Let Xt be any d-dimensional continuous process that takes values in an open connected domain 𝒪 in ℝd. In this paper, we give equivalent formulations of the conditional full support (CFS) property of Xt in 𝒪. We use them to show that the CFS property of X in 𝒪 implies the existence of a martingale M under an equivalent probability measure such that M lies in the ϵ>0 neighborhood of Xt for any given ϵ under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in
markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (2008), to processes with more general state space.

1. Introduction

We consider a financial market with d risky assets and a risk-free asset which is used as a numéraire and therefore assumed to be equal to one. We assume that the price processes of the d risky assets are given by an ℝd-valued process Yt=(Yt1,Yt2,…,Ytd), where Yti=eXti, 1≤i≤d, and the d-dimensional process Xt=(Xt1,Xt2,…,Xtd) is defined on a filtered probability space (Ω,ℱ,𝔽=(ℱt)t∈[0,T],P) and adapted to the filtration 𝔽 that satisfies the usual assumptions. We assume that there are transaction costs in the market and they are fully proportional in the sense that each cost is equal to the actual dollar amount being traded beyond the riskless asset, multiplied by a fixed constant. In the presence of such transaction costs, it is reasonable to assume that purchases and sales do not overlap to avoid dissipation of wealth. In general, in markets with proportional transaction costs trading strategies θt=(θt1,θt2,…,θtd) are given by the difference of two processes Lt=(Lt1,Lt2,…,Ltd) and Mt=(Mt1,Mt2,…,Mtd) representing respectively the cumulative number of shares purchased and sold up to time t, namely, θt=Lt-Mt. We are also required to start and end without any position in the risky assets to and this requirement corresponds to θ0=θT=0.

For each such trading strategy θt=Lt-Mt, the corresponding wealth process, after taking into account the incurred transaction costs, is given by
(1.1)Vt(θ)=∑i=1d∫0tθsidYsi-ϵ∑i=1d∫0tYsidVar(θi)s,
where Var(θi)s=Lsi+Msi is the total variation of θi in [0,s] for each 1≤i≤d and ϵ>0 is the proportion of the transaction costs. In our model (1.1), transaction costs between risky assets and cash are permitted and all transaction costs are charged to the cash account. Next, we introduce the class of trading strategies that we consider in this paper.

Definition 1.1.

An admissible trading strategy is a predictable ℝd-valued process θt=(θt1,θt2,…,θtd) of finite variation with θ0=θT=0 such that the corresponding wealth process Vt(θ) satisfies Vt(θ)≥-C for some deterministic C>0 and for all t∈[0,T].

In the next definition, we state the absence of arbitrage condition for the market.

Definition 1.2.

We say that the market (1,Yt1,Yt2,…,Ytd) does not admit arbitrage with ϵ-sized transaction costs if there is no admissible trading strategy θt=(θt1,θt2,…,θtd) such that the corresponding value process Vt(θ) satisfies
(1.2)P(VT(θ)>0)>0,P(VT(θ)≥0)=1.

The absence of arbitrage condition excludes trading strategies that enables the investors to have nonnegative payoff with the possibility of positive payoff with zero initial investment. The purpose of this note is to study the sufficient conditions on (Yt1,Yt2,…,Ytd) that ensure absence of arbitrage in the market (1,Yt1,Yt2,…,Ytd). It is clear that if the stock price process Yt=(Yt1,Yt2,…,Ytd) is a martingale under a measure Q that is equivalent to the original measure P, then the model (1.1) does not admit arbitrage. This can easily be seen from the fundamental theorem of asset pricing (see [1]) that states that martingale price processes do not admit arbitrage in frictionless markets (i.e., ϵ=0). In the absence of such martingale measure for Y, the existence of a process Y~t=(Y~t1,Y~t2,…,Y~td) which is a martingale under an equivalent measure Q and which has the following property:
(1.3)|Yti-Y~ti|≤ϵYti,fori=1,2,…,d,∀t∈[0,T],
also implies absence of arbitrage for the model (1.1). To see this simple fact, observe the following:
(1.4)VT(θ)=∑i=1d∫0Tθsid(Ysi-Y~si)+∑i=1d∫0TθsidY~si-ϵ∑i=1d∫0TYsidVar(θi)s=∑i=1d[∫0T(Ysi-Y~si)dθsi-ϵ∫0TYsidVar(θi)s]+∑i=1d∫0TθsidY~si.
Note that because of (1.3), we have ∑i=1d[∫0T(Ysi-Y~si)dθsi-ϵ∫0TYsidVar(θi)s]≤0 a.s. This implies that
(1.5)VT(θ)≤∑i=1d∫0TθsidY~si.
The financial interpretation of (1.5) is that trading at price process Y~t without transaction costs is always at least as profitable as trading at price process Yt with transaction costs. The martingale property of Y~t implies that trading on Y~t is arbitrage free, and therefore trading on Y with transaction costs is also arbitrage-free.

