An Evaluation for the Probability Density
of the First Hitting Time
Shih-Yu Shen1*, Yi-Long Hsiao2
1Department of Mathematics, National Cheng-Kung University, Tainan, Chinese Taipei 2Department of Finance, National Dong Hwa University, Hualien County, Chinese Taipei
Email: *shen@mail.ncku.edu.tw
Received March 7,2013; revised April 7, 2013; accepted April 14, 2013
Copyright © 2013 Shih-Yu Shen, Yi-Long Hsiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Let h t
be a smooth function, Bt a standard Brownian motion and hinf
;B h
the first hitting time. Inthis paper, new formulations are derived to evaluate the probability density of the first hitting time. If denotes
the density function of
,u x t
t
xB for th, then uxx2ut and u h
t t,
0. Moreover, the hitting time density is
th
d 1
,
2ux h t t . Applying some partial differential equation techniques, we derive a simple integral equation
for dh
t . Two examples are demonstrated in this article.Keywords: Brownian Motion; First Hitting Time; Heat Equation; Boundary Value Problem
1. Introduction
Since the publication of Black and Scholes’ [1], and Merton’s [2] papers in 1973, a stock price following a geometric Brownian motion becomes the standard model for the dynamics of a stock price. Therefore, the calcula-tions for the first hitting time get important in the field of finance recently [3].
Let h t
be a smooth function for t0, and Bt a standard Brownian motion. The first hitting
time
; h B t0
is defined as inf ;
t Bth t B
00
. It isknown that h has a continuous density [4]. Sometime
we call the function h t
or the curve xh t
the barrier. For a constant barrier, the result has been well- known for a long time. In this case, the density is2
3exp 2
2π
h h
t t
([5], Chapter 9). The distribution of the hitting time for non-constant barrier was considered by many authors; for example in [6-9]. In [6], Cuzick de-veloped an asymptotic estimate for the first hitting den-sity with a general barrier. In [7,8], the authors’ formulas contain an expected value of a Brownian function. The formulations in [7] are hard to see how the density for a
general barrier is evaluated. Although the expected value in [8] can be evaluated by solving a partial differential equation, using a numerical method to compute the value is still not easy. In [9], the density function for parabolic barriers was expressed analytically in terms of Airy func-tions. In this article, we derive new exact formulations for the hitting density with a general barrier. Thus, partial differential equation (PDE) techniques may be applied to evaluate the density function of the first hitting time. Let
,u x t be the probability density of
Btx t, h B00
;i.e. u x t
, dxP
Bt
x x, d ,x t
h B0 0
. It willbe shown that u x t
, is the solution of an initial-boundary value problem of a heat equation, and the hit-
ting density is
0
,
2 0
h
u h t t x
h
. Our derivation re-sults a simple integral equation for the density function. In Section 2, we show that the density function of the first hitting time can be evaluated though solving an ini-tial-boundary value problem of the heat equation. Then, the density function will be the solution of a simple inte-gral equation. In Section 3, a couple of examples are solved by PDE techniques to demonstrate the justifica-tion of the new method. The last secjustifica-tion is the conclu-sion.
2. The Boundary Value Problem
Let Bt, B00, a standard Brownian motion, h t
be asmooth function and h inf ;
t Bth t
0 0h
the first
hit-ting time. We consider first. Let p x t
, de-note the probability P
Btx t, h
, and u x t
, de- note the density function p x t
, . Surely,t
,
u x t ful-
fills the heat equation, uxx2 t ([10], p. 352). Never-theless, in this section, it will be proven with another way.
To derive the formulations for , we consider the
hitting problem at discrete times
, whereu
,u x t
t t1, , ,2 tn
i
t t i
n
. Let p xi
denote
, ; 1, 2,
i j
t t j
B x B h t j i
P , ,
and u xi
denote
d d p xi
x . Therefore,
1 ,
i
h t i
u x
G x ui d (1)where
2 2
, 1 2π e
x t
G x t
and t t n . Taking
the limit, we have
lim n ,
np x p x t (2) and
lim n ,
nu x u x t (3)
We will show that satisfies the heat equation.
Integrating Equation (1), we have
,
u x t
2 2 1 1 e d 2π x x h it i
p x u x
t
i d , (4)for x<h t
i1 . The probability difference between twosteps is
2 2 2 2 1 2 2 2 21 e d d
2π 1
e d d
2π
1 e d
2π 1
e d d .
