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DEWEY
HB31
.M415
working paper
department
of economics
ON THE IRRELEVANCE OF TRADE TIMING
Lones Smith
96-6
Feb.
1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ON THE IRRELEVANCE OF TRADE TIMING
Lones Smith
96-6
Feb.
1996
13 1996
96-6
On
the Irrelevance of Trade Timing*
Lones Smith^
Department of Economics
Massachusetts Institute of Technology
MA 02139
Cambridge,
February
1996
4,
Abstract
Given standard, transparent assumptions,
Street adage that 'timing
is
everything'.
I
this
paper questions the Wall
show that
for
a 'small' risk-neutral trader with private information that
dependent of the public information
is
an Arrow
is
security,
conditionally in-
exactly indifferent about the timing of
his trade: His expected return per dollar invested
is
a martingale. This
is
true
despite the fact that he expects the asset price itself to rise given favorable
information and
I
fall
given unfavorable information.
demonstrate the result
in generality
and point out that the Arrow security
assumption cannot be relaxed: With compound securities paying on two
there
is
states,
a (generically strict) preference to trade immediately, while for more
general assets, the result
is
ambivalent
— one may even wish to delay trading!
*I owe special thanks to Massimiliano Amarantefor proving a key inequality (A). I am also grateful to
Glenn Ellison for inquiring of an economic interpretation of the 'conditional martingale' used in an early
version of Smith and S0rensen (1996), and to Angel Serrat Tubert for helping me clarify my thoughts.
I have likewise profited from comments on the Usenet group sci. math. research
including Dan Cass
(S.J.F.C. in Rochester), and especially Drazen Pantic (University of Belgrade) for extending the proof
of the timing irrelevance proposition to nondiscrete measures. Finally, I thank Peter DeMarzo, Thomas
Piketty, Paul Samuelson, Peter Sorensen, Jiang Wiang, and Richard Zeckhauser for helpful feedback. I
acknowledge the NSF for financial support. Any errors and opinions expressed are mine alone.
^e-mail address: lonesQlones.mit.edu
—
1.
'Timing
is
everything'
has remarkably
little
is
to say
INTRODUCTION
a popular refrain on Wall Street. Yet the finance literature
about a topic that
is
deemed
of
supreme importance among
This paper attempts to redress this anomaly, by inquiring of the
applied practitioners.
As
purely informational motives for trade timing.
such,
I
When
trader with no strategic reasons to time his trade.
consider the problem of a small
his 'informational edge' over
is
the market maximized?
To
fix ideas,
consider the following hypothetical scenario.
A
corporate insider of a
biotech firm has gotten wind of an announcement next Friday of the unexpected
The
approval of his firm's soon-to-be major drug.
this information.
1
Of
insider naturally wishes to profit
FDA
from
course, in theory there are obscenely large arbitrage opportunities:
Indeed, by buying on slim enough margins, the insider can expect to reap tremendous
So suppose
profits.
that will ring truer
amount
of
for
whatever reason that he cannot buy on margin
when
I
consider only imperfect information. 2 Let
money with which
to invest,
that his purchases do not affect the price.
restriction
him have a given
by buying stocks, and assume he
When
—a
is
poor enough
should he purchase? Typical gut instincts
argue in favor of buying immediately. Indeed, at one obvious
level,
delay only sees the
depreciation of his informational advantage, as he expects the price to start rising. But
that
is
not the issue here, as the trader cares not so
much about
the asset price as
its
reciprocal, the return (or his 'bang per buck').
By
this yardstick,
I
now wish
to
make the
ent assumptions, so long as his timing
completely irrelevant^.
To
is
in
advance of the public announcement,
'low' states
Discounting
is
H,
initial asset price.
1.
He wishes buy
^his example
that
2
it
L, respectively,
ignored,
is
The
asset pays off 1 or zero in the 'high'
= 1 - T(L) € (0, 1) is given.
so that po = 7(H) is also the
and the prior chance 7(H)
and traders are assumed
risk neutral,
Let the insider discover his private information that the payoff will be
as
many
shares as his wealth will permit, and being 'small', this activity
for illustrative purposes only,
and neither endorses such trading
activity,
nor suggests
actually occurs.
