Statistics and Probability Letters 83 (2013) 1046–1053
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Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro
On the trivariate joint distribution of Brownian motion and
its maximum and minimum
ByoungSeon Choi a,∗ , JeongHo Roh b
a
Department of Economics, Seoul National University, Seoul, 151-742, Republic of Korea
b
Department of Physics, Seoul National University, Republic of Korea
article
info
Article history:
Received 17 October 2012
Accepted 17 December 2012
Available online 22 December 2012
Keywords:
Brownian motion
Maximum and minimum
Joint probability distribution
Jacobi’s triple product identity
abstract
The trivariate joint probability density function of Brownian motion and its maximum and
minimum can be expressed as an infinite series of normal probability density functions.
In this letter, we show that the infinite series converges uniformly, and satisfies the
Fokker–Planck equation. Also, we express it as a product form using Jacobi’s triple product
identity, and present some error bounds of a finite series approximation of the infinite
series.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Consider a standard Brownian motion {Wt |t ≥ 0} with W0 = 0. Denote its maximum and minimum, respectively, by
lt = min0≤s≤t Ws and ut = max0≤s≤t Ws . It is known that the trivariate joint distribution of (Wt , lt , ut ) is expressed as
P (a ≤ lt ≤ ut ≤ b, Wt ∈ dx) =
∞ 

√
1
2π t
k=−∞
− √
1
2π t

exp −

exp −
1
2t
1
2t
{x − 2k(b − a)}2

{x − 2b − 2k(b − a)}2

dx,
(1)
where a ≤ 0 ≤ b. This equation and its variants are found in the literature of probability such as Bachelier (1901,
pp. 192–194), Lévy (1948, p. 213), Darling and Siegert (1953), Cox and Miller (1965, p. 222), Freedman (1971, pp. 26–7),
Feller (1971, p. 341), Borodin and Salminen (2002, p. 174) etc. The purposes of this note are to show uniform convergence
of the infinite series of Eq. (2), to show that it is a solution to the Fokker–Planck equation, to present some approximations
of the trivariate joint probability density function, and to analyze their error bounds.
2. Uniform convergence
For a fixed t ∈ (0, ∞), define two sequences of functions {qk (x; t )|k = · · · , −1, 0, 1, . . .} and {rk (x; t )|k = · · · , −1, 0,
1, . . .}, respectively, by
∗
Corresponding author. Tel.: +82 2 880 6394; fax: +82 2 886 4231.
E-mail addresses: [email protected] (B. Choi), [email protected] (J. Roh).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2012.12.015
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
qk (x; t ) = √

1
exp −
2π t
1
2t

{x − 2k(b − a)}2 ,
1047
( a ≤ x ≤ b)
and


1
2
exp − {x − 2b − 2k(b − a)} ,
1
rk (x; t ) = √
2π t
(a ≤ x ≤ b).
2t
Clearly, qk (x; t ) and rk (x; t ) are positive for each k. Eq. (1) becomes
P (a ≤ lt ≤ ut ≤ b, Wt ∈ dx) =
∞

{qk (x; t ) − rk (x; t )} dx.
(2)
k=−∞
It can be shown that
qk+1 (x; t ) = β(x)α k qk (x; t ),
(k = · · · , −1, 0, 1, . . .),




4(b−a)2
and β(x) = exp 2t (b − a)(x − b + a) . It can be driven from Eq. (3) that
where α = exp − t
qk (x; t ) = β k (x)α
where q0 (x; t ) = √ 1
2π t
lim
qk+1 (x : t )
k→∞
qk (x; t )
 
k
2
q0 (x; t ),
(k = · · · , −1, 0, 1, . . .),
 


k
k(k−1)
exp − 2t1 x2 and 2 = 2 . Eq. (3) implies
(3)
(4)
= lim β(x)α k = 0.
k→∞
The ratio test indicates that
) converges for any x ∈ [a, b]. For each
k=1 qk (x; t
∞k ≥ 1, qk (x; t ) is increasing on [a, b],
∞
and then, 0 < qk (x; t ) ≤ qk (b; t ). Since
k=1 qk (b; t ) is convergent, the series
k=1 qk (x; t ) converges uniformly on the
compact set [a, b]. For details of this uniform convergence, readers may refer to Rudin (1976, p. 148). Similarly, we can
−1
∞
show that k=−∞ qk (x; t ) converges uniformly on the compact set [a, b]. Therefore, the series k=−∞ qk (x; t ) converges
uniformly on [a, b]. We know from Eq. (4) that
∞
∞

