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International Journal of Stochastic Analysis
VolumeΒ 2012Β (2012), Article IDΒ 687376, 14 pages
http://dx.doi.org/10.1155/2012/687376
Research Article

Consistent Price Systems in Multiasset Markets

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received 27 May 2012; Accepted 9 July 2012

Academic Editor: LukaszΒ Stettner

Copyright Β© 2012 Florian Maris and Hasanjan Sayit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋𝑑 be any d-dimensional continuous process that takes values in an open connected domain π’ͺ in ℝ𝑑. In this paper, we give equivalent formulations of the conditional full support (CFS) property of 𝑋𝑑 in π’ͺ. We use them to show that the CFS property of X in π’ͺ implies the existence of a martingale M under an equivalent probability measure such that M lies in the πœ–>0 neighborhood of 𝑋𝑑 for any given πœ– 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 numΓ©raire 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 π‘Œπ‘‘=(π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘), where π‘Œπ‘–π‘‘=𝑒𝑋𝑖𝑑, 1≀𝑖≀𝑑, and the 𝑑-dimensional process 𝑋𝑑=(𝑋1𝑑,𝑋2𝑑,…,𝑋𝑑𝑑) is defined on a filtered probability space (Ξ©,β„±,𝔽=(ℱ𝑑)π‘‘βˆˆ[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 πœƒπ‘‘=(πœƒ1𝑑,πœƒ2𝑑,…,πœƒπ‘‘π‘‘) are given by the difference of two processes 𝐿𝑑=(𝐿1𝑑,𝐿2𝑑,…,𝐿𝑑𝑑) and 𝑀𝑑=(𝑀1𝑑,𝑀2𝑑,…,𝑀𝑑𝑑) representing respectively the cumulative number of shares purchased and sold up to time 𝑑, namely, πœƒπ‘‘=πΏπ‘‘βˆ’π‘€π‘‘. We are also required to start and end without any position in the risky assets to and this requirement corresponds to πœƒ0=πœƒπ‘‡=0.

For each such trading strategy πœƒπ‘‘=πΏπ‘‘βˆ’π‘€π‘‘, the corresponding wealth process, after taking into account the incurred transaction costs, is given by 𝑉𝑑(πœƒ)=𝑑𝑖=1ξ€œπ‘‘0πœƒπ‘–π‘ π‘‘π‘Œπ‘–π‘ βˆ’πœ–π‘‘ξ“π‘–=1ξ€œπ‘‘0π‘Œπ‘–π‘ ξ€·πœƒπ‘‘Var𝑖𝑠,(1.1) where Var(πœƒπ‘–)𝑠=𝐿𝑖𝑠+𝑀𝑖𝑠 is the total variation of πœƒπ‘– in [0,𝑠] for each 1≀𝑖≀𝑑 and πœ–>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 πœƒπ‘‘=(πœƒ1𝑑,πœƒ2𝑑,…,πœƒπ‘‘π‘‘) of finite variation with πœƒ0=πœƒπ‘‡=0 such that the corresponding wealth process 𝑉𝑑(πœƒ) satisfies 𝑉𝑑(πœƒ)β‰₯βˆ’πΆ for some deterministic 𝐢>0 and for all π‘‘βˆˆ[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,π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘) does not admit arbitrage with πœ–-sized transaction costs if there is no admissible trading strategy πœƒπ‘‘=(πœƒ1𝑑,πœƒ2𝑑,…,πœƒπ‘‘π‘‘) such that the corresponding value process 𝑉𝑑(πœƒ) satisfies 𝑃𝑉𝑇𝑉(πœƒ)>0>0,𝑃𝑇(πœƒ)β‰₯0=1.(1.2)

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 (π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘) that ensure absence of arbitrage in the market (1,π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘). It is clear that if the stock price process π‘Œπ‘‘=(π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘) 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., πœ–=0). In the absence of such martingale measure for π‘Œ, the existence of a process ξ‚π‘Œπ‘‘ξ‚π‘Œ=(1𝑑,ξ‚π‘Œ2π‘‘ξ‚π‘Œ,…,𝑑𝑑) which is a martingale under an equivalent measure 𝑄 and which has the following property: |||π‘Œπ‘–π‘‘βˆ’ξ‚π‘Œπ‘–π‘‘|||β‰€πœ–π‘Œπ‘–π‘‘[],for𝑖=1,2,…,𝑑,βˆ€π‘‘βˆˆ0,𝑇,(1.3) also implies absence of arbitrage for the model (1.1). To see this simple fact, observe the following: 𝑉𝑇(πœƒ)=𝑑𝑖=1ξ€œπ‘‡0πœƒπ‘–π‘ π‘‘ξ‚€π‘Œπ‘–π‘ βˆ’ξ‚π‘Œπ‘–π‘ ξ‚+𝑑𝑖=1ξ€œπ‘‡0πœƒπ‘–π‘ π‘‘ξ‚π‘Œπ‘–π‘ βˆ’πœ–π‘‘ξ“π‘–=1ξ€œπ‘‡0π‘Œπ‘–π‘ ξ€·πœƒπ‘‘Var𝑖𝑠=𝑑𝑖=1ξ‚Έξ€œπ‘‡0ξ‚€π‘Œπ‘–π‘ βˆ’ξ‚π‘Œπ‘–π‘ ξ‚π‘‘πœƒπ‘–π‘ ξ€œβˆ’πœ–π‘‡0π‘Œπ‘–π‘ ξ€·πœƒπ‘‘Var𝑖𝑠+𝑑𝑖=1ξ€œπ‘‡0πœƒπ‘–π‘ π‘‘ξ‚π‘Œπ‘–π‘ .(1.4) Note that because of (1.3), we have βˆ‘π‘‘π‘–=1[βˆ«π‘‡0(π‘Œπ‘–π‘ βˆ’ξ‚π‘Œπ‘–π‘ )π‘‘πœƒπ‘–π‘ βˆ«βˆ’πœ–π‘‡0π‘Œπ‘–π‘ π‘‘Var(πœƒπ‘–)𝑠]≀0 a.s. This implies that 𝑉𝑇(πœƒ)≀𝑑𝑖=1ξ€œπ‘‡0πœƒπ‘–π‘ π‘‘ξ‚π‘Œπ‘–π‘ .(1.5) The financial interpretation of (1.5) is that trading at price process ξ‚π‘Œπ‘‘ without transaction costs is always at least as profitable as trading at price process π‘Œπ‘‘ with transaction costs. The martingale property of ξ‚π‘Œπ‘‘ implies that trading on ξ‚π‘Œπ‘‘ is arbitrage free, and therefore trading on π‘Œ with transaction costs is also arbitrage-free.

