An Evaluation for the Probability Density of the First Hitting Time ()
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 
be a smooth function for
, and 
a standard Brownian motion. The first hitting time 
is defined as
. It is known that 
has a continuous density [4]. Sometime we call the function 
or the curve 
the barrier. For a constant barrier, the result has been wellknown for a long time. In this case, the density is
([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 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 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 
be the probability density of
;
i.e.
. It will be shown that 
is the solution of an initialboundary value problem of a heat equation, and the hitting density is
. Our derivation results 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.
2. The Boundary Value Problem
Let Bt, 
, a standard Brownian motion, 
be a smooth function and 
the first hitting time. We consider 
first. Let 
denote the probability
, and 
denote the density function
. Surely, 
fulfills the heat equation, 
([10], p. 352). Nevertheless, in this section, it will be proven with another way. To derive the formulations for
, we consider the hitting problem at discrete times
, where
. Let 
denote
and 
denote
. Therefore,
(1)
where 
and
. Taking the limit, we have
(2)
and
(3)
We will show that 
satisfies the heat equation. Integrating Equation (1), we have
(4)
for
. The probability difference between two steps is
(5)
Using a substitution, 
, we have
where
Consequently,
(6)
Note that, if 
is continuous and bounded
([11], p. 9) and, therefore, if 
Let
The function 
is continuous and bounded. Moreover, 
is differentiable, since
Thus, when 
approaches 0, the first term of the right-hand side of Equation (6) is
. The second term is 0, because 
and, therefore, 
when 
is small. Letting 
approach 0, we have
(7)
Since
, differentiating both sides of Equation (7) with respect to
, we have the partial differential equation
(8)
The barrier 
is assumed to be differentiable, and, therefore, there exists an positive number 
not depending on 
such that
. Consider the probability density near the boundary 
(9)
where
. Note that
When
,
Consequently,
We have the boundary condition
(10)
Therefore, we have a proposition as follows:
Proposition 1
The density function 
is subject to the initialboundary value problem:
(11a)
(11b)
(11c)
where 
is the Dirac delta function. The initialboundary value problem is mathematically well-post. The hitting probability
,
(12)
Then, the hitting density 
(13)
Substituting Equations (8) and (10) into Equation (13), we have
Using integration by parts, we have
(14)
Similarly, if
, the hitting density will be
. There is an integral equation for the boundary values of a heat equation ([12], p. 219).
(15)
where 
and
is the Heaviside step function. For problem (11), the integral equation becomes
or
(16)
Equation (16) can be solved by a numerical method easily.
3. Examples
Example 1: Linear boundaries.
Let 
with
. The initial-boundary value problem (11) has a close-form solution. The solution, which is a Green’s function for the boundary
, is
Thus, the hitting density 
is
consistent with that in [8].
Example 2: First-passage time probability in an interval.
Let 
and 
be the density function for
. In this example, we evaluate the probability density of the first passage time
.
Let 
and 
be the density of
. Using 
and 
to denote the probability density
for 
and 
respectively, we have an initial-boundary value problem for both functions, 
and
. From the proposition, 
and 
fulfill the equations
and
The solutions are
and
The density function is
Thus
The density of 
has to be
(17)
where 
is the density function of
; i.e.
The integral in Equation (17) can be calculated.
where 
and
. Similarly,
Therefore, the hitting rate
In the special case of
,
The probability of that the Brownian motion process takes on the value 0 at least once in the interval 
is
This result is the same as in ([13], p. 191). In a simple case of
,
This is the well-known result of the density of the first hitting time for a constant barrier.
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 
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 
and 
be two standard Brownian motion, 
a smooth surface and
the first hitting time. If 
presents the probability density of 
for
, the two-dimensional formulations may be as follows,
(18a)
(18b)
(18c)
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.
NOTES