Thou Shalt Buy and Hold
y z x
Alb ert Shiryaev Zuo quan Xu Xun Yu Zhou
This Version August
Abstract
An investor holding a sto ck needs to decide when to sell it over a given investment
horizon It is tempting to think that she should sell at the maximum price over the
entire horizon which is however imp ossible to achieve A close yet realistic goal is to
sell the sto ck at a time when the exp ected relative error b etween the selling price and
the aforementioned maximum price is minimized This problem is investigated for a
Black Scholes market A sto ck go o dness index de ned to b e the ratio b etween
the excess return rate and the squared volatility rate is employed to measure the
quality of the sto ck It is shown that when the sto ck is go o d enough or to b e precise
the optimal strategy is to hold on to the sto ck selling only at the end when
of the horizon Moreover the resulting exp ected relative error diminishes to zero
when go es to in nity On the other hand one should sell the sto ck immediately if
< These results justify the widely accepted nancial wisdom that one should
buy and hold a sto ck if it is go o d that is
This research was motivated by the pap ers Shiryaev and Li and Zhou and was started
when Shiryaev was visiting Zhou in Hong Kong in May The formulation of the problem and partial
results were presented at Imp erial College London for the rst time in Octob er and subsequently at
the Quantitative Metho ds in Finance Conference in Sydney and the International Conference
on Mathematics of Finance and Related Applications in Hong Kong The full results do cumented in this
pap er had b een obtained by February and were presented at the seminars in Oxford Stanford and
ETH during March May as well as at the Second Annual Risk Management Conference in Singap ore
and the th Bachelier World Congress in London The pap er has b ene ted from comments of participants
at these conferences and seminars as well as those of the Editors in Chief for which the authors are
grateful The usual disclaimers applies Shiryaev acknowledges nancial supp ort from Grants RFBI
and YaF while Zhou acknowledges nancial supp ort from the RGC Earmarked
Grants CUHK and CUHK and a start up fund of the University of Oxford
y
Steklov Mathematical Institute Gubkina str Moscow Russia
z
Mathematical Institute The University of Oxford St Giles Oxford OX LB UK Email
xuz maths ox ac uk
x
Mathematical Institute The University of Oxford St Giles Oxford OX LB UK and De
partment of Systems Engineering and Engineering Management The Chinese University of Hong Kong
Shatin Hong Kong Email zhouxy maths ox ac uk
Key words Black Scholes market optimal stopping buy and hold sto ck go o dness
index value function
Intro duction
A conventional and widely accepted investment wisdom is the so called buy and hold rule
i e one should buy a go o d sto ck and leave it alone for a long time It is based on the
empirical observation that in the long run investing in a go o d company gives go o d rate
of return while the volatility which is short term in nature is insigni cant One of the
main theoretical foundations of the buy and hold rule is the e cient market hyp othesis
EMH if the market is e cient and the price is right at any given time then there is
no p oint to trade or to sell So you sell a sto ck b ecause something happ ens to you not
b ecause something happ ens to the market Another argument is based on transaction
costs including bid o er spread brokerage capital gains tax and so on Warren Bu ett
who clearly do es not b elieve in EMH b ecause his primary investment philosophy is to
nd under priced sto cks is a buy and hold advo cate b elonging to the transaction cost
scho ol Barb er and Odean examine the issue from the b ehavioural nance p oint
of view They argue that due to over con dence investors tend to trade excessively An
elab orative empirical study on US household investors reveals that active trading has led
to p o or p erformance hence trading is hazardous to your wealth
Perverse enough to our b est knowledge no well established dynamic investment mo del
has generally pro duced the pure buy and hold rule When transaction costs are ignored
the famous Merton p ortfolio Samuelson Merton via utility maximization re
quires trading continuously so as to keep a constant prop ortion in wealth among sto cks
