R 1Introduction [email protected] [email protected] ARBITRAGE-FREEMODELSINMARKETSWITHTRANSACTIONCOSTS

in PROBABILITY

ARBITRAGE-FREE MODELS IN MARKETS WITH TRANSACTION COSTS

HASANJAN SAYIT

Department of Mathematical Sciences, Worcester Polytechnic Institute email: [email protected]

FREDERI VIENS¹

Department of Statistics, Purdue University email: [email protected]

SubmittedMay 18, 2010, accepted in final formJuly 2, 2011 AMS 2000 Subject classification: 91G10, 91B25, 60G22

Keywords: Financial markets, arbitrage, transaction cost, sticky process, fractional Brownian mo- tion, time-change

Abstract

In the paper[7], Guasoni studies financial markets which are subject to proportional transaction costs. The standard martingale framework of stochastic finance is not applicable in these markets, since the transaction costs force trading strategies to have bounded variation, while continuous- time martingale strategies have infinite transaction cost. The main question that arises out of[7] is whether it is possible to give a convenient condition to guarantee that a trading strategy has no arbitrage. Such a condition was proposed and studied in[6]and[1], the so-called stickiness property, whereby an asset’s price is never certain to exit a ball within a predetermined finite time. In this paper, we define the multidimensional extension of the stickiness property, to handle arbitrage-free conditions for markets with multiple assets and proportional transaction costs. We show that this condition is sufficient for a multi-asset model to be free of arbitrage. We also show thatd-dimensional fractional Brownian models are jointly sticky, and we establish a time-change result for joint stickiness.

1 Introduction

In[7], a market with multiple risky assets and proportional transaction costs were studied. In the setting of [7], the market contains one risk free asset, used as a numeraire and hence assumed identically equal to 1, and drisky assets, given by anR^d−valued processY_t = (Y_t¹,Y_t²,· · ·,Y_t^d)

1SUPPORTED BY NSF GRANT DMS 0907321.

614

that is càdlàg (right- continuous with left-limits), adapted, and quasi-left continuous (i.e., Y_τⁱ = Y_τ−ⁱ , 1≤i≤d for all predictable stopping timesτ). Transaction costs are proportional and each unit of numeraire traded in the risky assets generates a transaction cost ofkunits that are charged to the riskless asset account.

Trading strategies are given by adapted, left-continuous,R^d−valued processesθ= (θt¹,θt²,· · ·,θt^d) that are of finite variation and satisfy the following admissibility condition:

V_t(θ) =

d

X

i=1

Z t

0

θsⁱd Y_sⁱ−

d

X

i=1

( Z t

0

kY_sⁱd|Dθⁱ|s+k|θtⁱ|Y_tⁱ)≥ −M a.s. (1) for some determistic M > 0 and all t ≥ 0. Here Dθⁱ is the derivative of θtⁱ in the sense of distribution, and |Dθⁱ|t is the total variation measure associated to Dθⁱ in [0,t]. In (1), the termPd

i=1

Rt

0kY_sⁱd|Dθⁱ|scorresponds to the cost of trading andPd

i=1k|θtⁱ|Y_tⁱ corresponds to the liquidation cost at the end of trading.

Definition 1. An admissible trading strategyθ is an arbitrage strategy if V_t(θ)≥0and P(V_t(θ)>

0)>0for some t>0.

Remark 1. Due to Proposition 2.5 of [7] and the quasi-left continuity assumption on the price processes, left-continuity of the trading strategiesθ can be relaxed to predictablity.