The process ξ‚π‘Œπ‘‘ is called consistent price systems (CPSs) for the price process π‘Œ. 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 πœ–>0. We say that ξ‚π‘Œπ‘‘ξ‚π‘Œ=(1𝑑,ξ‚π‘Œ2π‘‘ξ‚π‘Œ,…,𝑑𝑑) is an πœ–-consistent price system for π‘Œπ‘‘=(π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘), if there exists a measure π‘„βˆΌπ‘ƒ such that ξ‚π‘Œπ‘‘ is a martingale under 𝑄, and 1β‰€ξ‚π‘Œ1+πœ–π‘–π‘‘π‘Œπ‘–π‘‘[].≀1+πœ–,for𝑖=1,2,…,𝑑,βˆ€π‘‘βˆˆ0,𝑇(1.6)

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 𝑋𝑑=(𝑋1𝑑,𝑋2𝑑,…,𝑋𝑑𝑑) with state space ℝ𝑑 has the CFS property, then the exponential process π‘Œπ‘‘=(π‘Œ1𝑑,π‘Œ2𝑑,…,π‘Œπ‘‘π‘‘) admits πœ–-CPS for any πœ–>0. The proof of this result is based on a clever approximation of π‘Œ by a discrete process which is called random walk with retirement (see [10]). In this paper, we consider continuous processes 𝑋𝑑 with general state space π’ͺ, where π’ͺ 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 𝑋𝑑 with the state space π’ͺ has the corresponding CFS property, then for any given πœ–>0 there exists a martingale 𝑀𝑑=(𝑀1𝑑,𝑀2𝑑,…,𝑀𝑑𝑑), under an equivalent change of measure, such that ||π‘€π‘–π‘‘βˆ’π‘‹π‘–π‘‘||[].β‰€πœ–forany𝑖=1,2,…,𝑑andanyπ‘‘βˆˆ0,𝑇(1.7) By an abuse of language we call such 𝑀 a πœ–-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 π’ͺ 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 𝑋𝑑=(𝑋1𝑑,𝑋2𝑑,…,𝑋𝑑𝑑), π‘‘βˆˆ[0,𝑇] be a 𝑑-dimensional continuous process that takes values in an open connected domain π’ͺβŠ‚β„π‘‘. For simplicity of our discussion, we assume that 0∈π’ͺ. We also assume that the process 𝑋𝑑 is defined on a probability space (Ξ©,β„±,𝑃) and adapted to a filtration 𝔽=(ℱ𝑑)π‘‘βˆˆ[0,𝑇] that satisfies the usual assumptions in this space. Let 𝐢([𝑒,𝑣],π’ͺ) denote the set of continuous functions 𝑓 defined on the interval [𝑒,𝑣] and with values in π’ͺ and, for any π‘₯βˆˆβ„π‘‘, let 𝐢π‘₯([𝑒,𝑣],π’ͺ) denote the set of functions in 𝐢([𝑒,𝑣],π’ͺ) with 𝑓(𝑒)=π‘₯.

Definition 2.1. An adapted continuous process 𝑋𝑑 satisfies the CFS property in π’ͺ, if for any π‘‘βˆˆ[0,𝑇) and for almost all πœ”βˆˆΞ©, 𝑋SuppLawπ‘ ξ€Έπ‘ βˆˆ[𝑑,𝑇]βˆ£β„±π‘‘(πœ”)=𝐢𝑋𝑑(πœ”)([]𝑑,𝑇,π’ͺ),a.s.(2.1)
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 𝑋𝑑 be an adapted continuous process that takes values in an open and connected domain π’ͺβŠ‚β„π‘‘. We say that 𝑋𝑑 is linear sticky if for any π›Όβˆˆβ„π‘‘, πœ–>0, and any deterministic 0β‰€π‘ β‰€πœƒβ‰€π‘‡, 𝑃sup[]π‘‘βˆˆπ‘ ,πœƒ||π‘‹π‘‘βˆ’π‘‹π‘ ||ξƒ°βˆ’π›Ό(π‘‘βˆ’π‘ )<πœ–βˆ£β„±π‘ ξƒͺ>0,a.s.(2.2) on the set {π‘‹π‘ βˆˆβ‹‚π‘‘βˆˆ[0,πœƒβˆ’π‘ ](π’ͺβˆ’π›Όπ‘‘}.
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 𝑋𝑑 be an adapted continuous process that takes values in an open and connected domain π’ͺβŠ‚β„π‘‘. (a)We say that 𝑋𝑑 is 𝑓-sticky for π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑) if 𝑃sup[]π‘‘βˆˆπœ,𝑇||π‘‹π‘‘βˆ’π‘‹πœ||ξƒ°βˆ’π‘“(π‘‘βˆ’πœ)<πœ–βˆ£β„±πœξƒͺ>0,a.s.(2.3) on the set {π‘‹πœβˆˆβ‹‚π‘‘βˆˆ[0,π‘‡βˆ’πœ](π’ͺβˆ’π‘“(𝑑))} for any πœ–>0 and any stopping time 𝜏.(b)We say that 𝑋𝑑 is weak 𝑓-sticky for π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑) if 𝑃sup[]π‘‘βˆˆπ‘ ,𝑇||π‘‹π‘‘βˆ’π‘‹π‘ ||ξƒ°βˆ’π‘“(π‘‘βˆ’π‘ )<πœ–βˆ£β„±π‘ ξƒͺ>0,a.s.(2.4) on the set {π‘‹π‘ βˆˆβ‹‚π‘‘βˆˆ[0,π‘‡βˆ’π‘ ](π’ͺβˆ’π‘“(𝑑))} for any πœ–>0 and any deterministic time π‘ βˆˆ[0,𝑇].

Remark 2.4. It is clear that the CFS property of 𝑋 in π’ͺ is equivalent to the weak 𝑓-stickiness of 𝑋 for all π‘“βˆˆπΆ0([0,𝑇],π’ͺ). The linear stickiness of 𝑋𝑑 is seemingly weaker condition than the weak 𝑓-stickiness of 𝑋𝑑 for all π‘“βˆˆπΆ0([0,𝑇],π’ͺ). However, this is not the case and in Lemma 2.11 we will show that linear stickiness is equivalent to weak 𝑓-stickiness of 𝑋𝑑 for all π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑). This, in turn, implies that the linear stickiness property is equivalent to the CFS property.

Remark 2.5. When a process 𝑋𝑑 is 0-sticky as in (𝑏) in Definition 2.3, we say that 𝑋𝑑 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 𝑋𝑑 has paths that are as close as one wants to the path 𝑓(𝑑)+π‘‹πœ. As it was shown in [14], the 𝑓-stickiness holds for any π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑) for the process (𝐡𝐻1𝑑,𝐡𝐻2𝑑,…,𝐡𝐻𝑑𝑑), where 𝐡𝐻1𝑑,𝐡𝐻2𝑑,…,𝐡𝐻𝑑𝑑 are independent fractional Brownian motions with respective Hurst parameters 𝐻1,𝐻2,…,π»π‘‘βˆˆ(0,1). From [10], the 𝑓-stickiness also holds for any continuous Markov process with the full support property in 𝐢0([0,𝑇],ℝ𝑑) for any π‘“βˆˆπΆ0([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 𝑋𝑑=(𝑋1𝑑,𝑋2𝑑,…,𝑋𝑑𝑑) be a continuous process that takes values in a connected domain π’ͺ in ℝ𝑑. If 𝑋𝑑 is linear sticky, then 𝑋𝑑 admits CPSs for all πœ–>0.