The continuous time Markowitz mo del Richardson Zhou and Li or the recent
b ehavioural p ortfolio selection mo del Jin and Zhou b oth lead to similar continu
ously rebalancing optimal p ortfolios even in the Black Scholes case where there is only
one sto ck available When transaction costs are present trading is signi cantly reduced
in fact in this case there are buying selling and no trade regions and an optimal strategy
is to move the wealth pro cess to the no trade region as so on as p ossible and stay there as
far as p ossible Davis Panas and Zariphop oulou Liu and Lo ewenstein Dai
Xu and Zhou However such a strategy in general is still not exactly buy and hold
Consider an inno cent individual investor who has no knowledge whatso ever ab out
utility maximization or mean variance theories He just b ought a sto ck at the b eginning
1
Clearly buy and hold applies nontrivially only to a dynamic mo del For a single p erio d mo del every
strategy is buy and hold
2
See also the concluding section of this pap er and the intro duction section of Liu and Lo ewenstein
for an overview of the literature on this
of a year and for some reason e g to invest for a sp eci c event or an obligation must
sell it in one year s time Therefore he needs to decide the b est time to sell within the
year What would b e his criterion to evaluate the b est timing A simple and naive yet
natural and p erhaps dominating one for many small investors would b e to sell higher
But how high is higher It would b e ideal if the sto ck could b e sold exactly at the
maximum price over the entire year unfortunately this is mission imp ossible b ecause the
timing of the maximum price would b e known only at the end of the year Then how
ab out if he tries to sell at the price closest to the maximum This sounds sensible but
one needs to de ne precisely the meaning of the closeness An immediate measure
would b e the absolute di erence the absolute error or any p ower of the di erence
b etween the selling price and the maximum price except that such a measure would fail
to capture the scale of the prices themselves The scale will b e taken care of if we consider
the relative error In conclusion a natural criterion for the investor is to minimize the
exp ected relative error b etween the selling price and the maximum price
Motivated by the ab ove in this pap er we formulate and investigate the sto ck selling
problem for a Black Scholes market Although the relative error criterion involves the
maximum sto ck price over the entire investment horizon and hence is not adapted
we are able to transform the problem into a standard optimal stopping problem with a
terminal payo and an adapted state pro cess We intro duce a sto ck go o dness index
which is de ned to b e the ratio b etween the excess return rate and the squared volatility
rate to measure the quality of the sto ck Our main result is that one should sell at the
and sell end of the horizon if the go o dness index of the sto ck is greater than or equal to
immediately if the index is non p ositive The implication of this result is two fold
b eing a go o d or bad sto ck is precisely characterized by a critical level of the go o dness
index and the optimal selling strategy for a good sto ck is a pure buy and hold one
Moreover the optimal relative errors for b oth go o d and bad sto cks are explicitly derived
based on which sensitivity analysis on the market parameters is carried out In particular
it is shown that the exp ected relative error when one buys and holds diminshes to zero if
the sto ck go o dness index tends to in nity This implies that for a su ciently go o d sto ck
the buy and sell rule almost realizes selling at the ultimate maximum price
It is interesting to note that the buy and hold rule has long b een b elieved to b e the
antithesis of the market timing the notion that one can enter the market on the lows
and sell on the highs do es not work for small investors at least so it is b etter to just buy
3
Clearly the go o dness index is intimately related to the Sharp e ratio in fact it is the Sharp e ratio
further normalized by the volatility rate It turns out that this index plays a more prominent role in the
particular problem considered in this pap er
1
4
see the Indeed a stronger result holds that one should sell immediately if the index is less than