qk (x; t ) = q0 (x; t )
∞

β k (x)α
 
k
2
,
(5)
k=−∞
k=−∞
which can be represented by Jacobi’s triple product identity (see, e.g., Zwillinger, 2003, p. 48) as follows.
∞

qk (x; t ) = q0 (x; t )
k=−∞
∞ 


1−α
j


1+
j=1
1
β(x)
α
j

1 + β(x)α

j −1


.
(6)
It can be shown that
rk+1 (x; t ) = γ (x)α k rk (x; t ),
(k = · · · , −1, 0, 1, . . .),
 4

where γ (x) = β(x) exp − t b(b − a) . It can be driven from Eq. (7) that
rk (x; t ) = γ k (x)α
(7)
 
k
2
r0 (x; t ), (k = · · · , −1, 0, 1, . . .),
(8)



∞
where r0 (x; t ) = √ 1 exp − 2t1 [x − 2b]2 . Applying the same method as before, we can show that the series k=−∞ rk (x; t )
2π t
converges uniformly on [a, b], and that its sum is
∞

rk (x; t ) = r0 (x; t )
k=−∞
∞

γ k (x)α
 
k
2
,
(9)
k=−∞
which can be expressed as
∞

k=−∞
rk (x; t ) = r0 (x; t )
∞ 


1 − αj
j=1


1+
1
γ (x)
αj

1 + γ (x)α j−1



.
(10)
1048
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
Theorem 1. For integers M and N satisfying M + N ≥ 0, let
N

S−M ,N (x, t ) =
{qk (x; t ) − rk (x; t )} ,
k=−M
where a ≤ x ≤ b and t > 0. For a fixed t, as M → ∞ and N → ∞, S−M ,N (x, t ) converges uniformly to S−∞,∞ (x, t ) on the set
{a ≤ x ≤ b}, which is equal to
∞

{qk (x; t ) − rk (x; t )} =
k=−∞
∞

α
 
k
2

q0 (x; t )β k (x) − r0 (x; t )γ k (x) .

k=−∞
This infinite sum can be also expressed as
q0 (x; t )
∞ 


1 − αj


1+
j =1
1
β(x)
αj


1 + β(x)α j−1


− r0 (x; t )
∞ 


1 − αj


1+
j =1
1
γ (x)
αj


1 + γ (x)α j−1


. Consider the Fokker–Planck equation
∂ f (x, t )
1 ∂ 2 f (x, t )
=
.
∂t
2 ∂ x2
(11)
It is known (see, e.g., Cox and Miller, 1965, p. 222) that S−∞,∞ (x, t ) =
k=−∞ {qk (x; t ) − rk (x; t )} satisfies the
Fokker–Planck
equation
(11).
To
prove
it
minutely,
we
need
to
show
that
the
orders
of
summation and differential operators
∞
of k=−∞ {qk (x; t ) − rk (x; t )} can be exchangeable, i.e., the infinite series is differentiable term by term. However, as far as
the authors know, it has not been proven before. It can be easily shown that, for each integer k, qk (x; t ) and rk (x; t ) satisfy
the Fokker–Planck equation (11), i.e.,
∞
∂ qk (x; t )
1 ∂ 2 qk (x; t )
∂ rk (x; t )
1 ∂ 2 r k ( x; t )
=
and
=
.
(12)
∂t
2
∂ x2
∂t
2
∂ x2
Thus, qk (x; t ) − rk (x; t ) is also a solution to the Fokker–Planck equation (11), and so is a linear superposition S−M ,N (x, t ) for
any nonnegative integers M and N. It is shown in Appendix using uniform convergence that
1 ∂ 2 S−∞,∞ (x, t )
2
∂
x2
=
1 ∂2
2∂
x2