To show this result, one fix any πœ–>0 and define the following increasing sequence of stopping times associated with the process 𝑋: 𝜏0=0,πœπ‘›+1=inf𝑑β‰₯πœπ‘›ξ€½||π‘‹π‘‘βˆ’π‘‹πœπ‘›||β‰₯πœ–π‘›+1ξ€Ύβˆ§π‘‡,βˆ€π‘›β‰₯0,(2.5) with πœ–π‘›+1∢=πœ–βˆ§π‘‘(π‘‹πœπ‘›,πœ•π’ͺ)/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 𝑛β‰₯1 we define Δ𝑛=ξ‚»π‘‹πœπ‘›βˆ’π‘‹πœπ‘›βˆ’1whenπœπ‘›<𝑇,0otherwise.(2.6) Let 𝒒𝑛=β„±πœπ‘› for every 𝑛β‰₯0. Note that πœ–π‘› is bounded and π’’π‘›βˆ’1 measurable.

In the following, we use the notation Supp(Ξ”π‘›βˆ£π’’π‘›βˆ’1) to denote the smallest closed set of ℝ𝑑 that contains the values of the random variable 𝐸[Ξ”π‘›βˆ£π’’π‘›βˆ’1] with probability one. We use π΅π‘Ÿ(π‘₯) to denote the open ball in ℝ𝑑 with center π‘₯ and radius π‘Ÿ. When the center is 0, we simply write π΅π‘Ÿ. We first prove the following lemma.

Lemma 2.7. If 𝑋𝑑 is 𝑓-sticky in π’ͺ for all π‘“βˆˆπΆ0([0,𝑇]), then the process {Δ𝑛} in (2.6) satisfies the following three properties: ξ€·Ξ”(i)π‘ƒπ‘š=0,βˆ€π‘šβ‰₯π‘›βˆ£Ξ”π‘›ξ€Έ(ξ€·Ξ”=0=1;ii)Suppπ‘›βˆ£π’’π‘›βˆ’1ξ€Έ=0βˆͺπœ•π΅πœ–π‘›ξ€½Ξ”π‘Žπ‘™π‘šπ‘œπ‘ π‘‘π‘ π‘’π‘Ÿπ‘’π‘™π‘¦π‘œπ‘›π‘›βˆ’1ξ€Ύ,ξ€·Ξ”β‰ 0(iii)π‘ƒπ‘šξ€Έβ‰ 0,βˆ€π‘šβ‰₯1=0.(2.7)

Proof. Property (𝑖) is obvious since {Δ𝑛=0}={πœπ‘›=𝑇} and πœπ‘› is increasing. Property (𝑖𝑖𝑖) follows from the fact that almost surely each path of 𝑋𝑑 is contained in a compact set of π’ͺ and therefore minπ‘›πœ–π‘›(πœ”)>0 almost surely πœ”βˆˆΞ©. To prove property (𝑖𝑖), let us assume that 𝑃(πœπ‘›βˆ’1<𝑇)>0 and let πΊπ‘›βˆ’1 be any π’’π‘›βˆ’1 measurable set such that 𝑃(πΊπ‘›βˆ’1∩{πœπ‘›βˆ’1<𝑇})>0. Then, it is clear that there exist 𝑇′<𝑇, π‘¦βˆˆπ’ͺ and 𝜁>0 such that 𝜁<πœ–βˆ§π‘‘(𝑦,πœ•π’ͺ)/4 and 𝑃(πΊπ‘›βˆ’1∩{πœπ‘›βˆ’1<𝑇′}∩{π‘‹πœπ‘›βˆ’1∈𝐡𝜁(𝑦)})>0, where 𝑑(𝑦,πœ•π’ͺ) is the distance of 𝑦 with the boundary πœ•π’ͺ of π’ͺ. It is also clear that on the set πΊπ‘›βˆ’1∩{πœπ‘›βˆ’1<𝑇′}∩{π‘‹πœπ‘›βˆ’1∈𝐡𝜁(𝑦)} we have πœ–βˆ§3𝜁/2β‰€πœ–π‘›<2𝜁.