2
concluding section
and hold Our mo del and result however show that market timing attempting to sell
higher indeed leads to buy and hold
The initial imp etus to investigating this mo del was the combination of two pap ers by
Shiryaev and Li and Zhou which at the rst glance seemed to b e rather
unrelated The former investigates the quickest detection problem for a change of market
parameters whereas the latter reveals the high chance of a Markowitz mean variance
strategy hitting the exp ected return target The chemistry b etween the two pap ers has
nonetheless emerged somewhat unexp ectedly leading to the formulation and solution of
the present sto ck selling problem The mathematical analysis of the underlying optimal
stopping problem is related to Graversen Peskir and Shiryaev which app ears to
b e the rst pap er of its kind where the problem is to stop a Brownian motion so as to
minimize the squared error from the maximum
The remainder of the pap er is organized as follows The sto ck selling mo del is formu
lated and the main results presented in section Section is devoted to transforming
the mo del into a standard optimal stopping problem In sections and optimality for
various cases is proved using two di erent approaches while in section the optimal rel
ative errors are derived Section concludes with nal remarks Some technical details
are relegated to an app endix
A Sto ck Selling Problem
Consider a Black Scholes economy where there is a sto ck with an appreciation rate a and
a volatility rate along with a saving account with a continuously comp ounding
interest rate r The discounted sto ck price pro cess P follows
t
dP a r P dt P dB P
t t t t
on a standard ltered probability space F fF g P where B is a standard Brow
t t
nian motion with B under P Here fF g is the P augmentation of the ltration
t t
generated by B We can alternatively write
t B
t
P e
t
De ne the running maximum price pro cess where a r
M max P t
t s
6s6t
and a go o dness index of the sto ck
a r
An investor buys a share of the sto ck at time and for some reason she must sell the
sto ck by a pre sp eci ed date T The question is to determine the b est time to sell
In general we could de ne the following criterion for the optimal selling problem
P
Maximize E U
M
T
over T the set of all F stopping time T where U is certain utility function
t
In this pap er we investigate the sp ecial case when U is linear This is a particularly
interesting case b ecause it is equivalent to the following
M P
T
min E
T
M
T
which means that the investor wishes to minimize the exp ected relative error b etween the
discounted selling price and the discounted maximum price over the entire horizon
T
Before we solve let us consider with U x log x x This problem turns
out to have a simple solution To see this let for simplicity and write
B t B S max B
t
t t
s
6s6t
Then
P
sup E S R sup E log
T
M
T T
T
Thus the optimal stopping is trivially
T if
any time b etween T if
if
T if
any time b etween T if
if
So the solution is of a simple bang bang structure stop either at the b eginning or at the
Now if U is linear which is the problem we would like to solve in this pap er end
the ab ove argument fails and it is not clear what an optimal stopping time might b e In
this pap er via rather involved probabilistic analysis we shall solve the cases when
and and show that the optimal solution p ossesses similar bang bang structure
As a by pro duct Problem has bang bang solution for any p ower utility function
i h
M
5
T
inf E S Incidentally once may also consider a dual problem R inf E log
T T
T
P
R This problem has the same bang bang solution
U x x b ecause by mo difying appropriately the drift and volatility values the problem
is mathematically equivalent to the one with a linear utility function
Let denote the probability distribution function of a standard normal random
variable The main results of the pap er are as follows
Theorem i If then T is the unique optimal sel ling time to Problem
and either T or is optimal when Moreover when
the optimal expected relative error is given by
p p
T
e T T r
Furthermore r decreases in and increases in and
r
ii If then is the unique optimal sel ling time to Problem Moreover
the optimal relative error is given by
p p
T
T T r e
So when the sto ck go o dness index one should hold on to the sto ck selling only
