Consistent Price Systems in Multiasset Markets

Abstract

Let 𝑋_𝑡 be any d-dimensional continuous process that takes values in an open connected domain 𝒪 in ℝ^𝑑. In this paper, we give equivalent formulations of the conditional full support (CFS) property of 𝑋_𝑡 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 𝑋_𝑡 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 ℝ^𝑑-valued process 𝑌_𝑡=(𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡), where 𝑌^𝑖𝑡=𝑒^𝑋𝑖𝑡, 1≤𝑖≤𝑑, and the 𝑑-dimensional process 𝑋_𝑡=(𝑋^1𝑡,𝑋^2𝑡,…,𝑋^𝑑𝑡) is defined on a filtered probability space (Ω,ℱ,𝔽=(ℱ_𝑡)_{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 𝜃_𝑡=(𝜃^1𝑡,𝜃^2𝑡,…,𝜃^𝑑𝑡) are given by the difference of two processes 𝐿_𝑡=(𝐿^1𝑡,𝐿^2𝑡,…,𝐿^𝑑𝑡) and 𝑀_𝑡=(𝑀^1𝑡,𝑀^2𝑡,…,𝑀^𝑑𝑡) representing respectively the cumulative number of shares purchased and sold up to time 𝑡, namely, 𝜃_𝑡=𝐿_𝑡−𝑀_𝑡. We are also required to start and end without any position in the risky assets to and this requirement corresponds to 𝜃₀=𝜃_𝑇=0.

For each such trading strategy 𝜃_𝑡=𝐿_𝑡−𝑀_𝑡, the corresponding wealth process, after taking into account the incurred transaction costs, is given by 𝑉_𝑡(𝜃)=^𝑑_𝑖=1^𝑡0𝜃^𝑖𝑠𝑑𝑌^𝑖𝑠−𝜖^𝑑_𝑖=1^𝑡0𝑌^𝑖𝑠𝑑Var𝜃^𝑖_𝑠,(1.1) where Var(𝜃^𝑖)_𝑠=𝐿^𝑖𝑠+𝑀^𝑖𝑠 is the total variation of 𝜃^𝑖 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 ℝ^𝑑-valued process 𝜃_𝑡=(𝜃^1𝑡,𝜃^2𝑡,…,𝜃^𝑑𝑡) of finite variation with 𝜃₀=𝜃_𝑇=0 such that the corresponding wealth process 𝑉_𝑡(𝜃) satisfies 𝑉_𝑡(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,𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) does not admit arbitrage with 𝜖-sized transaction costs if there is no admissible trading strategy 𝜃_𝑡=(𝜃^1𝑡,𝜃^2𝑡,…,𝜃^𝑑𝑡) such that the corresponding value process 𝑉_𝑡(𝜃) satisfies 𝑃𝑉_𝑇(𝜃)>0>0,𝑃𝑉_𝑇(𝜃)≥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 (𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) that ensure absence of arbitrage in the market (1,𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡). It is clear that if the stock price process 𝑌_𝑡=(𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) 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^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) which is a martingale under an equivalent measure 𝑄 and which has the following property: |||𝑌_𝑖𝑡−𝑌^𝑖𝑡|||≤𝜖𝑌_𝑖𝑡,for𝑖=1,2,…,𝑑,∀𝑡∈[0,𝑇],(1.3) also implies absence of arbitrage for the model (1.1). To see this simple fact, observe the following: 𝑉_𝑇(𝜃)=^𝑑_𝑖=1^𝑇0𝜃^𝑖𝑠𝑑𝑌^𝑖𝑠−𝑌^𝑖𝑠+^𝑑_𝑖=1^𝑇0𝜃^𝑖𝑠𝑑𝑌^𝑖𝑠−𝜖^𝑑_𝑖=1^𝑇0𝑌^𝑖𝑠𝑑Var𝜃^𝑖_𝑠=^𝑑_𝑖=1^𝑇0𝑌^𝑖𝑠−𝑌^𝑖𝑠𝑑𝜃^𝑖𝑠−𝜖^𝑇0𝑌^𝑖𝑠𝑑Var𝜃^𝑖_𝑠+^𝑑_𝑖=1^𝑇0𝜃^𝑖𝑠𝑑𝑌^𝑖𝑠.(1.4) Note that because of (1.3), we have ∑^𝑑𝑖=1[∫^𝑇0(𝑌^𝑖𝑠−𝑌^𝑖𝑠)𝑑𝜃^𝑖𝑠−𝜖∫^𝑇0𝑌^𝑖𝑠𝑑Var(𝜃^𝑖)_𝑠]≤0 a.s. This implies that 𝑉_𝑇(𝜃)≤^𝑑_𝑖=1^𝑇0𝜃^𝑖𝑠𝑑𝑌^𝑖𝑠.(1.5) 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 𝑌_𝑡 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^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) is an 𝜖-consistent price system for 𝑌_𝑡=(𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡), if there exists a measure 𝑄∼𝑃 such that 𝑌_𝑡 is a martingale under 𝑄, and 11+𝜖≤𝑌𝑌^{𝑖𝑡𝑖𝑡}≤1+𝜖,for𝑖=1,2,…,𝑑,∀𝑡∈[0,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 𝑋_𝑡=(𝑋^1𝑡,𝑋^2𝑡,…,𝑋^𝑑𝑡) with state space ℝ^𝑑 has the CFS property, then the exponential process 𝑌_𝑡=(𝑌^1𝑡,𝑌^2𝑡,…,𝑌^𝑑𝑡) 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 𝑋_𝑡 with general state space 𝒪, where 𝒪 is any connected open set in ℝ^𝑑. 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 𝑋_𝑡 with the state space 𝒪 has the corresponding CFS property, then for any given 𝜖>0 there exists a martingale 𝑀_𝑡=(𝑀^1𝑡,𝑀^2𝑡,…,𝑀^𝑑𝑡), under an equivalent change of measure, such that ||𝑀^𝑖𝑡−𝑋^𝑖𝑡||≤𝜖forany𝑖=1,2,…,𝑑andany𝑡∈[0,𝑇].(1.7) 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 𝑋_𝑡=(𝑋^1𝑡,𝑋^2𝑡,…,𝑋^𝑑𝑡), 𝑡∈[0,𝑇] be a 𝑑-dimensional continuous process that takes values in an open connected domain 𝒪⊂ℝ^𝑑. For simplicity of our discussion, we assume that 0∈𝒪. We also assume that the process 𝑋_𝑡 is defined on a probability space (Ω,ℱ,𝑃) and adapted to a filtration 𝔽=(ℱ_𝑡)_{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∈ℝ^𝑑, let 𝐶_𝑥([𝑢,𝑣],𝒪) denote the set of functions in 𝐶([𝑢,𝑣],𝒪) with 𝑓(𝑢)=𝑥.