More
it is
illustrate this hopefully striking proposition, let's consider the
following simple two-period, two-state model.
and
point that under standard and transpar-
generally, with unfavorable information,
assume he
likewise faces a short sale constraint.
will not affect the price. If
he purchases immediately, his return per dollar will be 1/po-
he waits until the next period's public information
of a public signal a
H
whose outcome
or L, then the expect price
the insider's expected return
4
next price p\, namely
In this paper,
timing
is
I
'
.
.
.
,
higher: E\p\
is
is
in {cri,
an }
\
revealed,
namely the announcement
correlated with the state of the world
is
H]
unchanged! For
is
= ££=1 7{o k H)7(H
|
it
If
|
ok )
>p
3
But
.
equals the expected reciprocal of the
5
wish to explore the generality of this observation. Obviously, trade
not always moot. For one, the insider here has perfect information.
The paper
formulates and proves the timing irrelevance proposition for any partially informed trader
— so long as
his information is conditionally
the true state of the world.
assumption in the
It
independent of the public information given
must be underscored that
literature. It so
happens that the
this is the
standard regularity
on the asset being a pure
result turns
'Arrow security', paying a dividend of $1 in one state of the world, and
one ventures outside this domain,
In fact, with
compound bundles
I
can show that timing irrelevance no longer prevails.
of two
Arrow
a (generically
securities, there is
preference to trade immediately on any private information, while the result
for
more general
Let
me
securities:
I
briefly explore the
show by example that one may even wish
economics of
expectation, even though the price
is
is
my main
expected to
rise
result.
benefits
price
is
Why
is
is
strict)
ambivalent
to delay trading!
the return constant in
(with favorable information)? There
a simple perhaps well-known intuition that at least
more favorable than the
justifies
price process: Since the return
why
is strictly
the return process
convex in the
from the riskiness of the stochastic price movements. For instance,
if
price,
tomorrow, while the expected return
is flat
But why then does the
at 3.
3
This intuitive result follows from the Cauchy-Schwartz inequality (J2 a kb k ) 2
put a k = 7{H n a k )ly/7{^) and b k =
The identity sums
5
One might hazard
^).
over
all 'possible'
it
>
1/3
benefit exactly
< (£ a fc)C>*)» ^ we
public signals a k namely those with with 7((J k
,
is
the current
1/3, an equal chance of moving to 1/2 or 1/4 leaves the expected price at 3/8
4
Once
elsewhere.
D H) >
0.
that with a longer time horizon, the trader could learn from early public signals to
better time his trade; but this conjecture is false, for the basic two-period logic can just as well be iterated
into the future.
cancel out the expected rise in the price? Here,
is
I
have not found any deeper intuition than
suggested by a close reading of the above example: the vector of possible prices next
period
is
proportional to the vector of quotients of conditional and unconditional signal
— the dot product
probabilities, or 7(H\o}.) oc 7{ak\H)/7{ok)- Thus, the expected return
—
and the vector of conditional signal chances
of the return vector
signal distribution; since the expected return
is
unchanged
independent of the
is
for signals that are
pure noise,
the result follows. This intuition extends to the case of partial information in the paper.
The
finance literature has been very focused in
First, there is
American
call
what
it
has to say about trade timing. 6
a large and well-established literature on option pricing, especially for the
and put options. The exercise date of such an instrument
and one may optimally wish
to exercise
this is relevant only insofar as
it
it
before
is
expiration date.
its
a choice variable,
For
my
purposes,
represents a single-person decision problem dealing with
timing a discrete trade. But in two absolutely crucial respects, the option pricing problem
is
a different beast:
It is
concerned not with returns but with prices, and more crucially,
the interaction between private and public information
is
a non-issue for option pricing.
For this reason, timing work in Ross (1989) on the phenomenon of 'uncertainty resolution
irrelevancy',
is
also not relevant here
Second, there
— and my resolution date
is
assumed known, anyway.
a large and growing literature on the optimal timing of trades when
is
faced with strategic considerations. See, for instance, Kyle (1985) and (1989),
Pfleiderer (1989),
to
my
and more
thrust here
recently,
Smith (1995). This
— which concerns a 'small trader'
line of
work
is
clearly orthogonal
effect.