In the case when there is only one risky asset, the model (1) reduces to V_t(θ) =

Z t

0

θsd Y_s−k Z t

0

Y_sd|Dθ|s−kY_t|θt|. (2)

This model was studied in the recent papers[8,1]. In[8], the notion of stickiness (see definition 2.9 of[8]and also Proposition 1 of [1]) was introduced as a sufficient for no-arbitrage in the model (2). It was also shown that a large class of Markov processes and models with full support in the Wiener space are sticky. In[1]stickiness was further studied and other classes of sticky processes were provided. In this note, we introduce a condition, which we call joint stickiness, and show that it is sufficient for no-arbitrage in the model (1), see proposition 1. Then we show joint stickiness remains unchanged under composition with continuous functions, see proposition 2. As an example, we show the joint sticky property for independent fractional Brownian motions with possibily different Hurst parameters, see Proposition 3. Lastly, we show a time change result on joint stickiness and provide non-semimartingale joint sticky processes by using time change, see Proposition 4 and corollaries thereafter.

2 Main Results

LetX_t= (X_t¹,X²_t,· · ·,X_t^d)be a càdlàg process adapted to the filtrationF= (Ft)_t∈[0,T]. For anyF stopping timeτ≤T, letA^τ,ε_i ={sup_t∈[τ,T]|Xⁱ_t−X_τⁱ|< ε}for anyε >0.

Definition 2. We say that X_t= (X¹_t,X²_t,· · ·,X^d_t)is jointly sticky with respect toFif

P[∩^di=1A^τ,ε_i |F_τ]>0 a.s. (3)

for anyFstopping timeτ≤T, and anyε >0.

In the following proposition, we show that joint stickiness implies no arbitrage in the model (1).

Proposition 1. Let X = (X_t¹,X²_t,· · ·,X_t^d)be a jointly sticky, adapted, and càdlàg process . Then, the market (1,e^X0t,e^X1t,· · ·,e^Xdt)does not admit arbitrage with propotional transaction costs k for any k>0.

Proof. Fixk>0. Assumeθs= (θ_s¹,θ_s²,· · ·,θ_s^d)is an arbitrage strategy. Then there is t∈[0,T] such that V_t(θ)≥0 and P(V_t(θ)> 0)>0. Let τ=inf{s≥ 0 :θ_sⁱ 6= 0,i =1, 2,· · ·,d} ∧t. If τ= t almost surely, then the left-continuity of the paths and the definition of τimpliesθs =0 on[0,t]for almost allω, thusV_t(θ) =0 almost surely and this contradicts with the assumption P(V_t(θ)>0)>0. Therefore we assume that the eventA={τ <t}has positive probability. Let Y_sⁱ = e^Xsi, ˜Y_sⁱ =Y_τ∧ⁱ _s, and Z_sⁱ = Y_sⁱ−Y˜_sⁱ for all 1≤ i≤ d and alls ∈[0,t]. We can write (1) as following

V_s(θ) =

d

X

i=1

Zs

τ

θ_µⁱdY˜_µⁱ+

d

X

i=1

Zs

τ

θ_µⁱd Z_µⁱ−k

d

X

i=1

( Zs

τ

Y_µⁱd|Dθⁱ|_µ+|θsⁱ|Y_sⁱ). (4) onAfor anys∈[τ,t]. Observe thatPd

i=1

Rs

τθ_µⁱdY˜_µⁱ=0 and

d

X

i=1

Zs

τ

θ_µⁱd Z_µⁱ =

d

X

i=1

Z_sⁱθ_sⁱ−

d

X

i=1

Zs

τ

Z_µⁱd Dθ_µⁱ. onAfor anys∈[τ,t]. Thus (4) becomes

V_s(θ) =

d

X

i=1

(Z_sⁱθsⁱ−k|θsⁱ|Y_sⁱ)−

d

X

i=1

( Z^s

τ

Z_µⁱd Dθ_µⁱ+k Z^s

τ

Y_µⁱd|Dθⁱ|_µ) (5) LetA^τ,ε_i ={sup_s∈[τ,t]|X_sⁱ−X_τⁱ|< ε}for anyε >0. SinceX_s= (X_s¹,X_s²,· · ·,X_s^d)is jointly sticky, the eventA^ε₁=A∩(∩^di=1A^τ,ε_i )has postive probability for anyε >0. Observe that onA^ε₁,|Z_sⁱ| ≤(e^ε−1)Y_sⁱ for alls∈[τ,t]and for each 1≤i≤d. Therefore onA^ε₁, we have

