with many interacting agents
Ulrich Horst
Institut furMathematik
Humboldt-Universitatzu Berlin
Unter den Linden 6
D-10099Berlin
13th June 2001
Abstract
We consider anancial market model with alarge numberof in-
teracting agents. Investors are heterogeneous in their expectations
about the future evolution of an asset price process. Their current
expectation is based on the previous statesof their \neighbors" and
onarandomsignalaboutthe\moodofthemarket". Weanalyzethe
asymptoticsofbothaggregatebehaviourandassetprices. Wegivesuf-
cientconditionsforthedistributionofequilibriumpricestoconverge
toauniqueequilibrium, and provideamicroeconomicfoundation for
theuseofdiusionmodelsintheanalysisofnancialpriceuctuations.
Key Words: behavioral nance, diusion models, stochastic dierence
equations, interactingMarkovchains
AMS 2000 subject classication: 60K35,60J20, 60F05, 91B28
ThepresentpaperisbasedonmyPh.D.thesis. IamgratefultomysupervisorHans
Follmerfor numeroussuggestions and for many motivatinginputs. Thanks are dueto
PeterBank,DirkBecherer,PeterLeukert,AlexanderSchied,andChing-TangWuforin-
numerableandfruitfuldiscussions. FinancialsupportofDeutscheForschungsgemeinschaft
(SFB373,\QuanticationandSimulationofEconomicProcesses",Humboldt-Universitat
zuBerlin)isgratefullyacknowledged.
Inmathematical nance,the priceprocess of ariskyassert is usuallymod-
eledasthetrajectoryofastochasticprocessonsomeunderlyingprobability
space(;F;P). SuchaprobabilisticapproachwasrstinitiatedbyBachelier
(1900), who introducedBrownian motion asa model forprice uctuations
ontheParisstockexchange. Aspricesshouldstaypositive,geometricBrow-
nianmotionisnowwidelyusedasthebasicreferencemodelintheanalysis
of nancial price uctuations. Kreps (1982) showed that such a diusion
model can be justied asthe rational expectation equilibrium ina market
with highly rational agents who all believe in this kind of price dynamics,
and who instantaneously and rationally discount all available information
into thepresentprice. Fromthistheoreticalpoint ofviewlargeand sudden
priceuctuationsreectrationalchangesinthevaluationofanassetrather
thanirrationalshiftsinthesentiment of investors.
Traders, bycontrast,oftenconsidernancialmarketsasbeingmuchless
rational. Many practitioners believe that technical trading is possible,and
thatherd eects unrelatedto economic fundamentalscan cause bubblesor
crashes. In particular,they aretypicallyawareof thefact thatasset prices
maybedrivenbysuddenshiftsinthe\moodofthemarket". Inviewofsuch
market realities, it seems natural to regard priceprocesses asthe result of
aninteractionbetweenmanyagentswithboundedrationality. Inparticular,
oneshouldadmitimitationandcontagioneectsintheformationofagents'
expectations. When the set A of agents involved inthe formation of stock
prices becomes large, such an approach allows to bring in techniques from
the theory of interacting Markov processes or from Markov random eld
theory. An early attempt into this direction has been made by Follmer
(1974) whoconsidersa static modelof endogenouspreference formation.
In thispaperwe providea uniedprobabilisticframework formodeling
nancial markets where the demand for a risky asset results from the in-
teractionofa largenumberof traders. Followinganapproach suggestedby
Follmerand Schweizer (1993),we are goingto view the stockpriceprocess
asasequence of temporarypriceequilibria. We assumethat theexcess de-
mandof agent a2A inperiodtdependsonhis current individualstate x a
t
reecting, forexample, his expectation aboutthe future evolution of stock
prices. Suchadistinctionbetweendierenttypesofagentshasbeenamajor
conomic models; see, e.g. Brock and Hommes (1997), Kirman (1998), Lux
(1998) orLuxand Marchesi(1999). Fora givenconguration ofindividual
statesx=(x a
)
a2A
,thelawofagent a'snewstate dependsbothon thecur-
rent states(x b
)
b2N(a)
oftheagents inhissocialneighborhood N(a) andon
theaverage expectation in thewhole population, i.e., on the \mood of the
market". The mood of the market is described by the empirical distribu-
tion of individualagents' states or, more completely, by theempirical eld
R (x) associated with the conguration x. We consider a situation where
an individual agent has complete information about the actions chosen by
agentsinhisreferencegroup,butonlyhasincompleteinformationaboutthe
average action throughout the entire population. Individual trader do not
knowR (x) forsure,butonlyobserve arandom signal,e.g.,a stock market
index,whose distributiondependson R (x).
Inournancialmarketmodel,therandomevolutionof themood ofthe
market is the only component aecting the formation of temporary price
equilibria. The microscopic process fx
t g
t2N
which describes thestochastic
evolution of all the individualstates generates via the macroscopic process
fR (x
t )g
t2N
an endogenous random environment f%~
t g
t2N
for the evolution
of theassetprice process. Thedynamics of thestock priceprocess fp
t g
t2N
obeys a recursiverelationof theform
p
t+1
=F(%~
t+1
;p
t
) (t2N) : (1)
Ouraimisto stateconditionswhichensurethatthestockpricesbehave
asymptotically in a stable manner. To this end, we shall rst analyze the
long runbehaviour of the macroscopic process fR (x
t )g
t2N
which describes
thedynamicsofaggregatebehaviour. Usingresultsabouttheasymptoticbe-
haviour oflocally andgloballyinteractingMarkov chainsprovidedinHorst
(2000),weshowthat, inthelimitofaninnitesetA oftraders,thedynam-
icson thelevelof aggregate behaviour can be describedbyaMarkovchain
associated witha suitablerandomsystemwith completeconnections. This
allows us to state conditions on the behaviour of individual agents which
guarantee that themood ofthemarket settles down inthelong run. More
precisely,weplaceaquantitativeboundoftheeectsofsocialinteractionin
oureconomy, and show thatthe macroscopic processconverges inlawto a
uniqueequilibriumdistributioniftheinteractionbetweendierentagentsis
evolvesina stationary randomenvironment.
Inthesecondpartofthispaperwestudytheasymptoticsoftheinduced
equilibrium price process. In a rst step, we analyze an ergodic reference
model,i.e.,weassumethatthedrivingsequencef%~
t g
t2N
isstationaryander-
godic. Economically,thisamountsto asituationwherethemoodisalready
in equilibrium. We show that the asset price process becomes asymptot-
ically stationary if the destabilizing eects of the random environment on
thedynamicsofthepriceprocessisonaveragenottoostrong. Nonetheless,
thepriceuctuationsinournancialmarketmodelmay be highlyvolatile.
Thisfeaturemaybeinterpretedasthetemporaryoccurrenceofbubblesand
crashesina nancialmarketmodel whoseoverallbehaviouris ergodic.
From an economic point of view, however, such a stationarity assump-
tion on the driving sequence might be rather restrictive. Instead, it seems
more natural to investigate the asymptotic behaviour of asset prices un-
der the assumption the at the mood is out of equilibrium, i.e., under the
assumptionthattherandomenvironmentfortheevolution ofthepricepro-
cessisspeciedbyanon-stationarystochasticprocess. However, giventhat
themacroscopicprocesssettlesdowninthelongrun,itis desirabletohave
suÆcientconditionswhichensurethatassetpricesaredrivenintoastation-
ary regime. Assuming a simple log-linear structure for the excess demand
functionsleadsto a classof log-linearpriceprocessesof theform
logp
t+1
=f(~%
t+1 )logp
t +g(~%
t+1
) (t2N): (2)
Underasuitablemean-contraction conditionon thepricedynamicsfp
t g
t2N
speciedby (2) we show that stock prices converge to a stationary regime
ifthe mood of the market itself settles down inthe long run. Armed with
these resultsone could now try to analyze thestructure of theequilibrium
distribution,and toestimate, forexample,theasymptoticvariance ofstock
prices. Such an empirical analysis, however, is beyond the scope of this
paper,and isleft forfuture research.