at T This in turn implies that the sto ck must b e a go o d one The b etter the sto ck as
measured by the smaller the relative error the latter b eing sub ject to an upp er b ound
that is inversely prop ortional to In particular the error diminishes to zero when
go es to in nity This suggests that the buy and hold rule almost realizes selling at the
maximum price if the sto ck is su ciently go o d On the other hand if then one
should sell the sto ck immediately or short sell if p ossible This is a bad sto ck the investor
ought to get rid of as so on as p ossible
It is interesting to examine our results applied to some real data We take the one
in Mehra and Prescott based on S P The estimated parameters
are the following b oth are annual gures a r and In this
case by large margin Moreover if we take T year then it
follows from that r This means that if you buy and hold an S P
index fund for one year then statistically you are exp ected to achieve almost of
the maximum p ossible return
6
One might argue that it is not reasonable to mo del an index such as S P as a geometric
Brownian motion and hence the results in this pap er may not apply However the gures a r
and do app ear very plausible for a typical go o d sto ck On the other hand we exp ect that
our analysis and results in this pap er extend to the case of a market index which is nothing else than a
linear combination of geometric Brownian motions
An Equivalent Problem
To prove the part of the optimal selling times stated in Theorem we assume without
loss of generality that since by a change of time one can write B B where
t
t
B is a standard Browian motion
t
Recall the notation Problem is equivalent to with the following
stopping problem
B
e
sup E
S
T
e
T
S
T
is not The ab ove is not a standard optimal stopping problem b ecause the term e
F adapted To get around for any stopping time T we have
t
i h
max B B
B B
t
e e
S B
6t6T
E E E min e e
S max B
S
6t6T
t
T
e
maxfe e g
max B B
t
S B
6t6T
E E min e e F
oi h n
S
x
T
E E min e e
x S B
E G S B
where
oi h n
x S
T t
t x T G t x E min e e
Direct computations show for details see App endix A that when
x T t
T t
p
G t x e
T t
x T t x T t
x x
p p
e e
T t T t
and when
x T t
p
G t x x T t
T t
q
x T t
x T t
T t
x
T t
p
e e
T t
Equation implies that is actually a standard optimal stopping problem with
a terminal payo G and an underlying adapted state pro cess
X S B X
t
t t
the so called drawdown pro cess
In view of the dynamic programming approach we consider the following problem
E G t X V t x sup
t x t
T
T t
where X x under P with t x T given and xed and T in general
t t x s
denotes the set of all F stopping times s for s The original problem is
t
certainly
B
e
V sup E
S
T
e
T
It is well known that V satis es the following dynamic programming equation or
variational inequalities
minf LV V Gg t x T
V T x G T x x
V t t T
x
where the op erator L is de ned by
f t x Lf t x f t x f t x
xx t x
The holding region is therefore
C f t x T V t x G t x g
while the selling region is
D f t x T V t x G t x g
An optimal selling time is
inf ft T t S B D g
t t
So the problem b oils down to nding and or analyzing V
Noting that B has stationary indep endent increments and X is a Markovian pro cess
we may rewrite
x
V t x sup E G t X
6 6T t
where X under P is explicitly given as
x
X x S B t
t t
t
Optimal Selling Times When 1 and 0
In this section we derive optimal selling times for the cases when and
resp ectively via an approach that turns the problem with the terminal payo to one with
a running payo This approach is standard in the optimal stopping literature see e g
Peskir and Shiryaev To pro ceed note that
law
x
X jY j
where Y is the unique strong solution to the SDE
dY sign Y dt d B
t t t
Y x
The pro cess Y has an in nitesimal generator L de ned by By It o Tanaka s formula
we have
Z Z
s s
jY j x I Y du sign Y I Y dB Y
s u u u u
s
where Y is the lo cal time of Y at So
s
Z Z
s s
G t s jY j G t x LG t u jY j du G t u jY j sign Y dB
s u x u u u
Z
s
G t u jY j d Y
x u
u
Z
s
G t x H t u jY j du M
u s
where we have used the fact that G t for 6 t 6 T details in App endix C
x
and H and M are de ned resp ectively by
G t x H t x LG t x G t x G t x
xx t x
Z
s
M G t u jY j sign Y dB
s x u u u
Due to the de nition of G and b elow we have 6 G 6 G 6 so M is a
x
martingale and Problem can b e expressed as