|Z_sⁱθ_sⁱ| ≤(e^ε−1)|θ_sⁱ|Y_sⁱ (6) and

| Z s

τ

Z_µⁱd Dθ_µⁱ| ≤(e^ε−1) Zs

τ

Y_µⁱd|Dθⁱ|µ (7)

for alls∈[τ,t]. From (5) (6), and (7) we conclude that onA^ε₁ V_s(θ)≤(e^ε−1−k)

d

X

i=1

[|θsⁱ|Y_sⁱ+ Zs

τ

Y_µⁱd|Dθⁱ|_µ], ∀s∈[τ,t] (8)

Note thatPd

i=1|θtⁱ|Y_tⁱ+Rt

τY_sⁱd|Dθⁱ|s>0 almost surely onA(this follows from the definitions of Aandτ). Therefore from (7) it follows thatV_t(θ)<0 onA^ε₁⊂Awheneverε <l n(1+k). This contradicts with the assumption P(V_t(θ)≥0) =1, sinceP(A^ε₁)>0 for allε >0. This shows that θ can not be an arbitrage strategy. This completes the proof.

Example 1. Let L¹_t,L²_t,· · ·,L^d_t be a sequence of independent Lévy processes in[0,T]with respect to the filtrationF. Then L= (L¹_t,L²_t,· · ·,L^d_t)is jointly sticky with respect toF. To see this, letτbe any stopping time ofF. Let A^τ,ε_i ={sup_{t∈[0,T−τ]}|L_τ+tⁱ −Lⁱ_τ|< ε}for each1≤i≤d and for anyε >0.

Then we have

P(∩^di=1A^τ,ε_i |F_τ) =P(∩^di=1A^τ,ε_i ) =

d

Y

i=1

P(A^τ,ε_i )>0.

The first equality above follows from the independence of L_τ+tⁱ −Lⁱ_τ with F_τ for each1≤ i ≤ d, the second equality follows from the independence assumption on Lⁱ_t,i = 1, 2,· · ·,d, and the last inequality follows from stickiness of lévy processes (the stickiness of Lévy processes was shown in[8]).

Proposition 2. Let X_t = (X¹_t,X_t²,· · ·,X_t^d)be a jointly sticky process with respect to the filtrationF. Let{f₁,f₂,· · ·,f_d}be a family of real valued continuous functions onR^d. Let Y_tⁱ=f_i(X¹_t,X²_t,· · ·,X^d_t) for each i∈ {1, 2,· · ·,d}. Then the process Y= (Y_t¹,Y_t²,· · ·,Y_t^d)is also jointly sticky with respect to F.

Proof. Fix anyε >0. For any stopping timeτ≤T, letB_i^τ,ε={sup_t∈[τ,T]|Y_tⁱ−Y_τⁱ|< ε}for each i∈ {1, 2,· · ·,d}. We need to show

P[∩^d_iB_i^τ,ε|F_τ]>0 a.s. (9) and this is equivalent to showing P[A∩(∩^d_iB^τ,ε_i )]>0 for anyA∈ F_τwithP(A)>0. FixA∈ F_τ with P(A) > 0, and let M > 0 be such that the event A₀ = A∩ {−M ≤ X_τⁱ ≤ M, 1 ≤ i ≤ d} has positive probability. Note that A₀ ∈ Fτ. The set O = [−M−1,M+1]×[−M −1,M+ 1]× · · · ×[−M−1,M+1] is a closed bounded set inR^d. Since f₁,f₂,· · ·,f_d are continuous on R^d, they are uniformly continuous onO. Therefore, there is a δ0 > 0, such that for each 1≤i ≤d,|f_i(x)−f_i(y)|< εas long as x,y ∈Oand||x− y||< δ0. Letδ1=min(1,δ0)and letA^τ,δ_i ¹ ={sup_t∈[τ,T]|X_tⁱ−X_τⁱ|< δ1}. SinceX_t is jointly sticky, the setA₁=A₀∩(∩^di=1A^τ,δ_i ¹)has positive probability. OnA₁, we haveX_τ∈O,X_t ∈O, and||X_t−X_τ|| ≤δ1≤δ0for allt∈[τ,T]. ThereforeA₁⊂ ∩^d_iB^τ,ε_i . SinceA₁⊂A, we haveP[A∩(∩^diB^τ,ε_i )]>P(A₁)>0. This completes the proof.