Third, there has been work on market timing based on returns, but at a
heuristic level than here.
which uses
it is
CAPM
The
best paper of this genre
The next
I
have seen here
(APT)
rather than the arbitrage pricing theory
enlightening to read that literature knowing
section describes a plain vanilla
my main
model
in
Admati and
as
I
is
much more
Merton (1981),
have. If anything,
result.
which
I
can state and prove both the
timing irrelevance proposition for Arrow securities and a timing impatience proposition for
compound
securities
on two
states.
trading for a three state security!
6
I
An example
follows with a strict preference to delay
close with a conclusion
and a mathematical appendix.
See Duffie (1992) or Merton (1990) for references, a discussion of this point, and a glimpse at
attention the mainstream finance literature has paid to issues of timing.
little
how
THE FORMAL MODEL AND RESULTS
2.
To be
• Information Structure.
build on the
state space
now standard
Q,
is
assumed
analysis (unlike
sigma algebra on
space
variables
a
:
—)
later
is
done by treating E also as
admit general signal spaces.
tedious, so
I
I
shall instead
discrete,
shall therefore define S, the
all 'signal events'.
u =
Let
G
{0,cr)
The
17.
simply think of the signals as stochastic
8 mapping the payoff-relevant state space into the signal events
7 be
the
common
prior probability
trader with the starring role in this paper
and observes
r,
a public signal an G
SG
Beyond the primary
in particular that r
.
.
.
is initially
G
S about the state of the world
All signals r, o\, 02,
(I)
as the
my
inferences about 6. Finally, let
informative signal
x E. Think of
(or,
Bayesian traders observing a realization of a will be able to make
signal outcomes).
The small
=
fl
The
will also
somewhat
Q, is
components
— namely, the space of
S
shall
as the space of payoff irrelevant signals realizations.
While no harm
finite.
MS)
£
I
model of Milgrom and Stokey (MS) and partition the
(1982)
into two statistically-related
payoff relevant states, and
set
deliberately uncontroversial at this stage,
T
G
S. In
endowed with a private
periods n
realized. Crucially,
is
are independent, conditional
assertion that the private
measure on fL
on each
=
I
1, 2,
is
.
.
,
N,
assume that
G
state 6k
and public information
.
conditionally
independent, this also precludes any distracting incentives the trader might have to try to
learn from early public signal realizations
be the sigma-field generated by the
within
§„
is
S, or equivalently
also
first
and make inferences about future ones. Let §„
n signals
(<Xi,.
.
.
,
<x„).
Thus, (S„)
a progressively refined sub-sigma-field of
S.
is
a filtration
(So every event in
an event in S n+ i, for instance.)
• Payoffs and Prices.
Let the payoff function of the asset be n
:
1-4
E, realized and
observed in period iV (a known date), with no discounting until then. So the price of the
asset at the
is
end of period n
is
the expected asset payoff, or
the standard assumption that the asset
As
usual, the
random
is efficiently
pn
=
£[7r|S n ].
Embedded
priced given risk neutral traders.
must be
price process (p n ) corresponding to the filtration (S n )
a martingale (when adapted to (§„)).
That
is,
pm
here
=
£[p n |§m] for
all
n
>
m.
This
implies that there are no arbitrage opportunities on the basis of public information alone.
But given one's private information, one anticipates a period by period erosion
in one's
informational edge. Indeed, the following result
Lemma
Under assumption I, prices
(Prices)
not surprising (and
is
are expected to
move
its
proof omitted).
in the direction of
private information:
£(7r|S n ,T)|£[£(7r|S n+1 ,T)|S n ,r]
as the private information r
G
T
is
favorable or unfavorable, or £(7r|S n ,T)
^
£(7r|S n ).
So on price considerations alone, the individual ought to trade right away on his private
information, regardless of the informational structure.
than the market, then he
the purchase price
is
is
always
expected to
* Timing Irrelevance.
is
If
the trader
is
ever
more optimistic
by the conditional independence of the
so,
so
rise,
As noted
now
is
earlier,
signals.
But
the anticipated cheapest time to buy.
the preceding logic
is
not relevant
if
one
simply trying to invest a given amount. Rather, one cares about the return per dollar.
This
say,
is
true even for short-selling
and
— namely, buying negative units of the share worth $1
by buying the shares back
later netting out
In this general setting, the
after the asset value
is
realized.
most far-reaching timing irrelevance claim would hold that
the return of an investment of $1 in period n equals the expected payoff from deferring
this investment
one period, namely
£[7r|S n
This conjecture
is false.
