An Evaluation for the Probability Density of the First Hitting Time

Let be a smooth function, a standard Brownian motion and   h t t B     inf ; h B h       the first hitting time. In this 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 x B  for h t   , then 2 xx u t u  and     , 0 t t u h  . Moreover, the hitting time density is   t h d     1 , 2 x u h t t . Applying some partial differential equation techniques, we derive a simple integral equation for   h d t . Two examples are demonstrated in this article.

 

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].

 
x h t  the barrier.For a constant barrier, the result has been wellknown for a long time.In this case, the density is Chapter 9).The distribution of the hitting time for non-constant barrier was considered by many authors; for example in [6][7][8][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

 
, u x t be the probability density of   0 , 0 . It will be shown that   , u x t is the solution of an initialboundary value problem of a heat equation, and the hit-  .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 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.

The Boundary Value Problem
where the limit, we have and We will show that satisfies the heat equation.Integrating Equation (1), we have where The function   g v is continuous and bounded.More- Thus, when t  approaches 0, the first term of the right-hand side of Equation ( 6) is . The second term is 0, because , , differentiating both sides of Equation ( 7) with respect to x , we have the partial differential equation 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 where s   .Note that We have the boundary condition Therefore, we have a proposition as follows: Proposition 1 The density function is subject to the initialboundary value problem: where   x  is the Dirac delta function.The initialboundary value problem is mathematically well-post.The hitting probability Then, the hitting density Substituting Equations ( 8) and ( 10) into Equation ( 13), we have Using integration by parts, we have . There is an integral equation for the boundary values of a heat equation ( [12], p. 219). where

 
H t is the Heaviside step function.For problem (11), the integral equation becomes Equation ( 16) can be solved by a numerical method easily.

Examples
Example 1: Linear boundaries.Let

  h t a bt
  with .The initial-boundary value problem (11) has a close-form solution.The solution, which is a Green's function for the boundary 1 , e e e 2π x a x ab t t u x t t Thus, the hitting density  [8].
Example 2: First-passage time probability in an interval.
Let and be the density function for a In this example, we evaluate the probability density of the first passage time spectively, we have an initial-boundary value problem for both functions, and .From the proposition, and fulfill the equations The solutions are The density function is The density of a t   has to be The integral in Equation ( 17) can be calculated.
In the special case of The probability of that the Brownian motion process takes on the value 0 at least once in the interval   This r is th

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 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 ba n (16).
A similar resu for tw imensional problems may be expected and i

]
F. Black and M. Scholes, "The Pricing of Options and Equation (18) is established, the probability density of the first hitting time for two-dimensional Brownian motion m