First we show that 𝑃(Δ𝑛=0βˆ£π’’π‘›βˆ’1)>0 a.s. on {Ξ”π‘›βˆ’1β‰ 0}. To see this, define the following stopping time: ξ‚»πœπœ=π‘›βˆ’1onπΊπ‘›βˆ’1βˆ©ξ€½πœπ‘›βˆ’1<π‘‡ξ…žξ€Ύβˆ©ξ€½π‘‹πœπ‘›βˆ’1βˆˆπ΅πœξ€Ύ,(𝑦)𝑇otherwise.(2.8) The 0-stickiness of 𝑋 implies that 𝑃sup[]π‘‘βˆˆπœ,𝑇||π‘‹π‘‘βˆ’π‘‹πœ||ξƒͺ<𝜁,𝜏<𝑇>0.(2.9) But {supπ‘‘βˆˆ[𝜏,𝑇]|π‘‹π‘‘βˆ’π‘‹πœ|<𝜁,𝜏<𝑇}βŠ‚{Δ𝑛=0} and since πΊπ‘›βˆ’1 was an arbitrary π’’π‘›βˆ’1 measurable set with 𝑃(πΊπ‘›βˆ’1∩{πœπ‘›βˆ’1<𝑇})>0, we have 𝑃Δ𝑛=0βˆ£π’’π‘›βˆ’1ξ€Έξ€½Ξ”>0a.s.onπ‘›βˆ’1ξ€Ύβ‰ 0.(2.10) Next we show that πœ•π΅πœ–π‘›(πœ”)βŠ‚Supp(Ξ”π‘›βˆ£π’’π‘›βˆ’1)(πœ”) almost surely on {Ξ”π‘›βˆ’1β‰ 0}. Too see this, take any π‘₯βˆˆπœ•π΅1, 0<πœ–β€²<𝜁 and define π‘“βŽ§βŽͺ⎨βŽͺ⎩(𝑑)=6πœπ‘‘ξ€·π‘‡βˆ’π‘‡ξ…žξ€Έπ‘₯if0β‰€π‘‘β‰€π‘‡βˆ’π‘‡β€²2,3𝜁π‘₯otherwise,(2.11) and note that π‘ƒξ‚€πœ<𝑇,π‘‹πœβˆˆξ™π‘‘βˆˆ[𝜏,𝑇](π’ͺβˆ’π‘“(π‘‘βˆ’πœ))>0.(2.12) By 𝑓-stickiness of 𝑋, we obtain 𝑃sup[]π‘‘βˆˆπœ,𝑇||π‘‹π‘‘βˆ’π‘‹πœ||ξƒͺβˆ’π‘“(π‘‘βˆ’πœ)<πœ–β€²,𝜏<𝑇>0,(2.13) or equivalently 𝑃(π΄βˆ©πΊπ‘›βˆ’1)>0, where 𝐴=supξ€Ίπœπ‘‘βˆˆπ‘›βˆ’1ξ€»,𝑇||π‘‹π‘‘βˆ’π‘‹πœπ‘›βˆ’1ξ€·βˆ’π‘“π‘‘βˆ’πœπ‘›βˆ’1ξ€Έ||ξƒ°βˆ©ξ€½πœ<πœ–β€²π‘›βˆ’1<π‘‡ξ…žξ€Ύβˆ©ξ‚†π‘‹πœπ‘›βˆ’1βˆˆξ™π‘‘βˆˆ[πœπ‘›βˆ’1,𝑇]ξ€·ξ€·π’ͺβˆ’π‘“π‘‘βˆ’πœπ‘›βˆ’1.ξ€Έξ€Έ(2.14) We claim that π΄βŠ‚{Ξ”π‘›βŠ‚π΅2πœ–ξ…ž(πœ–π‘›π‘₯)}. Indeed, if πœ”βˆˆπ΄, we get ||π‘‹πœπ‘›βˆ’1+(π‘‡βˆ’π‘‡β€²)/2(πœ”)βˆ’π‘‹πœπ‘›βˆ’1||β‰₯|||𝑓(πœ”)π‘‡βˆ’π‘‡β€²2|||βˆ’||||π‘‹πœπ‘›βˆ’1+(π‘‡βˆ’π‘‡β€²)/2(πœ”)βˆ’π‘‹πœπ‘›βˆ’1ξ‚΅(πœ”)βˆ’π‘“π‘‡βˆ’π‘‡ξ…ž2ξ‚Ά||||β‰₯3πœβˆ’πœ–ξ…ž>πœ–π‘›(πœ”).(2.15) Hence {πœπ‘›<𝑇} on 𝐴. Also, for πœ”βˆˆπ΄ we have 𝑋0=π‘‘πœπ‘›βˆ’π‘‹πœπ‘›βˆ’1,πœ•π΅πœ–π‘›ξ€Έξ€·π‘“ξ€·πœβ‰₯π‘‘π‘›βˆ’πœπ‘›βˆ’1ξ€Έ,πœ•π΅πœ–π‘›ξ€Έβˆ’||π‘‹πœπ‘›βˆ’π‘‹πœπ‘›βˆ’1ξ€·βˆ’π‘“π‘‘βˆ’πœπ‘›βˆ’1ξ€Έ||>||π‘“ξ€·πœπ‘›βˆ’πœπ‘›βˆ’1ξ€Έβˆ’πœ–π‘›π‘₯||βˆ’πœ–ξ…ž,||π‘‹πœπ‘›βˆ’π‘‹πœπ‘›βˆ’1βˆ’πœ–π‘›π‘₯||≀||π‘“ξ€·πœπ‘›βˆ’πœπ‘›βˆ’1ξ€Έβˆ’πœ–π‘›π‘₯||+||π‘‹πœπ‘›βˆ’π‘‹πœπ‘›βˆ’1ξ€·βˆ’π‘“π‘‘βˆ’πœπ‘›βˆ’1ξ€Έ||<||π‘“ξ€·πœπ‘›βˆ’πœπ‘›βˆ’1ξ€Έβˆ’πœ–π‘›π‘₯||+πœ–β€²<2πœ–β€².(2.16) So for all πœ”βˆˆπ΄, 𝑑(Δ𝑛,πœ–π‘›π‘₯)<2πœ–β€². Since this is true for any small πœ–β€², π‘₯∈𝐡1 and any arbitrary π’’π‘›βˆ’1 measurable set with 𝑃(πΊπ‘›βˆ’1∩{πœπ‘›βˆ’1<𝑇})>0, we conclude that πœ•π΅πœ–π‘›(πœ”)βŠ‚Supp(Ξ”π‘›βˆ£π’’π‘›βˆ’1)(πœ”) almost surely on {Ξ”π‘›βˆ’1β‰ 0}. Note that the other direction πœ•π΅πœ–π‘›(πœ”)βˆͺ{0}βŠƒSupp(Ξ”π‘›βˆ£π’’π‘›βˆ’1)(πœ”) is clear from the definition of Δ𝑛.