Definition 2.1. An adapted continuous process 𝑋_𝑡 satisfies the CFS property in 𝒪, if for any 𝑡∈[0,𝑇) and for almost all 𝜔∈Ω, SuppLaw𝑋_𝑠_{𝑠∈[𝑡,𝑇]}∣ℱ_𝑡(𝜔)=𝐶_{𝑋𝑡(𝜔)}([𝑡,𝑇],𝒪),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 𝑋_𝑡 be an adapted continuous process that takes values in an open and connected domain 𝒪⊂ℝ^𝑑. We say that 𝑋_𝑡 is linear sticky if for any 𝛼∈ℝ^𝑑, 𝜖>0, and any deterministic 0≤𝑠≤𝜃≤𝑇, 𝑃sup_{[]𝑡∈𝑠,𝜃}||𝑋_𝑡−𝑋_𝑠−𝛼(𝑡−𝑠)||<𝜖∣ℱ_𝑠>0,a.s.(2.2) on the set {𝑋_𝑠∈⋂_{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 𝑋_𝑡 be an adapted continuous process that takes values in an open and connected domain 𝒪⊂ℝ^𝑑. (a)We say that 𝑋_𝑡 is 𝑓-sticky for 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑) if 𝑃sup_{[]𝑡∈𝜏,𝑇}||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||<𝜖∣ℱ_𝜏>0,a.s.(2.3) on the set {𝑋_𝜏∈⋂_{𝑡∈[0,𝑇−𝜏]}(𝒪−𝑓(𝑡))} for any 𝜖>0 and any stopping time 𝜏.(b)We say that 𝑋_𝑡 is weak 𝑓-sticky for 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑) if 𝑃sup_{[]𝑡∈𝑠,𝑇}||𝑋_𝑡−𝑋_𝑠−𝑓(𝑡−𝑠)||<𝜖∣ℱ_𝑠>0,a.s.(2.4) on the set {𝑋_𝑠∈⋂_{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 𝑋_𝑡 is seemingly weaker condition than the weak 𝑓-stickiness of 𝑋_𝑡 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 𝑋_𝑡 for all 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑). This, in turn, implies that the linear stickiness property is equivalent to the CFS property.