I
=
,T]/p n
shall
n
is
simply too
— namely, one
in
rich.
which n
=
What
l 6o
,
securities are linear combinations of
The reason
must do
I
—
1 if
Arrow
Proposition 1 (Timing Irrelevance)
amount
\S n ,T]
]
is
its failure,
restrict focus to
is
9
=
O,
and
securities.
some
will point out
a pure Arrow security
state 8
£ ©• Such a
otherwise. General
With
this motivation,
Under assumption I, a
to invest is indifferent about
and
that the range of the space of payoff
the 'indicator function' on
simple security awards a payoff n(0)
with a fixed
,T]/pn+1
soon explore the nature of
that the inequality can go either way.
functions
£[£[7r|S n+1
how he times
compound
we have
partially informed trader
his trades of
an Arrow
security:
(P(0o|Sn,T)/(P(0o|Sn)
=
£[y(0o|§n + i,T)/:P(0o|S„ + i) |S„,T].
(1)
The
proposition
proved in the appendix, but right here
is
space E, where the argument
is less
general but certainly
assume a discrete signal
more transparent.
=
about the multiple periods, and simply denote S
forget
shall
I
{Si, 52,
.
.
.
,
In particular,
Sm }•
Thus, S
and mutually exclusive partition of S, one of whose elements
totally exhaustive
Then given information t E T,
publicly observed next period.
period for an Arrow security
Iff
coincides with
its
will
a
be
the expected return next
current return, since
= Y,V{Sm\T)V{H\Sm ,T)/7{H\Sm
E[P{H\Sm ,T)/y(H\Sn )\T\
is
)
m
=
VPK m y{Sm \H,T)?{H\T)l'S>{S
\H)9(H)/nS
^
= V9K
^
ml
{
{
nSm
'
° ml{T)]
rn
\T)
m)
nSm\H)?(H\T)/nSm \T)
9(Sm \H)1>(H)/nsm )
= ?(H\T)/7(H)Y 7(Sm)
/
m
= 9(H\T)/7{H)
where
all
steps are simplifications except the middle equality, which
conditional independence, namely
The
of
T
given
of state
state
H
Sm
Sm )/7(H\Sm
.
The
)
is
now more
next period
T
H, or T(5rn H',T)/y(5m |T)
J
involved.
really a
is
vector of such premia
given information
is
proportional to the signal likelihood ratio
•
y(Sm )/J>(Sm \H). So
= 7(Sm \H),
the dot product of the informational
premium vector and
this time, provided conditional
this reduces to
T(5m )/0 (5m |T)
5
— and
the conditional likelihood vector
independent of the signal distribution, as in the introduction.
• Trading Impatience with Two-State Securities.
conditional independence assumption 'clearly' drives the result.
showing that the Arrow security restriction
is
Proposition 2 (Trading Impatience)
securities,
one has a
One might argue
I
now
that the
refute such logic,
by
absolutely crucial. For two state securities,
the natural intuition that traders prefer to trade at once
Arrow
public signal-dependent
measure of an informational premium
independence holds, or 7(Sm \H,T)
is
The
divided by the unconditional signal likelihood ratio of
|
(7(Sm \T))
the statement of
7(Sm \H,T) = 7(Sm \H).
intuition in the introduction
return 7(H\T,
is
is valid.
For (generic) assets that bundle together two
(strict) incentive to trade
immediately on any information.
Proof:
Let the
compound
For simplicity,
security
let's restrict
pay out 7r(^)
in state 0(, with tt(8i)
^
n(6 2 )7
Rephrasing the desired
focus to discrete signal spaces.
highest immediately.
we wish
to
show that
This statement
is
simultaneously captured for both favorable information (stock purchase)
result,
the per dollar return exceeds one,
if
and unfavorable information
by the twin
(short sale)
it is
inequalities,
£[7r|r]/pog£MT,8]M|l
These
initial
and
=
final returns are (resp.) £[tt\T]/ Pq
c[v\T,S]/pi
=
2^-y{Sm\T)
^
.
J2e ?(0e\T)n(6e ) / J^t
r—
.
(2)
W) ^) and
7
.