Now, define πœ–π‘›, Δ𝑛, 𝑛β‰₯0 as above and let 𝑀𝑛=𝑋0+βˆ‘π‘›π‘–=1Δ𝑖, 𝑛β‰₯0. The ℝ𝑑-valued process 𝑀𝑛=∢(𝑀1𝑛,𝑀2𝑛,…,𝑀𝑑𝑛) will be used to construct CPSs for 𝑋𝑑. Next, we prove a lemma that shows that all of 𝑀𝑖𝑛, 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,𝒒𝑛)}+βˆžπ‘›=0 is a martingale.

Proof. For any 𝑛β‰₯0, let πœ‡π‘› be the regular conditional probability of Δ𝑛 with respect to π’’π‘›βˆ’1 and let Ω𝑛={πœ”βˆˆΞ©βˆ£Supp(Ξ”π‘›βˆ£π’’π‘›βˆ’1)(πœ”)=0βˆͺπœ•π΅πœ–π‘›}. Let π‘˜ be any strictly increasing convex function defined on ℝ with values in (0,+∞) such that π‘˜(𝑑)=𝑑 for every 𝑑β‰₯1. Define πΊπ‘›βˆΆΞ©π‘›Γ—β„π‘‘β†’β„π‘‘ as follows: 𝐺𝑛(ξ€œπœ”,𝛼)=β„π‘‘π‘˜(𝛼⋅π‘₯)π‘₯π‘‘πœ‡π‘›(πœ”,β‹…).(2.17) Obviously for each 𝑛, 𝐺𝑛(β‹…,π‘Ž) is π’’π‘›βˆ’1 measurable and convex with respect to 𝛼. As a consequence, for any fixed πœ”βˆˆΞ©π‘›, Im(𝐺𝑛(πœ”,β‹…)), the image of the function 𝐺𝑛(πœ”,β‹…) is convex. We first prove that for every 𝑛β‰₯1 and πœ”βˆˆΞ©π‘›: lim|𝛼|β†’βˆžπΊπ‘›π›Ό(πœ”,𝛼)β‹…|𝛼|=+∞.(2.18) By the way of contrary, assume that this is not true, for some 𝑛β‰₯1 and πœ”βˆˆΞ©π‘›. Then, there exists a sequence (π›Όπ‘š)π‘šβ‰₯1 with |π›Όπ‘š|β†’βˆž such that 𝐺𝑛(πœ”,π›Όπ‘š)β‹…(π›Όπ‘š/|π›Όπ‘š|) is bounded above. We can assume that (π›Όπ‘š/|π›Όπ‘š|) converges to some 𝛼 (this is a bounded sequence and therefore has a convergent subsequence). We have πΊπ‘›ξ€·πœ”,π›Όπ‘šξ€Έβ‹…π›Όπ‘š||π›Όπ‘š||β‰₯ξ€œπ›Όπ‘šβ‹…π‘₯≀0π‘˜ξ€·π›Όπ‘šξ€Έπ›Όβ‹…π‘₯π‘šβ‹…π‘₯||π›Όπ‘š||π‘‘πœ‡π‘›(+ξ€œπœ”,β‹…)(π›Όπ‘šβ‹…π‘₯)/|π›Όπ‘š|>πœ–π‘›(πœ”)/4π‘˜ξ€·π›Όπ‘šξ€Έπ›Όβ‹…π‘₯π‘šβ‹…π‘₯||π›Όπ‘š||π‘‘πœ‡π‘›ξ€œ(πœ”,β‹…)β‰₯βˆ’π‘˜(0)+2𝛼⋅π‘₯>πœ–π‘›(πœ”)ξ€·π›Όπ‘šξ€Έβ‹…π‘₯2||π›Όπ‘š||π‘‘πœ‡π‘›(πœ”,β‹…),(2.19) for big enough π‘š. Therefore, we can conclude that ∫2𝛼⋅π‘₯>πœ–π‘›(πœ”)((π›Όπ‘šβ‹…π‘₯)2/|π›Όπ‘š|)β‹…(1/|π‘Žπ‘š|)π‘‘πœ‡π‘›(πœ”,β‹…) converges to 0 as |π‘Žπ‘š|β†’+∞, which will imply after passing to the limit that πœ‡π‘›(πœ”,{2𝛼⋅π‘₯>πœ–π‘›}) and this is a contradiction. From this it follows easily that 0∈Im(𝐺𝑛(πœ”,β‹…)). If 0βˆ‰Im(𝐺𝑛(πœ”,β‹…)), then using the geometric form of Hahn Banach theorem, there exists a unit vector π›½βˆˆβ„π‘‘ such that βˆ«β„π‘‘π‘˜(𝛼⋅π‘₯)𝛽π‘₯π‘‘πœ‡π‘›(πœ”,β‹…)<0 for every π›Όβˆˆβ„π‘‘. Therefore, limsupπ‘‘β†’βˆžβˆ«β„π‘‘π‘˜(𝑑𝛽⋅π‘₯)𝛽π‘₯π‘‘πœ‡π‘›(πœ”,β‹…)≀0. But ξ€œβ„π‘‘π‘˜(𝑑𝛽⋅π‘₯)𝛽π‘₯π‘‘πœ‡π‘›(πœ”,β‹…)=𝐺𝑛(πœ”,𝑑𝛽)⋅𝑑𝛽||||,𝑑𝛽(2.20) and so it contradicts (2.18).
Next, we want to show that Im(𝐺𝑛(πœ”,β‹…)) is closed. Let π‘ŽβˆˆIm(𝐺𝑛(πœ”,β‹…)), so there exists a sequence (π›Όπ‘š)π‘šβ‰₯1 such that 𝐺𝑛(πœ”,π›Όπ‘š)β†’π‘Ž. But then |π›Όπ‘š| is unbounded, and therefore this contradicts (2.18). So based on the continuity of 𝐺𝑛(πœ”,β‹…), π‘ŽβˆˆIm(𝐺𝑛(πœ”,β‹…)).
Therefore, we conclude that for any 𝑛β‰₯1 and πœ”βˆˆΞ©π‘›, there exists an 𝛼𝑛(πœ”)βˆˆβ„π‘‘, unique, as a consequence of the strict monotonicity of π‘˜, such that 𝐺𝑛(πœ”,𝛼𝑛(πœ”))=0.   𝐺𝑛 being continuous with respect to 𝛼 and π’’π‘›βˆ’1 measurable with respect to πœ”, it follows that 𝛼𝑛 is π’’π‘›βˆ’1 measurable. We extend 𝛼𝑛 with 1 outside Ω𝑛 and define: 𝑍𝑛=π‘˜ξ€·π›Όπ‘›β‹…Ξ”π‘›ξ€Έ1{Δ𝑛≠0}ξ€·π‘˜ξ€·π›Ό2𝐸𝑛⋅Δ𝑛1{Δ𝑛≠0}βˆ£π’’π‘›βˆ’1ξ€Έ+1{Δ𝑛=0}ξ€·Ξ”2𝑃𝑛=0βˆ£π’’π‘›βˆ’1ξ€Έ.(2.21) It is easy to check that 𝑍𝑛 satisfies πΈξ€·π‘π‘›βˆ£π’’π‘›βˆ’1𝐸𝑍=1,π‘›Ξ”π‘›βˆ£π’’π‘›βˆ’1ξ€Έ=0.(2.22) Let 𝐿𝑛=βˆπ‘›π‘–=1𝑍𝑖 and 𝐿=limπ‘›β†’βˆžπΏπ‘›. Note that this limit exists almost surely since 𝐿𝑛+1=𝐿𝑛 a.s. on {Δ𝑛=0} and {Δ𝑛=0}β†—Ξ©. From (2.22), we get πΈξ€·πΏπ‘›βˆ£π’’π‘›βˆ’1ξ€Έ=πΏπ‘›βˆ’1πΈξ€·πΏπ‘›π‘€π‘›βˆ£π’’π‘›βˆ’1ξ€Έ=πΏπ‘›βˆ’1π‘€π‘›βˆ’1,(2.23) which shows that (𝐿𝑛)𝑛β‰₯1 and (𝑀𝑛𝐿𝑛)𝑛β‰₯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 𝐸(𝐿)=𝐸limπ‘›β†’βˆžπΏ1{Δ𝑛=0}ξ‚Ά=limπ‘›β†’βˆžπΈξ€·πΏ1{Δ𝑛=0}ξ€Έ=limπ‘›β†’βˆžπΈξ€·πΏπ‘›1{Δ𝑛=0}ξ€Έ=1βˆ’limπ‘›β†’βˆžπΈξ€·πΏπ‘›1{Δ𝑛≠0}ξ€Έ=1βˆ’limπ‘›β†’βˆžπΈξ€·πΈξ€·πΏπ‘›1{Δ𝑛≠0}βˆ£π’’π‘›βˆ’11ξ€Έξ€Έ=1βˆ’2limπ‘›β†’βˆžπΈξ€·πΏπ‘›βˆ’11{Ξ”π‘›βˆ’1β‰ 0}ξ€Έ=1βˆ’limπ‘›β†’βˆžξ‚€12𝑛=1.(2.24) Combining Fatou's lemma with the equation 𝐸(𝐿𝑛)=𝐸(𝐿)=1, we obtain 𝐸(πΏβˆ£π’’π‘›)=𝐿𝑛. Also, πΈξ€·π‘€π‘›πΏβˆ£π’’π‘›βˆ’1𝐸𝑀=πΈπ‘›πΏβˆ£π’’π‘›ξ€Έβˆ£π’’π‘›βˆ’1𝑀=πΈπ‘›πΏβˆ£π’’π‘›βˆ’1ξ€Έ=π‘€π‘›βˆ’1πΏπ‘›βˆ’1𝑀=πΈπ‘›βˆ’1πΏβˆ£π’’π‘›βˆ’1ξ€Έ.(2.25) 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 𝑀𝑖𝑛 is uniformly integrable for each 1≀𝑖≀𝑑. In particular, 𝐸𝑄(sup𝑛β‰₯0|𝑀𝑖𝑛|)<∞ for each 𝑖=1,2,…,𝑑.