In our nal section we study a diusion approximation of the discrete-
time price process fp
t g
t2N
. Under simplifying assumption on excess de-
mand functions of the agents, the sequence of temporary price equilibria
can be approximated in law by a diusion process fP
t g
t0
in continuous
time. Thisresult providesanothermicroeconomicfoundation fortheuse of
ofthemarketisalreadyinequilibrium,i.e.,iftheassetpriceprocessevolves
in a stationary and ergodic random environment, then we nd ourselves
in the setting analyzed in Follmer and Schweizer (1993), who obtained a
continuous-time model from a sequence of suitably specied discrete-time
processes evolving in an exogenously given random environment; see also
Follmer (1994). Proving a functional central limit theorem for stochastic
processes evolving in a non-stationary random environment, we are able
to extend the Follmer-Schweizer model by (i) analyzing a situation were
thedrivingsequence isderived endogenously,and (ii)byreplacingthesta-
tionarityassumptionon themoodofthemarketbyanasymptoticstability
condition. WeshowthatthediusionlimitfP
t g
t0
convergestoastationary
process whoseinvariant distributioncan begiven inclosedform.
Therest of thispaperisorganized asfollows. InSection2weintroduce
ournancial market model. In Sections 3 and 4 we describe the dynamics
of individualand aggregate behaviour,respectively. Section 5analyzes the
asymptoticbehaviour ofthe discrete-time assetprice process. In Section 6
we passto a diusionlimitincontinuoustime. Section7 concludes.
2 The Microeconomic Model
Letus describea temporary equilibriummodel forthe priceevolution of a
speculative asset. We consider a nancial market model with a countably
innite set A of economic agents trading a single risky asset. In reaction
to a proposed stock price p in period t, each agent a 2 A forms an excess
demand z a
t
(p). Individual excess demand at time t is to be thought of as
thedierencebetweenindividualdemandandindividualinitialendowment
in period t. The actual stock price p
t
at date t will be determined by the
equilibriumconditionof zero total excess demand,and sothepriceprocess
willbegivenbyasequenceoftemporarypriceequilibriafp
t g
t2N
. Weassume
thattheexcess demand ofan individualagent a2A takes theform
z a
t
(p)=z(p;p a
t
): (3)
Here, p a
t
denotes an individual reference level of an agent a at date t, e.g.,
his price expectation for the following period t+1. We shall assume that
agentsareheterogeneousintheirindividualreferencelevels. Moreprecisely,
at any given date t 2 N, the value p
t
depends on the present individual
state x a
t
of agent a2A, on thepreviousequilibriumpricep
t 1
,and on the
proposedpricep. Thus,it takes theform
p a
t
(p)=g(x a
t
;p
t 1
;p) (4)
forsome measurablefunctiong:CR 2
!R. Here
C:=fc
1
;:::;c
N g
isa xedset of individualstates.
Example2.1 (\Fundamentalists and Chartists") Let us consider a nan-
cial market model where the individual excess demand function takes the
log-linear form
z(p;p a
t
)=logp a
t
logp (5)
asin Follmer and Schweizer (1993). We putC=f 1;0;+1g and consider
a model with optimistic (x a
t
= +1) and pessimistic (x a
t
= 1) information
traders (\fundamentalists") andwith chartists (x a
t
=0).
We assume that a fundamentalist's subjective perception logL+x a
t of
the current value of the assetuctuates aroundsome long run fundamental
value logL. Independent of the proposed price, his expectation is based on
the idea that the nextpricewill movecloser to hisactual benchmark forthe
fair value of the stock. Moreprecisely, his reference level takes the form
logp a
t
=logp
t 1 +c
f
(logL+x a
t
logp
t 1
) (c
f
>0): (6)
The chartist, on the other hand, takes the proposed price as a serious
signal about the future evolution of the security price and replaces L in (6)
by p. Thus,his priceexpectation takes the form
logp a
t
=logp
t 1 +c
c
(logp logp
t 1
) (c
c
>0): (7)
In equilibrium, i.e., for p = p
t
, a chartist forecasts the future evolution of
the asset price process from past observations.
In our model, the dynamics of the price process will be induced by an
underlyingmicroscopicprocess fx
t g
t2N
=f(x a
t )
a2A g
t2N
whichdescribesthe
t t2N
valuesin thecongurationspace
E :=C A
=fx=(x a
)
a2A :x
a
2Cg:
We shall view E as a compact metric space which is equipped with the
producttopologyand denote byE theproduct--eldonE.
Let usnowconcentrate on the resulting stockpriceprocess fp
t g
t2N . In
a rst step, we considera situation where only nitely many investors are
active on the market. To this end, we x a sequence fA
n g
n2N
of nite
subsetsofA satisfyingA
n
"A asn"1. Ifonlythetraders inA
n
areactive
onthemarket,theequilibriumstockpricep n
t
atdatetisdeterminedbythe
market clearing condition of zero total excess demand, i.e., p n
t
is given by
an implicitsolutionof theequation
X
a2A
n z
a
t (p
n
t
)=0: (8)
In view of (3) and (4), and in terms of per capita excess demand, we can
rewrite(8) as
Z
C z(p
n
t
;g(c;p n
t 1
;p n
t ))%
n
(x
t
)(dc) =0: (9)
Here,
% n
(x
t ):=
1
jA
n j
X
a2A
n Æ
x a
t
()2M(C)
denotestheempiricaldistribution ofthestates assumed bythe traders a2
A
n
inperiodt, and M(C) istheclass of allprobabilitymeasureson C.
For any n 2 N, the implicit equation (9) denes the sequence of tem-
porary equilibrium prices fp n
t g
t2N
in the case of a nite set A
n
of traders
whoare involved intheformationof equilibriumprices. Inorder to extend
the construction of equilibrium prices to the innite system of all agents,
we are going to restrict our state space E to the classE
0
of congurations
x=(x a
)
a2A
2E whichadmit theweak limit
%(x):= lim
n!1
% n
(x):
E
0 :=
(
x2E:9 lim
n!1 1
jA
n j
X
a2A
n Æ
x a
()2M(C) )
:
Thus, forx2E
0
theempiricaldistribution%(x) associatedwiththecong-
urationx exists. We willcall%(x
t
) themoodof themarketat timet.
Even though we consider an economy with a countably innite set of
agents,itseemsreasonabletoviewthesetofagentswhoaredirectlyinvolved
intheformationoftheequilibriumpriceatdatetasarandomsubsetA
t A
ofrepresentativeagents. By sayingthatA
t
isa setof representative agents
we mean that the empirical distribution %~
t
of the states assumed by the
tradersinA
t
isa randomvariable whoseconditional law
~
Q(%(x
t
);): (10)
isdescribed bya stochastic kernel
~
Qfrom M(C) to M(C).
We are now ready to dene the equilibrium price p
t
at date t. In the
limitof an innite set of agents, p
t
will be given as an implicit solution of
theequation
Z
C z(p
t
;g(c;p
t 1
;p
t ))%~
t
(dc)=0 (11)
which may be viewed as the limiting form of (9). In order to obtain a
uniquely denedstock priceprocess, we impose thefollowing conditionon
theexcess demandfunctionz:
Assumption 2.2 There exists a measurable function F :M(C)R ! R
such that the implicitequation
Z
C z(p
;g(c;p;p
))~% (dc)=0
admits a unique solution p
=F(% ;~ p)for any pair (% ;~ p)2M(C)R.