Example 2. Let B = (B¹_t,B²_t,· · ·,B_t^d)be d−dimensional Brownian motion. Then the process X = (|B¹_t|¹³,|B¹_t +B²_t|¹³,· · ·,|B¹_t +B^d_t|¹³)is not a semimartingale; see Theorem 72 on page 221 of[10]. However, X is jointly sticky thanks to Proposition 2 and Example 1.

The following corollary extends the Proposition 1 in[1].

Corollary 1. If the process X_t= (X¹_t,X²_t,· · ·,X_t^d)is jointly sticky, then for any real valued continuous function g:R^d→R, the process Y_t=g(X¹_t,X_t²,· · ·,X_t^d)is sticky.

In the following Proposition shows that any finite sequence of independent fractional Brownian motions with possibily different Hurst parameters is jointly sticky.

Proposition 3. Let B^H_tⁱ =Rt

−∞[(t−s)^Hi−12−1_{s≤0}(−s)^Hi−12]d B⁽_tⁱ⁾,i=1, 2,· · ·,d be a sequence of independent fractional Brownian motions in[0,T]with respective Hurst parameters H₁,H₂,· · ·,H_d∈ (0, 1). Then for any deterministic continuous functions f₁,f₂,· · ·,f_d on [0,T], the process B = (B^H_t¹+f₁(t),B^H_t²+f₂(t),· · ·,B^H_t^d+f_d(t))is jointly sticky .

Proof. Let

Ω ={ω∈C(R):ω(0) =0 and∀t∈R, lim

s→t

ω(t)−ω(s) q

|t−s|l o g(_|t−s|¹ )=0},

Btheσ−algebra of subsets ofΩthat is generated by the cylinder sets andPthe Wiener measure on (Ω,B). Let (Ω⁽ⁱ⁾,B⁽ⁱ⁾,P⁽ⁱ⁾),i = 1, 2,· · ·,d be d copies of (Ω,B,P). With slight abuse of notation, we denote byPthed−dimensional Wiener measureP⁽¹⁾×P⁽²⁾× · · · ×P^(d)on(Ω^d,B^d), where(Ω^d,B^d)is the product space of(Ω⁽ⁱ⁾,B⁽ⁱ⁾),i=1, 2,· · ·,d. Without loss of generality, we assume that for each 1≤i≤d,B^H_tⁱ is defined on(Ω^d,B^d,P)by the improper Riemann-Stieltjes integrals

B^H_tⁱ(ω) = Zt

−∞

[(t−s)^Hi−12−1_{s≤0}(−s)^Hi−12]dω⁽ⁱ⁾(s), t≥0. (10) whereω= (ω⁽¹⁾,ω⁽²⁾,· · ·,ω^(d))∈Ω^d (see the proof of Theorem 4.3 of[3]). LetF^B= (F_t^B)_t∈[0,T] be the filtration given by

F_t^B=∨^d_i=1σ{B^H_sⁱ: 0≤s≤t}.

Then B_t^H1,B^H_t²,· · ·,B^H_t^d are independent fractional Brownian motions in the filtered probability space(Ω^d,B^d,F^B= (F_t^B)t∈[0,T],P). LetF_t^Ωd=

∨^d_i=1σ{{ω∈Ω^d:ω⁽ⁱ⁾(s_j)≤a_j,j=1, 2,· · ·n}:−∞<s_j≤t,a_j∈R,n∈N}

Then(ω⁽¹⁾(t),ω⁽²⁾(t),· · ·,ω^(d)(t))isd−dimensional Brownian motion in the filtered probability space (Ω^d,B^d,F^Ωd = (Ft^Ωd)t∈[0,T],P). It is clear thatF_t^B ⊂ F_t^Ωd,t ≥0, thereforeF^B stopping times are alsoF^Ωd stopping times.

