Applied Mathematics, 2013, 4, 792-796
http://dx.doi.org/10.4236/am.2013.45108 Published Online May 2013 (http://www.scirp.org/journal/am)
An Evaluation for the Probability Density
of the First Hitting Time
Shih-Yu Shen1*, Yi-Long Hsiao2
1
Department of Mathematics, National Cheng-Kung University, Tainan, Chinese Taipei
Department of Finance, National Dong Hwa University, Hualien County, Chinese Taipei
Email: *[email protected]
2
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  h  inf  ; B  h   the first hitting time. In
this paper, new formulations are derived to evaluate the probability density of the first hitting time. If u  x, t  denotes
the density function of x  Bt for t   h , then u xx  2ut and u  h  t  , t   0 . Moreover, the hitting time density
1
u x  h  t  , t  . Applying some partial differential equation techniques, we derive a simple integral equation
2
for d h  t  . Two examples are demonstrated in this article.
d h  t  is
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 calculations for the first hitting time get important in the field of
finance recently [3].
Let h  t  be a smooth function for t  0 , and B 
Bt ; t  0 a standard Brownian motion. The first hitting
time  h is defined as inf t ; Bt  h  t  B0  0 . It is
known that  h has a continuous density [4]. Sometime
we call the function h  t  or the curve x  h  t  the
barrier. For a constant barrier, the result has been wellknown for a long time. In this case, the density is
h
 h2 
exp    ([5], Chapter 9). The distribution of the
2πt 3
 2t 


hitting time for non-constant barrier was considered by
many authors; for example in [6-9]. In [6], Cuzick developed an asymptotic estimate for the first hitting density 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
*
Corresponding author.
Copyright © 2013 SciRes.
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 functions. 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  Bt  x, t  h B0  0;
i.e. u  x, t  dx  P  Bt   x, x  dx  , t   h B0  0. It will
be shown that u  x, t  is the solution of an initialboundary value problem of a heat equation, and the hith  0  
ting density is
u  h  t  , t  . Our derivation re2 h  0  x
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 initial-boundary value problem of the heat equation. Then,
the density function will be the solution of a simple integral equation. In Section 3, a couple of examples are
solved by PDE techniques to demonstrate the justification of the new method. The last section is the conclusion.
AM
S.-Y. SHEN, Y.-L. HSIAO
pi 1  x   pi  x 
2. The Boundary Value Problem
Let Bt, B0  0 , a standard Brownian motion, h  t  be a
smooth function and  h  inf t ; Bt  h  t  the first hitting time. We consider h  0   0 first. Let p  x, t  denote the probability P  Bt  x, t   h  , and u  x, t  de
note the density function
p  x, t  . Surely, u  x, t  fult
fills the heat equation, u xx  2ut ([10], p. 352). Nevertheless, in this section, it will be proven with another way.
To derive the formulations for u  x, t  , we consider the
hitting problem at discrete times t1 , t2 , , tn  , where
t
ti  i . Let pi  x  denote
n



P Bti  x, Bt j  h  t j  ; j  1, 2,, i ,
h ti 
ui 1  x    G  x,   ui   d
where G  x,    1 2πt e

 x  2
  

lim pn  x   p  x, t 
x
 
v2

1
e 2 t ui  x  dxdv
2πt
where
 x,
a v  
v  h  ti  ,

a  v  u  x   ui  x  v 
v
 
e 2 t  i
dxdv
v
t 2πt
v2
v2

v
1 x
 
e 2 t a v ui  x  dxdv
v 
t 2πt

(1)
(2)
v2


 
v2

v
1 x
e 2 t  ui  x  dxdv
v a v 
t 2πt
Note that, if g  v  is continuous and bounded

