(t) + z - 2r - 2p(t - r) ]exp- 2(z - r)
xexp [S(t)+ z-2r -tp(t- )]2 u [S'(t)+ p]
Sexp
2(t - ) 22
xexp f2[S(t) - r] Erf [S(t) + z-2r + (t - )
xxp1V2(t-Er)

Evaluation of first-passage-time probability densities 33
g(r)
0.25 '
0.20
0.15
0.10
0.05
2 4 6 8 TIME
Figure 1. The oscillating curve is the first-passage-time p.d.f. g(') through S(t) = 3 + 0-1 cos (2.rt) for the
Wiener process having infinitesimal variance equal to 2 and drift y = 0 starting on a reflecting boundary at
x = 0. The first-passage-time p.d.f. through the mean threshold (S(t)) = 3 has also been plotted.
g(r)
0.35
0.28
0.21
0.14
0.076 8 TI
2 4 6 8 TIME
Figure 2. As in Figure 1 except that now M
= 0.5.
g(r)
0.15
0.12-
0.09-
0.06-
0.03 -
2 4 6 8 TIME
Figure 3. As in Figure 1 but with M
= -0-5.

34 V. GIORNO, A. G. NOBILE, L. M. RICCIARDI AND S. SATO
A numerical
evaluation
of g[S(t), t I
xo, to]
can then be obtained
from (1.12), (5.13)
and (5.14) by means of the algorithm
of Buonocore
et al. [1]. Figures
1-3 show the
first-passage-time
p.d.f. through the boundary
S(t)= 3 + 0.1 cos (2xrt)
for Wiener
processes
originating
in a reflecting
boundary
at r = 0.
We remark
that the examples
discussed in this section involve diffusion
processes
all satisfying
conditions
(3.1) or (4.2). In Section 6 an example will be discussed in
which
condition
(3.1) does not hold.
6. Process with linear drift and linear infinitesimal
variance
Let X(t) be the diffusion
process
with infinitesimal
moments
(Feller [3])
(6.1) Al(x) =px + q, A2(x) = 2rx,
with p, q, r E R and r > 0. The diffusion interval is I = [0, +oo). Hereafter we shall
assume q > 0 so that the point 0 is either a regular
or an entrance boundary.
As
proved in Giorno et al. [5], the expression
of the transition
p.d.f. of X(t) in the
presence
of a zero-flux
condition
at zero is given by
f(r)(x, t I
Z, )
r)=[eP <-) - 1] exp ze" -
(6.2) r[ -- (q-r)2rp
p xze
X[xe-
P
-
T)
(q-r)/2r
I[
2p
V
xz
eP
t -
cz
e
*
zLr[eP
'-')
-
1]
p
0;
f(r)(x, t z, r) = 1 (X\(qr)/2r exp{ +>
(6.3)x, r(t
-
e
r(t
- )
(6.3)
[ 2V ]
Ir[(t
- r)' p =0.
Using (1.9), we obtain
Wr)[S(t)z, t z, ] =[e _ exp p[S(t) +
zeP(t-]p
S(t)zpt)(q-r)2r
(6.4)
x[S'(t)pS(t)eP(t-) )[2pVS(t)_zept_-
(6.4) x S'(t) eP('T) -1 + k(t)I lr-1 r[eP(t -
1- ]
p
V/S(t)ze"t- I [2pVS
(t)zePt-}) P
+ ep"-) - 1 q/rL r[e(-)- 1] 1, p =;
W(r)[S(t), ti z, r]= r(t-
1) exp1
r(t)+z-
) (q-r)/2r ( S(t) + k(t)
(
r(t-r) r(t-) z(t
(6.5)
2lir [zS$]+ \7St) iq/r,[2p"S7) = 0.
Cay(1dn
tty- r(tt
- ()
Clearly, (6.1) do not satisfy condition (4.2) so that the procedure of Section 4 cannot