Now, letτbe any stopping time ofF^Band let

A^τ,ε_i ={sup_{t∈[0,T−τ]}|B^H_τ+ⁱ_t−B_τ^Hi+f_i(τ+t)−f_i(τ)|< ε}

for each 1≤i≤dand for anyε >0. To show the jointly stickiness ofB, we need to show P(∩^d_i=1A^τ,ε_i |F_τ^B)>0 a.s.

However, sinceF_τ^B⊂ F_τ^Ωd, it is sufficient to show

P(∩^di=1A^τ,ε_i |F_τ^Ωd)>0 a.s. (11) We divide the proof of (11) into two steps.

(A)For eachω(s)∈Ω^d set

π⁽₁ⁱ⁾ω(s) =ω⁽ⁱ⁾(s)1_{(−∞,τ(ω)]}(s),s∈R, π⁽₂ⁱ⁾ω(s) =ω⁽ⁱ⁾(τ(ω) +s)−ω⁽ⁱ⁾(τ(ω)),s≥0, for all 1≤i≤d. For each 1≤i≤d, let

Ω⁽ⁱ⁾1 ={π⁽ⁱ⁾1 ω:ω∈Ω^d}, Ω⁽ⁱ⁾2 ={π⁽ⁱ⁾2 ω:ω∈Ω^d}

and letB₁⁽ⁱ⁾andB₂⁽ⁱ⁾be theσ−algebras generated by the cylinder sets ofΩ⁽₁ⁱ⁾andΩ⁽₂ⁱ⁾respectively.

Also let Ωi = Ω⁽¹⁾_i ×Ω⁽²⁾_i × · · · ×Ω⁽_i^d),i = 1, 2 and letBi = B_i⁽¹⁾× B_i⁽²⁾× · · · × B_i^(d),i = 1, 2.

It is clear that π⁽₁ⁱ⁾ : Ω^d → Ω⁽₁ⁱ⁾ isF_τ^Ωd measurable for each 1 ≤ i ≤ d, hence the map π1 : Ω^d→Ω1given byπ1ω= (π⁽¹⁾₁ ω,π⁽²⁾₁ ω,· · ·,π⁽₁^d)ω)isF_τ^Ωd measurable (for notational simplicity we writeπ1ω:=ω1 = (ω⁽¹⁾₁ ,ω⁽²⁾₁ ,· · ·,ω⁽₁^d))) . Also it follows from Theorem 6.16 of[13] that π2ω:= (π⁽¹⁾₂ ω(s),π⁽²⁾₂ ω(s),· · ·,π⁽₂^d)ω(s))isd−dimensional Brownian motion independent from F_τ^Ωd. Define a mapτ⁰:Ω1→Rbyτ⁰(ω1):=τ(ω), whereω∈Ω^d is such thatω1=π1ω(note that ifω⁰,ω∈Ω^d andπ1ω⁰ =π1ω, thenτ(ω) =τ(ω⁰), sinceτisF_τ^Ωd measurable). Then for eachω∈Ω^d, we can write

(B_τ+^Hi_t−B_τ^Hi)(ω) = (12)

Zτ⁰(π1ω)

−∞

[(τ⁰(π1ω) +t−s)^Hi−12−(τ⁰(π1ω)−s)^Hi+12]dπ⁽1ⁱ⁾ω(s) +

Z t

0

(t−s)^Hi−12dπ⁽₂ⁱ⁾ω(s)

Note thatτ⁰(π1(ω)) =τ(ω), thereforeτ⁰(π1(·))isF_τ^Ωd measurable.