1
e
2πt
h i 
pi 1  x    
x
 x  2
2 t
ui   d dx ,
(4)
pi 1  x   pi  x 
h  ti 
  

1
e
2πt

 ui  x  
x
h  ti 
x
  
x

  h t
 i
 x  2
2 t
1
e
2πt

1
e
2πt
 x  2

1
e
2πt
2 t
2 t
d dx
ui  x  d dx .
Using a substitution, v  x   , we have
Copyright © 2013 SciRes.
([11], p. 9) and, therefore, if g  v    1

lim 
v2
2v  t
e g  v  dv  g   0  .
t πt
Let
ui  x  ,

g v  1 a
 a v v  v  ui  x  dx ,
v
if v  0
if v  0.
1
 g  v   ui  x  
v
1 a v 
 lim 2 a v vui  x   ui  x  dx
 
v 0 v
x
1
 lim 2 x vui  x   ui  x  dx
v 0 v
 1 d
1
v2
 lim 2  ui  x   O v 3  
ui  x 
v 0 v
2

 2 dx
g   0   lim
 ui    ui  x   d dx
 x  2
2 t
 x  2
v2
1 t
e g  v  dv  g  0  ,
πt
The function g  v  is continuous and bounded. Moreover, g  v  is differentiable, since
ui   d dx

t 0 
t 0 
for x < h  ti 1  . The probability difference between two
steps is
x
lim 
(3)
We will show that u  x, t  satisfies the heat equation.
Integrating Equation (1), we have
(6)

v
1 a v 
e 2 t a v vui  x  dxdv

v  
t 2πt


lim un  x   u  x, t 
if v  x  h  ti  .
pi 1  x   pi  x  t
and
n 
if v  x  h  ti 
Consequently,
the limit, we have
n 
v2

1

e 2 t  ui  x   ui  x  v   dxdv
2πt
  a v
and t  t n . Taking
2 t
a v 


d
pi  x  . Therefore,
dx
and ui  x  denote
793
(5)
v 0
 
Thus, when t approaches 0, the first term of the
AM
S.-Y. SHEN, Y.-L. HSIAO
794
1 d
ui  x  . The
2 dx
second term is 0, because x  h  ti  and, therefore,
a  v   x when v is small. Letting t approach 0,
we have
right-hand side of Equation (6) is 

1
p  x, t   u x  x, t  .
t
2
where   x  is the Dirac delta function. The initialboundary value problem is mathematically well-post.
The hitting probability ph  t  ,
h t 
ph  t   1   u  x, t  dx.
Then, the hitting density d h  t 
(7)

Since
p  x, t   u  x, t  , differentiating both sides
t
of Equation (7) with respect to x , we have the partial
differential equation
1
ut  x, t   u xx  x, t  .
2
(8)
d
ph  t 
dt
h t  
d
  h  t  u  h  t  , t   
u  x, t  dx.
t
dt
dh  t  
h t 
d h  t    
 
e

s 2 t
2
 h ti  st  2
2 t

1
e
2πt
h ti 

ui    d
 h ti  2 
e


2 t
(9)
2 s  h ti   
2

ui    d ,


1
dh  t    ux  h  t  , t  .
2
lim 

1
e
2πt
When s  t  
(14)
Similarly, if h  0   0 , the hitting density will be
1
u x  h  t  , t  . There is an integral equation for the
2
boundary values of a heat equation ([12], p. 219).
h 0 
u  h  t  , t    u  , 0  G   , 0, h  t  , t  d
where s   . Note that
t 0 h ti 
1
u xx  x, t  dx.
2
Using integration by parts, we have
ui 1  h  ti   st 

1
e
2πt
 0 u x  h   ,  G   h   , , h  t  , t 
t
 h ti  2

2 t
f   d 
h  ti 1   h  ti 
t
(13)
Substituting Equations (8) and (10) into Equation (13),
we have
The barrier h  t  is assumed to be differentiable, and,
therefore, there exists an positive number  not depending on t such that h  ti 1   h  ti   t . Consider the probability density near the boundary h  ti 1 
h  ti 
(12)
(15)
 u  h   ,  G  h   , , h  t  , t  d ,
1
f  h  ti  
2
where G  ,  , x, t  
1