∂
{qk (x; t ) − rk (x; t )} =
∂t
k=−∞
∞


∞


{qk (x; t ) − rk (x; t )} =
k=−∞
∂ S−∞,∞ (x, t )
.
∂t
(13)
Also, it is shown as in Appendix that, for any t ∈ (0, ∞),
S−∞,∞ (a, t ) = 0
S−∞,∞ (b, t ) = 0.
and
Theorem 2. The infinite series S−∞,∞ (x, t ) =
and t > 0, i.e.,
(14)
∞
k=−∞
{qk (x; t ) − rk (x; t )} satisfies the Fokker–Planck equation on a < x < b
∂ S−∞,∞ (x, t )
1 ∂ 2 S−∞,∞ (x, t )
=
,
∂t
2
∂ x2
and the boundary conditions are S−∞,∞ (a, t ) = 0 and S−∞,∞ (b, t ) = 0 for any t > 0.
3. Approximation and error bound
It can be shown that
rk (x; t ) = η(x)δ k qk (x; t ),

where δ = exp −
4b(b−a)
t

(k = · · · , −1, 0, 1, . . .),
(15)
 2b

and η(x) = exp t (x − b) . Eqs. (3) and (15) imply the following equalities hold for each
integer k;
qk (x; t ) − rk (x; t ) = 1 − η(x)δ k qk (x; t ) =



1
η(x)

δ −k − 1 rk (x; t ),
rk (x; t ) − qk+1 (x; t ) = η(x)δ k − β(x)α k qk (x; t ).


(16)
(17)
It is clear that, for any x ∈ [a, b],
0 < α < δ < 1,
0 < β(x) < η(x) ≤ 1,
0 < γ (x) < 1.
(18)
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
1049
The following inequalities hold by Eqs. (16)–(18);
· · · ≥ qk (x : t ) ≥ rk (x : t ) ≥ qk+1 (x; t ) ≥ rk+1 (x; t ) ≥ · · · ,
(k = 1, 2, . . .),
(19)
· · · ≥ rk+1 (x; t ) ≥ qk+1 (x; t ) ≥ rk (x : t ) ≥ qk (x : t ) ≥ · · · ,
(k = −1, −2, . . .).
(20)
For any integers M and N satisfying M + N ≥ 0, define the error term for an approximation S−M ,N (x, t ) of S−∞,∞ (x, t )
by ϵ−M ,N (x, t ) = S−∞,∞ (x, t ) − S−M ,N (x, t ). For positive integers m and n, define the following functions;
RN ,0 (x, t ) = qN +1 (x; t ),
N +n

RN ,n (x, t ) = qN +1 (x; t ) −
{rk (x; t ) − qk+1 (x; t )} ,
k=N +1
R−M ,0 (x, t ) = r−M −1 (x; t ),
−
M −1
R−M ,−m (x, t ) = r−M −1 (x; t ) −
{qk (x; t ) − rk−1 (x; t )} .
k=−M −m
The following inequalities hold by Eqs. (19) and (20).
RN ,0 (x, t ) ≥ RN ,1 (x, t ) ≥ RN ,2 (x, t ) ≥ · · · ≥ RN ,∞ (x, t ) > 0,
(21)
R−M ,0 (x, t ) ≥ R−M ,−1 (x, t ) ≥ R−M ,−2 (x, t ) ≥ · · · ≥ R−M ,−∞ (x, t ) > 0.
(22)
Eq. (16) implies that

 ∂ qk (x; t )
∂
{qk (x; t ) − rk (x; t )} = −η′ (x)δ k qk (x; t ) + 1 − η(x)δ k
∂x
∂x


2b
1
= − δ k η(x)qk (x; t ) −
1 − η(x)δ k {x − 2k(b − a)} qk (x; t ).
t
t


Let dk (x) = 2bη(x)δ k + 1 − η(x)δ k {x − 2k(b − a)}. Then, Eq. (23) can be written as
(23)
∂
qk (x; t )
{qk (x; t ) − rk (x; t )} = −
dk (x).
∂x
t
It can be shown that dk (x) < 0 if k > K+ = max
(24)


, 4bt(lnb−2a) and x ∈ [a, b]. Hence, qk (x; t ) − rk (x; t ) is increasing on
[a, b] for k > K+ . Also, for any k > 0, qk (x; t ) is increasing on [a, b]. Thus, the following proposition holds:
N > K+ , n ≥ 0,
3b
2(b−a)
and a ≤ x ≤ b ⇒ RN ,n (a, t ) ≤ RN ,n (x, t ) ≤ RN ,n (b, t ).
It can be also shown that dk (x) < 0 if k < K− = min