(B)LetA^τ,ε_i be as in (2) for each 1≤i≤d. For eachω1= (ω⁽¹⁾₁ ,ω⁽²⁾₁ ,· · ·,ω⁽₁^d))inΩ1, define hⁱ_t(ω1):=

Zτ⁰(ω1)

−∞

[(τ⁰(ω1) +t−s)^Hi−12−(τ⁰(ω1)−s)^Hi+12]dω⁽ⁱ⁾₁ (s) +f(τ⁰(ω1) +t)−f(τ⁰(ω1)),

and for eachω2= (ω⁽¹⁾2 ,ω⁽²⁾2 ,· · ·,ω^(d)2 )∈Ω2define H_tⁱ(ω2):=

Z ^t

0

(t−s)^Hi−12dω⁽ⁱ⁾2 (s) Then from (12) and the definition ofτ⁰, it follows that

[B_τ+^Hi_t−B_τ^Hi+f_i(τ+t)−f_i(τ)](ω) =hⁱ_t(π1ω) +Hⁱ_t(π2ω),t≥0. (13) Define

Cⁱ:={(ω1,ω2)∈Ω1×Ω2: sup

t∈[0,T−τ⁰(ω1)]|hⁱ_t(ω1) +H_tⁱ(ω2)|< ε}

for each 1≤i≤d. Sincehⁱ_t+Hⁱ_t is continuous process inΩ1×Ω2,CⁱisB1× B2measurable for each 1≤i≤d. Sinceπ1isF_τ^Ωd measurable andπ2is independent fromF_τ^Ωd, from Proposition A.2.5 of[4], for almost everyω∈Ω^d we have

E[1_∩d

i=1Cⁱ(π1,π2)|F_τ^Ωd](ω) =φ(π1ω) (14) whereφ(ω1) = E1_∩d

i=1Cⁱ(ω1,π2). From (13) and the definitions of Cⁱ andA^τ,ε_i , it is clear that 1_Ci(π1ω,π2ω) =1_A^τ,ε

i (ω)for each 1≤i≤dandω∈Ω^d. Therefore E[1_∩^d

i=1A^τ,ε_i |F_τ^Ωd](ω) =E[1_∩^d

i=1Cⁱ(π1,π2)|F_τ^Ωd](ω) =φ(π1ω).

for eachω∈Ω^d. In the following, we will show thatφ(ω1)>0 for eachω1∈Ω1. To see this, note that the random variables 1Cⁱ(ω1,π1),i=1, 2,· · ·,dare independent for each fixedω1∈Ω1(this follows from the independence of the Brownian motionsπ⁽ⁱ⁾₂ ω,i=1, 2,· · ·,dand the definitions ofH_tⁱ). Therefore, we haveφ(ω1) =E1C¹(ω1,π2)E1C²(ω1,π2)· · ·E1C^d(ω1,π2). Let

B^ε_i(ω1) ={ω∈Ω^d:(ω1,πiω)∈Cⁱ} for each 1 ≤ i ≤ n. Then, we have 1_Ci(ω1,πi) =1_Bε

i(ω1) for each 1≤ i ≤ d. This shows that φ(ω1) = P(B₁^ε)×P(B₂^ε)× · · · ×P(B^ε_d). Therefore, it is sufficient to show P(B_i^ε)> 0 for each 1≤i≤d. Note that

B^ε_i(ω1) ={ω∈Ω^d: sup

t∈[0,T−τ⁰(ω1)]|hⁱ_t(ω1) + Zt

0

(t−s)^Hi−12dπiω < ε}.

If τ⁰(ω1) = 0, then B^ε_i(ω1) = Ω^d, so P(B^ε_i(ω1)) > 0. If τ⁰(ω1) > 0, then sinceπi(ω) is a Brownian motion andhⁱ_t(ω1)is a deterministic continuous function for eachω1, from the results in[6,8,11], it follows thathⁱ_t(ω1) +Rt

0(t−s)^Hi−12dπiωhas full support in C[0,τ⁰(ω1)]. This, in turn, implies that B^ε_i(ω1)has positive probability for eachi. Therefore φ(ω1)> 0 for each ω1∈Ω1. Now, the result follows from (14).This completes the proof.