Remark 2.5. When a process 𝑋_𝑡 is 0-sticky as in (𝑏) in Definition 2.3, we say that 𝑋_𝑡 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 𝑋_𝑡 has paths that are as close as one wants to the path 𝑓(𝑡)+𝑋_𝜏. As it was shown in [14], the 𝑓-stickiness holds for any 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑) for the process (𝐵^𝐻1_𝑡,𝐵^𝐻2_𝑡,…,𝐵^𝐻𝑑_𝑡), where 𝐵^𝐻1_𝑡,𝐵^𝐻2_𝑡,…,𝐵^𝐻𝑑_𝑡 are independent fractional Brownian motions with respective Hurst parameters 𝐻₁,𝐻₂,…,𝐻_𝑑∈(0,1). From [10], the 𝑓-stickiness also holds for any continuous Markov process with the full support property in 𝐶₀([0,𝑇],ℝ^𝑑) for any 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑).

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 𝑋_𝑡=(𝑋^1𝑡,𝑋^2𝑡,…,𝑋^𝑑𝑡) be a continuous process that takes values in a connected domain 𝒪 in ℝ^𝑑. If 𝑋_𝑡 is linear sticky, then 𝑋_𝑡 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: 𝜏₀=0,𝜏_𝑛+1=inf_{𝑡≥𝜏𝑛}||𝑋_𝑡−𝑋_𝜏𝑛||≥𝜖_𝑛+1∧𝑇,∀𝑛≥0,(2.5) with 𝜖_𝑛+1∶=𝜖∧𝑑(𝑋_𝜏𝑛,𝜕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 Δ_𝑛=𝑋^𝜏𝑛0−𝑋_𝜏𝑛−1wohtheenrw𝜏_𝑛is<e.𝑇,(2.6) Let 𝒢_𝑛=ℱ_𝜏𝑛 for every 𝑛≥0. Note that 𝜖_𝑛 is bounded and 𝒢_𝑛−1 measurable.

In the following, we use the notation Supp(Δ_𝑛∣𝒢_𝑛−1) to denote the smallest closed set of ℝ^𝑑 that contains the values of the random variable 𝐸[Δ_𝑛∣𝒢_𝑛−1] with probability one. We use 𝐵_𝑟(𝑥) to denote the open ball in ℝ^𝑑 with center 𝑥 and radius 𝑟. When the center is 0, we simply write 𝐵_𝑟. We first prove the following lemma.

Lemma 2.7. If 𝑋_𝑡 is 𝑓-sticky in 𝒪 for all 𝑓∈𝐶₀([0,𝑇]), then the process {Δ_𝑛} in (2.6) satisfies the following three properties: ((ii)i)𝑃SupΔp_𝑚=Δ_𝑛0,∀𝑚≥𝑛∣Δ_𝑛=0=1;(iii)𝑃Δ_𝑚∣𝒢_𝑛−1=0∪𝜕𝐵_𝜖𝑛𝑎𝑙𝑚𝑜𝑠𝑡𝑠𝑢𝑟𝑒𝑙𝑦𝑜𝑛Δ_𝑛−1≠0,≠0,∀𝑚≥1=0.(2.7)

