Asymptotic Exponential Arbitrage and Utility-based Asymptotic Arbitrage in Markovian Models of ...

15 Pages·2014·0.19 MB·English

Asymptotic Exponential Arbitrage
and Utility-based Asymptotic Arbitrage in
Markovian Models of Financial Markets∗
Martin Le Doux Mbele Bidima† Mikl´os Ra´sonyi ‡
4 June 23, 2014
1
0
2

n
Abstract
u
J
Consider adiscrete-timeinfinitehorizon financialmarket modelinwhich thelogarithm of
0
2 the stock price is a time discretization of a stochastic differential equation. Under conditions

differentfromthosegivenin[10],weprovetheexistenceofinvestmentopportunitiesproducing
]
C
an exponentially growing profit with probability tending to 1 geometrically fast. This is
O
achieved using ergodic results on Markov chains and tools of large deviations theory.
.
h
Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its rela-
t
a
m tionship to thefirst part of thepaper.
[
Keywords: Asymptotic exponential arbitrage, Markov chains, large deviations, expected utility.

1
v
2
1 Introduction
1
3
5
. Intheclassicaltheoryoffinancialmarkets,absenceofarbitrage(risklessprofit)ischaracterized
6
0 by the existence of suitable “pricing rules”: risk-neutral (i.e. equivalent martingale) measures for
4
1 the discountedpriceprocessofthe riskyasset. Thisresultisoftenreferredtoas“the fundamental
:
v theorem of asset pricing”.
i
X
Further developments of arbitrage theory encompass the so-called “large financial markets”
r
a (see[7],[6]andthe referencestherein). Inthesepapers the followingcommonfeatureofnumerous
models is highlighted: on each finite time horizon T > 0, there is no arbitrage opportunity but
whenT tendstoinfinity,onemayrealizerisklessprofitinthelongrun. Suchtradingopportunities
are referred to as “asymptotic arbitrage”.
An important tool that can be used for the study of asymptotic arbitrage is the theory of
large deviations (see [2]), as proposed in [6]. More recently, in [10] we presented the discrete-time
versions of some results in [6] about asymptotic arbitrage and, in this framework, we extended
∗Theauthorsthanktherefereeandtheassociateeditorforextremelyconstructiveandhelpfulreports.
†UniversityofYaound´eI,Cameroon,e-mail: [email protected]
‡MTA Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary and University of Edinburgh, U.K., e-mail:
[email protected]
1
them by studying “asymptotic exponential arbitrage with geometrically decaying probability of
failure”,i.e. we discussedthe possibility for investorsto realizean exponentially growingprofiton
their long-terminvestments while controlling (at a geometrically decaying rate) the probability of
failingtoachievesuchaprofit. Someoftheseresultsweresubsequentlyprovedforcontinuous-time
models in [4]. In the present paper we prove results similar to Theorem 5 of [10] using different
arguments and technical tools (the large deviation results of [9] instead of those in [8]). In this
way we manage to cover some well-known models for asset prices which were untractable in the
setting of [10], see Examples 3.14, 3.15 below. We recall now the setting of [10].
Consider a financial market in which two assets are traded: a riskless asset (a bank account
or a risk-free bond) with interest rate set to 0, i.e. with price normalized to B := 1 at all times
t
t N; and a single risky asset (such as stock) whose (discounted) price is assumed to evolve as
∈
S :=exp(X ), t N, (1)
t t
∈
where the logarithm of the stock price, X , is an R-valued stochastic process governed by the
t
discrete time difference equation
X X =µ(X )+σ(X )ε , t 1, (2)
t t−1 t−1 t−1 t
− ≥
starting from a constant X R. Here µ, σ : R R are measurable functions (determining the
0
∈ →
drift and volatility of the stock) and (εt)t∈N is an R-valued sequence of i.i.d. random variables
representing the random driving process of the stock price evolution.
Notethatthelog-priceprocessX isclearlya(discrete-time)Markovchaininthe(uncountable)
t
state space R (see pp. 211–228 in [1]). We suppose that its evolution is modelled on a filtered
probability space (Ω, ,F,P), where F := ( t)t∈N and t := σ(Xs,0 s t), is the natural
F F F ≤ ≤
filtrationofthe log-priceprocessX ofthe stock. Inthe sequelEdenotesexpectationwithrespect
t
to the probability P.