In the following Proposition we show a time change result on joint stickiness.

Proposition 4. Let X_t = (X¹_t,X²_t,· · ·,X^d_t)be a continuous process adapted to the filtrationF. Let V_t be a nondecreasing continuous process such that for each t, V_t isFstopping time. Then we have the following

(i) X is jointly sticky with respect to F if and only if for any stopping time τ ≤ T of F and any δ > 0, the stopping time τ1 = inf{t ≥ τ : |X_tⁱ−X_τⁱ| ≥ δ, 1 ≤ i ≤ d} ∧T satisfies P(τ1=T|F_τ)>0a.s.

(ii) If X is jointly sticky with respect toF, then the time changed process Y_t=X_V

t∧T= (X¹_V

t∧T,X²_V

t∧T,· · ·,X_V^d

t∧T) is jointly sticky with respect to the filtrationG= (Gt)t∈[0,T], whereGt=FV_t∧T.

Proof. Proof of (i): Assume X is jointly sticky. To show P(τ1 = T|F_τ) > 0, we need to show P(A∩ {τ1=T})>0 for anyA∈ F_τ withP(A)>0. LetA^τ,

δ 2

i ={sup_t∈[τ,T]|X_tⁱ−X_τⁱ|< ^δ₂}. Since X is jointly sticky, the eventA₁=A∩(∩^d_i=1A^τ,

δ 2

i )has positive probability. OnA₁we clearly have τ1=T and so P(A∩ {τ1=T})>0. To show the other direction, letτ≤T be any stopping time andA∈ F_τ be any event withP(A)>0. For anyε >0, letA^τ,ε_i ={sup_t∈[τ,T]|Xⁱ_t−X_τⁱ|< ε}for eachi. We need to show the eventA₁=A∩(∩^diA^τ,ε_i )has positive probability. Letτ1=inf{t≥τ:

|Xⁱ_t−X_τⁱ| ≥ ^ε₂, 1≤i≤d} ∧T. Since P(τ1=T|Fτ)>0 almost surely, the eventA∩ {τ1=T}has positive probability. By the definition ofτ1we haveA∩ {τ1=T} ⊂A₁and therefore P(A₁)>0.

This completes the proof.

Proof of (ii): Denote Y_tⁱ = X_Vⁱ

t∧T for each i. Let τ ≤ T be any stopping time of G. For any δ >0, let τ1=inf{t≥τ:|Y_tⁱ−Y_τⁱ| ≥δ, 1≤i ≤d} ∧T. Due to part(i)above, we only need to showP(τ1= T|G_τ)>0 almost surely. This is equivalent to showing that for anyA∈ G_τ with

P(A)>0, P(A∩ {τ1= T})>0. To see this, letτ0=inf{t ≥τ:|Y_tⁱ−Y_τⁱ| ≥ ^δ₄, 1≤i ≤d} ∧T. Letτ^A=τ1A+T1_Ω/Aandτ^A₀=τ01A+T1_Ω/A, then both ofτ^Aandτ^A₀areGstopping times. Since τ^A< τ^A₀ onA, there exists a deterministic numberk such thatA₁={τ^A< k< τ^A₀}has positive probability. Note thatA₁⊂AandA₁∈ Gk. Sinceτ0>k> τonA₁, by the definition ofτ0, for each 1≤i≤dwe have

sup_t∈[τ,k]|Y_tⁱ−Y_τⁱ| ≤δ

4 (15)

onA₁. Letθ=inf{t≥V_k∧T:|Xⁱ_t−X_Vⁱ

k∧T| ≥^δ₄, 1≤i≤d} ∧T. SinceX is jointly sticky, the event A₂=A₁∩ {θ =T}has positive probability. Sinceθ=TonA₂, for each 1≤i≤dwe have

sup

t∈[k,T]|X_Vⁱ

t∧T−X_Vⁱ

k∧T| ≤δ

4 (16)

onA₂. From (15) and (16), for each 1≤i≤dwe have

sup_t∈[τ,T]|Y_tⁱ+Y_τⁱ| ≤sup_t∈[τ,k]|Y_tⁱ−Y_τⁱ|+ sup

t∈[k,T]|Y_tⁱ−Y_kⁱ|< δ. onA₂. This shows thatτ1=T onA₂. This completes the proof.