Proof. For any 1≀𝑖≀𝑑, set π‘€π‘–βˆ—=sup𝑛β‰₯0|𝑀𝑖𝑛| and observe that on {Ξ”π‘˜β‰ 0,Ξ”π‘˜+1=0} we have π‘€π‘–βˆ—β‰€|𝑋𝑖0|+π‘˜πœ–. Observing that 𝑄(Ξ”π‘˜β‰ 0)=𝑄(Ξ”π‘˜β‰ 0βˆ£Ξ”π‘˜βˆ’1β‰ 0)⋯𝑄(Ξ”1β‰ 0βˆ£Ξ”0β‰ 0)𝑄(Ξ”0β‰ 0) and that 𝑄(Ξ”π‘˜β‰ 0βˆ£Ξ”π‘˜βˆ’1β‰ 0)=1/2 we obtain the following: πΈπ‘„ξ€·π‘€π‘–βˆ—ξ€Έ=βˆžξ“π‘˜=0πΈπ‘„ξ‚€π‘€π‘–βˆ—1{Ξ”π‘˜β‰ 0}∩{Ξ”π‘˜+1=0}ξ‚β‰€βˆžξ“π‘˜=0ξ€·||𝑋𝑖0||𝑄Δ+π‘˜πœ–ξ€·ξ€½π‘˜β‰ 0,Ξ”π‘˜+1=0ξ€Ύξ€Έ<∞.(2.26)

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 𝑋𝑑 be an adapted continuous process with state space π’ͺ and π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑). Then, 𝑋𝑑 is weak 𝑓-sticky if and only if it is 𝑓-sticky.

Proof. Let us show first that for any π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑) weak 𝑓-stickiness implies 𝑓-stickiness. Suppose for a contradiction that 𝑋𝑑 is weak 𝑓-sticky but not 𝑓-sticky. Then there exists a stopping time 𝜏 with 𝑃(𝜏<𝑇)>0, and an πœ–>0 such that π‘ƒξ‚€πœ<𝑇,π‘‹πœβˆˆξ™π‘‘βˆˆ[0,π‘‡βˆ’πœ]𝑃(π’ͺβˆ’π‘“(𝑑))>0,sup[]π‘‘βˆˆπœ,𝑇||π‘‹π‘‘βˆ’π‘‹πœ||ξƒͺβˆ’π‘“(π‘‘βˆ’πœ)<πœ–,𝜏<𝑇=0.(2.27) Since π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑), there exists a 𝛿>0 such that for all 𝑑, π‘ βˆˆ[0,𝑇], |π‘‘βˆ’π‘ |<𝛿 implies |𝑓(𝑑)βˆ’π‘“(𝑠)|<πœ–/3. In addition, we can find 𝑑1, 𝑑2∈[0,𝑇), 0<𝑑2βˆ’π‘‘1<𝛿, and 0<πœβ‰€πœ–/3 such that 𝑃𝑑1β‰€πœ<𝑑2,π‘‹πœβˆˆξ™π‘‘βˆˆ[𝜏,𝑇]ξ€·π’ͺπœξ€Έξ‚βˆ’π‘“(π‘‘βˆ’πœ)>0,(2.28) where π’ͺ𝜁={π‘₯∈π’ͺ/𝑑(π‘₯,πœ•π’ͺ)>𝜁}.
For each π‘žβˆˆπΌ=β„šβˆ©[𝑑1,𝑑2), let π΄π‘žβˆΆ=𝐴∩{𝑑1β‰€πœ<π‘ž}∩{supπ‘‘βˆˆ[𝜏,π‘ž]|π‘‹π‘‘βˆ’π‘‹πœ|<𝜁}, where ξ‚†π‘‘π΄βˆΆ=1β‰€πœ<𝑑2,π‘‹πœβˆˆξ™π‘‘βˆˆ[𝜏,𝑇]ξ€·π’ͺπœξ€Έξ‚‡βˆ’π‘“(π‘‘βˆ’πœ).(2.29) Since 𝑃(𝐴)>0 and ⋃𝐴=π‘žβˆˆπΌπ΄π‘ž, there exists a π‘žβˆ—βˆˆπΌ such that 𝑃(π΄π‘žβˆ—)>0. Note that π΄π‘žβˆ—βˆˆβ„±π‘žβˆ— and π΄π‘žβˆ—βŠ‚β‹‚π‘‘βˆˆ[0,π‘‡βˆ’π‘žβˆ—](π’ͺβˆ’π‘“(𝑑)). Hence, since 𝑋𝑑 is weak 𝑓-sticky, we obtain π‘ƒξƒ©π΄π‘žβˆ—βˆ©ξƒ―sup[π‘žπ‘‘βˆˆβˆ—],𝑇||π‘‹π‘‘βˆ’π‘‹π‘žβˆ—ξ€·βˆ’π‘“π‘‘βˆ’π‘žβˆ—ξ€Έ||<πœ–3ξƒ°ξƒͺ>0.(2.30) Let πΆπ‘žβˆ—=π΄π‘žβˆ—βˆ©{supπ‘‘βˆˆ[π‘žβˆ—,𝑇]|π‘‹π‘‘βˆ’π‘‹π‘žβˆ—βˆ’π‘“(π‘‘βˆ’π‘žβˆ—)|<πœ–/3}. Then we claim that πΆπ‘žβˆ—βŠ‚ξƒ―sup[]π‘‘βˆˆπœ,𝑇||π‘‹π‘‘βˆ’π‘‹πœ||ξƒ°βˆ’π‘“(π‘‘βˆ’πœ)<πœ–βˆ©{𝜏<𝑇},(2.31) which contradicts (2.27). Indeed, if πœ”βˆˆπΆπ‘žβˆ—, then for π‘‘βˆˆ[𝜏,π‘žβˆ—] we have ||π‘‹π‘‘βˆ’π‘‹πœ||<||π‘‹βˆ’π‘“(π‘‘βˆ’πœ)π‘‘βˆ’π‘‹πœ||+||𝑓||<πœ–(π‘‘βˆ’πœ)3+πœ–3<πœ–,(2.32) by the definition of π΄π‘žβˆ— and the choice of 𝛿. We will show also that |π‘‹π‘‘βˆ’π‘‹πœβˆ’π‘“(π‘‘βˆ’πœ)|<πœ– on πΆπ‘žβˆ— whenever π‘‘βˆˆ[π‘žβˆ—,𝑇]: ||π‘‹π‘‘βˆ’π‘‹πœ||≀||π‘‹βˆ’π‘“(π‘‘βˆ’πœ)π‘žβˆ—βˆ’π‘‹πœξ€·βˆ’π‘“(π‘‘βˆ’πœ)+π‘“π‘‘βˆ’π‘žβˆ—ξ€Έ||+||π‘‹π‘‘βˆ’π‘‹π‘žβˆ—ξ€·βˆ’π‘“π‘‘βˆ’π‘žβˆ—ξ€Έ||≀||π‘‹πœβˆ’π‘‹π‘žβˆ—||+||π‘“ξ€·π‘‘βˆ’π‘žβˆ—ξ€Έ||+||π‘‹βˆ’π‘“(π‘‘βˆ’πœ)π‘‘βˆ’π‘‹π‘žβˆ—ξ€·βˆ’π‘“π‘‘βˆ’π‘žβˆ—ξ€Έ||<πœ–3+πœ–3+πœ–3=πœ–.(2.33) Thus, weak 𝑓-stickiness implies 𝑓-stickiness. Since the opposite direction is obvious, the proposition is proved.