x
V t x sup E G t X
6 6T t
sup E G t jY j
6 6T t
Z
G t x sup E H t u jY j du
u
6 6T t
Z
x
G t x sup E du H t u X
u
6 6T t
A lengthy calculation see App endix B shows that
H t x G t x G t x
x
recall we are assuming that In this case Now if then a r
noting that G 6 by the monotonicity of G in x we have
x
H t x G t x G t x > t x T
x
and the inequality is strict if This shows that an optimal stopping time to
Problem is T t and it is the unique optimal solution if In particular
the conclusion applies to the original problem with T
On the other hand if then Noting that
i h i h
x x x x S S
T t T t
g e G t x e E minfe e g E minf e
is strictly increasing with resp ect to x we have
x
e G t x
or G t x G t x
x
x
Thus
H t x G t x G t x G t x G t x G t x
x x
This indicates that the unique optimal stopping time to Problem is and
V t x G t x In particular the conclusion is valid for the original problem
1 Optimal Selling Time When
While the approach in the previous section is rather direct we have yet to cover the case
for which we now use a di erent technique to tackle when
Prop osition If then
V t x G t x t T x
whenever and
V t x G t x t T x
whenever
7
As will b e seen shortly the case which has b een already solved in the previous section can
also b e solved by the approach presented in this section
is equivalent to Also as b efore we assume herein Proof Note that
that without loss of generality The pro of consists of several steps
Step We rst consider the case when In this case we are to show that
x x
E G T X G x x and E G T X G x for x
T T
x
where X is de ned by To this end taking and in the general expression
x
x X
T
derived in App endix D we have of E G T X E e
T
x T x T
T x T x x
p p
e e E G T X
T
T T
On the other hand it follows from with that
x T x x
T x x
p p p
G x e e e
T T T
De ne
x x
g x e E G T X G x
T
x T x T
T x T
p p
e e
T T
x T x x
T x
p p p
e x
T T T
Then
x T
T x x
p
x g x e e
T
However it is straightforward to check that g proving
Step In this step we show that when
x
G x x E G T X
T
Indeed by de nitions
h i h i
x x S B x S
T T T
E G T X E e G x E e
T
Applying Girsanov s theorem we have
h i
x x S B x S
T T T
E G T X G x E e e
T
i h
B x S
T T
e E e
h i
Q Q Q
Q
x S B T B
T T T
e E e e
h i
B x S T B
T T T
e e E e
Q
t B is a standard Brownian motion under probability Q with where B B
t
t
T B
T
d P This leads to d Q e
T x x S B B
T T T
e E G T X G x E e e e
T
Hence
T x x S B B
T T T
E G T X G x E e e B e e
T
T
The desired inequality then follows from
Step By the arbitrariness of T and x in the strict inequality we can
prove in exactly the same way that
E G T X G t x t T x
t x T
where X x under P provided that Thus follows from the fact that
t t x
V t x E G T X Similarly is implied by
t x T
Now we return to the pro of of Theorem If or then the unique
optimality of T follows immediately from the preceding prop osition in view of the
de nition of the holding region
by the arbitrariness of T and x in we can prove in If or
exactly the same way that
E G T X G t x x and E G T X G t x for x
t x T t x T
Then for any T we have
x x
E G T X jF > G X x
T
or
x x
E G T X > E G X x
T
Since T is arbitrary in the ab ove we conclude
x x
V x sup E G X E G T X x
T
T
In particular applying we obtain
V E G T X G
T
This implies that b oth T and are optimal
1
8
In exactly the same way we can show that V t G t t T meaning that when
2
one should either sell at the end or sell whenever the drawdown state x in particular at time
Optimal Relative Errors
is a good sto ck Based on the proved results so far a sto ck with the go o dness index
b ecause one should hold on to it until the end and one with is a bad one since
one should sell it immediately In this section we complete the pro of of Theorem by
deriving the optimal exp ected relative errors for b oth a go o d and a bad sto ck Since we
need to investigate the sensitivity of the optimal relative errors in we allow to vary
instead of assuming throughout this section
Go o d sto ck
Recall
a r t B
t
P e M max P
t t u
6u6t
The joint probability density function of P M is given by
t t
ln m s ln m s
p
exp ln s t f t s m
sm t
t
where s 6 m m > and
a r
see e g p or p Now we compute for any t and y
Z Z
s y
P
t
P y f t s m dm d s
M
t
y s
Z Z
s y
ln m s ln m s
p
exp ln s t dm ds
sm t
t
y s
Z Z
s y
ln m s ln m s
t
p
dm ds exp s e
m t
t
s y
Z Z
s y
ln m s t
t
p