Proof. Property (𝑖) is obvious since {Δ_𝑛=0}={𝜏_𝑛=𝑇} and 𝜏_𝑛 is increasing. Property (𝑖𝑖𝑖) follows from the fact that almost surely each path of 𝑋_𝑡 is contained in a compact set of 𝒪 and therefore min_𝑛𝜖_𝑛(𝜔)>0 almost surely 𝜔∈Ω. To prove property (𝑖𝑖), let us assume that 𝑃(𝜏_𝑛−1<𝑇)>0 and let 𝐺_𝑛−1 be any 𝒢_𝑛−1 measurable set such that 𝑃(𝐺_𝑛−1∩{𝜏_𝑛−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(𝐺_𝑛−1∩{𝜏_𝑛−1<𝑇′}∩{𝑋_𝜏𝑛−1∈𝐵_𝜁(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 𝐺_𝑛−1∩{𝜏_𝑛−1<𝑇′}∩{𝑋_𝜏𝑛−1∈𝐵_𝜁(𝑦)} we have 𝜖∧3𝜁/2≤𝜖_𝑛<2𝜁.
First we show that 𝑃(Δ_𝑛=0∣𝒢_𝑛−1)>0 a.s. on {Δ_𝑛−1≠0}. To see this, define the following stopping time: 𝜏=𝜏^𝑛−1𝑇ootnhe𝐺rwi_𝑛−1se∩.𝜏_𝑛−1<𝑇^∩𝑋_𝜏𝑛−1∈𝐵_𝜁(𝑦),(2.8) The 0-stickiness of 𝑋 implies that 𝑃sup_{[]𝑡∈𝜏,𝑇}||𝑋_𝑡−𝑋_𝜏||<𝜁,𝜏<𝑇>0.(2.9) But {sup_{𝑡∈[𝜏,𝑇]}|𝑋_𝑡−𝑋_𝜏|<𝜁,𝜏<𝑇}⊂{Δ_𝑛=0} and since 𝐺_𝑛−1 was an arbitrary 𝒢_𝑛−1 measurable set with 𝑃(𝐺_𝑛−1∩{𝜏_𝑛−1<𝑇})>0, we have 𝑃Δ_𝑛=0∣𝒢_𝑛−1>0a.s.onΔ_𝑛−1≠0.(2.10) Next we show that 𝜕𝐵_{𝜖𝑛(𝜔)}⊂Supp(Δ_𝑛∣𝒢_𝑛−1)(𝜔) almost surely on {Δ_𝑛−1≠0}. Too see this, take any 𝑥∈𝜕𝐵₁, 0<𝜖′<𝜁 and define 𝑓(𝑡)=⎧⎪⎨⎪⎩3𝜁𝑇𝑥6−𝜁𝑇𝑡^𝑥iofth0e≤rw𝑡is≤e,𝑇−2𝑇′,(2.11) and note that 𝑃𝜏<𝑇,𝑋_𝜏∈_{𝑡∈[𝜏,𝑇]}(𝒪−𝑓(𝑡−𝜏))>0.(2.12) By 𝑓-stickiness of 𝑋, we obtain 𝑃sup_{[]𝑡∈𝜏,𝑇}||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||<𝜖′,𝜏<𝑇>0,(2.13) or equivalently 𝑃(𝐴∩𝐺_𝑛−1)>0, where 𝐴=∩𝜏sup_{𝜏𝑡∈𝑛−1,𝑇}||𝑋_𝑡−𝑋_𝜏𝑛−1−𝑓𝑡−𝜏_𝑛−1||<𝜖′_𝑛−1∩𝑋_𝜏𝑛−1<𝑇^∈_{𝑡∈[𝜏𝑛−1,𝑇]}𝒪−𝑓𝑡−𝜏_𝑛−1.(2.14) We claim that 𝐴⊂{Δ_𝑛⊂𝐵_2𝜖(𝜖_𝑛𝑥)}. Indeed, if 𝜔∈𝐴, we get ||𝑋≥|||𝑓𝑇−_{𝜏𝑛−1+(𝑇−𝑇}^′_)/2(𝑇𝜔′)−𝑋_𝜏𝑛−1(𝜔)||2|||−||||𝑋^{𝜏𝑛−1+(𝑇−𝑇′)/2}(𝜔)−𝑋_𝜏𝑛−1(𝜔)−𝑓𝑇−𝑇^≥3𝜁−𝜖_2||||>𝜖_𝑛(𝜔).(2.15) Hence {𝜏_𝑛<𝑇} on 𝐴. Also, for 𝜔∈𝐴 we have 0=𝑑𝑋_𝜏𝑛−𝑋_𝜏𝑛−1,𝜕𝐵_𝜖𝑛>≥𝑑||𝑓𝑓𝜏_𝑛𝜏_𝑛−𝜏_𝑛−1,𝜕𝐵_𝜖𝑛−||𝑋_𝜏𝑛−𝑋_𝜏𝑛−1−𝑓𝑡−𝜏_𝑛−1||||𝑋_𝜏𝑛−𝜏_𝑛−1−𝜖_𝑛𝑥||−𝜖^,−𝑋_𝜏𝑛−1−𝜖_𝑛𝑥||≤<||||𝑓𝑓𝜏𝜏_𝑛𝑛−𝜏_𝑛−1−𝜖_𝑛𝑥||+||𝑋_𝜏𝑛−𝑋_𝜏𝑛−1−𝑓𝑡−𝜏_𝑛−1||−𝜏_𝑛−1−𝜖_𝑛𝑥||+𝜖′<2𝜖′.(2.16) So for all 𝜔∈𝐴, 𝑑(Δ_𝑛,𝜖_𝑛𝑥)<2𝜖′. Since this is true for any small 𝜖′, 𝑥∈𝐵₁ and any arbitrary 𝒢_𝑛−1 measurable set with 𝑃(𝐺_𝑛−1∩{𝜏_𝑛−1<𝑇})>0, we conclude that 𝜕𝐵_{𝜖𝑛(𝜔)}⊂Supp(Δ_𝑛∣𝒢_𝑛−1)(𝜔) almost surely on {Δ_𝑛−1≠0}. Note that the other direction 𝜕𝐵_{𝜖𝑛(𝜔)}∪{0}⊃Supp(Δ_𝑛∣𝒢_𝑛−1)(𝜔) is clear from the definition of Δ_𝑛.