Trading strategies in this marketare assumed F-predictable [0,1]-valuedprocesses(π ) (i.e.
t t≥1
π is assumed -measurable) and no short-selling or borrowing are allowed. This means that,
t t−1
F
at each time t, investors allocate a proportion π [0,1] of their overall wealth to the stock while
t
∈
the rest remains in the bank account. Hence, given any such strategy, the corresponding wealth
process Vπ of an investor obeys the dynamics
t
Vπ S
t =(1 π )+π t , for all t 1, (3)
Vπ − t tS ≥
t−1 t−1
where Vπ :=V >0 is the investor’s initial capital.
0 0
Definition 1.1. (Definitions 3 and 4 of [10]) Let π be a trading strategy.
t
i)We say that π is an asymptotic exponential arbitrage (AEA) in thewealth model (3) if there
t
is a constant b>0 such that, for all ǫ>0, there is a time t N satisfying
ǫ
∈
P(Vπ ebt) 1 ǫ, for all time t t . (4)
t ≥ ≥ − ≥ ǫ
2
ii) We say that π generates an asymptotic exponential arbitrage (AEA) with geometrically
t
decaying probability of failure (GDPF) if there are constants b>0, and c>0 such that,
P(Vπ ebt) 1 e−ct for all large enough t 1. (5)
t ≥ ≥ − ≥
Clearly, AEA with GDPF implies AEA. The second kind of asymptotic arbitrage above is
muchmorestringentthanthefirstone. Indeed,in(4)above,thereisnovisiblerelationshipbetween
the tolerance level ǫ and the elapsed time t from which the investor starts realizing exponentially
ǫ
growing profit; one may need to wait for a very long time to achieve a desired tolerance level.
The concept of AEA with GDPF removes this drawback by allowing investors to control, at a
geometricallydecayingrate,the probabilityoffailingto achieveanexponentiallygrowingprofitin
the long term.
We recall the main results of [10] here. Under the following main assumptions: boundedness
of the drift and volatility functions µ and σ; σ being bounded away from 0 on compacts and
exponentialintegrabilityoftheε s,weprovedTheorem2(resp. Theorem4)of[10]ontheexistence
t
ofAEA(resp. AEAwithGDPF)inthewealthmodel(3). Theorem5of[10]providedergodicity-
related conditions on X which ensured AEA with GDPF.
t
In sections 2 and 3 below we continue to consider the same models as in (1), (2), (3). Under
a new set of conditions on µ, σ and (εt)t∈N (see (A1),(A2),(A3),(A4) below), which are neither
stronger nor weaker than the corresponding conditions in [10] recalled above, we show again the
existenceofAEAwithGDPF (seeTheorem2.3below),usingclassicallargedeviationstechniques
from [2], Markov chains tools from [11] and ergodicity results on Markov chains from [9]. More-
over,the tradingstrategiesgeneratingthose arbitrageopportunities willbe explicitly constructed;
a contribution we already obtained in [10] under different conditions, but it was absent from the
inspiringcontinuous-timework[6]. Togetthoseexplicitarbitrageopportunities wewillbe consid-
ering only stationary Markovian strategies, that is; strategies (π ) where π = π(X ), t 1,
t t≥1 t t−1
≥
for some fixed measurable function π :R [0,1].
→
Insection4,wewilldiscusstheconceptof“utility-based”asymptoticarbitrage,thatis,asymp-
totic arbitrage linked to von Neumann-Morgenstern expected utilities (see Chapter 2 of [5]). An
optimal investment for an economic agent with utility function U and time horizon T is π with
t
finalportfoliovalueVπ forwhichtheexpectedutility EU(Vπ)ismaximal. Wedonotfocusonthe
T T
construction of optimal strategies but rather on ones that provide (rapidly) increasing expected
utilities for the agent as the time horizon tends to infinity. More precisely, we wish to treat ques-
tionslike: forpowerutilitiesU,andgivenanAEAstrategyπ asin(4),willtheinvestor’sexpected
t
utility EU(Vπ) tend to the highest available utility U( )? If so, how fast such a convergencewill
t ∞
takeplace? Conversely,ifanagentpursuesatradingstrategyπ suchthather/hisexpectedutility
t
hasaconvergencerateestimate,willπ generateAEA(withGDPF)? Weprovidepartialanswers
t
to these questions in Proposition 4.1, Theorem 4.2 and Theorem 4.4 below.