The process Y~t is called consistent price systems (CPSs) for the price process Y. The origin of CPSs is due to [2] and the name consistent price system first appeared in [3]. In the following, we write down the formal definition of CPSs.

Definition 1.3.

Let ϵ>0. We say that Y~t=(Y~t1,Y~t2,…,Y~td) is an ϵ-consistent price system for Yt=(Yt1,Yt2,…,Ytd), if there exists a measure Q~P such that Y~t is a martingale under Q, and
(1.6)11+ϵ≤Y~tiYti≤1+ϵ, fori=1,2,…,d,∀t∈[0,T].

The existence of such pricing functions is a central question in markets with proportional transaction costs and their existence was extensively studied in the past literature. For example, the papers [4, 5] studied CPSs for semimartingale models and the papers [6–11] studied CPSs for non-semi-martingale models. Other papers that studied similar problems include [4, 8, 12–17]. Particularly, the recent paper [10] introduced a general condition, conditional full support (CFS), for price processes and showed that if a continuous process Xt=(Xt1,Xt2,…,Xtd) with state space ℝd has the CFS property, then the exponential process Yt=(Yt1,Yt2,…,Ytd) admits ϵ-CPS for any ϵ>0. The proof of this result is based on a clever approximation of Y by a discrete process which is called random walk with retirement (see [10]). In this paper, we consider continuous processes Xt with general state space 𝒪, where 𝒪 is any connected open set in ℝd. Unlike the original paper [10], where the random walk with retirement is constructed by using geometric grids, in this paper we choose to work on arithmetic grid. As a consequence, we show that if the process Xt with the state space 𝒪 has the corresponding CFS property, then for any given ϵ>0 there exists a martingale Mt=(Mt1,Mt2,…,Mtd), under an equivalent change of measure, such that
(1.7)|Mti-Xti|≤ϵforanyi=1,2,…,dandanyt∈[0,T].
By an abuse of language we call such M a ϵ-consistent price system for the process X. To achieve this goal, we first provide a few of equivalent formulations of the CFS property. We use these equivalent formulations in the proof of our result. The advantage is that with our approach the proofs become more transparent and also it enables us to state some stronger results than the original paper. For example, our Lemma 2.10 is a stronger result than the corresponding result in [10] that states that the CFS property is equivalent to the so-called strong CFS property which is stated in terms of stopping times.

Our main result in this paper is Theorem 2.6 which states that the CFS property of X in any open connected domain 𝒪 implies the existence of CPSs. To prove this result, we first prove Lemmas 2.7, 2.8, 2.9, 2.10, and 2.11. In Lemma 2.7, we show that the CFS property implies the necessary properties of a random walk with retirement (see [10] for the formal definition of random walk with retirement). In Lemma 2.8, we prove that our approximating discrete time process is a martingale under an equivalent martingale measure. The proof of this Lemma gives an alternative and elementary proof for the corresponding result in the paper [10]. In Lemma 2.9, we prove that the approximating discrete time process is in fact a uniformly integrable martingale. The proof of this lemma is standard and similar to the corresponding proofs of the papers [10, 11]. In Lemma 2.10, we show the equivalence of the f-stickiness with the weak f-stickiness for each given f. In Lemma 2.11, we show that the CFS property is equivalent to the seemingly weaker linear stickiness property.

2. Main Results

Let Xt=(Xt1,Xt2,…,Xtd), t∈[0,T] be a d-dimensional continuous process that takes values in an open connected domain 𝒪⊂ℝd. For simplicity of our discussion, we assume that 0∈𝒪. We also assume that the process Xt is defined on a probability space (Ω,ℱ,P) and adapted to a filtration 𝔽=(ℱt)t∈[0,T] that satisfies the usual assumptions in this space. Let C([u,v],𝒪) denote the set of continuous functions f defined on the interval [u,v] and with values in 𝒪 and, for any x∈ℝd, let Cx([u,v],𝒪) denote the set of functions in C([u,v],𝒪) with f(u)=x.