Example2.3 Let us return to the log-linear demand structure described
by (5) together with (6) and (7). For a given conguration x
t 2 E
0 , the
empirical distribution %~
t
is described by the proportions %~ +
t
;%~
t
and %~ 0
t of
optimistic and pessimistic fundamentalists andof chartists, respectively. In
pliesthe following linear structure for the evolution of the logarithmic stock
price process:
logp
t+1
=f(~%
t+1 )logp
t +g(%~
t+1
); (12)
where
f(%~
t ):=
1 c
f (%~
+
t
~
%
t ) c
c
~
% 0
t
1 c
c
~
% 0
t
; g(%(x
t )):=
c
f
(logL+%~ +
t
~
%
t )
1 c
c
~
% 0
t
: (13)
In Section 5.2 we will state conditions which guarantee that the sequence
flogp
t g
t2N
convergestoastationary regime ifthemood ofthemarketsettles
down as t ! 1. Observe, however, that for c
c
> 1, the mappings f and
g have a singularity. If the fraction of chartists who are actually involved
in the formation of price equilibria comes close to the critical value %~= 1
c
c ,
thenthe priceprocessbecomeshighly volatile. Aslight changein thecurrent
proportion of chartists mayhave a lasting eecton assetprices.
GiventhatourAssumption2.2holdstrue,thestockpriceprocessfp
t g
t2N
isdened bytherecursive relation
p
t+1
=F(%~
t+1
;p
t
) (t2N) : (14)
In our setting, the evolution of the empirical distribution of agents' states
istherefore the onlycomponent aecting the formationof temporaryprice
equilibria. The process fx
t g
t2N
generates { via the aggregate quantities
f%(x
t
)g andf~%
t g
t2N
{anendogenousrandomenvironmentfortheevolution
of the stock price process. Our goal is to analyze the long run behaviour
of the asset price process fp
t g
t2N
. To this end, we have to analyze the
asymptotics of thedriving sequence f~%
t g
t2N
. In the next section, we begin
bydescribingthedynamicsofthemicroscopicprocessfx
t g
t2N
. InSection4
we state conditions on the behaviour of individual agents which guarantee
thatthe mood of themarket settlesdown inthelong run. In thiscase the
price process asymptotically evolves in a stationary random environment.
In Section 5 we show that this yields weak convergence of the asset price
process iftheexcess demandfunctionstake thelog-linearform(5).
Let us now specify the dynamics of the microscopic process fx
t g
t2N . We
assume that, ineach period t, any agent a2A chooses his next state x a
t+1
atrandomaccording tosome probabilitylawwhichdependsonthepresent
congurationofindividualsstatesx
t
=(x a
t )
a2A
. Moreprecisely,theprocess
fx
t g
t2N
willbe describedbya Markovchain,
(x;dy)= Y
a2A
a
(x;dy a
);
on asuitablesubsetof thecongurationspace E=C A
.
The dependence of the probabilitylaw a
on the present conguration
can have both a local and a global component. Local dependence in the
choice of an individual agent a 2 A refers to dependence of a
on some
set of neighbors b 2 N(a). In particular, introducing the notion of local
interaction requires to endow the countable set A with the structure of a
graph,where theagents are thenodesand where interactive linksbetween
certain pairsof agents exist. Since we shallalso admita global component
intheinteraction, we willneedto considerergodicaveragesover thewhole
graph of agents. Therefore, we limit ourselves to the case best understood
so far, where A carriesa lattice structure, and assume that the agents are
located onthed-dimensionalinteger lattice, i.e., A :=Z d
.
The neighborhood orreferencegroup N(a) associatedwithagent a2A
isgiven bytheset
N(a):=fb2A :ja bjl<1g:
Here, jj denotes the Euclidean distance on R d
and l 2 N is a xed \in-
teraction radius". In terms of such peer groups we can model situations
wherethecurrentreference levelp a
t+1
ofagent a,e.g.,hisexpectationabout
thefuture evolution of the stockprice, is inuencedbythe previous states
(x b
t )
b2N(a)
of his neighbors. Note that in our model any agent aects the
next state of just 2 dl
other persons, and so no individualtrader is able to
inuencethemood of thewholemarketinone singleperiod. In thissense,
we considera nancial marketmodel withmany smallinvestors.
In real nancial markets, the behaviour of an individual trader is, of
course,notonlyinuencedbytheactionschosenbytheagents inhis refer-
encegroup,butalso dependsonthecurrentmoodofthemarket,i.e.,onthe
typicallynothave completeinformationabouttheempiricaldistribution of
individualagents' states ina given period. Instead, itseems reasonable to
assumethattheyonlyhave incompleteinformationabout%(x
t
)inthesense
thattheyreceiveacommonnoisysignals
t
abouttheaverageexpectationat
date t. Forexample, s
t
could be a signal aboutthe fraction ofchartists or
optimisticinformationtraderswhoareactive onthemarketinperiodtand
may be revealed by,e.g., a stock market index. Thus, itmakes immediate
sensetointroducean additionaldependenceon signalsabout\global prop-
erties"of thecurrentconguration, e.g.,a dependenceon signalsaboutthe
moodofthemarket, into theinteraction. The followingsimplevotermodel
illustratesthisapproach.
Example3.1 Let us put C = f0;1g. We assume that an inidividual in-
vestor reacts both to the currents states of his neighbors and to a random
signal s2 [0;1] about the average action throughout the entire population.
For any xed signal s, an agent chooses his next state according to a tran-
sitionlaw a
s
fromE
0
to C which is described by the convex combination
a
s
(x;1)=p(x a
)+m a
(x)+s: (15)
Here, p(x a
)measures the dependenceof agent a'snewstate on hiscurrent
one,andm a
(x)istheproportionof`1' intheneighborhood N(a). Moreover,
s 2 [0;1] denotes a commonly known random signal about the empirical
average
m(x):= lim
n!1 1
jA
n j
X
a2A
n x
a
associated with the conguration x 2 E
0
. The conditional law Q(m(x
t );)
of the signal s
t
given the average m(x
t
) is described by a signal kernel Q
on [0;1]. Due to the linear structure of the transition probabilities a
s , the
law of large numbers shows that, for any given signal sequencefs
t g
t2N , the
process fm(x
t )g
t2N
satises almost surelythe deterministicdynamics
m(x
t+1
)=u(m(x
t );s
t
):=fm(x
t
)p(1)+(1 m(x
t
))p(0)g+m(x
t )+s
t :
In this model, the sequenceof empirical averages fm(x
t )g
t2N
may therefore
be viewed as a Markov chain on the state space [0;1]. In Sections 4 and 5,
fm(x
t )g
t2N
andtheinducedpriceprocessfp
t g
t2N
convergeinlawtoaunique
equilibrium.
The next exampleshows thatthedynamics ofthesequence fm(x
t )g
t2N
typically cannotbedescribed byaMarkov chain.
Example3.2 Considerthefollowinggeneralizationofthevotermodel(15).
Forx2E
0
ands2[0;1],theindividualtransitionprobabilitiesaredescribed
by a measurable mapping g
s :C
jN(a)j
![0;1]in the sensethat
a
s
(x;1) =g
s
fx b
g
b2N(a)
: (16)
Typically,wecannotexpectthatthereexistafunctionu:[0;1][0;1]![0;1]
such that m(x
t+1
)=u(m(x
t );s
t
). Nevertheless,we will showthat the mood
of the marketsettles down in the longrun ifthe dependence of the mapping
g
s on x
b
(b2N(a)) isnot too strong; cf.Example 4.6.
Due to thelocal dependenceof theindividualtransition laws a
on the
current conguration, the dynamics of the mood of the market in general
cannot be described by a Markov chain. In order to analyze the long run
behaviour of aggregatebehaviourin ournancialmarket model,we need a
more general mathematical framework which we are now going to specify.