(
T,inSm\ei)y(dt )ir(et )/7{sm )
^
^
With only
yt
=
M
=
it suffices
to
{p m } and {qm } that each
+
-
2/2
Zl+
Z2
Z/i
nsrn\0iWt\TM9t)3: t nsm\6iWt)
=
pm
=
show that
for
sum
we have
,
to
ft
(1
1,
-
\
ar)g
any x 6
\Pmyi
m)
PmZ\
771
(0, 1), y,, Z{
=
(1
0,
/
is
,
s
-
x)(y2 /z2 )
.
,,
(1
/
is
irrelevant exactly
'7
'(h)
=
two Arrow securities have the same
• the
two Arrow securities are informationally indistinguishable: 7(B m \8i)
timing
is irrelevant,
initial return: 3>(0i|T)/
more general compound
securities, the result
example showing that with three
7
Compound
and with two-state
states,
Otherwise, the states can be merged.
one
securities,
may go
may
=
\r
- x)(y
2 /z 2
• the
securities,
=
1.
?{0i),
So to
r t.\
\
)
(A)
thus established in the appendix.
proof of (a) also shows that timing
* Timing Ambivalence with General
x
tP(S m |^ 2 ),
and nonnegative weights
+ qm y2 >
^ *(yiM) +
+ qm Z2 §
This nifty dual inequality happens to be new, and
Remark. The
'
+
>
(3)
[
e)
9(& m \8i), Qm
7(9e )n(8e ). Notice that x(yi/zi)
s
>v
+
| 2^{xPm
*—*
/
nSm \eeP(Ge)n(e
J2 e
2 states, simply define
7{6e \T)Tr(ee ), and ze
establish (2),
?: e
either way.
in fact
I
now
either:
7(0 2 \T)/7{6 2 )
Securities.
impatience
when
<C>
is
=
7(§m\8 2 )
With Arrow
the rule.
With
provide a minimal
wish to delay trading, pending
8
observation of the public information. Let the payoffs instates 6\, 6 2 and #3 be n(8\)
,
n(9 2 )
=
7(6
=
X
)
trader
=
=
1/48, 7(8 2 \T)
The
£[tt]
=
=
40/96, 7(8 3 )
=
(1/96)96
+
=
96/11
=
=
25/48, 7(9 3 \T)
55/96 are chosen to yield a neat
(40/96)9.6
[(1/48)96
+
=
+
+
public signal realization between the two periods
public beliefs towards state 6 2 and
*
it is
A
Arrow
securities, the
away from
845/768 which very
hood
A„
n
> m. A
ratios, if
closely related
and
ir:
then A m
=
is
is
=
|
So,
state.
9
all
n
For instance,
> m.
8
The
9
See, for instance,
this
.
a
£[p„ \p m ] for
law of iterated expectations
,
Sn
]
then £[II n
if
qn
= T(H
|
Il
m =
]
II
m
\
So,
,
Sn and
)
this the conditional martingale
one plus the likelihood ratio
is
the return on
this latter martingale is formally identical to the timing irrel-
paper was indeed inspired by this conditional martingale.
nature of the numbers reflects the difficulty
Chow, Robbins and Siegmund (1971).
slightly exotic
.
=
pm
in
In a theoretical exploration of herd-
(ir) the unconditional martingale. Since
And
.
i.e.
by
APT
by lineage to the
a martingale,
£[7r
role played
Samuelson (1965). Namely,
in
it
and
evance proposition.
(3)
revealed before buying.
is
also related
Smith and S0rensen (1996) have suggestively dubbed
H,
assumed
So observing S\ tips
Beyond the key
ing,
the asset paying $1 in state
1/2).
11/10
classic martingale stochastic process is the set of likeli-
£[A n X m ,H] for
|
n„
(it) If
one conditions on the true
= (l—qn )/Qn
either So or S\ with
is
APT is a consequence of the
applied to the ultimate asset payoff
for all
=
slightly exceeds the initial return of 1.1,
timing irrelevance proposition
stylized level, the
+ 5 + 4)/10 =
.
Ross (1976) or the primal version of
n > m. At a
10
and 8 3 Plugging the above data into
8\
risk-neutral economy, the sequence (p n ) of prices
all
initial asset price
+4+5=
1
(2
and (1/2,3/4,
Link to the Arbitrage Theory of Pricing.