Definition 2.1.

An adapted continuous process Xt satisfies the CFS property in 𝒪, if for any t∈[0,T) and for almost all ω∈Ω,
(2.1)Supp(Law[(Xs)s∈[t,T]∣Ft(ω)])=CXt(ω)([t,T],O),a.s.

The CFS condition requires that, at any given time, the conditional law of the future of the process, given the past, must have the largest possible support. An equivalent formulation of this property is given in the following definition.

Definition 2.2.

Let Xt be an adapted continuous process that takes values in an open and connected domain 𝒪⊂ℝd. We say that Xt is linear sticky if for any α∈ℝd, ϵ>0, and any deterministic 0≤s≤θ≤T,
(2.2)P({supt∈[s,θ]|Xt-Xs-α(t-s)|<ϵ}∣Fs)>0,a.s.
on the set {Xs∈⋂t∈[0,θ-s](𝒪-αt}.

The equivalence of the CFS and the linear stickiness properties will be established in Lemmas 2.10 and 2.11. We also need the following definition.

Definition 2.3.

Let Xt be an adapted continuous process that takes values in an open and connected domain 𝒪⊂ℝd.

We say that Xt is f-sticky for f∈C0([0,T],ℝd) if
(2.3)P({supt∈[τ,T]|Xt-Xτ-f(t-τ)|<ϵ}∣Fτ)>0,a.s.
on the set {Xτ∈⋂t∈[0,T-τ](𝒪-f(t))} for any ϵ>0 and any stopping time τ.

We say that Xt is weak f-sticky for f∈C0([0,T],ℝd) if
(2.4)P({supt∈[s,T]|Xt-Xs-f(t-s)|<ϵ}∣Fs)>0,a.s.
on the set {Xs∈⋂t∈[0,T-s](𝒪-f(t))} for any ϵ>0 and any deterministic time s∈[0,T].

Remark 2.4.

It is clear that the CFS property of X in 𝒪 is equivalent to the weak f-stickiness of X for all f∈C0([0,T],𝒪). The linear stickiness of Xt is seemingly weaker condition than the weak f-stickiness of Xt for all f∈C0([0,T],𝒪). However, this is not the case and in Lemma 2.11 we will show that linear stickiness is equivalent to weak f-stickiness of Xt for all f∈C0([0,T],ℝd). This, in turn, implies that the linear stickiness property is equivalent to the CFS property.

Remark 2.5.

When a process Xt is 0-sticky as in (b) in Definition 2.3, we say that Xt is jointly sticky and this property was studied in the recent paper [14]. The f-stickiness roughly means that starting from any stopping time τ on, the process Xt has paths that are as close as one wants to the path f(t)+Xτ. As it was shown in [14], the f-stickiness holds for any f∈C0([0,T],ℝd) for the process (BtH1,BtH2,…,BtHd), where BtH1,BtH2,…,BtHd are independent fractional Brownian motions with respective Hurst parameters H1,H2,…,Hd∈(0,1). From [10], the f-stickiness also holds for any continuous Markov process with the full support property in C0([0,T],ℝd) for any f∈C0([0,T],ℝd).

The following is the main result of this paper. This result is an extension of the main result in [10] to processes with more general state space. We use [10] as a road map in the proof of this result.

Theorem 2.6.

Let Xt=(Xt1,Xt2,…,Xtd) be a continuous process that takes values in a connected domain 𝒪 in ℝd. If Xt is linear sticky, then Xt admits CPSs for all ϵ>0.

To show this result, one fix any ϵ>0 and define the following increasing sequence of stopping times associated with the process X:
(2.5)τ0=0,τn+1=inft≥τn{|Xt-Xτn|≥ϵn+1}∧T,∀n≥0,
with ϵn+1:=ϵ∧d(Xτn,∂𝒪)/2. One should mention that the paper [10] defined the corresponding stopping times in a slightly different way, see the proof of Theorem 1.2 in [10].

In addition, for each n≥1 we define
(2.6)Δn={Xτn-Xτn-1 whenτn<T,0otherwise.
Let 𝒢n=ℱτn for every n≥0. Note that ϵn is bounded and 𝒢n-1 measurable.

In the following, we use the notation Supp(Δn∣𝒢n-1) to denote the smallest closed set of ℝd that contains the values of the random variable E[Δn∣𝒢n-1] with probability one. We use Br(x) to denote the open ball in ℝd with center x and radius r. When the center is 0, we simply write Br. We first prove the following lemma.