ds d exp s e
t
t
y s
Z
ln s s ln s y
t
p
exp s exp ds e
t t
t
y
Z Z
ln s ln s
t
p
e s exp ds y d s s exp
t t
t
y y
Z Z
u u
t
p
u du y exp u du exp e
t t
t
ln y ln y
t ln y t ln y
p p
y
t t
9
There is a typ o in p there should b e r instead of r
Consequently
Z
P P
t t
E y dy P
M M
t t
Z
t ln y t ln y
p p
y dy
t t
t ln y t ln y
Z Z Z Z
p p
t t
u u
p p
e du dy e du dy y
p
p
Z Z Z Z
t t u
t e
u u
p p
e dy du e y dy du
p
t t u
e
p
Z Z
t
p
t u u t u
p p
du e du e
p
t
p
Z
t
p
t t u u
p
e du e
p p
t
e t t
i h
P
t
strictly decreases in t Indeed if then by Next we prove that E
M
t
Lemma E
p
P
t
t t
p
e e t E
t M
t
t
t t t
p p
e e e
t t
If 6 then
p
P
t
t t
p
E t e e
t M
t
t
This establishes the strict monotonicity in t On the other hand when straight
forward computation leads to
P P
t t
lim E lim E
t t
M M
t t
we have the following Hence when or
M P
t t
E t
M
t
Moreover if > then
t ln y t ln y P
t
p p
y P y
M
t t
t
t ln y ln y
p
> y y
t
As a result
Z
P P
t t
P y dy > E
M M
t t
Meanwhile
t ln y t ln y P
t
p p
y y P
M
t t
t
t ln y
t ln y ln y ln y
t
p p
y e 6
t
t
Hence
Z
P P
t t
E P y dy 6
M M
t t
an optimal selling time is Returning to Problem we have proved that when
h i
P
T
T Hence the corresp onding optimal relative error is r E Noting
M
T
and we complete the pro of of Theorem i
Bad sto ck
First for y >
ln y P M y P max s B
t s
s t
ln y t ln y t
p p
y
t t
when Thus Notice that
Z Z
dy P y dy P M E
t
M M y
t t
Z Z Z
ln y ln y
u t u t
p
exp du y du d y exp
t t
t
Z Z Z Z
ln y ln y
u t u t
p p
du dy du dy y exp exp
t t
t t
Z Z Z Z
u
e
u t u t
p p
d y du d y du exp y exp
t t
u
t t
e
Z Z
u t u t
p
du exp u d u exp
t t
t
Z Z
u t u t
p
du exp u du exp
t t
t
p p
t
e t t
Now for Problem the optimal relative error is
M P
T
E r E
M M
T T
which is after some easy manipulations
Conclusions
This pap er formulates a nite horizon sto ck selling mo del as one minimzing the exp ected
relative error b etween the selling price and the maximum price over the horizon It is
shown that one should hold on to the sto ck until the end if the sto ck go o dness index is
while one should sell immediately if the index is no greater than Our no less than
results justify the classical maxim that one should buy and hold a go o d sto ck Indeed
the very fact that our mo del is able to pro duce the buy and hold rule suggests in turn
that the criterion prop osed in this pap er may b e a sensible one that warrants further
investigations in more general settings
It is intriguing to compare our result to that of Merton Merton s p ortfolio
in the Black Scholes setting without consumption for a utility function u x x
where is exactly our stipulates that the sto ck to wealth ratio should b e kept as
sto ck go o dness index also called the Merton line Hence Merton s strategy degenerates
to buy and hold only in a very exceptional case when which requires a co ordination
b etween the risk attitude of the agent and the market opp ortunities Of course such a
requirement is impractical b ecause all the parameters are estimates prone to sometimes
large errors
This brings ab out another advantage of our mo del Unlike with other standard p ort
folio selection mo dels including Merton s our optimal solutions are insensitive to the
market parameters Indeed our de nition of a go o d as well as a bad sto ck involves a
range of the parameters instead of sp eci c values for them As demonstrated by the S P
is satis ed by a large margin which would accom example the criterion that
is mo date su cient level of errors In general statistical terms verifying whether
much easier than estimating the value of itself Hence the notorious mean blur problem
is hardly an issue in our mo del esp ecially for very go o d sto cks
Finally one may have by now noticed that the case when the go o dness index is b etween
and has b een left unsolved in this pap er This gap can b e lled by a partial
di erential equation PDE approach and the result is that one should sell immediately if
For the pure PDE argument we refer to a companion work Dai Jin Zhong and
Zhou where a buying decision is incorp orated in addition to the selling one We
have chosen to present the probabilistic approach here at the cost of having a gap for
two reasons First it was indeed the approach we had liked and employed since the very
b eginning of our research Second it is our view that while solving the case