Lemma 2.11. Let 𝑋𝑑 be a continuous adapted process with state space π’ͺ. Then 𝑋𝑑 is linear sticky if and only if 𝑋𝑑 is 𝑓-sticky for all π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑).

Proof. We only need to show that linear stickiness implies the weak 𝑓-stickiness for each π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑). Fix any π‘“βˆˆπΆ0([0,𝑇],ℝ𝑑),π‘ βˆˆ[0,𝑇], πœ–0>0. We need to show that 𝑃sup[]π‘‘βˆˆπ‘ ,𝑇||π‘‹π‘‘βˆ’π‘‹π‘ ||βˆ’π‘“(π‘‘βˆ’π‘ )<πœ–0ξƒ°βˆ£β„±π‘ ξƒͺ>0,a.s.(2.34) on the set 𝐡=∢{π‘‹π‘ βˆˆβ‹‚π‘‘βˆˆ[0,π‘‡βˆ’π‘ ](π’ͺβˆ’π‘“(𝑑))}. To do this, for any π΄βˆˆβ„±π‘  with 𝑃(𝐴∩𝐡)>0, we need to show that π‘ƒξƒ©ξƒ―π΄βˆ©π΅βˆ©sup[]π‘‘βˆˆπ‘ ,𝑇||π‘‹π‘‘βˆ’π‘‹π‘ ||βˆ’π‘“(π‘‘βˆ’π‘ )<πœ–0ξƒ°ξƒͺ>0.(2.35) Define 𝑍(πœ”)=infπ‘Ÿβˆˆ[0,π‘‡βˆ’π‘ ]𝑑(𝑋𝑠(πœ”)+𝑓(π‘Ÿ),πœ•π’ͺ) for any πœ”βˆˆπ΄βˆ©π΅. From the definition of 𝐡, it is clear that 𝑍>0 a.s. on 𝐴∩𝐡. Let β„Ž>0 be a constant such that the set 𝐡0={𝑍β‰₯β„Ž} has positive probability. Note that 𝐡0βˆˆβ„±π‘  and 𝐡0βŠ‚π΄βˆ©π΅. In the following, we show that 𝑃𝐡0βˆ©ξƒ―sup[]π‘‘βˆˆπ‘ ,𝑇||π‘‹π‘‘βˆ’π‘‹π‘ ||βˆ’π‘“(π‘‘βˆ’π‘ )<πœ–0ξƒ°ξƒͺ>0.(2.36) Let πœ–=min(πœ–0,β„Ž) and set 𝑑0=0, and define π‘‘π‘˜ξ‚†=inf𝑑β‰₯π‘‘π‘˜βˆ’1∢||𝑑𝑓(𝑑)βˆ’π‘“π‘˜βˆ’1ξ€Έ||β‰₯πœ–4ξ‚‡βˆ§(π‘‡βˆ’π‘ ),(2.37) for π‘˜β‰₯1. Let 𝑁 be the smallest positive integer such that 𝑑𝑁=π‘‡βˆ’π‘ . For each π‘˜β‰₯1, define 𝑔(𝑑) on [π‘‘π‘˜βˆ’1,π‘‘π‘˜] to be equal to the linear function that connects the two points 𝑓(π‘‘π‘˜βˆ’1) and 𝑓(π‘‘π‘˜). We can assume that 𝑔𝑑(𝑑)=π‘“π‘˜βˆ’1ξ€Έ+π›Όπ‘˜βˆ’1𝑑𝑑,onπ‘˜βˆ’1,π‘‘π‘˜ξ€»,(2.38) for some constant vector π›Όπ‘˜βˆ’1βˆˆβ„π‘‘ for each π‘˜β‰₯1. It is clear that sup[]π‘‘βˆˆ0,π‘‡βˆ’π‘ ||𝑓||β‰€πœ–(𝑑)βˆ’π‘”(𝑑)2.(2.39) Because of (2.39), to show (2.36) we only need to show 𝑃𝐡0βˆ©ξƒ―sup[]π‘‘βˆˆπ‘ ,𝑇||π‘‹π‘‘βˆ’π‘‹π‘ ||<πœ–βˆ’π‘”(π‘‘βˆ’π‘ )2ξƒ°ξƒͺ>0.(2.40)
For each π‘˜=0,1,2,…,π‘βˆ’1, let π΅π‘˜+1=π΅π‘˜βˆ©ξƒ―supξ€Ίπ‘‘βˆˆπ‘ +π‘‘π‘˜,𝑠+π‘‘π‘˜+1ξ€»||π‘‹π‘‘βˆ’π‘‹π‘ ||<πœ–βˆ’π‘”(π‘‘βˆ’π‘ )2π‘βˆ’π‘˜ξƒ°.(2.41) Note that 𝐡𝑁 is contained in the event in (2.40). Therefore, it is sufficient to prove that 𝐡𝑁 has positive probability. When π‘‘βˆˆ[𝑠+π‘‘π‘˜,𝑠+π‘‘π‘˜+1], we have 𝑔(π‘‘βˆ’π‘ )=𝑓(π‘‘π‘˜)+π›Όπ‘˜(π‘‘βˆ’π‘ )=𝑓(π‘‘π‘˜)+π›Όπ‘˜π‘‘π‘˜+π›Όπ‘˜[π‘‘βˆ’(𝑠+π‘‘π‘˜)]=𝑔([(𝑠+π‘‘π‘˜)βˆ’π‘ ])+π›Όπ‘˜[π‘‘βˆ’(𝑠+π‘‘π‘˜)]. Therefore, we have the following relation: ξƒ―supξ€Ίπ‘‘βˆˆπ‘ +π‘‘π‘˜,𝑠+π‘‘π‘˜+1ξ€»||π‘‹π‘‘βˆ’π‘‹π‘ ||<πœ–βˆ’π‘”(π‘‘βˆ’π‘ )2π‘βˆ’π‘˜ξƒ°βŠƒξ‚»π‘‹π‘ +π‘‘π‘˜βˆ’π‘‹π‘ βˆ’π‘”ξ€·ξ€Ίξ€·π‘ +π‘‘π‘˜ξ€Έ<πœ–βˆ’π‘ ξ€»ξ€Έ2π‘βˆ’π‘˜+1ξ‚Όβˆ©ξƒ―supξ€Ίπ‘‘βˆˆπ‘ +π‘‘π‘˜,𝑠+π‘‘π‘˜+1ξ€»||π‘‹π‘‘βˆ’π‘‹π‘ +π‘‘π‘˜βˆ’π›Όπ‘˜ξ€Ίξ€·π‘‘βˆ’π‘ +π‘‘π‘˜||<πœ–ξ€Έξ€»2π‘βˆ’π‘˜+1ξƒ°.(2.42) By the definition of π΅π‘˜ and the above relation, it is easy see that π΅π‘˜+1βŠƒπ΅π‘˜βˆ©ξƒ―supξ€Ίπ‘‘βˆˆπ‘ +π‘‘π‘˜,𝑠+π‘‘π‘˜+1ξ€»||π‘‹π‘‘βˆ’π‘‹π‘ +π‘‘π‘˜βˆ’π›Όπ‘˜ξ€Ίξ€·π‘‘βˆ’π‘ +π‘‘π‘˜||<πœ–ξ€Έξ€»2π‘βˆ’π‘˜+1ξƒ°.(2.43) On π΅π‘˜ we have 𝑑𝑋𝑠+π‘‘π‘˜ξ€Έξ€·π‘‹,πœ•π’ͺβ‰₯𝑑𝑠+𝑔𝑠+π‘‘π‘˜ξ€Έξ€Έξ€Έξ€·π‘‹βˆ’π‘ ,πœ•π’ͺβˆ’π‘‘π‘ +𝑔𝑠+π‘‘π‘˜ξ€Έξ€Έβˆ’π‘ ,𝑋𝑠+π‘‘π‘˜ξ€Έ>πœ–2βˆ’πœ–2π‘βˆ’π‘˜+1β‰₯πœ–4,||𝑋𝑠+π‘‘π‘˜βˆ’ξ€Ίπ‘‹π‘ +π‘‘π‘˜+π›Όπ‘˜ξ€·ξ€·π‘‘βˆ’π‘ +π‘‘π‘˜||=||π›Όξ€Έξ€Έξ€»π‘˜ξ€·ξ€·π‘‘βˆ’π‘ +π‘‘π‘˜||=||𝑑𝑔(𝑑)βˆ’π‘”π‘˜ξ€Έ||β‰€πœ–4,(2.44) for each π‘˜=0,1,…,π‘βˆ’1 and for all π‘‘βˆˆ[𝑠+π‘‘π‘˜,𝑠+π‘‘π‘˜+1]. From this, we conclude that π΅π‘˜βŠ‚{𝑋𝑠+π‘‘π‘˜βˆˆβ‹‚π‘‘βˆˆ[𝑠+π‘‘π‘˜,𝑠+π‘‘π‘˜+1](π’ͺβˆ’π›Όπ‘˜[π‘‘βˆ’(𝑠+π‘‘π‘˜)]}. Now, from the linear stickiness and the fact that 𝑃(𝐡0)>0 we conclude 𝑃(𝐡𝑁)>0. This completes the proof.