3
2 Main theorem on AEA with GDPF
We denote by λ the Lebesgue measure on (R). We assume throughout this paper that the
B
Markov chain X satisfies the following conditions:
t
(A ) The random variables ε s have a (common) density γ with respect to λ, and this density
1 t
is bounded and bounded away from 0 on each compact in R.
(A ) The drift µ is locally bounded. The volatility σ is positive, bounded away from zero on
2
each compact and it is (globally) bounded.
(A ) We impose the mean-reverting drift condition
3
x+µ(x)
limsup| | <1.
x
|x|→∞ | |
(A ) We assume the following integrability property for the law of the ε s:
4 t
κ>0 such that E eκε21 =:I < , (6)
∃ ∞
(cid:0) (cid:1)
and Eε =0 holds1.
1
Remark 2.1. These conditions aresimilarto those of[10]. The maindifference is that µ was
assumedto be a boundedfunction in[10]while itmaybe unboundedinthe presentpaper. Inthis
way we accomodate e.g. autoregressive processes (see Examples 3.14 and 3.15 below), which did
not fit the setting of [10]. While we relax boundedness of µ, we need the integrability condition
(A ) on ε , which is more stringent than the ones in [10]. Furthermore, (A ) is a much stronger
4 t 3
ergodicity condition on the Markov chain X than that of Theorem 5 in [10]. Hence our main
t
result (Theorem 2.3) does not generalize [10] but rather complements it.
Remark 2.2. Analogously to [6], where the exponential of an Ornstein-Uhlenbeck process
was considered, our conditions imply that the log-price X is ergodic (in a strong sense). It may
t
be argued on ecomonetric grounds that the price increments X /X rather than X should be
t t−1 t
assumed ergodic. Just like in [10], we opted for the present setting in order to be consistent with
[6]. Very similar arguments could be used to prove analogous results for the case where X /X
t t−1
is assumed to be an ergodic Markov chain. We do not pursue this route here.
Consider the following condition:
(RC ) The set R+ := x R µ(x)>0 satisfies λ(R+)>0. (7)
+
{ ∈ | }
We interpret R+ as representing all states of the stock log-prices X whose “drift” is positive.
t
Thus (RC ) means that the set of states x from which there is a “bright future” (i.e. there is an
+
upwardtrendforthestockprice)haspositiveLebesguemeasure. Thisisrathernatural: notethat
short-selling is prohibited in our model hence negative market trends cannot be taken advantage
of. We now state the main result of the present article.
1Thisisnotarestrictionofgenerality. IfwehadEε1=m,wecouldreplaceµ(x)byµ′(x):=µ(x)+σ(x)mand
εt byε′t:=εt−mandinthiswaygetbacktothecaseEε1=0.
4
Theorem 2.3. Assume that (A ) (A ) and (RC ) hold. Then the Markovian strategy
1 4 +
−
π+ :=1 (X ) produces an AEA with GDPF.
t R+ t−1
The proof will be presented at the end of the next section, after appropriate preparations.
3 Large deviation estimates
Considerthe R2-valuedauxiliaryprocessΦ :=(X ,X ), t 0,consistingoftwoconsecutive
t t−1 t
≥
values of the log-price process X , where X is an arbitrarily chosenconstant. We present below
t −1
a set of preliminary results.
Proposition 3.1. The process Φ is a Markov chain with state space R2.
t
Proof. We derive this from [1] pp. 211-228,where the Markovpropertyof any (discrete-time)
processY inaPolishstatespaceS isprovedwhenY =m(Y ,ξ )with(ξ ) asequenceofi.i.d.
t t+1 t t+1 t t
random variables independent of Y and valued in some measurable space S′ and m:S S′ S
0
× →
a measurable function. Clearly, X = m(X ,ε ) for t N with m(x,y) := x+µ(x)+σ(x)y,
t+1 t t+1
∈
x,y R. It follows that we have Φ = (X ,X ) = (X ,m(X ,ε )) = F(Φ ,ξ ), where
t+1 t t+1 t t t+1 t t+1
∈
ξ :=(0,ε ) and F is the measurable function defined on S S′ :=R2 R2 by F((x,y),(a,b)):=
t t
× ×
(y,m(y,b)). Since the ε s are i.i.d. and independent of X , the ξ s are also i.i.d. and independent
t 0 t
from Φ0, showing the result. (cid:4)
Notice that
P(x,A):=P(X AX =x)= p(x,y)dy, x R, A (R),
1 0
∈ | Z ∈ ∈B
A
where
1 y µ(x) x
p(x,y):= γ − − ,
σ(x) (cid:18) σ(x) (cid:19)
and this function is bounded away from 0 on each compact in R2, by (A ) and (A ).
