In this paper, we are concerned with Reflected Geometric Brownian Motion (RGBM) with two barriers. And the stationary distribution of RGBM is derived by Markovian infinitesimal generator method. Consequently the first passage time of RGBM is also discussed.
1. Introduction
We consider a finite-capacity fluid queue, the level of which at time 
is denoted by
. And 
satisfies the following differential equation:
(1)
This model shows fluid arrives into this queue at rate 
and leaves the queue at rate
. This fluid level can be also varied by a local variance function 
and a standard Brownian motion
. 
and 
are nondecreasing processes, interfering only when 
hits a or d and make 
vary between a and d.
In particular, when 
and
,
and 
disappear. Then the process 
becomes Geometric Brownian Motion. So we call 
determined by (1) Reflected Geometric Brownian Motion(RGBM).
Speak precisely, we are concerned with RGBM 
with two barriers a and d (d > a > 0), which is defined by
(2)
where 
is a standard Brownian Motion, 
and 
are constants and satisfy
.
Moreover, the processes L and U are uniquely determined by the following property [1,2]:
1) Both L and U are continuous nondecreasing processes with
;
2) L and U increase only when 
and
; respectively, i.e.,
According to the theory of stochastic differential equation, (2) is equivalent to
(3)
Such a process is a regenerative Markov process with state space [a,d] compact. Then it has a unique stationary distribution [1,3,4]. In the coming section, our objective is to derive the stationary distribution and give an expression for the Laplace Transform of the first passage time of RGBM 
by the method in [5-7].
2. Main Results on RGBM
2.1. On the Stationary Distribution of RGBM
In this section, we firstly give a Lemma on the stationary distribution of the reflected process 
with two-sided barriers and omit its proof.
Lemma 2.1 Let Z be the RGBM defined by (2) (or(3)). Then, as a Markov process, the stationary distribution 
of the process must satisfy the following equation
(4)
where 
and 
which denotes the space of all bounded continuous functions having twicely continuous derivatives on [a,d].
Proof. See similar argument in [1].
Suppose 
be a probability distribution on [a,d] and satisfies that
(5)
for
.
Define 
then by (4) and (5) it is equivalent to the following equation (Note that
)
(6)
where 
and 
On one hand, 
and 
can be computed by the same method in [5].
Proposition 2.1 Choose 
and
, then they respectively satisfy the following equations,
Then we have
Proof. A straightforward calculation.
On the other hand, since 
satisfies that for all
,
By twice integral changes, the above equation becomes that
i.e.,
(7)
Assume that
, satisfying that
, 
, and 
satisfies 
and
, then it follows from (7) that
(8)
Summarizing the discussion, we get the following theorem.
Theorem 2.1 
is the solution of
Then for all 
satisfying
, 
, (5) holds, i.e.,
.
Furthermore (5) holds for all
. This implies that 
is a stationary distribution of the corresponding Markov process
.
Remark 2.1 This theorem is a standard application of renewal theorems, so we sketch its proof.
Thus 
is the density of the stationary distribution of RGBM. Finally we will give an expression for the Laplace transform of the first passage time of RGBM.
2.2. On the First Passage Time of RGBM
In this section, we consider equation (2). Let
, define the first passage time by 
with the usual convention
. On the other hand, suppose
, for
, define a operator
Finally we are going to give the expression of the Laplace transform of
.
Theorem 2.2. For 
and
, then
(10)
(11)
where
and
Proof. Let
for
. Then applying 
formula for
, we have
(12)
The last equation holds, for 
and 
increase only when 
and
. Let 
be a stopping time and
. It follows from martingale optional theorem, that
(13)
In particular, take 
for
, and note that
and
Then
(14)
and
(15)
Replace 
by 
in (14) and by 
in (15), we immediately get (10) and (11) by
, 
and
. Thus the Proof of the theorem is completed.
3. Conclusions
This paper studies Reflected Geometric Brownian Motion (RGBM) with two barriers. Both the stationary distribution and Laplace transform of the first passage time of RGBM are derived. The studies for RGBM not only have practical significance, but also give an important result in theory of stochastic process.
4. Acknowledgements
This research is supported by the National Natural Science foundation of China (Grant No.70671074) and the Research Foundation of Tianjin university of Science and technology (Grant No.20080207). The authors would like to thank an anonymous referee for his constructive comments and suggestions on the first version of the manuscript.
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