2π i i i i i x x h t
t i x x t i x x h t
t i i x x t i h t
p x p x
u x
t
u x x
t
u u x
t
u x x
t dx
(5)Using a substitution, v x , we have
2 2 1 2 2 1 e d 2π 1e d d
2π i i v a v t i i v x t i a v
p x p x
u x u x v x v t
u x x v t
d . where
,
if
, if
i
i i
x v x h t
a v
v h t v x h t
Consequently,
2 2 2 2 1 2 2 2 2 e d 2π 1e d d
2π 1
e d d
2π
1
e d d
2π
i i
v
a v i i
t v x t i a v v a v t i a v v v
x t
i a v
p x p x t
u x u x v
v d x v v t t v
u x x v
v
t t
v
u x x v
v
t t
v
u x x v
v t t
(6)Note that, if g v
is continuous and bounded
2
0 1
lim e d 0 ,
π
v t t
g v v g
t
([11], p. 9) and, therefore, if g v
1
2
0 2
lim e d 0 .
π
v t t
v
g v v g
t t
Let
, i1 d , if 0.
i a
i a v v
u x v
g v
v u x x v
v
f 0The function g v
is continuous and bounded. More- over, g v
is differentiable, since
0 2 0 2 0 2 3 2 0 1 0 lim 1 lim d 1 lim d 1 1 lim2 2 d
i v
a v
i i
a v v v x i i x v v i i v
g g v u x
v
u x u x x
v
u x u x x
v
v
u x O v u x
x v
dright-hand side of Equation (6) is 1 d
2 d u xi
x
. The
second term is 0, because xh t
i and, therefore,when
a v x v is small. Letting approach 0,
we have
t
, 1
,2 x
p x t u x t
t
. (7)
Since p x t
, u x t
t
,
, differentiating both sidesof Equation (7) with respect to x, we have the partial
differential equation
, 1
,2
t xx
u x t u x t . (8)
The barrier is assumed to be differentiable, and,
therefore, there exists an positive number not
de-pending on such that
h t
t
h t
i1 h ti t.Con-sider the probability density near the boundary
i1h t
2
2 2
1
2
2 2
2 2
1
e d
2π 1
e e e
2π
i i
i i
i
i i
h t s t h t
t i
h t s h t
s t h t
t
i
u h t s t
u t
u t
d ,
(9)
where s. Note that
2
2 0
1 1
lim e d
2 2π
i i
h t t
i h t
t t f f h t
When
h t
i1 h tis t
t
,
1 1
0
1
lim .
2
i i i i
t u h t u h t
Consequently,
0
0
0
1
lim lim .
2 n
n n
t u h t n u h t
We have the boundary condition
,
0u h t t . (10)
Therefore, we have a proposition as follows:
Proposition 1
The density function is subject to the initial-
boundary value problem:
, u x t
, 2
,xx t
u x t u x t ,
,
(11a)
,
0u h t t (11b)
,0
,u x x (11c)
where
x is the Dirac delta function. The initial-boundary value problem is mathematically well-post. The hitting probability p th
,
1 h t
,h
p t
u x t d . (12) xThen, the hitting density dh
t
d d
d , ,
d
h h
h t
d t p t
t
h t u h t t u x t x
t t
d .(13)
Substituting Equations (8) and (10) into Equation (13), we have
1
, d . 2h t
h xx
d t u x t x
Using integration by parts, we have
1
, .2
h x
d t u h t t
(14)Similarly, if h
0 0, the hitting density will be
1 ,
2ux h t t
, d ,
. There is an integral equation for the
boundary values of a heat equation ([12], p. 219).
0
0
, , 0 , 0, , d
, , ,
, , , ,
h
t x
u h t t u G h t t
u h G h h t t
u h G h h t t
(15)
where
2
2
1
, , , e
2π
x t
G x t H t
t
and
H t is the Heaviside step function. For problem (11),
the integral equation becomes
0,0, ,
0t2 h
, ,
, d
0,G h t t
d G h h t t or
2
2
2 2
0
2 e d 1 e
2π 2π
h t h
. x
t t
t h
d
t t
(16)Equation (16) can be solved by a numerical method easily.