Lemma 2.7.

If Xt is f-sticky in 𝒪 for all f∈C0([0,T]), then the process {Δn} in (2.6) satisfies the following three properties:
(2.7)(i)P(Δm=0,∀m≥n∣Δn=0)=1;(ii)
Supp
(Δn∣Gn-1)=0∪∂Bϵnalmostsurely on{Δn-1≠0},(iii)P(Δm≠0,∀m≥1)=0.

Proof.

Property (i) is obvious since {Δn=0}={τn=T} and τn is increasing. Property (iii) follows from the fact that almost surely each path of Xt is contained in a compact set of 𝒪 and therefore minnϵn(ω)>0 almost surely ω∈Ω. To prove property (ii), let us assume that P(τn-1<T)>0 and let Gn-1 be any 𝒢n-1 measurable set such that P(Gn-1∩{τn-1<T})>0. Then, it is clear that there exist T′<T, y∈𝒪 and ζ>0 such that ζ<ϵ∧d(y,∂𝒪)/4 and P(Gn-1∩{τn-1<T′}∩{Xτn-1∈Bζ(y)})>0, where d(y,∂𝒪) is the distance of y with the boundary ∂𝒪 of 𝒪. It is also clear that on the set Gn-1∩{τn-1<T′}∩{Xτn-1∈Bζ(y)} we have ϵ∧3ζ/2≤ϵn<2ζ.

First we show that P(Δn=0∣𝒢n-1)>0 a.s. on {Δn-1≠0}. To see this, define the following stopping time:
(2.8)τ={τn-1onGn-1∩{τn-1<T′}∩{Xτn-1∈Bζ(y)},Totherwise.
The 0-stickiness of X implies that
(2.9)P(supt∈[τ,T]|Xt-Xτ|<ζ,τ<T)>0.
But {supt∈[τ,T]|Xt-Xτ|<ζ,τ<T}⊂{Δn=0} and since Gn-1 was an arbitrary 𝒢n-1 measurable set with P(Gn-1∩{τn-1<T})>0, we have
(2.10)P(Δn=0∣Gn-1)>0a.s.on{Δn-1≠0}.
Next we show that ∂Bϵn(ω)⊂Supp(Δn∣𝒢n-1)(ω) almost surely on {Δn-1≠0}. Too see this, take any x∈∂B1, 0<ϵ′<ζ and define
(2.11)f(t)={6ζt(T-T′)xif0≤t≤T-T′2,3ζxotherwise,
and note that
(2.12)P(τ<T,Xτ∈⋂t∈[τ,T](O-f(t-τ)))>0.
By f-stickiness of X, we obtain
(2.13)P(supt∈[τ,T]|Xt-Xτ-f(t-τ)|<ϵ′,τ<T)>0,
or equivalently P(A∩Gn-1)>0, where
(2.14)A={supt∈[τn-1,T]|Xt-Xτn-1-f(t-τn-1)|<ϵ′}∩{τn-1<T′}∩{Xτn-1∈⋂t∈[τn-1,T](O-f(t-τn-1))}.
We claim that A⊂{Δn⊂B2ϵ′(ϵnx)}. Indeed, if ω∈A, we get
(2.15)|Xτn-1+(T-T′)/2(ω)-Xτn-1(ω)|≥|f(T-T′2)|-|Xτn-1+(T-T′)/2(ω)-Xτn-1(ω)-f(T-T′2)|≥3ζ-ϵ′>ϵn(ω).
Hence {τn<T} on A. Also, for ω∈A we have
(2.16)0=d(Xτn-Xτn-1,∂Bϵn)≥d(f(τn-τn-1),∂Bϵn)-|Xτn-Xτn-1-f(t-τn-1)|>|f(τn-τn-1)-ϵnx|-ϵ′,|Xτn-Xτn-1-ϵnx|≤|f(τn-τn-1)-ϵnx|+|Xτn-Xτn-1-f(t-τn-1)|<|f(τn-τn-1)-ϵnx|+ϵ′<2ϵ′.
So for all ω∈A, d(Δn,ϵnx)<2ϵ′. Since this is true for any small ϵ′, x∈B1 and any arbitrary 𝒢n-1 measurable set with P(Gn-1∩{τn-1<T})>0, we conclude that ∂Bϵn(ω)⊂Supp(Δn∣𝒢n-1)(ω) almost surely on {Δn-1≠0}. Note that the other direction ∂Bϵn(ω)∪{0}⊃Supp(Δn∣𝒢n-1)(ω) is clear from the definition of Δn.