1 2
For z R2 and A (R2), let Qt(z,A):=P(Φ AΦ =z) be the t-step transitionkernel of
t 0
∈ ∈B ∈ |
the chain Φ .
t
We note that, for t 2 and A (R2),
≥ ∈B
Qt((u,v),A)= 1 (a ,a )p(v,a )p(a ,a )...p(a ,a )da ...da , u,v R. (8)
A t−1 t 1 1 2 t−1 t 1 t
ZRt ∈
Let λ denote the Lebesgue measure on (R2).
2
B
Proposition 3.2. The Markov chain Φ is ψ-irreducible, i.e. there is a non-trivial measure
t
ψ such that ψ(A)>0 implies that for all z, Qt(z,A)>0 for some t.
Proof. It suffices to check that λ is such a measure. If λ (A)>0 then for t=2 we get from
2 2
(8) that
Q2((u,v),A)= 1 (a ,a )p(v,a )p(a ,a )da da >0
A 1 2 1 1 2 1 2
ZR2
5
since p(v,a1)p(a1,a2) is (everywhere) positive. (cid:4)
We recalltwo definitions from Chapter 5 of [11] in our specific setting. A set C R2 is called
2
⊂
small if
Qt(x,A) µ(A) for all x C , A (R2)
2
≥ ∈ ∈B
with some non-trivial measure µ. The chain Φ is aperiodic if, for some small set C and corre-
t 2
sponding measure µ, the greatest common divisor of the set
E := n 1:for all x C , Qn(x,A) δ µ(A) for some δ >0 ,
C2 { ≥ ∈ 2 ≥ n n }
is 1.
Proposition 3.3. If C is a compact interval in R then C := R C is a small set for the
2
×
Markov chain Φ , and this chain is aperiodic.
t
Proof. It suffices to show that for all u R and v C,
∈ ∈
Qi((u,v),A) c λ (A (C C))
i 2
≥ ∩ ×
for i = 2,3 and appropriate constants c ,c > 0 since this implies 2,3 E . This is true by (8)
2 3 ∈ C2
with
c = inf p(v,a ) inf p(a ,a ), c = inf p(a ,a ) inf p(v,a ) inf p(a ,a )λ(C).
2 1 1 2 3 2 3 1 1 2
v,a1∈C a1,a2∈C a2,a3∈C v,a1∈C a1,a2∈C
(cid:4)
Now we need certain moment estimates.
Lemma 3.4. The random variable ε in (6) of Assumption (A ) satisfies the following prop-
1 4
erty: there is c>0 such that for every real number a 1 we have
≥
E ea|ε| eca2. (9)
≤
(cid:0) (cid:1)
Proof. Set ξ := ε . Then we have
1
| |
P eaξ >x = P exp κ log(eaξ) 2 >exp κ logx 2
a a
(cid:16) (cid:16) h i (cid:17) (cid:16) h i (cid:17)(cid:17)
(cid:0) (cid:1) 2
Iexp κ log(x)/a by Markov’s inequality
=≤ I(1)(κ(cid:16)/a−2)lo(cid:0)gx, (cid:1) (cid:17)
x
see (6) for the definition of I. Since the exponent (κ/a2)logx > 2 provided that x > e2a2/κ, we
have E eaξ = ∞P eaξ > x dx e2a2/κ +I ∞ 1/x2dx. The last integral is less than
0 ≤ exp(2a2/κ)
∞1/x(cid:0)2dx,(cid:1)whicRh is fi(cid:0)nite, thu(cid:1)s we conclude theRproof of (9) by taking c=c +(2/κ)with c >0
1 1 1
R
large enough. (cid:4)
The proof of Theorem 2.3 will be based on results from [9]. In order to apply the results of
that paper we will need to verify that the MarkovchainΦ satisfies condition (DV3+) below. We
t
formulate this condition only in the case where the state space is Rd.
We say that a ψ-irreducible and aperiodic Markov chain Z with transition law R = R(x,A)
t
satisfies condition (DV3+) if
6
(i) There are measurable functions V,W : Rd [1, ) and a small set C such that for all
→ ∞
x Rd,
∈
log(e−VReV)(x) δW(x)+b1 (x) (10)
C
≤−
for some δ,b>0.