To start with,we introduce the familyof shift-transformations
a
(a 2A)
on E denedby(
a
x)(b) =x a+b
.
Denition 3.3 (i) Wedenote by M(E) the class of all probability mea-
sures on E. A probability measure 2 M(E) will also be called a
randomeld.
(ii) A random eld 2 M(E) is called homogeneous, if is invariant
underthe shift maps (
a )
a2A . By
M
h
(E):=f2M(E) : =Æ
a
for all a2Ag
we denote the class of all homogeneous random elds on E.
(iii) Ahomogeneousrandomeld2M
h
(E)iscalledergodic,ifsatises
a 0-1-law on the -eld of all shift invariant events. The class of all
ergodic probability measures on E isdenoted by M
e (E).
For agiven n2N we putA
n
:=[ n;n] \A; and denote byE
e
theset
ofall congurationx2E suchthat theempiricaleld R (x), denedasthe
weak limit
R (x):= lim
n!1 1
jA
n j
X
a2A
n Æ
a x
();
exists and belongs to M
e
(E). The empirical eld R (x) carries all macro-
scopic information carried in the conguration of individual states x =
(x a
)
a2A 2E
e
. In particular,theempiricaldistribution
%(x)= lim
n!1 1
jA
n j
X
a2A
n Æ
x a
();
i.e., the mood of the market associated with the conguration x 2 E
e , is
givenasthe one-dimensionalmarginaldistributionof R (x).
Considertheproductkernel
s
onE governedbythetransitionlaws a
s
in(16). Proposition4.1 below shows that the measure
s
(x;)(x 2E
e ) is
concentratedon theset E
e
and thatthe empiricalaverage satises
m(y)=u(R (x);s):=
Z
s
(x;1)R (x)(dz)
s
(x;)-a.s. (17)
Thus, we have to consider the full dynamics of the sequence of empirical
eldsfR (x
t )g
t2N
even if, as in Example3.2, the behaviour of agent a2 A
depends on R (x) only on the empiricalaverage m(x). Theorem 4.2 below
showsthat,incontrasttothesequenceofempiricaldistributionsf%(x
t )g
t2N ,
the dynamics of the sequence of empirical eld fR (x
t )g
t2N
can indeed be
described by a Markov chain. Our aim is to formulate conditions on the
transition laws a
, i.e., on the behaviour of individualagents, which guar-
antee thatthesequence ofempiricaleldsfR (x
t )g
t2N
converges inlawto a
unique equilibrium distribution. In case we will say that the mood of the
marketsettlesdown inthelongrun.
To this end, we shall now be more specic about the structure of the
transition probabilities a
. We assume that theinteractive inuenceof the
present conguration x 2E
e
on agent a is felt both throughthe local sit-
uation(x b
)
b2N(a)
inhis neighborhood,and througha randomsignal about
theaverage situationthroughout theentire population A. The average sit-
uationis describedbytheempiricaleldR (x)associatedwithx2E
e . The
Q(R (x);) (18)
of the signal s given the empirical eld R (x) is specied by a stochastic
kernelQfrom M
h
(E) to S,whereS isa nitesignal space.
1
ThekernelQ
willbe calledthesignal kernel.
We also assume that interaction between dierent agents is spatially
homogeneous. This means that all traders react in the same manner both
to theactions previously chosen bytheagents intheir reference groupand
to the signal about aggregate behaviour. Thus, for a xed signal s 2 S
andcongurationx2E,theprobabilitythatagent a2A switchesto state
c2C inthefollowingperiodis given by
a
s
(x;c)=
s (
a
x;c); (19)
where
s
(x;)is astochastickernelfrom ES to C.
Assumption 3.4 Theprobabilitylawsf
s (x;)g
x2E
satisfyaspatialMarkov
property of order l in their dependence on the present conguration:
s (
a
x;)=
s (
a
y;) if
a x=
a
y on N(a):
Economically,this condition meansthat, forany xed signalaboutthe mood
of themarket, the newstateof an agent onlydependson the previous states
of this neighbors.
Letusnowxasignal s2S and acongurationx2E. It follows from
ourAssumption3.4and from(19) that
s
(x;):=
Y
a2A
s (
a x;)
denes a Feller kernel on the conguration space E which is spatially ho-
mogeneous:
Z
E f(y)
s (
a
x;dy)= Z
E f(
a y)
s
(x;dy) (20)
1
Theassumption that S is nite merely simplies notation. Ouranalysis also goes
throughundertheassumptionthat(S;S)isanarbitrarymeasurablespace.
s
togetherwith thesignalkernelQ determinea stochastickernel
(x;):=
Z
S
s
(x;)Q(R (x);ds) (21)
from E
e
to E. Infact, Proposition4.1 below shows that may beviewed
as a stochastic kernel on the conguration space E
e
. In contrast to
s ,
however,thekerneltypicallydoesnothavetheFellerproperty,duetothe
macroscopic dependence on the present conguration x via the empirical
eldR (x). Thus,wecannotapplythemethodinVasserstein(1969)inorder
to study the long run behaviour of the processes fx
t g
t2N
and fR (x
t )g
t2N .
Instead, we will useresults about the asymptotic behaviour of locally and
globallyinteractingMarkovprocessesrecentlyreportedinFollmerandHorst
(2001) andHorst (2001a).
4 The Dynamics of Aggregate Behaviour
Inthissection we aregoing to formulate conditionson theindividualtran-
sition laws
s
and on the signal kernel Q which guarantee that the mood
of the market settles down in the long run.
2
In a rst step, we use a law
of largenumbersfor therandom elds
s
inorder to view asa stochas-
tic kernel on the conguration space E
e
. For the proof we refer to Horst
(2001a),Proposition3.1.
Proposition 4.1 For all congurations x2 E
e
, and for any signal s2 S,
we have
s (x;E
e
) = 1. For
s
(x;)-a.e. y 2 E
e
, the empirical eld R (y)
takes the form
R (y)=u(R (x);s):=
Z
E
s
(y;)R (x)(dy):
In view of the this Proposition, we use E
e
as the state space for our
microscopicprocessfx
t g
t2N
. WedenotebyP
x
thedistributionoftheMarkov
chainfx
t g
t2N
withinitialstatex2E
e
. Sinceacongurationx2E
e
induces
2
Theasymptoticsof the Markov chain is explicitly analyzedinHorst (2001a). In
ordertokeepthepresentpaperself-contained,wesummarizesomeoftheresults,butomit
theproofs.
t t2N x
a.s.the macroscopic process fR (x
t )g
t2N
with state space M
e
(E). The law
of the random variable R (x
t+1
) depends on the microscopic conguration
x
t
inperiodt onlythroughtheempiricaleldR (x
t
). Thus,itis easilyseen
thatthefollowingresult holdstrue.
Theorem 4.2 (i) Under the measure P
x
the macroscopic process is a
Markov chain on the state space M
e
(S) withinitial value R (x).
(ii) For any given initial conguration x 2 E
e
, and for each xed signal
sequencefs
t g
t2N
our macroscopic process satises
R (x
t+1
)=u(;s
t
)ÆÆu(;s
1
)Æu(R (x);s
0
) P
x
-a.s. (22)
for any t2N.
Since, conditioned on the environment fs
t g
t2N
, the macroscopic pro-
cess followsalmost surelya deterministicdynamics,we proposea \random
systemwithcompleteconnections" (henceforth RSCC)asa suitablemath-
ematical framework for analyzing the dynamics of aggregate behaviour in
ournancialmarketmodel. Letusrecall thenotionof aRSCC.
Denition 4.3 Let (M
1
;d
M
1
) be a metric space and (M
2
;M
2
) be a mea-
surable space. Let Z denote a stochastic kernel from M
1 to M
2
, and let
v : M
1
M
2
! M
1
be a measurable mapping. Following Iosefescu and
Theodorescu (1968), we call the quadruple
:=((M
1
;d
M1 );(M
2
;M
2 );Z ;v)
arandomsystem withcompleteconnections.