— either in
=
optimal to wait until the public information
strictly
prior beliefs
in posterior beliefs T(9i \T)
(ll/24)(96/ll)]/10
respective state-dependent chances (1/2, 1/4, 1/2)
and so
=
(55/96)(96/ll)
96,
11/24. This yields the neat expected return per dollar
(25/48)9.6
yields an expected return of
Common
8.73, respectively.
assumed to have information r G T, resulting
is
£[7r|T]/po
and n(8 3 )
9.6,
1/96, 7(8 2 )
Po
The
=
48/5
=
of constructing the example.
CONCLUDING REMARKS
3.
• Apologetics.
Almost as firm as the
new
skepticism of simple
this
proposition
is
disbelief; (ii) 'of course it
nontrivially true, but the question posed
paper has addressed the
fails
(i)
no
first
and very natural
(the
lunch exists,
is
must be
my
claim
is
most natural,
is
economists'
true'; or (Hi) yes, the
not relevant.
I
believe that the
response by proof, and the second by showing
absent the Arrow security restriction. At any rate,
For the third reaction,
free
appear surprising. Reactions to earlier versions of
results that
paper have thus been
belief that
I
I
feel
how
the result
the question itself
is
novel.
humble: This paper merely explores a transparent
feel)
thought experiment in financial timing. Are
there purely informational reasons that private information rationally be traded
upon
without haste, or for otherwise timing one's trade?
As with any economic model,
role played
this
paper
is
meant
to sharpen our understanding of the
by the driving assumptions. Previous work on endogenous timing of trades
has either focused on models of pure public information, or confounded strategic effects
with pure timing considerations.
'small trader'.
The key
rather than prices,
is
finite
I
have sidestepped strategic
effects
by focusing on a
wealth assumption, which suggested the focus on returns
only a natural corollary of 'smallness'. But surely any timing model
ought to be robust to the simple calculus of when best to invest a dollar. Since a small
trader's information
is
not impacted in the price, this also precludes applying insights from
Radner (1979), that the equilibrium
will
be
fully revealing
the Arrow-Debreu securities are traded. 10 This
is
concerning the states for which
reasonable as
I
am
simply studying the
single-person decision problem.
* Lessons of the Paper.
This paper has been a simple economic exploration of how
the interaction of private with public information affects the timing of trades. Within the
maintained small trader world, the
conditionally independent. This
is
timing papers but also in classics
alternative
is
critical
assumption is that private and public signals are
almost universally posited, not only in most strategic
like
Grossman and
that private and public information
is
Stiglitz (1980).
The
tautological
conditionally correlated.
While by
no means technically appealing, many plausible information structures produce just
10
But see Geanakoplos (1990)
for further discussion.
this
result.
For instance,
if
distinct traders learn of different
profitability of the firm is the
sum
components of a
firm,
and
if
the
of the profitability of the separate parts, then given the
value of the firm, the separate signals are 'conditionally correlated'.
A
curious application of the paper
One may wish
is in
the problem of optimal design of a security:
noncompound Arrow
to choose a
security simply because
it
assures there
are no perverse incentives for the delay of informational revelation.
Smith (1995) has recently studied endogenous trading
One
lesson of this paper
is
that
'frenzies' in
a a strategic setting.
— correlated information or compound securities aside —
strategic considerations (or other large trader effects) are the only rational explanations
attempts to 'time one's trade'.
for
Finally,
I
feel
the inviting mathematical similarity between the arbitrage pricing theory
and the timing irrelevance proposition and the pivotal
role of
Arrow
securities in each
may
speak to a deeper underlying connection, and merits further investigation.
A.
APPENDIX
• General Proof of Proposition
addresses concerns that
my
This proof
1.
is
for completeness, but also
application of conditional independence
may
take special ad-
vantage of a discrete public signal space.
Let
SjT
be the sigma-algebra
y(SnT|0 o ,SnW0o|Sn) =
S£
S n +i, we have
SDT6
/
JsnT
{Tn5|5 G
S
fc
}.