(ii) There exists t > 0 such that, for each r < W , there is a measure β with β (eV) <
0 ∞ r r
k k ∞
and
P (Z A and Z has not quitted C (r) before t +1) β (A) (11)
x t0 ∈ t W 0 ≤ r
for all x C (r) and A (Rd), where C (r)= y Rd : W(y) r .
W W
∈ ∈B { ∈ ≤ }
We now recall the results of [9] which we will need in the sequel. Let W : Rd [1, ) such
0
→ ∞
that
W (x)
lim sup 0 1 =0. (12)
r→∞x∈Rd(cid:16)W(x) {W(x)>r}(cid:17)
Next, consider the Banachspace LW0 := g :Rd C:sup |g(x)| < , equipped with the norm
∞ { → x W0(x) ∞}
g :=sup g(x)/W (x).
k kW0 x| | 0
Theorem 3.5. Let Z satisfy (DV3+) with unbounded W. Then Z admits an invariant
t t
probability measure ν, the limit
t
1
Λ(g):= lim lnE [exp( g(Z ))]
z n
t→∞ t
nX=1
exists and it is finite for all g ∈LW∞0 and for all initial values Z0 =z (and it is independent of z).
Fix g0 ∈LW∞0. The function θ →Λ(g0+θg) is analytic in θ with Taylor-expansion
1
Λ(g +θg)=Λ(g )+θν(g)+ θ2σ2(g)+O(θ3),
0 0
2
where σ2(g):=lim (1/t)var(g(Z )+...+g(Z )).
t→∞ 0 t−1
Proof. This follows from Theorems 1.2 and 4.3 of [9].
Let us define Λ (θ):=Λ(θg) for θ R. Denote
g
∈
Λ∗(x):=sup(θx Λ (θ)), x R,
g θ∈R − g ∈
the Fenchel-Legendre conjugate of Λ ().
g
·
Corollary 3.6. Under the conditions of the previous Theorem, if σ2(g) > 0 then Λ∗(x) > 0
g
for all x=ν(g).
6
Proof. Λ is analytic, a fortiori, it is differentiable. Λ (0) = 0 by the definition of Λ. From
g g
the Taylor expansion of the preceding Theorem, Λ′(0) = ν(g) and Λ′′(0) = σ2(g) > 0 so we get
g g
that Λ∗(ν(g)) = ν(g) 0 Λ (0) = 0. By the definition of a conjugate function we always have
g × − g
Λ∗(x) 0 x Λ (0)=0 for all x R. It follows that ν(g) is a global minimiser for Λ∗. By the
g ≥ × − g ∈ g
7
differentiability of Λ , Λ∗ is strictly convex on its effective domain. This implies that the global
g g
minimiser ν(g) for Λ∗g is unique. This uniqueness implies that Λ∗g(x)>0 for all x6=ν(g). (cid:4)
In order to apply these results to our long-terminvestmentproblems we need to establish that
Φ satisfies (DV3+). First we prove a related statement about X .
t t
Proposition 3.7. The Markov chain X satisfies the “drift condition” (10) for d = 1,
t
R(x,A) = P(x,A) with a suitable compact interval C R and V(x) = W(x) = 1+qx2 with
⊂
a suitable q >0.
Proof. Recall that ReV(x):= eV(y)R(x,dy), for all x R. We have to show
∈
R
PeV(x) eV(x)−δW(x)+b1C(x) for all x R (13)
≤ ∈
for suitably small q >0 and C =[ K,K] with K suitably large.
−
Since PeV(x)= E eV(X1) X0 =x =E eV(x+µ(x)+σ(x)ε1) , it follows from (13) that we need
|
(cid:0) (cid:1) (cid:0) (cid:1)
to show,
E e1+q(x+µ(x))2+2q(x+µ(x))σ(x)ε1+qσ2(x)ε21 e(1−δ)V(x)+b1C(x) for all x R. (14)
≤ ∈
(cid:0) (cid:1)
To get this, it is sufficient to prove the two claims below:
Claim 1: For all x with x >K with K large enough we have
| |
E e1+q(x+µ(x))2+2q(x+µ(x))σ(x)ε1+qσ2(x)ε21 e(1−δ)(1+qx2) (15)
≤
(cid:0) (cid:1)
Claim 2: For “small” x (i.e. x K we have
| |≤
supE e1+q(x+µ(x))2+2q(x+µ(x))σ(x)ε1+qσ2(x)ε21 <G(K), (16)
x∈C (cid:0) (cid:1)
for some positive constant G(K).