Now, define ϵn, Δn, n≥0 as above and let Mn=X0+∑i=1nΔi, n≥0. The ℝd-valued process Mn=:(Mn1,Mn2,…,Mnd) will be used to construct CPSs for Xt. Next, we prove a lemma that shows that all of Mni, 1≤i≤d are in fact uniformly integrable martingales under an equivalent change of measure. The proof of this lemma uses Lemma 3.1 of [10] as a road map (see also Proposition 2.2.14 of [18]).

Lemma 2.8.

There exists a measure Q equivalent to P under which the ℝd-valued discrete process {(Mn,𝒢n)}n=0+∞ is a martingale.

Proof.

For any n≥0, let μn be the regular conditional probability of Δn with respect to 𝒢n-1 and let Ωn={ω∈Ω∣Supp(Δn∣𝒢n-1)(ω)=0∪∂Bϵn}. Let k be any strictly increasing convex function defined on ℝ with values in (0,+∞) such that k(t)=t for every t≥1. Define Gn:Ωn×ℝd→ℝd as follows:
(2.17)Gn(ω,α)=∫Rdk(α⋅x)xdμn(ω,⋅).
Obviously for each n, Gn(·,a) is 𝒢n-1 measurable and convex with respect to α. As a consequence, for any fixed ω∈Ωn, Im(Gn(ω,·)), the image of the function Gn(ω,·) is convex. We first prove that for every n≥1 and ω∈Ωn:
(2.18)lim|α|→∞Gn(ω,α)⋅α|α|=+∞.
By the way of contrary, assume that this is not true, for some n≥1 and ω∈Ωn. Then, there exists a sequence (αm)m≥1 with |αm|→∞ such that Gn(ω,αm)·(αm/|αm|) is bounded above. We can assume that (αm/|αm|) converges to some α (this is a bounded sequence and therefore has a convergent subsequence). We have
(2.19)Gn(ω,αm)⋅αm|αm|≥∫αm⋅x≤0k(αm⋅x)αm⋅x|αm|dμn(ω,⋅)+∫(αm⋅x)/|αm|>ϵn(ω)/4k(αm⋅x)αm⋅x|αm|dμn(ω,⋅)≥-k(0)+∫2α⋅x>ϵn(ω)(αm⋅x)2|αm|dμn(ω,⋅),
for big enough m. Therefore, we can conclude that ∫2α·x>ϵn(ω)((αm·x)2/|αm|)·(1/|am|)dμn(ω,·) converges to 0 as |am|→+∞, which will imply after passing to the limit that μn(ω,{2α·x>ϵn}) and this is a contradiction. From this it follows easily that 0∈Im(Gn(ω,·))¯. If 0∉Im(Gn(ω,·))¯, then using the geometric form of Hahn Banach theorem, there exists a unit vector β∈ℝd such that ∫ℝdk(α·x)βxdμn(ω,·)<0 for every α∈ℝd. Therefore, limsupt→∞∫ℝdk(tβ·x)βxdμn(ω,·)≤0. But
(2.20)∫Rdk(tβ⋅x)βxdμn(ω,⋅)=Gn(ω,tβ)⋅tβ|tβ|,
and so it contradicts (2.18).

Next, we want to show that Im(Gn(ω,·)) is closed. Let a∈Im(Gn(ω,·))¯, so there exists a sequence (αm)m≥1 such that Gn(ω,αm)→a. But then |αm| is unbounded, and therefore this contradicts (2.18). So based on the continuity of Gn(ω,·), a∈ Im(Gn(ω,·)).