3
(i) Given an initial value 2 M
1
, a RSCC induces two stochastic pro-
cessesf
t g
t2N
andf
t g
t2N
onthecanonicalprobability space(
^
;
^
F;
^
P
)
takingvalues in M
1
and in M
2
, respectively, by
t+1
=v(
t
;
t ) and
^
P
(
t 2j
t
;
t 1
;
t 1
;
t 2
;:::)=Z(
t
;):
3
Werefer the readertothe books ofIosefescu andTheodorescu (1968) and Norman
(1972) for adetaileddiscussionof RSCCs. Underthe dierentname \iteratedfunction
systems", this class of processes is also studied in, e.g., Barnsley, Demko, Elton, and
Geronimo(1988)andLasotaandYork(1994).
Here,
0
= P
-a.s. These processes arecalled the associatedMarkov
process and the signalsequence, respectively.
(ii) Wesay thata randomsystemwithcompleteconnections isadistance-
diminishingmodel, if the transformation v :M
1
M
2
!M
1
satises
the contraction condition
d
M
1
(v(;);v(
^
;))d
M
1 (;
^
)
for some constant <1.
Let f
t g
t2N
be a Markov chain associated with a random system with
completeconnections=((M
1
;d
M1 );(M
2
;M
2
);Z ;v). Foranyxed signal
sequence f
t g
t2N
,we have that
t+1
=v(;
t
)ÆÆv(;
1
)Æu(;
0 )
^
P
-a.s. (23)
Letusnowxaninitialcongurationx2E
e
. Inviewof (20)and(23), our
macroscopicprocessmaybeviewedastheMarkovchainwithstartingpoint
R (x)associated withtherandomsystem withcompleteconnections
:=((M
h
(E);d);(S;S);Q;u); (24)
wherethemapping u:M
h
(E)S !M
h
(E) is denedby
u(R ;s):=
Z
E
s
(x;)R (dx);
and where d denotes the metric introduced in Follmer and Horst (2001)
whichinducestheweaktopologyonM
h
(E);seealso(2.53)inHorst(2000).
4
Indeed,letusdenoteby(f
t g
t2N
;(
^
P
% )
%2M
h (E)
)theMarkovchainonM
h (E)
N
associated with
. Aneasy inductionargumentshowsthat
P
x [fR (x
t )g
t2N
2B]=
^
P
R(x) [f
t g
t2N
;2B] (25)
forallB inB N
,theBorel--eldonM
h (E)
N
. Inorderto guaranteeasymp-
totic stability of our macroscopic process, it is therefore enough to state
4
ObservethatwearegoingtoviewthemacroscopicprocessasaMarkovchainonthe
compactmetricspace(M
h
(E);d)eventhoughR (x
t )2M
e
(E)forallt2N . Wereferto
Horst(2001a)foradetaileddiscussionofthisissue.
t 2N
auniqueequilibrium. Dueto Theorem4.1.2 inNorman(1972), theprocess
f
t g
t2N
converges inlawwhenevertherandomsystem
isdistancedimin-
ishingin thesenseof Denition4.3 (ii), and given thatthe signalkernel Q
satisesthefollowingconditions.
Assumption 4.4 (i) The signal kernel Q from M
h
(E) to S satises a
uniformLipschitz condition in the sense that
sup
s2S;6=
^
jQ(;s) Q(
^
;s)
d(;
^
)
L<1: (26)
(ii) Wehave that inf
;i Q(;s
i )>0.
Remark 4.5 Economically, our Assumption 4.4 (ii) states that the agents
receiveevery signal withsmall butpositiveprobability, regardless of the pre-
vailingmood of the market. Both parts (i) and (ii) of the above assumption
exclude situations where the traders have complete information about the
aggregate behaviour throughout the entirepopulation.
Let usnow state a conditionon theindividualtransition laws
s which
guarantees that the random system
is distance diminishing. To this
end, we dene, for any pair (a;s) 2 A S, a vector r s
a
= (r s
a;b )
b2A with
components
r s
a;b
=sup
1
2 k
s
(x;)
s
(y;)k:x=y oa b
(b2A);
wherek
s
(x;)
s
(y;)k denotesthetotal variationdistanceof thesigned
measure
s
(x;)
s
(y;)onthesetC. Sinceweassumethattheinteraction
betweendierentagents isspatiallyhomogeneous, itis easilyseen that
r s
a;b
=r s
a b;0
(a;b2A) :
Economically, the quantity r s
a;0
measures, fora xed signal s2 S, the de-
pendenceofthenewstateofagent0onthecurrentstateofagenta. Inorder
to ensure convergence of the mood of the market, we place a quantitative
Weak SocialInteractionprevailsif
:=sup
s X
a r
s
a;0
<1; (27)
i.e., if the dependence of an agent's action on the current conguration is
nottoo strong. Our weak socialinteractionassumptionexcludessituations
where an agents imitates with probability one the behaviour of any of his
neighbors.
Beforewe provethemainresultofthissection,weconsider asimpleex-
ample,whereourweaksocialinteractionassumptioncanindeedweveried.
Example4.6 Let us x a conguration x 2 E
e
and a signal s 2 S, and
assume that the probability that an agent switches to state `1' is given by
(16). Suppose that the mapping g
s : C
jN(a)j
! [0;1] is dierentiable, and
denote the partial derivative withrespect tox b
by g b
s . If
max
s X
a2N(0) max
n
g a
s
(x b
)
b2N(0)
: x b
2C o
<1;
then our weaksocial interactioncondition (27) is satised.
We shallnowstateconditionsonthebehaviourofindividualagentsand
on thesignal kernelQ which guarantee asymptotic stabilityon thelevel of
aggregate behaviour.
Theorem 4.7 Suppose that our Assumption 4.4 holds true, and that the
weak social interaction condition (27) is satised. Then the macroscopic
process fR (x
t )g
t2N
converges in law to a unique stationary distribution
.
Here
isa probability measures on M
h (E).
Proof: Due to Lemma 4.9 in Horst (2001a), our weak social interaction
conditionyields
ju(;s) u(
^
;s)jd(;
^
) (;
^
2M
h
(E); s2S):
Thus, therandomsystemwith completeconnections
introducedin (24)
is distance diminishingin the sense of Denition 4.3 (ii). In view of (25),
our assertion therefore follows from Theorem 4.1.2 in Norman (1972) as
(M
h
(E);d)is acompact metricspace. 2
Wearenowgoingtoanalyzethelongrunbehaviourofstockpricesinour-
nancialmarketmodel. Inordertosimplifynotation,andinviewof(25),we
shallfromnowon assumethattherandomenvironmentf~% g
t2N
fortheevo-
lution of the asset price process fp
t g
t2N
is generated by the Markov chain
(f
t g
t2N
;(
^
P
)
2M
h (E)
) associated with the random system with complete
connections
introducedin(24). More precisely,we considera pricepro-
cessfp
t g
t2N
whichisdenedbytherecursiverelation(14),butassumethat
theconditional law
~
Q(
t
;) (28)
oftherandomvariable%~
t
giventherandomeld
t
isdescribedbyastochas-
tic kernel
~
Q from M
h
(E) to M(C). Throughout this section, we assume
thattheassumptionsstated inTheorem4.7aresatised.