,
2π  t   
e

 x  2
2 t  
H  t    and
H  t  is the Heaviside step function. For problem (11),
the integral equation becomes
1
lim ui 1  h  ti 1    ui  h  ti   .
2
t 0
Consequently,
G   0, 0, h  t  , t   0 2d h   G   h   , , h  t  , t  d  0,
t
n
1
lim un  h  tn    lim   u0  h  t0   .
n   2 
t 0
or
We have the boundary condition
u  h  t  , t   0.
t
(10)
Therefore, we have a proposition as follows:
Proposition 1
The density function u  x, t  is subject to the initialboundary value problem:
u xx  x, t   2ut  x, t  ,
(11a)
u  h  t  , t   0,
(11b)
u  x, 0     x  ,
(11c)
Copyright © 2013 SciRes.
0
2
2π  t   
e

 h t  h  2
2 t  
d h   d 
x2
1  2t
e . (16)
2πt
Equation (16) can be solved by a numerical method
easily.
3. Examples
Example 1: Linear boundaries.
Let h  t   a  bt with a  0 . The initial-boundary
value problem (11) has a close-form solution. The
solution, which is a Green’s function for the boundary
x  a  bt , is
AM
S.-Y. SHEN, Y.-L. HSIAO
 x  2 a 2
x2

1   2t
2 ab
e  e e 2t
u  x, t  
2πt 


.


dt   
Thus, the hitting density d h  t  is
1 
u  h t  , t 
2 x
 a  bt 2
 a bt 2

1  a  bt  2t
2 ab  a  bt 
e
e 2t

e
t
2 2πt  t

2π  t  t0 
e
   a 2
2 t  t 0 
.






  dt    u0    d  a dt   u0   d ,
a
(17)
where u0   is the density function of Bt0   ; i.e.
2
 a bt 2

1
u0   
e 2 t0 .
2πt0
2t

Let  a     inf  ;  t0 , B  a Bt0  



2

u  x, t   2 u  x , t  t  t0
2
x
t

2  x  a 
a
 
2π

1
π

a2
where  
dx
 x   a 2  a 2
2   t  t0  2 t

dx

 a2
 ,

 a2
 ,
t0
1 
and  
. Similarly,
2t0
t



Therefore, the hitting rate
 x  2 a  
   x  

 e 2  t  t0   e 2  t  t 0 
2π  t  t0  

2
1

 H a  x


and
 x  2 a   2
   x  2

2
t
t
2 t t



e
0
 e  0
u  x, t ;   

2π  t  t0  


 H  x  a.


The density function is
 1  l
 2 x u  a, t ;   ,
d    
 1  u r  a, t ;   ,
 2 x
  x  a  e

2

1
a

e 2t e   π a 1  erf
π 2t t
a2
The solutions are
1
e
 x  a 2   x 2
2  t  t0  2 t0
a
 t  t0  t 0  t  t 0 
da  t  
2
 t  t0  t0  t  t0 


2
1

e 2t e  a  π a 1  erf
π 2t t
u  x, t 0     x    .
Copyright © 2013 SciRes.
a

and
Thus
 dt   u0   d
a dt   u0   d
u  a, t   0, t  t0

t
The integral in Equation (17) can be calculated.
and dt  
be the density of  a  t . Using u l  x, t ;   and u r  x, t ;  
to denote the probability density

P Bt  x, t   a Bt0   for   a and   a rex
spectively, we have an initial-boundary value problem
for both functions, u l and u r . From the proposition,
u l and u r fulfill the equations
r
 t  t0 

d a  t    dt    u0    d
consistent with that in [8].
Example 2: First-passage time probability in an interval.
Let  a  inf  ;  t0 , B  a and d a  t  be the density function for  a  t . In this example, we evaluate the
probability density of the first passage time d a  t  .
u l  x, t ;   
 a
The density of  a  t has to be


a
e

t 2πt
795
if   a
if   a.
1
2t 3
e

a2
2t
 1  a 2
a

e
erf

π
π



 
 a 2  .
In the special case of a  0 ,
da  t  
t0
1
.