Proof of Claim 1. Using Assumption (A ), for x large enough, there is a small δ > 0
3
| |
such that (x+µ(x))2 (1 4δ)x2. Since 1 δ(1+qx2) for x large enough, it follows that
≤ − ≤ | |
e1+q(x+µ(x))2 e(1−3δ)(1+qx2).
≤
By(A )thereisM >0suchthat,forallx,σ(x) M. IfwechooseqsuchthatqM2 <κ/2then
2
≤
it is enough to show that E e2q|x+µ(x)|M|ε1|+(κ/2)ε21 e2δqx2. By the Cauchy-Schwarz inequality,
≤
(cid:0) (cid:1)
it suffices to prove
E e4q|x+µ(x)|M|ε1| E eκε21 e2δqx2 (17)
q q ≤
(cid:0) (cid:1) (cid:0) (cid:1)
By (6), the second term on the left-hand side of (17) is the constant √I. This is smaller than
eδqx2 for large enough x. So, since again by (A ), 4q x+µ(x)M 4qM x for x large enough,
3
| | | | ≤ | | | |
it remains to show E e4qM|x||ε1| eδqx2 for large x, or, equivalently,
q ≤ | |
(cid:0) (cid:1)
E e4qM|x||ε1| e2δqx2 for large x. (18)
≤ | |
(cid:0) (cid:1)
ApplyingLemma3.4,theleft-handsideof (18)issmallerthane16cq2M2|x|2 forsomefixedconstant
c > 0. Hence, if one chooses q small enough such that 16q2M2c < 2δq and qM2 < κ/2 then (18)
holds, showing Claim 1.
8
Proof of Claim 2. By Assumption (A ), µ is bounded above on any compact C = [ K,K]
2
−
by some positive constant A = A(K) and the function x (x+µ(x))2 is also bounded on C by
7→
some positive constant B =B(K). Applying Cauchy-Schwarz Inequality and (6),
E e1+q(x+µ(x))2+2q(x+µ(x))σ(x)ε1+qσ2(x)ε21 E e1+qB+2q(K+A)M|ε1|+(κ/2)ε21
≤
(cid:0) (cid:1) e((cid:0)1+qB) E e4q(K+A)M|ε1| E(cid:1) eκε21
≤ q q
(cid:0) (cid:1) (cid:0) (cid:1)
= e(1+qB)√I E e4q(K+A)M|ε1|
q
(cid:0) (cid:1)
We then choose K large enough such that 4q(K +A)M 1 and we get, by Lemma 3.4, that
≥
for all x C =[ K,K],
∈ −
E e1+q(x+µ(x))2+2q(x+µ(x))σ(x)ε+qσ2(x)ε2 e(1+qB)√I e16c′q2(K+A)2M2,
≤
(cid:0) (cid:1) p
for a fixed constant c′ > 0. This holds for all x C, hence (16) holds true when taking the
∈
supremum over C of the left-hand side of this latter inequality. (cid:4)
Proposition 3.8. The Markov chain Φ satisfies (DV3+)(i).
t
Proof. We followProposition4.1of[9]anddeducethis statementfromProposition3.7above.
Recall V(x) = W(x) = 1+qx2, C and δ > 0 from that Proposition. Take C := R C. For
2
×
x,y R define V (x,y):=V(y)+(δ/2)W(x) and W (x,y):=(1/2)(W(x)+W(y)). Then
2 2
∈
loge−V2QeV2(x,y) = V(y) (δ/2)W(x)+log eδ2W(y)+V(z)P(y,dz)
− − ZR
V(y) (δ/2)W(x)+(δ/2)W(y)+[V(y) δW(y)+b1 (y)]
C
≤ − − −
δW (x,y)+b1 (x,y),
≤ − 2 C2
showing that (10) is true with V ,W . As C has been shown to be small in Proposition 3.3, we
2 2 2
conclude. (cid:4)
Proposition 3.9. The chain Φ satisfies condition (DV3+)(ii) as well.
t
Proof. Consider V (x,y),W (x,y), defined in the previous Proposition. We choose t := 2,
2 2 0
and let r < W = .
∞
k k ∞
It suffices to prove existence of a measure β on (R2) such that,
r
B
β (eV2)< and Q2 (x,y),D C (r) β (D), (19)
r W r
∞ ∩ ≤
(cid:0) (cid:1)
for all (x,y) C (r) and all D (R2).