a−4b
2(b−a)
, − 4bt(lnb−2a) −
1
2

(25)
and x ∈ [a, b]. Hence, rk (x; t ) − qk (x; t ) is
decreasing on [a, b] for k < K− . Also, for any k < 0, rk (x; t ) is decreasing on [a, b]. Thus, the following proposition holds:
M > −K − , m ≥ 0 ,
a ≤ x ≤ b ⇒ R−M ,−m (b, t ) ≤ R−M ,−m (x, t ) ≤ R−M ,−m (a, t ).
and
(26)
We now summarize properties of the error bounds of S−M ,N (x, t ) as follows.
Theorem 3. For M > −K− , N > K+ , m ≥ 0, n ≥ 0, x ∈ [a, b], and t > 0, the following inequalities hold:
RN ,n (a, t ) ≤ RN ,n (x, t ) ≤ RN ,n (b, t ),
R−M ,−m (b, t ) ≤ R−M ,−m (x, t ) ≤ R−M ,−m (a, t ),
RN ,n (a, t ) − R−M ,−m (a, t ) ≤ RN ,n (x, t ) − R−M ,−m (x, t ) ≤ RN ,n (b, t ) − R−M ,−m (b, t ),
RN ,∞ (a, t ) − R−M ,−∞ (a, t ) ≤ ϵ−M ,N (x, t ) ≤ RN ,∞ (b, t ) − R−M ,−∞ (b, t ),
−R−M ,−m (a, t ) ≤ −R−M ,∞ (a, t ) ≤ ϵ−M ,N (x, t ) ≤ RN ,∞ (b, t ) ≤ RN ,n (b, t ),




ϵ−M ,N (x, t ) ≤ max R−M ,−m (a, t ), RN ,n (b, t ) . Jacobi’s triple product identity representation in Theorem 1 implies that
S−∞,∞ (x, t ) = JL (x, t ) + O α L ,
 
(27)
where another approximation JL (x, t ) of S−∞,∞ (x, t ) is defined by
q0 (x; t )
L 


j =1
1 − αj


1+
1
β(x)
αj

1 + β(x)α j−1



− r0 (x; t )
L 


j =1
1 − αj


1+
1
γ (x)
αj

1 + γ (x)α j−1



.
1050
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
Example 1. We know that, for N ≥ 0,
RN ,0 (b, t ) = qN +1 (b; t ) = β N +1 (b)α

N +1
2

q0 (b; t ) ≤ α

N +1
2

√
1
2π t
≤ αN
2 /2
√
1
2π t
.
Hence, the following proposition holds.
N >

1
b−a

 √
t
− ln ϵ 2π t ⇒ RN ,0 (b, t ) < ϵ.
2
Also, we know that, for M ≥ 1,
R−M ,0 (a, t ) = r−M −1 (a; t ) = γ −M −1 (a)α

≤
3/2
α
γ ( a)
M +1
α


−M −1

M +2
−3(M +1)/2
2
2

r0 (a; t )
1
2
r0 (a; t ) ≤ α (M −1) /2 √
,
2π t
where the first inequality holds because


α 3/2
2
< 1.
=
exp
(
b
−
a
)
a
γ M + 1 ( a)
t
Hence, the following proposition holds.
M −1>
1
b−a

 √

t
− ln ϵ 2π t ⇒ R−M ,0 (a, t ) < ϵ.
2
Theorem 3 implies the following proposition.
min {M − 1, N } >
1

b−a
 √



t
− ln ϵ 2π t ⇒ ϵ−M ,N (x, t ) < ϵ.
2
As an example, let a = −1, b = 2, t = 2, x = 0.5 and ϵ = 10−15 , then
1
b−a