Therefore, we conclude that for any n≥1 and ω∈Ωn, there exists an αn(ω)∈ℝd, unique, as a consequence of the strict monotonicity of k, such that Gn(ω,αn(ω))=0. Gn being continuous with respect to α and 𝒢n-1 measurable with respect to ω, it follows that αn is 𝒢n-1 measurable. We extend αn with 1 outside Ωn and define:
(2.21)Zn=k(αn⋅Δn)1{Δn≠0}2E(k(αn⋅Δn)1{Δn≠0}∣Gn-1)+1{Δn=0}2P(Δn=0∣Gn-1).
It is easy to check that Zn satisfies
(2.22)E(Zn∣Gn-1)=1,E(ZnΔn∣Gn-1)=0.
Let Ln=∏i=1nZi and L=limn→∞Ln. Note that this limit exists almost surely since Ln+1=Ln a.s. on {Δn=0} and {Δn=0}↗Ω. From (2.22), we get
(2.23)E(Ln∣Gn-1)=Ln-1E(LnMn∣Gn-1)=Ln-1Mn-1,
which shows that (Ln)n≥1 and (MnLn)n≥1 are martingales under P. We thus get E(Ln)=E(Z1)=1, and Fatou's lemma gives E(L)≤1. We will show that E(L)=1. We have
(2.24)E(L)=E(limn→∞L1{Δn=0})=limn→∞E(L1{Δn=0})=limn→∞E(Ln1{Δn=0})=1-limn→∞E(Ln1{Δn≠0})=1-limn→∞E(E(Ln1{Δn≠0}∣Gn-1))=1-12limn→∞E(Ln-11{Δn-1≠0})=1-limn→∞(12)n=1.
Combining Fatou's lemma with the equation E(Ln)=E(L)=1, we obtain E(L∣𝒢n)=Ln. Also,
(2.25)E(MnL∣Gn-1)=E(E(MnL∣Gn)∣Gn-1)=E(MnL∣Gn-1)=Mn-1Ln-1=E(Mn-1L∣Gn-1).
Hence, L is the density of a measure Q under which our discrete process Mn is a martingale. And since L>0 (Ln>0 for all n), Q is equivalent to P.

Lemma 2.9.

Under the measure Q of Lemma 2.8 the process Mni is uniformly integrable for each 1≤i≤d. In particular, EQ(supn≥0|Mni|)<∞ for each i=1,2,…,d.

Proof.

For any 1≤i≤d, set M*i=supn≥0|Mni| and observe that on {Δk≠0,Δk+1=0} we have M*i≤|X0i|+kϵ. Observing that Q(Δk≠0)=Q(Δk≠0∣Δk-1≠0)⋯Q(Δ1≠0∣Δ0≠0)Q(Δ0≠0) and that Q(Δk≠0∣Δk-1≠0)=1/2 we obtain the following:
(2.26)EQ(M*i)=∑k=0∞EQ(M*i1{Δk≠0}∩{Δk+1=0})≤∑k=0∞(|X0i|+kϵ)Q({Δk≠0,Δk+1=0})<∞.

The two lemmas above uses the f-stickiness. The f-stickiness is seemingly stronger condition than the weak f-stickiness since it involves stopping times. However, the next Lemma 2.10 shows that, in fact, these two conditions are equivalent.

Lemma 2.10.

Let Xt be an adapted continuous process with state space 𝒪 and f∈C0([0,T],ℝd). Then, Xt is weak f-sticky if and only if it is f-sticky.

Proof.

Let us show first that for any f∈C0([0,T],ℝd) weak f-stickiness implies f-stickiness. Suppose for a contradiction that Xt is weak f-sticky but not f-sticky. Then there exists a stopping time τ with P(τ<T)>0, and an ϵ>0 such that
(2.27)P(τ<T,Xτ∈⋂t∈[0,T-τ](O-f(t)))>0,P(supt∈[τ,T]|Xt-Xτ-f(t-τ)|<ϵ,τ<T)=0.
Since f∈C0([0,T],ℝd), there exists a δ>0 such that for all t, s∈[0,T], |t-s|<δ implies |f(t)-f(s)|<ϵ/3. In addition, we can find t1, t2∈[0,T), 0<t2-t1<δ, and 0<ζ≤ϵ/3 such that
(2.28)P(t1≤τ<t2,Xτ∈⋂t∈[τ,T](Oζ-f(t-τ)))>0,
where 𝒪ζ={x∈𝒪/d(x,∂𝒪)>ζ}.