W
∈ ∈B
LetH denotetheprojectionofC (r)onthefirstcoordinate(whichisthesameasitsprojection
W
on the second coordinate). By (A ) and (A ), the function p(x,y) is bounded on H H by a
1 2
×
constant J. Hence
Q2 (x,y),D C (r) p(y,a )p(a ,a )da da J2λ (D C (r))=:β (D).
W 1 1 2 1 2 2 W r
∩ ≤Z ≤ ∩
(cid:0) (cid:1) D∩CW(r)
Finally, it is clear that βr(eV2) < as it is the Lebesgue-integral of a continuous function on a
∞
compact of R2. (cid:4)
9
Corollary 3.10. The Markov chain Φ has an invariant probability measure ν equivalent to
t
λ .
2
Proof. Theorem 3.5 implies that Φ has an invariant probability measure, say, ν.
t
Furthermore, from (8), P(Φ Φ = (x,y)) is λ -absolutely continuous for each (x,y) R2,
2 0 2
∈ ·| ∈
hence we get ν λ . On the other hand, from the definitions of recurrent and positive chains
2
≪
on pages 186 and 235 of [11], it follows by Proposition 10.1.1 and Theorem 10.4.9 of the same
reference that ν ψ, where ψ is a maximal irreducibility measure. Hence ψ λ by Proposition
2
∼ ≫
4.2.2 (ii) in [11], so we get ν λ2. It follows that ν λ2, as required. (cid:4)
≫ ∼
We nowproceedto aproper investigationofasymptotic arbitrageexponentialopportunities in
the wealth model (3). Inspecting again the dynamics of the investor’s wealth process Vπ in this
t
model, for any Markovianstrategy π , we may express it in the form
t
t t f(Φ )
Vπ =V exp f(Φ ) =V exp t n=1 n , for all t 1, (20)
t 0 (cid:0)nX=1 n (cid:1) 0 (cid:16) P t (cid:17) ≥
where the function f is defined by
f(x,y):=log (1 π(x))+π(x)exp(y x) , x,y R, (21)
− − ∈
(cid:0) (cid:1)
and Φ =(X ,X ), t N, is the Markovchainin consideration. We will need to insure that, for
t t−1 t
∈
any Markovian strategy π , the sequence of random variables log(Vπ/V )= t f(Φ ) satisfies
t t 0 n=1 n
P
a large deviation principle (LDP) hypotheses. That is, we will need that the limit Λ (θ) :=
f
limt→∞ 1t logE(eθPtn=1f(Φn)) exists, for each θ ∈ R, with Λf satisfying the remaining conditions
in G¨artner-Ellis Theorem as stated in Theorem 2.3.6 in [2].
Define the function W :R2 [1, ) by W (x,y):=1+ x + y , for all x,y R. Clearly, W
0 0 0
→ ∞ | | | | ∈
satisfies (12) with d=2 and W =W .
2
Lemma 3.11. The function f belongs to the space LW0.
∞
Proof. Forallx,y R,sinceπ(x) [0,1],wehave1 π(x)+π(x)exp(y x) 1+exp(y x).
∈ ∈ − − ≤ −
It follows that f(x,y) x + y +1.
≤| | | |
On the other hand, for 0 a 1/2, we have 1 a+aexp(y x) 1/2. And for a>1/2, we
≤ ≤ − − ≥
have1 a+aexp(y x) (1/2)exp(y x). Tosumup,weobtainthatf(x,y) log(1/2) x y
− − ≥ − ≥ −| |−| |
for all x,y R.
∈
Hence f(x,y) c(1+ x + y ), for some constant c>0, and the claim follows. (cid:4)
| |≤ | | | |
Proposition 3.12. Let π be any Markovian strategy in the wealth model (3). Then Λ (θ):=
t f
limt→∞ 1t logE(X−1,X0) eθPtn=1f(Φn) , θ ∈ R is a well-defined analytic function so the averages
1log(Vπ/V ) = 1 t (cid:0)f(Φ ) satisf(cid:1)y a large deviations estimate with good rate function Λ∗ (the
t t 0 t n=1 n f
P
convex conjugate of Λ ).
f
Proof. By Theorem 3.5, Λ verifies the conditions of the G¨artner-Ellis Theorem 2.3.6 in [2]
f
(analyticity implies essential smoothness). Applying this theorem we conclude. (cid:4)
10