An Evaluation for the Probability Density of the First Hitting Time

Full text

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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 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 denotes

the density function of

 

,

u x t

t

xB for th, then uxx2ut and u h

 

t t,

0. Moreover, the hitting time density is

 

t

h

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 t0, and B

t a standard Brownian motion. The first hitting

time

; h B t0

 is defined as inf ;

t Bth t B

 

00

. It is

known 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 is

2

3exp 2

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 B00

;

i.e. u x t

 

, dxP

Bt

x x, d ,x t

h B0 0

. It will

be 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.

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2. The Boundary Value Problem

Let Bt, B00, a standard Brownian motion, h t

 

be a

smooth function and  h inf ;

t Bth t

 

 

0 0

h

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, uxx2 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

, where

u

,

u x t

t t1, , ,2tn

i

t t i

n

 . Let p xi

 

denote

 

, ; 1, 2,

i j

t t j

Bx Bh t ji

P , ,

and u xi

 

denote

 

d d p xi

x . Therefore,

 

 

   

1 ,

i

h t i

ux

 G xui  d (1)

where

 

 2 2

, 1 2π e

x t

G x t

  and  t t n . Taking

the limit, we have

 

 

lim n ,

np xp x t (2) and

 

 

lim n ,

nu xu 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 i

t i

p x u x

t          

 

i d , (4)

for x<h t

 

i1 . The probability difference between two

steps is

 

 

   

 

 

     

 

 

   

 

2 2 2 2 1 2 2 2 2

1 e d d

2π 1

e d d

1 e d

2π 1

e d d .

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 1

e d d

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 1

e d d

2π 1

e d d

1

e d d

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

 

 

 

   

, i

1 d , if 0.

i a

i a v v

u x v

g v

v u x x v

v        

f 0

The 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 lim

2 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                      

d

(3)

right-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 vx 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 sides

of Equation (7) with respect to x, we have the partial

differential equation

 

, 1

 

,

2

t xx

u x tu 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

 

 

i1

h t

 

    

 

        

 

2

2 2

1

2

2 2

2 2

1

e d

2π 1

e e e

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 ti

s t

t

 

 

 ,

 

 

1 1

0

1

lim .

2

i i i i

t uh tu 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

 

,

0

u 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 tu x t ,

,

(11a)

 

,

0

u 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) x

Then, 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 . 2

h 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

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,

Gh t t

dGh   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-boundary

value problem (11) has a close-form solution. The solution, which is a Green’s function for the boundary

0

a

(4)

 

  2 2 2 2 2 2 1

, e e e

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 , 2

1 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 0

inf ; ,

a t Ba

    

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, 0

u a ttt

and

,0

.

u x t  x

The solutions are

      

2 2 0 0 2 2 2 0 1

, ; e e

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 .

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 d

t t t t

         

The density of  a t has to be

 

   

   

   

0 0 0 d d d a t a

t 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 0

1 e .

t u t    

The integral in Equation (17) can be calculated.

   

 

     

 

     

 

2 2 0 0 2 2 0 2 2 0 2 2

0 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 d

1 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 a0,

 

0

0 1

.

π 2 π

a

t

d t

t tt t t

 

The probability of that the Brownian motion process takes on the value 0 at least once in the interval

 

t0,t is

0

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 t00,

 

2 2 3 2π a t e . a t a

(5)

the 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

htt

the first hitting t u x y t

, ,

presents the

probabil-ity densprobabil-ity of

1, 2

 

,

t t

B Bx y for th, the

two-di-mensional formulations may be as follows,

2 xx yy

uuut

,

(18a)

 

, , ,

0

u 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  xByB (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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[10] L. Breiman, “Probability,” SIAM, Philadelphia, 1992.

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Springer-Verlag, New York, 1982.

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