π 2t t π t  t0 t
The probability of that the Brownian motion process
takes on the value 0 at least once in the interval  t0 ,t 
is
 t 
t0
t
2
t0 π   t d  π arccos  t0  .
0


This result is the same as in ([13], p. 191). In a simple
case of t0  0 ,
da  t  
a
2πt 3
e

a2
2t
.
AM
S.-Y. SHEN, Y.-L. HSIAO
796
Corporate Liabilities,” Journal of Political Economy, Vol.
81, No. 3, 1973, pp. 637-659. doi:10.1086/260062
This is the well-known result of the density of the first
hitting time for a constant barrier.
[2]
R. C. Merton, “Theory of Rational Option Pricing,” Bell
Journal of Economics and Management Science, Vol. 4,
No. 1, 1973, pp. 141-183. doi:10.2307/3003143
[3]
C. Profeta, B. Roynette and M. Yor, “Option Prices as
Probabilities,” Springer, New York, 2010.
doi:10.1007/978-3-642-10395-7
[4]
B. Ferebee, “The Tangent Approximation to One-Sided
Brownian Exit Densities,” Z. Wahrscheinlichkeitsth, Vol.
61, No. 3, 1982, pp. 309-326. doi:10.1007/BF00539832
[5]
G. R. Grimmett and D. R. Stirzaker, “Probability and
Random Processes,” Oxford University Press, New York,
1982.
[6]
J. Cuzick, “Boundary Crossing Probabilities for Stationary Gaussian Processes and Brownian Motion,” Transactions of the American Mathematical Society, Vol. 263, No.
2, 1981, pp. 469-492.
doi:10.1090/S0002-9947-1981-0594420-5
[7]
J. Durbin, “The First-Passage Density of a Continuous
Gaussian Process to a General Boundary,” Journal of Applied Probability, Vol. 22, No. 1, 1985, pp. 99-122.
doi:10.2307/3213751
[8]
P. Salminen, “On the First Hitting Time and the Last Exit
Time for a Brownian Motion to/from a Moving Boundary,” Advances in Applied Probability, Vol. 20, No. 2,
1988, pp. 411-426. doi:10.2307/1427397
[9]
A. Martin-Löf, “The Final Size of a Nearly Critical Epidemic and the First Passage Time of a Wiener Process to
a Parabolic Barrier,” Journal of Applied Probability, Vol.
35, No. 3, 1998, pp. 671-682.
doi:10.1239/jap/1032265215
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
problem 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 progress in the evaluation of hitting time density. Two examples 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 applied to evaluate the hitting time distribution with any
smooth barrier numerically by using the integral Equation (16).
A similar result for two-dimensional problems may be
expected. Let Bt1 and Bt2 be two standard Brownian
motion, h  x, t   y a smooth surface and

 h  inf t ; h  Bt1 , t   Bt2

the first hitting time. If u  x, y, t  presents the probability density of Bt1 , Bt2   x, y  for t   h , the two-dimensional formulations may be as follows,
(18a)
u xx  u yy  2ut


u  x, h  x, t  , t   0,

 
(18b)

u  x, y, 0    x  B01  y  B02 .
(18c)
[10] L. Breiman, “Probability,” SIAM, Philadelphia, 1992.
doi:10.1137/1.9781611971286
As long as Equation (18) is established, the probability
density of the first hitting time for two-dimensional
Brownian motion may be evaluated by an analytical or
numerical method.
[11] J. Kevorkian, “Partial Differential Equations: Analytical
Solution Techniques,” Wadsworth, Belmont, 1990.
REFERENCES
[13] S. M. Ross, “Stochastic Processes,” John Wiley & Sons,
New York, 1983.
[1]
[12] F. John, “Partial Differential Equations,” 4th Edition,
Springer-Verlag, New York, 1982.
F. Black and M. Scholes, “The Pricing of Options and
Copyright © 2013 SciRes.
AM