For each q∈I=ℚ∩[t1,t2), let Aq:=A∩{t1≤τ<q}∩{supt∈[τ,q]|Xt-Xτ|<ζ}, where
(2.29)A:={t1≤τ<t2,Xτ∈⋂t∈[τ,T](Oζ-f(t-τ))}.
Since P(A)>0 and A=⋃q∈IAq, there exists a q*∈I such that P(Aq*)>0. Note that Aq*∈ℱq* and Aq*⊂⋂t∈[0,T-q*](𝒪-f(t)). Hence, since Xt is weak f-sticky, we obtain
(2.30)P(Aq*∩{supt∈[q*,T]|Xt-Xq*-f(t-q*)|<ϵ3})>0.
Let Cq*=Aq*∩{supt∈[q*,T]|Xt-Xq*-f(t-q*)|<ϵ/3}. Then we claim that
(2.31)Cq*⊂{supt∈[τ,T]|Xt-Xτ-f(t-τ)|<ϵ}∩{τ<T},
which contradicts (2.27). Indeed, if ω∈Cq*, then for t∈[τ,q*] we have
(2.32)|Xt-Xτ-f(t-τ)|<|Xt-Xτ|+|f(t-τ)|<ϵ3+ϵ3<ϵ,
by the definition of Aq* and the choice of δ. We will show also that |Xt-Xτ-f(t-τ)|<ϵ on Cq* whenever t∈[q*,T]:
(2.33)|Xt-Xτ-f(t-τ)|≤|Xq*-Xτ-f(t-τ)+f(t-q*)|+|Xt-Xq*-f(t-q*)|≤|Xτ-Xq*|+|f(t-q*)-f(t-τ)|+|Xt-Xq*-f(t-q*)|<ϵ3+ϵ3+ϵ3=ϵ.
Thus, weak f-stickiness implies f-stickiness. Since the opposite direction is obvious, the proposition is proved.

Lemma 2.11.

Let Xt be a continuous adapted process with state space 𝒪. Then Xt is linear sticky if and only if Xt is f-sticky for all f∈C0([0,T],ℝd).

Proof.

We only need to show that linear stickiness implies the weak f-stickiness for each f∈C0([0,T],ℝd). Fix any f∈C0([0,T],ℝd),s∈[0,T], ϵ0>0. We need to show that
(2.34)P({supt∈[s,T]|Xt-Xs-f(t-s)|<ϵ0}∣Fs)>0,a.s.
on the set B=:{Xs∈⋂t∈[0,T-s](𝒪-f(t))}. To do this, for any A∈ℱs with P(A∩B)>0, we need to show that
(2.35)P(A∩B∩{supt∈[s,T]|Xt-Xs-f(t-s)|<ϵ0})>0.
Define Z(ω)=infr∈[0,T-s]d(Xs(ω)+f(r),∂𝒪) for any ω∈A∩B. From the definition of B, it is clear that Z>0 a.s. on A∩B. Let h>0 be a constant such that the set B0={Z≥h} has positive probability. Note that B0∈ℱs and B0⊂A∩B. In the following, we show that
(2.36)P(B0∩{supt∈[s,T]|Xt-Xs-f(t-s)|<ϵ0})>0.
Let ϵ=min(ϵ0,h) and set t0=0, and define
(2.37)tk=inf{t≥tk-1:|f(t)-f(tk-1)|≥ϵ4}∧(T-s),
for k≥1. Let N be the smallest positive integer such that tN=T-s. For each k≥1, define g(t) on [tk-1,tk] to be equal to the linear function that connects the two points f(tk-1) and f(tk). We can assume that
(2.38)g(t)=f(tk-1)+αk-1t,on[tk-1,tk],
for some constant vector αk-1∈ℝd for each k≥1. It is clear that
(2.39)supt∈[0,T-s]|f(t)-g(t)|≤ϵ2.
Because of (2.39), to show (2.36) we only need to show
(2.40)P(B0∩{supt∈[s,T]|Xt-Xs-g(t-s)|<ϵ2})>0.

For each k=0,1,2,…,N-1, let
(2.41)Bk+1=Bk∩{supt∈[s+tk,s+tk+1]|Xt-Xs-g(t-s)|<ϵ2N-k}.
Note that BN is contained in the event in (2.40). Therefore, it is sufficient to prove that BN has positive probability. When t∈[s+tk,s+tk+1], we have g(t-s)=f(tk)+αk(t-s)=f(tk)+αktk+αk[t-(s+tk)]=g([(s+tk)-s])+αk[t-(s+tk)]. Therefore, we have the following relation:
(2.42){supt∈[s+tk,s+tk+1]|Xt-Xs-g(t-s)|<ϵ2N-k}⊃{Xs+tk-Xs-g([(s+tk)-s])<ϵ2N-k+1}∩{supt∈[s+tk,s+tk+1]|Xt-Xs+tk-αk[t-(s+tk)]|<ϵ2N-k+1}.
By the definition of Bk and the above relation, it is easy see that
(2.43)Bk+1⊃Bk∩{supt∈[s+tk,s+tk+1]|Xt-Xs+tk-αk[t-(s+tk)]|<ϵ2N-k+1}.
On Bk we have
(2.44)d(Xs+tk,∂O)≥d(Xs+g((s+tk)-s),∂O)-d(Xs+g((s+tk)-s),Xs+tk)>ϵ2-ϵ2N-k+1≥ϵ4,|Xs+tk-[Xs+tk+αk(t-(s+tk))]|=|αk(t-(s+tk))|=|g(t)-g(tk)|≤ϵ4,
for each k=0,1,…,N-1 and for all t∈[s+tk,s+tk+1]. From this, we conclude that Bk⊂{Xs+tk∈⋂t∈[s+tk,s+tk+1](𝒪-αk[t-(s+tk)]}. Now, from the linear stickiness and the fact that P(B0)>0 we conclude P(BN)>0. This completes the proof.