Example 3. Let B^H_t¹,B^H_t²,· · ·,B^H_t^d be a sequence of independent fractional Brownian motions with respect to the filtrationF= (Ft)t≥0. Letνt be any bounded time change. Then the process X_t = (B^H_ν¹

t ,B_ν^H2

t ,· · ·,B^H_ν^d

t )is jointly sticky with respect to the filtration(Fνt)t∈[0,T]. To see this, let M be such thatνt ≤ M almost surely for all t ∈[0,T]. From Proposition 3, B = (B^H_t¹,B_t^H2,· · ·,B^H_t^d)is jointly sticky for the filtration(Ft)t∈[0,M]. Then, from part(ii)of Proposition (4), we conclude X is jointly sticky with respect to(F_νt)_t∈[0,T].

References

[1] Bayraktar, E., Sayit, H.: “On the stickiness property ” Accepted to Quantitative Finance.

[2] Bayraktar, E., Sayit, H.: “No Arbitrage Conditions For Simple Trading Strategies ”The Annals of Finance, Forthcoming.

[3] Cheridito, P: “Arbitrage in fractional Brownian motion models"’,Finance Stoch.7(4), 533–

553, 2003.MR2014249

[4] Lamberton, D., Lapeyre, B.: “Stochastic calculus applied to finance"’. Chapman & Hall (1996).MR1422250

[5] Delbaen, F., Schachermayer, W.: “A General Version of the Fundamental Theorem of Asset Pricing” Math. Ann.300, 1994, 463–520.MR1304434

[6] Guasoni, P.: “No Arbitrage with Transaction Costs, with Fractional Brownian motion and Beyond”Mathematical Finance, Vol. 16, Issue 2, 2006, 469-588.MR2239592

[7] Guasoni, P.: “Optimal Investment with Transaction Costs and Without Semimartingales”The Annals of Applied Probability, 12(2002), No. 4, 1227-1246.MR1936591

[8] Guasoni, P.; Rasonyi, M; Schachermayer, W.: "‘Consistent Price Systems and Face-Lifting Pricing under Transaction Costs"’Annals of Applied Probability, 18 (2008), no. 2, 491-520.

MR2398764

[9] Guasoni, P.; Rasonyi, R.; Schachermayer, W.: “The fundamental Theorem of Asset Pricing for Continuous Processes under Small Transaction Costs"’Annals of Finance, forthcoming.

Annals of Applied Probability, 18 (2008), no. 2, 491-520.MR2398764

[10] Protter, P.: “Stochastic Integration and Differential Equations", Second Edition, Version 2.1, Springer-Verlag, Heidelberg, 2005.MR2273672

[11] Cherny, A. S.: “Brownian moving averages have conditional full support"’, The annals of Applied Probability, 18(5), 1825-1830, 2008.MR2432181

[12] Jarrow, R., Protter, P., Sayit, H.: “No Arbitrage Without Semimartingales” The annals of applied probability, 19(2009), No.2, 596-616.MR2521881

[13] Karatzas, I, Shreve E., S.:Brownian Motion and Stochastic Calculus, Second Edition, Springer- Verlag, Heidelberg, 1991.MR1121940

[14] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Third Edition, Springer- Verlag, Heidelberg, 1999.MR1725357

[15] Rogers, L.C.G.: “Arbitrage with fractional Brownian motion”Math. Finance7, 1997, 95–105.

MR1434408