3. Examples
Example 1: Linear boundaries.
Let h t
a bt with . The initial-boundaryvalue problem (11) has a close-form solution. The solution, which is a Green’s function for the boundary
0
a
2 2 2 2 2 2 1, e e e
2π
x a x
ab
t t
u x t
t .
Thus, the hitting density dh
t is
2 2 2 2 2 2 2 1 , 21 e e e
2 2π
e 2π
a bt a bt
ab
t t
a bt t
u h t t x a bt a bt t t t a t t
consistent with that in [8].
Example 2: First-passage time probability in an
inter-val.
Let and be the den-
sity function for
a 0inf ; ,
a t B a
t
da
t . In this example, we evaluate the
probability density of the first passage time da
t .Let a
inf
; t B0, a Bt0
and
t
d
be the density of a t
. Using l
, ;u x t
and ur
x t, ;
to denote the probability density
Bt x t, a Bt0
x
P for a and a re-
spectively, we have an initial-boundary value problem
for both functions, and . From the proposition,
and fulfill the equations l
u ur
l
u ur
2
0 2u x t, 2 u x t, t t
t x
, 0, 0u a t tt
and
,0
.u x t x
The solutions are
2 2 0 0 2 2 2 0 1, ; e e
2π
x x a
t t t t
l
u x t H a x
t t and
2 2 0 0 2 2 2 0 1, ; e e .
2π
x x a
t t t t
r
u x t H x a
t t
The density function is
1
, ; , if
2
1 , ; , if .
2 l
t
r
u a t a
x d
u a t a
x Thus
2 0 2 0 0 e . 2π a t t t a dt t t t
The density of a t has to be
0 0 0 d d d a t at a t
d t d u
d u d u
,
(17)where u0
is the density function of Bt0 ; i.e.
2 0 2 0 01 e .
2π t u t
The integral in Equation (17) can be calculated.
2 2 0 0 2 2 0 2 2 0 2 20 0 0
2 2
0 0 0
2 2 d 2 e d 2π 1 e d π 1
e e π 1 erf ,
π 2 a
t
x a x
a t t t
x a a
a t t t
a a t d u x a x t t t t t
x a x
t t t t t
a a t t
where t0
t
and
0 1
2t
. Similarly,
2 2 0 2 2 d1 e e π 1 erf ,
π 2 t a a a t d u a a t t
Therefore, the hitting rate
2
2 2
2 3
1 e 1 e erf
π π 2 a a t a a
d t a
t .
In the special case of a0,
00 1
.
π 2 π
a
t
d t
t t t t t
The probability of that the Brownian motion process takes on the value 0 at least once in the interval
t0,t is0
0 0
0 π
π t t
This r is th
2
d arccos .
t t
t t
esult e same as in ([13], p. 191). In a simple
case of t00,
2 2 3 2π a t e . a t athe density of the first hitting time for a constant barrier.
problem
rrier numerically by using the integral Equa-tio
lt o-d
. Let d Brownian
motion, a sm h surface an
ime. If
This is the well-known result of
4. Conclusions
The proposition proposed in Section 2 may offer a simple way to evaluate the density of the hitting time with a general barrier by solving an initial-boundary value
of the heat equation. The density function
,u x t locally satisfies the heat equation is well-known
[10]. The main contribution of this paper is the derivation of the boundary condition (10). This result makes pro-gress in the evaluation of hitting time density. Two ex-amples with exact solutions are demonstrated in Section 3. Even though the examples may be solved by other method, the new formulations in this paper can be ap-plied to evaluate the hitting time distribution with any smooth ba
n (16).
A similar resu for tw imensional problems may be
expected and 2
t
B be two standar
1
t
B
,t h x y oot d
1 2
inf ;t h B t, B
h t t
the first hitting t u x y t
, ,
presents theprobabil-ity densprobabil-ity of
1, 2
,t t
B B x y for th, the
two-di-mensional formulations may be as follows,
2 xx yy
u u ut
,
(18a)
, , ,
0u x h x t t
ay be evaluated by an analytical or numerical method.
] F. Black and M. Scholes, “The Pricing of Options and
(18b)
1
2
0 0
, ,0 .
u x y xB yB (18c)
As long as Equation (18) is established, the probability density of the first hitting time for two-dimensional Brownian motion m
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