Assumption
?(T\8 ,S n )9{S\6 ,& n )V(8
\& n )
=
I justifies y(SnTndo\S n =
)
7{T\8 Q ,S n )?(S n 6
S„+i, and thus
O>(0o|Cu).<&
= [ h>dJ>
JsnT
= y(Snrn0o |S»)
=
y(T\e ,s n )?{sne
=
y(T\eo ,s n
\B n )
o \s n
)dy
)J'?(0
=
?(T\6Q ,S n )
10
o \8 n+1 )dV,
J 7(0
\8 n ).
For
since
SE
S n +i- Hence
/
As 5 G S n +i
is
y{&o\Bl+1 )i T dy
=
arbitrary and 35 (^ |Sn+i)
y{T\e Q s„)
,
is
?(e
J
\s n+1 )
d?.
an S„+i -measurable function, this
is
equivalent
to
WolCi)
n T|S»+i]
= 3Wo,S„M0 o |S B+ i).
by definition of a conditional expectation. Dividing both sides by 7(8o\S n+ i) yields 11
«,
,
mfl
,
P(«+
e.[7(e \sl+1 )i T \B n ]
__
i)Ir
Sn+i
T(^o|S n +i)
since (P(#o|Sn+i)
a S n+ i-measurable function. Taking expectations,
is
I
have
^•j-XSfef*
Bayes rule to the LHS, and use the definition of a conditional expectation
Finally, apply
on the RHS, to get 7{6
\S n ,T)/?(e \S n )
• Proof of Inequality ( A)
holds.
Assume
Next, set
^=
E
PrnVl
p* l
PmZl
first
step
is
to prove
a=y,p<
we
\i
rather illustrative, as
f°r otherwise
it
shows when equality
double equality must hold in (A).
define
+ gm3/2
+ qm Z2
5 = VJ qv
and
gmj/l
X
PmZ\
+ Qm V2
+qm Z2
+ y2) / (zi + Z2) ^ xA + (l — x)B ^ x(yi/zi) + (1 — x) (1/2/^2)that B > A, with B > A generically. Too see this, write
then we wish to show that
The
If
E.[9(6 \SZ+1 )/9(6 \S n+1 ) |S„,T], as required.
The proof is
.
WLOG that 3/2/^2 > IJil z
A = j/2/^2 — 2/i/^i-
=
(yi
+ gmj/gCj/l/fl +
Pm2l + 9m ^2
PmiVl/ Zl)Z!
A)z2
=
2/iM
+A
E
Pmgm*2
PmZ\
and likewise
2
B=
11
Note that
CP(^o|S^+1
)
>
since
y /z l
l
is finite
+ A^2
QmZ2
PmZ\
+ qm z2
and includes no zero chance
11
states.
+ qm Z2
Hence,
S-A=A
(qm
>
m
-Pm)qmZ2
A V"^/
=A
> 9m - Pm)
PmZl+qm Z2
(qm
2
-Pm)
x
__
^
final equality
'
Z2
+
\Z\+Z2
- p m )z\Z2
z 2 )(p Tn Zi+qTn z2
(qm
{zi+
)
ZlZ 2
(zi+z2 )(p m zi+qm z2
where the
"
(
)
holds since Y^ m P™
=
^m Q™-
So
B>
A, unless pm
= qm
for all
m.
Simple calculation reveals that the
critical
=
threshold x
x
=
z2 /(z\
equality in (a). There are two immediate consequences: First, since
that
B>
+ z2
)
yields double
we have
just
shown
A, we have
y±±yi^ xA+ {i-x)B
<
zi+z2
as
x ^
yi/zi
x.
Second, because
B >
A, and
and y2 /z2 we must have y2 /z2
,
A
>B >
and
B
are clearly each weighted averages of
A> y\jz\.
It is
then intuitive, and also easily
verified, that
xA +
for
x ^
- x)B |
x{ yi / Zl )
+
(1
-
x)(y2 /z2 )
x.
Finally,
in (A)
(1
x
=
x corresponds
can only hold
if
A =
7{e 2 \T)iy{e 2 ) or y{S m \0i)
=
to 7r(#i)
or
pm
7(S m \e 2 )
=
n(6 2 ), which has been ruled out. Thus, equality
= qm
for all
for all
12
m, as
m. This reduces to 7(6i\T)/7(6i)
required.
=
MIT LIBRARIES
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13
Date Due
AWL 8 M996
Lib-26-67