In Section5.1we consideran ergodicreference model. We shall assume
thattheinitialdistributionoftheMarkovchainf
t g
t2N
isgivenbyitsunique
invariantdistribution
. Underthelaw
^
P
()
:=
R
^
P
()
(d)theprocess
f
t g
t2N
is stationary and ergodic. In economic terms this amounts to a
situationwherethemoodofthemarketisalready inequilibrium. Imposing
asuitablemean-contractionconditiononthetransformationF weshowthat
the sequence fp
t g
t2N
governed by (14) converges pathwise to a stationary
process fP
t g
t2N
inthesensethat
^
P
[lim
t!1 jp
t P
t
j=0]=1: (29)
Nonetheless, the price uctuations in our nancial market model may be
highly volatile. Consider, for example the log-linear dynamics (12) and
assume that c
c
> 1. In periods where the fraction of chartists who are
involved in the formation of equilibrium prices comes close to the critical
value 1
c
c
, the process fp
t g
t2N
may become highly unstable. Economically,
this feature can be interpreted as the temporary occurrence of bubbles or
crashesina nancialmarketmodel whoseoverallbehaviouris ergodic.
In Section 5.2, we concentrate on a nancial market model where the
sequenceoftemporarypriceequilibriaisgoverned byalog-linearstochastic
dierenceequation oftheform
logp
t+1
=f(~%
t+1 )logp
t +g(~%
t+1
) (t2N):
butsettlesdowninthelongrun. Westate conditionsonthenon-stationary
randomenvironment f~%
t g
t2N
whichensurethat thepriceprocess converges
to astationary regime.
5.1 EquilibriumPrices in a Stationary Environment
Letusanalyzethelongrunbehaviouroftheassetpriceprocessinanergodic
referencemodel. WeassumethattheMarkovchainf
t g
t2N
whichdescribed
the stochastic evolution of the aggregate behaviour already starts in its
probabilistic equilibrium
; see Theorem 4.7. In this case, the sequence
of temporaryprice equilibriafp
t g
t2N
is driven by a stationary and ergodic
random environment f%~
t g
t2N
, and so we can apply a result provided in
Borovkov(1998)inordertostudytoasymptotics oftheassetpriceprocess.
We denote by p the initial price at time t = 0, put %~ 0
n := (%~
1
;:::;%~
n )
andassume thatthe followingconditionsconcerningtheiterates
F(%~ 0
n
;p):=F(%~
n
;)ÆÆF(%~
2
;)ÆF(%~
1
;p) (n2N):
aresatised;see also Borovkov(1998), Chapter2, Section8.
Assumption 5.1 (i) For some p
0
2 R and for each Æ > 0, there exists
N =N(Æ) such that, for all n1, we have
^
P
[jp
0 F(%~
0
n
;p
0
)j>N]<Æ:
Thus,the sequence fp
t g
t2N
isassumed to bebounded in probability.
(ii) ThefunctionF =F(% ;~ p)iscontinuousinpandthereexistsaninteger
r1, a number >0, and a measurable function c:R r
! R
+ such
that
jF(%~ 0
r
;p
1
) F(%~ 0
r
;p
2
)j c(%~ 0
r )jp
1 p
2
j (p
1
;p
2 2R)
1
r
^
E
logc(%~ 0
r
) ;
where
^
E
denotes the expectation with respect to the measure
^
P
.
(iii) Thesequenceflogc(%~
jr
;::: ;%~
jr+r )g
j2N
satisesthestronglawoflarge
numbers.
condition for the sequence fp
t g
t2N
governed by (14). The next theorem
follows from Theorem 12.2 in Borovkov (1998). In the case of a linear
transformationF,itreduces to Theorem1 inBrandt(1986).
Theorem 5.2 SupposethatAssumption5.1issatised. Thenthereexistsa
stationaryprocessfP
t g
t2N
suchthat, foranystarting pointpofthesequence
fp
t g
t2N
, we have
^
P
[lim
t!1 jP
t p
t
j=0]=1:
If the initial value p has the same distribution as the random variable P
0 ,
then the price process fp
t g
t2N
is stationary.
Forournancialmarketmodel,andundertheassumptionthatthemood
of the market is already in equilibrium,Theorem 5.2 provides a boundfor
the aggregate eect of interaction between dierent traders which ensures
that the inducedprice uctuations are asymptotically stationary. Beyond
this bound the price process may become highly transient. A continuous-
timeanalogueofTheorem5.2isformulatedinFollmerandSchweizer(1993),
whereitisshownthatthetrajectoriesinasimplelog-linearmodelmayeither
tendto zeroorgooto innitywithpositiveprobabilityifthedestabilizing
eectsof theenvironmentare onaverage too strong.
Example5.3 Consider the log-linear demand structure described by (5)
together with(6)and(7). Inthiscase,themarketclearingconditionimplies
the log-linear price dynamics of the form
logp
t+1
=f(%~
t+1 )logp
t +g(%~
t+1 );
where the mappings f;g:M(C)!R aregiven by (13). If the conditions
^
E
logjf(%^
0
)j<0 and
^
E
(logjg(%^
0 )j)
+
<1
are satised, then the price process converges pathwise to a stationary pro-
cess. Such a log-linear pricedynamics isanalyzed in detail in Section 5.2.
Example5.4 (Generalized auto-regression) Suppose that the sequence of
equilibrium prices fp
t g
t2N
obeys the recurrence relation
p
t+1
=G(f(%~
t+1 )
^
G(p
t
)+g(%~
t+1 ));
where G;G:R !R areLipschitz continuous functions, i.e.,
j
^
G(p
1 )
^
G(p
2 )j
^
kjp
1 p
2
j; jG(p
1
) G(p
2
)jkjp
1 p
2 j:
As a result of Theorems 8.4 and 12.2 in Borovkov (1998), the sequence
fp
t g
t2N
satises parts (i) and(ii) of our Assumption5.1 if
logj
^
kkj+
^
E
logjf(%~
0
)j<0 and
^
E
(logjg(~%
0 )j)
+
<1:
5.2 EquilibriumPrices in a Non-Stationary Environment
Let us now concentrate on a nancial market model where both the indi-
vidual excess demands functions and the reference levels take a log-linear
form; see for instance (5) together with (6) and (7). In such a situation,
thedynamicsofourpriceuctuationsisdescribedbyastochasticsequence
fp
t g
t2N
whichobeys thelog-linearrecursiverelation
logp
t+1
=f(~%
t+1 )logp
t +g(~%
t+1
) (t2N): (30)
Our goal is to analyze the asymptotic behaviour of asset prices under the
assumptionthatthe moodof themarketis outofequilibriumbutbecomes
stationary inthelongrun.
In order to make this more precise, note rst that the environment
f~%
t g
t2N
fortheevolutionofthepriceprocessisstationaryandergodicunder
thelaw
^
P
. Next, we introducethe -elds
^
F
t
:=(%~
s
:st)
anddenoteby
^
T :=
T
t2N
^
F
t
thetail-eldgeneratedbythesequencef~% g
t2N .
We saythat theprocess f~% g
t2N
has anice asymptoticbehaviour,if
lim
t!1 sup
k
^
P
^
P
k
^
Ft
=0: (31)
Here,k
^
P
^
P
k
^
Ft
denotesthetotalvariationofthesignedmeasure
^
P
^
P
on
^
F
t
. Since the total variation distance is continuous along decreasing -
algebras,we have
lim
t!1 k
^
P
^
P
k
^
Ft
=k
^
P
^
P
k
^
T :
Thus, (31)impliesthat P
=P
on T,and sotheasymptoticbehaviour of
anicedrivingsequence f~%
t g
t2N
isthesameunder
^
P
andunder
^
P
. Inthis
sensethesequence f~%g
t2N
becomesstationary inthe longrun.
Lemma 5.6belowshowsthat theprocess f%~
t g
t2N
has anice asymptotic
behaviour wheneverthe Markovchainf
t g
t2N
converges inlawto a unique
equilibriumandifthestochastickernel
~
Qintroducedin(28)satisesamild
regularity condition. Using this result, we show in Theorem 5.7, that our
nancialpriceuctuations behave asymptoticallyina stablemanner ifthe
destabilizingeectsoftherandomenvironmentonthedynamicsoftheprice
process areon average not toostrong.