Proof of Theorem 2.6. By Lemmas 2.8 and 2.9, there exists an equivalent probability measure π‘„βˆΌπ‘ƒ such that (𝑀𝑖𝑛,𝒒𝑛)𝑛β‰₯0 is a uniformly integrable martingale for each 1≀𝑖≀𝑑. Let π‘€π‘–βˆž=limπ‘›β†’βˆžπ‘€π‘–π‘›. For each π‘‘βˆˆ[0,𝑇], set 𝑀𝑖𝑑=𝐸𝑄[π‘€π‘–βˆžβˆ£β„±π‘‘]. Observe that ξ‚‹π‘€π‘–πœπ‘›=𝐸𝑄[π‘€π‘–βˆžβˆ£β„±πœπ‘›]=𝑀𝑖𝑛, and 𝑀𝑖𝑑=𝐸𝑄[ξ‚‹π‘€π‘–πœπ‘›βˆ£β„±π‘‘] on the set {πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›} for all 𝑛β‰₯0. Thus the following equation holds: ξ‚€ξ‚‹π‘€π‘–π‘‘βˆ’π‘‹π‘–π‘‘ξ‚1{πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›}=πΈπ‘„π‘€ξ€Ίξ€·π‘–π‘›βˆ’π‘‹π‘–π‘‘ξ€Έ1{πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›}βˆ£β„±π‘‘ξ€»,𝑛β‰₯1.(2.45) We write π‘€π‘–π‘›βˆ’π‘‹π‘–π‘‘=(π‘€π‘–π‘›βˆ’π‘‹π‘–πœπ‘›)+(π‘‹π‘–πœπ‘›βˆ’1βˆ’π‘‹π‘–π‘‘)+(π‘‹π‘–πœπ‘›βˆ’π‘‹π‘–πœπ‘›βˆ’1). Note that each of π‘€π‘–π‘›βˆ’π‘‹π‘–πœπ‘›, π‘‹π‘–πœπ‘›βˆ’1βˆ’π‘‹π‘–π‘‘, and π‘‹π‘–πœπ‘›βˆ’π‘‹π‘–πœπ‘›βˆ’1 takes values in (βˆ’πœ–,πœ–) on the set {πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›}. Therefore, we have ξ‚‹π‘€βˆ’3πœ–β‰€|π‘–π‘‘βˆ’π‘‹π‘–π‘‘|≀3πœ– on the set {πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›}. Since β‹ƒβˆžπ‘›=1{πœπ‘›βˆ’1β‰€π‘‘β‰€πœπ‘›}=Ξ©, we conclude that |||ξ‚‹π‘€βˆ’3πœ–β‰€π‘–π‘‘βˆ’π‘‹π‘–π‘‘|||≀3πœ–.(2.46) Since πœ–>0 is arbitrary, the claim follows.

Example 2.12. Let 𝐡𝐻1𝑑,𝐡𝐻2𝑑,…,𝐡𝐻𝑑𝑑 be a sequence of independent fractional Brownian motions with respective Hurst parameters 𝐻1,𝐻2,…,π»π‘‘βˆˆ(0,1). Let π‘“π‘–βˆΆβ„β†’(π‘Žπ‘–,𝑏𝑖) be a homeomorphism for each 𝑖=1,2,…,𝑑, where (π‘Žπ‘–,𝑏𝑖) is an open interval in the real line. Then the new process (𝑓1(𝐡𝐻1𝑑),𝑓2(𝐡𝐻2𝑑),…,𝑓𝑑(𝐡𝐻𝑑𝑑)) admits CPSs for each πœ–>0. This can be easily seen from the CFS property of the process (𝐡𝐻1𝑑,𝐡𝐻2𝑑,…,𝐡𝐻𝑑𝑑) which was shown in [14] and the fact that the map 𝑓(π‘₯)=(𝑓1(π‘₯),𝑓2(π‘₯),…,𝑓𝑑(π‘₯)) is a homomorphism from ℝ𝑑 to (π‘Ž1,𝑏1)Γ—(π‘Ž2,𝑏2)Γ—β‹―Γ—(π‘Žπ‘›,𝑏𝑛).

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