 √

t
− ln ϵ 2π t = 1.9228.
2
Thus, we may choose M = 3 and N = 2. The asymptotic values are as follows.
(−M , N ) S−M ,N (0.5, 2)
(0, 0)
0.071034228403985
(−1, 1)
0.054397289152109
(−2, 2)
0.054397288013575
(−3, 3)
0.054397288013575
(−M , N )
(−1, 0)
(−2, 1)
(−3, 2)
(−4, 3)
S−M ,N (0.5, 2)
0.054355942725271
0.054397288013575
0.054397288013575
0.054397288013575
We know from the above table that a pair of orders (−M , N ) = (−2, 1) is good enough to obtain a finite series approximate
value with absolute error less than 10−15 .
Let L = ln α/ ln ϵ . Then, L = 1.9188. The asymptotic values Jl (0.5, 2) of Jacob’s triple product identity representation are
as follows.
l
Jl (0.5, 2)
0 0.071034228403985
1
0.054397289405489
2 0.054397288013575
3
0.054397288013575
4 0.054397288013575
5
0.054397288013575
l
Jl (0.5, 2)
We know from the above table that an order L = 2 is good enough to obtain an approximate value of Jacob’s triple product
identity representation with absolute error less than 10−15 .
4. Conclusion
In this note, it is shown that the infinite series of the trivariate joint probability density function of Brownian motion and
its maximum and minimum converges uniformly, and that it satisfies the Fokker–Planck equation. Also, the joint density
function is represented through Jacobi’s triple product identity. Moreover, some properties of error bounds to approximate
the infinite series by a finite series are presented.
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
1051
Acknowledgment
The authors wish to acknowledge a reviewer and an associate editor for their comments and suggestions. Financial
support from the Institute for Research in Finance and Economics of Seoul National University is gratefully acknowledged.
Appendix
For a real number m, let fm (x, t ) = √ 1
exp − 2t1 (x − m)2 . Then, the following holds;

2π t

t − (x − m)2
∂ fm
=−
fm (x, t ),
∂t
2t 2
∂ 2 fm
(x − m)4 − 6t (x − m)2 + 3t 2
=
fm (x, t ),
∂t2
4t 4
∂ 2 fm
t − (x − m)2
=
−
fm (x, t ),
∂ x2
t2


(x − m) 3t − (x − m)2
∂ 3 fm
=
fm (x, t ).
∂ x3
t3
(A.1)
(A.2)
(A.3)
(A.4)
√
3t
Proof of Eq. (13). Let t be a positive constant. If k > b2+
, then Eq. (A.4) implies that
(b−a)
∂ 2 qk (x;t )
∂ x2
∂ 3 qk (x;t )
∂ x3
is positive on [a, b], and that
is increasing on [a, b]. Thus, we know that, for any x ∈ [a, b],
∂ 2 qk (x; t )
∂ 2 qk (b; t )
t − {b − 2k(b − a)}2
<
=
−
qk (b; t ),
∂ x2
∂ x2
t2
0<
(A.5)
where the equality holds by Eq. (A.3). Eq. (3) implies
− t −{b−2(kt+21)(b−a)} qk+1 (b; t )
2
lim
− t −{b−2kt 2(b−a)} qk (b; t )
2
k→∞
The ratio test and Eq. (A.3) indicate that
= 0.
∞
k=1
(A.6)
∂ 2 qk (b;t )
∂ x2
converges. Thus, Eq. (A.6) implies
on the compact set [a, b]. Similarly, we can show that
so does
∂ 2 qk (x;t )
.
k=−∞
∂ x2
∞
∂2
∂ x2

∞


qk (x; t )
=
k=−∞

∞


rk (x; t )
k=−∞

 
2
3
k=1
∂ 2 qk (x;t )
∂ x2
converges uniformly
converges uniformly on the compact set [a, b], and
∞

∂ 2 qk (x; t )
.
∂ x2
k=−∞
=
∞
k=−∞
(A.7)
∂ 2 rk (x;t )
∂ x2
converges uniformly on the compact set [a, b], and that
∞

∂ 2 rk (x; t )
.
∂ x2
k=−∞
Let x ∈ (a, b) be fixed. For m > 0, Eq. (A.2) implies that the equation
and t = t2,m = 1 −
∞
Therefore, the following equality holds.
Using the same method we can show that
∂2
∂ x2
∂ 2 qk (x;t )
k=−∞
∂ x2
 −1
(A.8)
∂ 2 fm (x,t )
∂t2
 

= 0 holds at t = t1,m = 1 + 23 (x − m)2




(x − m)2 . Therefore,  ∂ fm∂(tx,t )  has its supremum at one of the points {0, t1 , t2 , ∞}. Eq. (A.1) implies
the following equalities;




t − (x − m)2 
 ∂ fm 
 = lim
f (x, t ) = 0,
t →0 ∂ t 
t →0
2t 2




t − (x − m)2 
 ∂ fm 


= lim
f (x, t ) = 0,
lim
t →∞  ∂ t 
t →∞
2t 2
lim 
(A.9)
(A.10)
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B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053