Assumption 5.5 The stochastic kernel
~
Q from (M
h
(E);d) to M(C) in-
troduced in (28) satises a uniform Lipschitz condition:
k
~
Q (;)
~
Q(
^
;)kLd(;
^
):
Letusnowestablishthefollowingresultabouttheasymptoticbehaviour
ofthedrivingsequence f%~
t g
t2N .
Lemma5.6 Suppose that ourAssumptions4.4and 5.5aresatised. Then
the random environment f%~
t g
t2N
forthe evolution of the assetpriceprocess
has a niceasymptotic behaviourin the sense of (31).
Proof: Letusdenote byB theBorel--eldon(M
h
(E);d). Ina rststep,
we aregoing to establishtheexistence of aconstant L<1 such that
^
P
[f~%
0
;:::;%~
t g2B]
^
P
~
[f~%
0
;:::;%~
t g2B]
Ld(;
~
) (32)
for all t 2 N and B 2 t
i=1
B. To this end, we denote by L
Q
and L
~
Q the
Lipschitz constants for the stochastic kernels Q and
~
Q, respectively, and
introduce thequantity
t
:=sup
B sup
6=
~
^
P
[f~%
0
;%~
2
;:::;%~
t g2B]
^
P
~
[f~%
0
;%~
2
;::: ;%~
t g2B]
d(;
~
)
:
Dueto thecontraction propertyofthetransformationuestablishedinThe-
orem 4.7 and because of the uniform Lipschitz conditions imposed on the
stochastickernelQ and Q, we havethat
^
P
[f~%
0
;:::;%~
t+1 g2B]
^
P
~
[f~%
0
;:::;%~
t+1 g2B]
sup
B
^
P
[~%
0 2B]
^
P
~
[~%
0 2B]
+sup
B
^
Q(;B)
^
Q(
~
;B)
sup
B;s
^
P
u(;s) [f~%
0
;:::;%~
t g2B]
^
P
u(
~
;s) [f~%
0
;:::;%~
t g2B]
(L
~
Q +L
Q
+
t )d(;
~
);
and so
t (L
~
Q +L
Q )
X
i2N
i
L
~
Q +L
Q
1
:
Inparticular,(32) holdswithL= L
~
Q +L
Q
1 .
Let us denote by U t
the t-fold iteration of the transition operator U
associated withtheMarkov chain f
t g
t2N . Since
k
^
P
^
P
k
^
Ft
=sup
B
(U
t
^
P
[f~%
t g
t2N
2B])() Z
P
~
[f~%
t g
t2N 2B]
(d
~
)
and because the mapping 7! P
[f~%
t g
t2N
2 B] from M
h
(E) to [0;1] is
Lipschitz continuous withLipschitz constant L<1,wecan applyLemma
2.1.57inIosefescuandTheodorescu(1968): Thereareconstants
L<1and
<1 such that
sup
;B
(U
t
^
P
[f~%
t g
t2N
2B])() Z
M
h (E)
P
~
[f~%
t g
t2N 2B]
(d
~
)
L t
:
Thisyieldsourassertion. 2
Inourpresentsetting,thelogarithmicstockpriceprocessisdescribedby
alinearrecursive stochasticequation ina non-stationaryenvironment. Un-
dertheassumption that theenvironment has a niceasymptotic behaviour,
the asymptotics of such processes is analyzed in Horst (2001b). These re-
sults allow us to introduce a bound for the aggregate eect of interaction
betweendierent traderswhichensuresthatthepriceprocessisdriveninto
equilibriumwheneverthemoodofthemarketitselfsettlesdowninthelong.
t t2N
linear relation (30) and that our Assumptions 4.4 and 5.5 are satised. If
the random variables f(%~
0
) and g(%~
0
) satisfy
^
E
logjf(%~
0
)j<0 and
^
E
(log
jg(%~
0 )j)
+
<1; (33)
then there existsa unique probability measure on R N
such that the shifted
sequencefp
t+T g
t2N
convergesin distributionto as T !1.
Proof: Due to (31) and (33), thesequence f(f(%~
t );g(%~
t ))g
t2N
is \nice" in
the sense of Denition 2.1 in Horst (2001b). Thus, our assertion follows
fromTheorem2.4 inHorst(2001b). 2
6 Continuous-Time Asset Price Processes
Inthissection,weshall againassume thatthe dynamicsof thelogarithmic
priceprocesscanbedescribedbylinearrecursivestochasticequation. Under
mild technical assumptionson the drivingsequence f%~
t g
t2N
we willobtain
a continuous-time assetprice process fP
t g
t0
by passageto the limit from
thediscrete-timeequilibriumpriceprocess fp
t g
t2N
denedrecursivelyby
logp
t+1
logp
t
=f(%~
t+1 )logp
t +g(%~
t+1
) (t2N): (34)
TheconvergenceconceptweuseisweakconvergenceontheSkorohoodspace
D d
ofallR d
-valuedright-continuousfunctionswithleftlimitson[0;1), en-
dowed with the weak topology. A similar approach was carried out by
Follmerand Schweizer(1993) whopassedto a continuous-time model from
a sequence of suitably specied discrete-time processes evolving in an ex-
ogenously given stationary and ergodic random environment. We extend
the Follmer-Schweizer model by (i) analyzing a situation were the driving
sequence is derived endogenously, and (ii)byreplacingthe stationarity as-
sumptiononthe moodof themarketbyan asymptoticstabilitycondition.
To this end, we consider in Section 6.1 a sequence of discrete-time
stochastic processes fP n
g
n2N , P
n
= fP n
t g
t2N
, dened recursively by the
linearrelation
P n
t+1 P
n
t
= 1
p
n A
t P
n
t +
1
p
n B
t
(t;n2N)
t t t2N
non-stationary driving sequence f(A
t
;B
t )g
t2N
which allow us to derive a
convergence result for the processes fP n
g
n2N
. This will be achieved by
applyinganinvarianceprincipletothenon-stationary continuous-time pro-
cessesX n
and Y n
which arespeciedby
X n
t :=
1
p
n [nt]
X
i=0 A
i
and Y n
t :=
1
p
n [nt]
X
i=0 B
i
: (35)
Armedwith these results, we establish in Section6.2 a Black-Scholes type
approximationfortheassetpriceprocess(34)inasituationwherethemood
ofthemarketsettlesdownin thelongrun.
In the sequel itwill be convenient to denote byLaw(X;P) the lawof a
randomvariableX under themeasure P.
6.1 AFunctionalCentralLimitTheoremfor Non-Stationary
Sequences
Letf(A
t
;B
t )g
t2N
asequenceofR 2
-valuedrandomvariablesdenedonsome
probability space (;F;P). For any n 2 N, we consider a discrete-time
process fP n
t g
t2N
given bythelinearrelation
P n
t+1 P
n
t
= 1
p
n A
t P
n
t +
1
p
n B
t
: (36)
IffZ n
t g
t2N
isanydiscrete-timeprocess,weidentifyZ n
withthecontinuous-
time process Z n
t
:= Z n
[nt]
(t 0) whose paths are right-continuous. In
termsofthequantitiesX n
and Y n
denedin(35), ourstochasticdierence
equation (36)is equivalentto thestochastic dierentialequation
dP n
t
=P n
t dX
n
t +dY
n
t
: (37)
If the driving sequence f(A
t
;B
t )g
t2N
is stationary and ergodic under the
law P, and under what Follmer and Schweizer (1993) call \standard as-
sumptions"onthetwosourcesofrandomnessfA
t g
t2N
andfB
t g
t2N
,onecan
applyan invariance principleto thesequences X n
and Y n
(n2N) dened
by (35)and assume that theprocess f(X n
;Y n
)g
n2N
is \good" in thesense
ofthefollowingdenition.