may b e mathematically interesting for the sake of completeness it is not as signi cant
and interesting nancially The reason is that the de nition of a go o d sto ck or
that the excess return rate is greater than or equal to half of the squared volatility is so
generous that it covers many of the sto cks commonly p erceived as go o d Indeed as
by a large margin is shown in section the S P has a go o dness index greater than
With a large set of go o d sto cks as p er our de nition available it is less interesting to
consider sto cks outside of this set
10
In fact the cases solved in this pap er could also b e treated by the PDE approach
11
Let us take the excess return rate to b e p a a very mo dest one for a typical go o d sto ck which
was estimated to b e the equity premium based on S P data in Mehra and Prescott Then
any sto ck whose annual volatility is less than will b e quali ed as a go o d sto ck according to
our de nition
App endix
A Function G
Here we derive the explicit expression of the function G de ned by
h n o i
x S
T t
G t x E min e e
Z
z x
e dP S 6 z e P S 6 x t T x
T t T t
x
Noting
z T t z T t
z
p p
e P S 6 z
T t
T t T t
we have
Z Z
z T t
z T t
z z
T t
p p
e d e e d z
T t
T t
x x
x T t
T t
p
e
T t
Assume for now that Then
Z
z T t
z z
p
e d e
T t
x
Z Z
z T t z T t
z z
p p
e d z e d
T t T t
x x
Z
z T t x T t
z x
p p
e d e
T t T t
x
Z
z T t
z
p
e d
T t
x
x T t x T t
x T t
p p
e e
T t T t
Hence
Z
z x
G t x e dP S 6 z e P S 6 x
T t T t
x
Z Z
z T t z T t
z z z
p p
e d e e d
T t T t
x x
x T t x T t
x x
p p
e e
T t T t
x T t
T t
p
e
T t
x T t x T t
x x
p p
e e
T t T t
This is The case when can b e dealt with similarly leading to Denoting
by G t x to highlight the dep endence on it is not hard to verify that in fact
lim G t x G t x
B Equations for H
We prove for the case when the other case b eing similar Write
x T t
T t
p
e G t x
T t
x T t x T t
x x
p p
e e
T t T t
A t x B t x C t x
Then
x T t
T t
T t
p
T t e A t x e
x
x T t
x T t
T t
T t
p
A t x e T t e
xx
T t
A t x x T t
x
x T t
T t
T t
p
A t x A t x e x T t T t e
t
A t x A t x x T t
x
Hence
A t x A t x A t x
xx t x
A t x A t x x T t A t x A t x x T t
x x x
A t x A t x
x
Next
x T t
x
T t
p
e T t B t x B t x e
x
x T t
x
T t
p
x T t B t x e T t e
t
B t x B t x x T t
x
x T t
x T t
x
T t
p
e T t B t x B t x e
xx x
T t
x T t
x
T t
p
e e T t
B t x B t x B t x x T t
x x
B t x B t x
x
x T t B t x x T t B t x
x
leading to
B t x B t x B t x
t x xx
x T t B t x B t x B t x
x x
x T t B t x x T t B t x
x
B t x B t x
x
Finally
x T t
x
T t
p
C t x C t x e T t e
x
x T t
x T t
x
T t
p
T t e C t x C t x e
xx x
T t
x T t
x
T t
p
e T t e
C t x C t x C t x x T t
x x
x T t
x
T t
p
e e T t x T t C t x
t
C t x C t x x T t
x
So
C t x C t x C t x x T t C t x C t x C t x
xx x x t x
C t x C t x C t x x T t
x x
C t x C t x
x
Since G is a linear combination of A B and C we conclude
C Pro of of G (t 0+) = 0
x
Again we only prove for the case when Following the calculation in the previous
subsection we have
A t x B t x C t x G t x
x x x x
x T t
T t
T t
p
T t e e
x T t
x
T t
p
e T t B t x e
x T t
x
T t
p
C t x e T t e
B t x C t x
Hence
G t B t C t
x
x
X
T
D Calculation of E e
x
We rst study the distribution function of X x S B t x For any
t t
t
y
x S B y
t t
P x S B 6 y P e 6 e
t t
Z Z Z Z
y
se
y B x S
t t
x y
e 6 e P e e f t s m dm ds f t s m dm ds
fe m6se g
x y
s e s
Z Z
y
se
ln m s ln m s
p
exp ln s t dm ds
sm t
x y
t
e s
Z Z
y
se
ln m s ln m s
t
p
s dm ds e exp
m t
x y
t
e s
Z Z
y
se
ln m s t
t
p
ds e d exp s
t
x y
t
e s
Z
y
ln se ln s s
t
p
exp d s e exp s
t t
x y
t
e
Z Z
y
ln se ln s
t
p
ds d s s exp s exp e
t t
x y x y
t
e e
Z Z
u u
y u t u
p
du e e exp du e e exp
t t
t
x y x y
Z Z
u t u t
y
p
du e exp du exp
t t
t
x y x y
x y t x y t
y
p p
e
t t
Therefore
Z
x
x
X y
T
E e e dP X 6 y
T
Z
x y T x y T
y y
p p
e e d
T T
Z
x y T T y
p
dy exp
T
t
Z Z
x y T x y T
y y y y
p p
e e d d e e
T T
Z
x T T x y T
T y x T
p p
e d e
T T
Z
x y T
y
p
d e
T
x T x T T
x T T
p p
e
T T
Z
x y T
y
p
d e
T
x T x T T
x T T
p p
e
T T
x T T
x T
p
e
T
Now supp ose then r
x T x T
x
T x X
T
p p
e E e
T T
x T
x T
p
e
T
E A Lemma
Lemma E If x then
x x
x
p p
x e e
x x
Proof This is evident by
Z Z
x x
t t x
t
p p p
x e dt e dt e
x x
and
Z Z
x x
t t
x
p p
x e dt e dt
t
Z
x
x t x
x x
p p p
x dt x x e e e
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