Proof of Theorem <xref ref-type="statement" rid="thm1">2.6</xref>.

By Lemmas 2.8 and 2.9, there exists an equivalent probability measure Q~P such that (Mni,𝒢n)n≥0 is a uniformly integrable martingale for each 1≤i≤d. Let M∞i=limn→∞Mni. For each t∈[0,T], set M~ti=EQ[M∞i∣ℱt]. Observe that M~τni=EQ[M∞i∣ℱτn]=Mni, and M~ti=EQ[M~τni∣ℱt] on the set {τn-1≤t≤τn} for all n≥0. Thus the following equation holds:
(2.45)(M~ti-Xti)1{τn-1≤t≤τn}=EQ[(Mni-Xti)1{τn-1≤t≤τn}∣Ft],n≥1.
We write Mni-Xti=(Mni-Xτni)+(Xτn-1i-Xti)+(Xτni-Xτn-1i). Note that each of Mni-Xτni, Xτn-1i-Xti, and Xτni-Xτn-1i takes values in (-ϵ,ϵ) on the set {τn-1≤t≤τn}. Therefore, we have -3ϵ≤|M~ti-Xti|≤3ϵ on the set {τn-1≤t≤τn}. Since ⋃n=1∞{τn-1≤t≤τn}=Ω, we conclude that
(2.46)-3ϵ≤|M~ti-Xti|≤3ϵ.
Since ϵ>0 is arbitrary, the claim follows.

Example 2.12.

Let BtH1,BtH2,…,BtHd be a sequence of independent fractional Brownian motions with respective Hurst parameters H1,H2,…,Hd∈(0,1). Let fi:ℝ→(ai,bi) be a homeomorphism for each i=1,2,…,d, where (ai,bi) is an open interval in the real line. Then the new process (f1(BtH1),f2(BtH2),…,fd(BtHd)) admits CPSs for each ϵ>0. This can be easily seen from the CFS property of the process (BtH1,BtH2,…,BtHd) which was shown in [14] and the fact that the map f(x)=(f1(x),f2(x),…,fd(x)) is a homomorphism from ℝd to (a1,b1)×(a2,b2)×⋯×(an,bn).

DelbaenF.SchachermayerW.A general version of the fundamental theorem of asset pricingJouiniE.KallalH.Martingales and arbitrage in securities markets with transaction costsSchachermayerW.The fundamental theorem of asset pricing under proportional transaction costs in finite discrete timeCvitanicJ.PhamH.TouziN.A closed-form solution to the problem of super-replication under transaction costsLeventalS.SkorohodA. V.On the possibility of hedging options in the presence of transaction costsBenderC.SottinenT.ValkeilaE.Pricing by hedging and no-arbitrage beyond semimartingalesChernyA. S.General arbitrage pricing model: transaction costsGuasoniP.No arbitrage under transaction costs, with fractional Brownian motion and beyondJakub\.enasP.LeventalS.RyznarM.The super-replication problem via probabilistic methodsGuasoniP.RásonyiM.SchachermayerW.Consistent price systems and face-lifting pricing under transaction costsMarisF.MbakopE.SayitH.Consistent price systems for bounded processesBayraktarE.SayitH.On the stickiness propertyBayraktarE.SayitH.On the existence of consistent price systemssubmittedSayitH.ViensF.Arbitrage-free models in markets with transaction costsChernyA. S.Brownian moving averages have conditional full supportGuasoniP.RasonyiR.SchachermayerW.The fundamental theorem of asset pricing for continuous processes under small transaction costsGasbarraD.SottinenT.Van ZantenH.Conditional full support of Gaussian processes with stationary incrementsKabanovY. M.SafarianM.