Denition 6.1 (DuÆe and Protter (1992)) A sequence fZ g
n2N
of semi-
martingales dened on probability spaces ( n
;F n
;P n
) is called \good" if,
for any sequence fH n
g
n2N
of c adl a g adapted processes, the convergence
Law((Z n
;H n
);P n
) w
!Law ((Z ;H);P) (n!1)
implies the convergence
Law
Z n
;H n
; Z
H n
dZ n
;P n
w
!Law
Z ;H;
Z
H dZ
;P
:
Here w
! denotes weak convergenceof probability measures.
Letussummarizes some resultsfrom Follmerand Schweizer(1993).
Proposition 6.2 (i) Suppose that the driving sequence f(A
t
;B
t )g
t2N is
stationary and ergodic and that EA
0
= EB
0
= 0. Under \standard
assumptions"on thetwo sourcesofrandomness fA
t g
t2N
andfB
t g
t2N ,
thesequencef(X n
;Y n
)g
n2N
introduced in (35)convergesin lawtothe
Gaussian martingale (X;Y) = V W. Here, W = (W
1
;W
2
) denotes
a two-dimensional standard Brownian motion and V isa suitable de-
terministic 22 dispersion matrix.
(ii) Ifthesequencef(X n
;Y n
)g
n2N
is\good"inthesenseofDenition6.1,
the process f(X n
;Y n
;P n
)g
n2N
converges in law to (X;Y;P). Here,
P =fP
t g
t0
is the uniquestrong solutionof the stochastic dierential
equation
dP
t
=P
t dX
t +dY
t
: (38)
Thatis, P =fP
t g
t0
is the pathwise solutionof a SDEof the form
dP
t
=P
t dW
t +d~
~
W
t
whereW,
~
W arestandard Brownianmotionswithcorrelation %. If >
0, then the diusion limit fP
t g
t0
converges to a stationary process,
andits invariant distribution can begiven in closed form.
We arenowgoingto establisha\non-stationary" versionofProposition
6.2. We obtain a convergence result for the processes fP n
g
n2N
given that
the driving sequence f(A
t
;B
t )g
t2N
is out of equilibrium, but has a nice
asymptoticbehaviour.
Assumption 6.3 (i) There exists a probability measure P on (;F)
such that the environment f(A
t
;B
t )g
t2N
is stationary and ergodic un-
derthe law P
.
(ii) Theasymptoticbehaviouroftheenvironmentf(A
t
;B
t )g
t2N
isthesame
underP and underP
, i.e.,
kP P
k
T
= lim
t!1
kP P
k
^
F
t
=0
where
^
F
t
:=((A
s
;B
s
):st) andT :=
T
t2N
^
F
t
isthe tail-eld gen-
erated by the sequence f(A
t
;B
t )g
t2N .
(iii) We have E
A
0
= E
B
0
= 0, where E
denotes the expectation with
respect to the law P
.
(iv) UnderthelawP
aninvarianceprinciplecanbeappliedtothesequence
f(X n
;Y n
)g
t2N
given by (35).
Below, wewillshowthatthedrivingsequencef(f(%~
t );g(%~
t ))g
t2N
forthe
assetpriceprocessdenedby(34)satisesparts(i),(ii)and(iv)oftheabove
assumptionwheneverourAssumptions4.4and5.5aresatised. Inthiscase
thedrivingsequence also satisesone ofthe \standardassumptions". This
allowsustoobtainadiusionapproximationfortheassetpriceprocess(34).
Theorem 6.4 If the driving sequence f(A
t
;B
t )g
t2N
satises Assumption
6.3, then the sequence of processes fZ n
g
n2N
= f(X n
;Y n
)g
n2N
converges
in distribution to the Gaussian martingale (X;Y) = V W. Here, W =
(W
1
;W
2
) is a 2-dimensional standard Brownian motion under the law P
and V isa deterministic volatility matrix.
Proof: Inorder to verify ourassertion, we proceedinseveral steps.
1. Dueto ourAssumption6.3, we knowthat
Law(Z n
;P
) w
!Law(V W;P
) (n!1):
2. We shall now use the assumption that the asymptotic behaviour of
the driving sequence f(A
t
;B
t )g
t2N
is the same under P
and under
theoriginal measure P inorder to show that the sequencesfX n
g
n2N
andfY g
n2N
satisfyaninvarianceprincipleP. Moreprecisely,we are
goingto verify that
Law (Z n
;P) w
!Law(V W;P
) (n!1): (39)
To this end, let f
n g
n2N
be a sequence of real numbers such that
n
" 1 and such that
n
= p
n ! 0 as n ! 1. For a given \time
horizon"T >0,andforeachn2N, weintroducethetwo-dimensional
process f e
Z n
t g
0tT
given by
e
Z n
t :=
(
1
p
n P
[nt]
i=
n (A
i
;B
i ) if
n
p
n
tT
0 otherwise.
We denotebyd
0
(;)and B
D
theSkorohood metric 5
and theBorel--
eldon thespace D
R
2[0;T],respectively. Notethat
d
0 (Z
n
; e
Z n
)
n
p
n
1
n n
X
i=0 jA
i j;
1
n n
X
i=0 jB
i j
!
: (40)
SinceP=P
onthetail-eldgenerated bythesequence f(A
t
;B
t )g
t2N
and because the environment f(A
t
;B
t )g
t2N
is ergodic under the law
P
;theseries
1
n
n
X
i=0 jA
i j and
1
n
n
X
i=0 jB
i j
areP-andP
-almostsurelyconvergentasn!1. Sincelim
n!1
n
p
n
=
0weobtainthat
lim
n!1 d
0 (Z
n
; e
Z n
)=0 P-a.s.and P
-a.s. (41)
Observe now that the event f e
Z n
2 Bg (B 2 B
D
) belongs to the -
algebra
^
F
n
. Since theenvironment has a nice asymptoticbehaviour
inthesenseof(31), there existsasequence fc
n g
n2N ,c
n
#0asn!1
such that
sup
B
P[
e
Z n
2B] P
[ e
Z n
2B]
c
n
: (42)
5
Forthedenitionofd0see, e.g.,Billingsley(1968),p.113.
3. Let us now denote by Q the law of the Gaussian martingale V W
underthemeasure P
and xaQ
-continuousset B 2B
D
. ByStep 1
above we know that
lim
n!1 P
[Z n
2B]=Q
[B]:
Thus, due to (41) and due to Theorem 4.2 in Billingsley(1968), we
havethat
lim
n!1 P
[ e
Z n
2B]=Q
[B]:
Using(42)we see that
lim
n!1 P[
e
Z n
2B]=Q
[B]:
Therefore,(41) and Theorem4.2inBillingsley(1968) implythat
Law (Z n
;P) w
!Law(V W;P
) (n!1): (43)
2
Remark 6.5 It is straightforward to extend the above theorem to the case
E
A
0 6=0,E
B
0
6=0. For notational convenience,however,weshallrestrict
our attentionto the caseE
A
0
=E
B
0
=0.
Let us assume that the non-stationary driving sequence f(A
t
;B
t )g
t2N
dened on (;F;P) is \good" in the sense of DuÆe and Protter (1992)
and satises our Assumption6.3. In this case it follows from Theorem 6.4
andProposition6.2thatthesequenceofprocessesfP n
g
n2N
denedby(37)
converges in distribution to the unique strong solution of the stochastic
dierentialequation
dP
t
=P
t dX
t +dY
t :
Armedwith these results we are now readyto show how the discrete-time
assetpriceprocessfp
t g
t2N
inanancialmarketmodelwithlog-linearexcess
demandfunctionscanindeedbeapproximatedinlawbyadiusionprocess.