 ∂ f x, t  
 t − (x − m)2 

1 ,m 
 m
 1,m
f x, t1,m 

 = 




∂t
2t12,m


√
−1
2/3
1

,
= √ 
√ 
√ 5/2 exp
|
x
−
m|3
2
1
+
2
/
3
2 2π 1 + 2/3




 ∂ f x, t  
 t − (x − m)2 

2 ,m 
 m
 2,m
f x, t2,m 

 = 




∂t
2t22,m


√
−1
1
2/3

= √ 
.
√ 
√ 5/2 exp
|x − m|3
2 1 − 2/3
2 2π 1 − 2/3
(A.11)
(A.12)
Eqs. (A.9)–(A.12) imply that there exists a constant c ∈ (0, ∞) satisfying


 ∂ fm (x, t ) 
1
≤c

.
(A.13)
max
0<t <∞ 
∂t 
|x − m|3
∞

∂ fm (x,t )
−3
Since
< ∞ by the integral test, Eq. (A.13) implies ∞
converges uniformly on the set
m=−∞ |x − m|
m=1
∂t
{0 < 
t < ∞}. For this uniform convergence property, readers may refer to Rudin (1976, p. 152). Similarly, it can be shown
∞
−1
∂ fm (x,t )
∂ fm (x,t )
converges uniformly on the set {0 < t < ∞}. So does
. This uniform convergence
that
m=−∞
m=−∞
∂t
∂t
implies
∞

∂ fk (x, t )
∂
=
∂t
∂t
k=−∞

∞


f k ( x, t ) .
(A.14)
k=−∞
We know from Eq. (A.14) that the following equalities hold:
∞

∂ qk (x; t )
∂
=
∂
t
∂
t
k=−∞

∞

∂ r k ( x; t )
∂
=
∂t
∂t
k=−∞


∞

qk (x; t ) ,
(A.15)
k=−∞
∞


rk (x; t ) .
(A.16)
k=−∞
Eqs. (A.7), (A.8), (A.15) and (A.16) complete the proof.
Proof of Eq. (14). It can be shown that β(b)γ (b) = α . Thus,
1
β(b)
α j + β(b)α j−1 = γ (b)α j−1 +
1
γ (b)
αj.
(A.17)
Since q0 (b; t ) = r0 (b; t ), Eq. (A.16) and Theorem 1 yield S−∞,∞ (b, t ) = 0.
It can be shown that
∞

r k ( a; t ) = r 0 ( a; t )
k=−∞
∞

γ k (a)α
 
k
2
= r0 (a; t )
k=−∞
= r 0 ( a; t )
∞

∞

γ k−1 (a)α
= q0 (a; t )
k=−∞
k−1
2

k=−∞
β 1−k (a)α 2(k−1) α

k−1
2

= q0 (a; t )
k=−∞
∞


∞

β −k (a)α

−k

2
k=−∞
β k (a)α
 
k
2
=
∞

qk (a; t ),
(A.18)
k=−∞
where the first equality holds by Eq. (8), the third does by β(a)γ (a) = α 2 , and the fourth does by γ0 (a; t )β(a) = q0 (a; t )α .
Eq. (A.18) implies S−∞,∞ (a, t ) = 0. It completes the proof. References
Bachelier, L., 1901. Théorie mathématique du jeu. Ann. Sci. Éc. Norm. Supér. Ser. 3 18, 143–209.
Borodin, A.N., Salminen, P., 2002. Handbook of Brownian Motion – Facts and Formulae, second ed. Birkhäuser, Basel.
Cox, D.R., Miller, H.D., 1965. The Theory of Stochastic Processes. Chapman and Hall Ltd., London.
Darling, D.A., Siegert, A.J.F., 1953. The first passage problem for a continuous Markov process. Ann. Math. Stat. 24, 624–639.
Feller, W., 1971. An Introduction to Probability Theory and its Applications, second ed. John Wiley & Sons, Inc.
B. Choi, J. Roh / Statistics and Probability Letters 83 (2013) 1046–1053
Freedman, D., 1971. Brownian Motion and Diffusion. Holden-Day, San Francisco.
Lévy, P., 1948. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.
Rudin, W., 1976. Principles of Mathematical Analysis, third ed. McGraw-Hill, New York.
Zwillinger, D., 2003. CRC Standard Mathematical Tables and Formulae. Chapman & Hall/CRC, Boca Raton.
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