Now, define 𝜖_𝑛, Δ_𝑛, 𝑛≥0 as above and let 𝑀_𝑛=𝑋₀+∑^𝑛𝑖=1Δ_𝑖, 𝑛≥0. The ℝ^𝑑-valued process 𝑀_𝑛=∶(𝑀^1𝑛,𝑀^2𝑛,…,𝑀^𝑑𝑛) will be used to construct CPSs for 𝑋_𝑡. Next, we prove a lemma that shows that all of 𝑀^𝑖𝑛, 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 ℝ^𝑑-valued discrete process {(𝑀_𝑛,𝒢_𝑛)}^+∞𝑛=0 is a martingale.

Proof. For any 𝑛≥0, let 𝜇_𝑛 be the regular conditional probability of Δ_𝑛 with respect to 𝒢_𝑛−1 and let Ω_𝑛={𝜔∈Ω∣Supp(Δ_𝑛∣𝒢_𝑛−1)(𝜔)=0∪𝜕𝐵_𝜖𝑛}. 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 𝐺_𝑛∶Ω_𝑛×ℝ^𝑑→ℝ^𝑑 as follows: 𝐺_𝑛(𝜔,𝛼)=_ℝ𝑑𝑘(𝛼⋅𝑥)𝑥𝑑𝜇_𝑛(𝜔,⋅).(2.17) Obviously for each 𝑛, 𝐺_𝑛(⋅,𝑎) is 𝒢_𝑛−1 measurable and convex with respect to 𝛼. As a consequence, for any fixed 𝜔∈Ω_𝑛, Im(𝐺_𝑛(𝜔,⋅)), the image of the function 𝐺_𝑛(𝜔,⋅) is convex. We first prove that for every 𝑛≥1 and 𝜔∈Ω_𝑛: lim_|𝛼|→∞𝐺_𝑛(𝜔,𝛼)⋅𝛼|𝛼|=+∞.(2.18) By the way of contrary, assume that this is not true, for some 𝑛≥1 and 𝜔∈Ω_𝑛. Then, there exists a sequence (𝛼_𝑚)_𝑚≥1 with |𝛼_𝑚|→∞ such that 𝐺_𝑛(𝜔,𝛼_𝑚)⋅(𝛼_𝑚/|𝛼_𝑚|) is bounded above. We can assume that (𝛼_𝑚/|𝛼_𝑚|) converges to some 𝛼 (this is a bounded sequence and therefore has a convergent subsequence). We have 𝐺_𝑛𝜔,𝛼_𝑚⋅𝛼^𝑚||𝛼_𝑚||≥_{𝛼𝑚⋅𝑥≤0}𝑘𝛼_𝑚⋅𝑥𝛼^𝑚⋅𝑥||𝛼_𝑚||𝑑𝜇^𝑛+(𝜔,⋅)_{(𝛼𝑚⋅𝑥)/|𝛼𝑚|>𝜖𝑛(𝜔)/4}𝑘𝛼_𝑚⋅𝑥𝛼^𝑚⋅𝑥||𝛼_𝑚||𝑑𝜇^𝑛≥−𝑘(0)+_{2𝛼⋅𝑥>𝜖𝑛(𝜔)}(𝜔,⋅)𝛼_𝑚⋅𝑥²||𝛼_𝑚||𝑑𝜇^𝑛(𝜔,⋅),(2.19) for big enough 𝑚. Therefore, we can conclude that ∫_{2𝛼⋅𝑥>𝜖𝑛(𝜔)}((𝛼_𝑚⋅𝑥)²/|𝛼_𝑚|)⋅(1/|𝑎_𝑚|)𝑑𝜇_𝑛(𝜔,⋅) converges to 0 as |𝑎_𝑚|→+∞, which will imply after passing to the limit that 𝜇_𝑛(𝜔,{2𝛼⋅𝑥>𝜖_𝑛}) and this is a contradiction. From this