Evaluation
of first-passage-time
probability
densities 35
be used. However, in this case a direct calculation leads to the expressions
of the
functions
W(r)[S(t), t I
z, r] and k(t) such that the kernel of integral
equation (1.11)
is continuous
at r = t. Indeed, we have the following
result.
Theorem
6.1. Let S(t) e C2[to, T]. Then:
(6.6) lim W(r)[s(t),
t I
S(r), ] = 0
rTt
iff
(6.7) k(t) = - pS(t) + q - S'(t)].
For brevity,
we omit the proof.
We conclude with a special case of the process considered in this section, that
leads us to a new closed form result for the first-passage-time p.d.f. when S(t)
reduces
to an arbitrary
constant
S. Let p = 0, q = 3r/2, S(t) = S and X(0) xo
> S.
Making use of well-known properties of the Bessel function (Gradshteyn and
Ryzhik
[6]):
(6.8) II/2(z) = (eZ
- e-),
1 [1 )]
(6.9) I3/2(z)
- + e-z (ez - e-z)
from (6.5) with the choice (6.7) we obtain
(6.10) W(r)(S,
t I
z, r)
= 'I1(S,
t I
z, ) + P2(S, t I
z, r),
where we have set
1 S [
-_
]2
(6.11) 1,(S,
tl z, )=
1-•r / exp -
r-_)12
2 xrr(t-r) Lt-,r(t-r)/r(t-r)
1 S
[V•
+
V]2
(6.12) T 2(S, t ] z, r) = 1
_
+
(
S
-exp +-
r.t
2V rr(t--r) Lt- (t-r)/ r(t - )
The following
theorem
then holds.
Theorem
6.3. For
xo > S one has
(6.13) g(r)(S,
t I
xo,
to)
= 2T,(S,
t I
xo,
to)
where IPl(S, t I
xo, to) is obtained
from (6.11).
Proof. We note that IIl(S, t I
S, r) = 0, for all t, r (r < t). From Equation
(1.11),
we thus have
g(r')(s, t I
xo, to)= 2[WI(S, t I
xo, to) + 2(S, t I
xo, to)]
(6.14) - drW2(S, tI S, r)g(r)(s, r xo, to).
tO

36 V. GIORNO, A. G. NOBILE, L. M. RICCIARDI AND S. SATO
To prove (6.13), we substitute
g(r)(S,
r xo, to) with 2WI(S,
r I
xo, to) in the integral
on the right-hand
side of (6.14). This integral
can be explicitly
calculated.
Indeed,
with the change of variable (r - to)/(t - r) = p one finds
2 drW2(S,
t S, r)WI(S, xo,to)= S 1- Sexp (V- o)
to I rr r(t - to)
1
_ ((fs-Vo)2 4Sp 1
x
120
dp
(p-
+
Vpi)
exp
(VS-
_)Sp
(t to) PP+Pexp r(t - to)p r(t - to)
= t2(S,
tI
xo,
to).
Equation
(6.14) thus identifies with (6.13), which
proves the theorem.
Note that, differently
from the closed form
solutions
of Section 5, the closed form
result (6.13) holds even though the kernel of the integral equation for the
first-passage-time
p.d.f. is non-vanishing
for all t and r.
Acknowledgements
Research carried out under CNR contracts
Nos. 86.02127, 86.02115, 87.01005,
and under
MPI financial
support.
References
[1] BUONOCORE, A., NOBILE,
A. G. AND
RICCIARDI,
L. M. (1987) A new integral equation for the
evaluation of first-passage-time probability densities. Adv. Appl. Prob. 19, 784-800.
[2] FELLER,
W. (1936) Zur Theorie der stochastichen Prozesse. Math. Ann. 113, 113-160.
[3] FELLER,
W. (1951) Two singular diffusion processes. Ann. Math. 54, 173-182.
[4] FORTET,
R. (1943) Les fonctions al6atoires du type de Markoff associ6es A
certaines equations
lin6aires aux d6riv6es partielles du type parabolique. J. Math. Pures Appl. 22, 177-243.
[5] GIORNO,
V., NOBILE,
A. G., RICCIARDI, L. M. AND
SACERDOTE,
L. (1986) Some remarks on the
Rayleigh process. J. Appl. Prob. 23, 398-408.
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