it follows easily that 0∈Im(𝐺_𝑛(𝜔,⋅)). If 0∉Im(𝐺_𝑛(𝜔,⋅)), then using the geometric form of Hahn Banach theorem, there exists a unit vector 𝛽∈ℝ^𝑑 such that ∫_ℝ𝑑𝑘(𝛼⋅𝑥)𝛽𝑥𝑑𝜇_𝑛(𝜔,⋅)<0 for every 𝛼∈ℝ^𝑑. Therefore, limsup_𝑡→∞∫_ℝ𝑑𝑘(𝑡𝛽⋅𝑥)𝛽𝑥𝑑𝜇_𝑛(𝜔,⋅)≤0. But _ℝ𝑑𝑘(𝑡𝛽⋅𝑥)𝛽𝑥𝑑𝜇_𝑛(𝜔,⋅)=𝐺_𝑛(𝜔,𝑡𝛽)⋅𝑡𝛽||𝑡𝛽||,(2.20) and so it contradicts (2.18).
Next, we want to show that Im(𝐺_𝑛(𝜔,⋅)) is closed. Let 𝑎∈Im(𝐺_𝑛(𝜔,⋅)), so there exists a sequence (𝛼_𝑚)_𝑚≥1 such that 𝐺_𝑛(𝜔,𝛼_𝑚)→𝑎. But then |𝛼_𝑚| is unbounded, and therefore this contradicts (2.18). So based on the continuity of 𝐺_𝑛(𝜔,⋅), 𝑎∈Im(𝐺_𝑛(𝜔,⋅)).
Therefore, we conclude that for any 𝑛≥1 and 𝜔∈Ω_𝑛, there exists an 𝛼_𝑛(𝜔)∈ℝ^𝑑, unique, as a consequence of the strict monotonicity of 𝑘, such that 𝐺_𝑛(𝜔,𝛼_𝑛(𝜔))=0.   𝐺_𝑛 being continuous with respect to 𝛼 and 𝒢_𝑛−1 measurable with respect to 𝜔, it follows that 𝛼_𝑛 is 𝒢_𝑛−1 measurable. We extend 𝛼_𝑛 with 1 outside Ω_𝑛 and define: 𝑍_𝑛=𝑘𝛼^𝑛⋅Δ_𝑛1_{Δ𝑛≠0}2𝐸𝑘𝛼_𝑛⋅Δ_𝑛1_{Δ𝑛≠0}∣𝒢_𝑛−1+2𝑃Δ1_𝑛=0∣^{Δ𝑛=0}𝒢_𝑛−1.(2.21) It is easy to check that 𝑍_𝑛 satisfies 𝐸𝐸𝑍𝑍_𝑛∣𝒢_𝑛−1=1,_𝑛Δ_𝑛∣𝒢_𝑛−1=0.(2.22) Let 𝐿_𝑛=∏^𝑛𝑖=1𝑍_𝑖 and 𝐿=lim_𝑛→∞𝐿_𝑛. Note that this limit exists almost surely since 𝐿_𝑛+1=𝐿_𝑛 a.s. on {Δ_𝑛=0} and {Δ_𝑛=0}↗Ω. From (2.22), we get 𝐸𝐿_𝑛∣𝒢_𝑛−1=𝐿_𝑛−1𝐸𝐿_𝑛𝑀_𝑛∣𝒢_𝑛−1=𝐿_𝑛−1𝑀_𝑛−1,(2.23) which shows that (𝐿_𝑛)_𝑛≥1 and (𝑀_𝑛𝐿_𝑛)_𝑛≥1 are martingales under 𝑃. We thus get 𝐸(𝐿_𝑛)=𝐸(𝑍₁)=1, and Fatou's lemma gives 𝐸(𝐿)≤1. We will show that 𝐸(𝐿)=1. We have 𝐸(𝐿)=𝐸lim_𝑛→∞𝐿1_{Δ𝑛=0}=lim_𝑛→∞=1−lim_𝑛→∞𝐸𝐿1_{Δ𝑛=0}=lim_𝑛→∞𝐸𝐿_𝑛1_{Δ𝑛=0}=1−1𝐸𝐿_𝑛1_{Δ𝑛≠0}=1−lim_𝑛→∞𝐸𝐸𝐿_𝑛1_{Δ𝑛≠0}∣𝒢_𝑛−12lim^𝑛→∞𝐸𝐿_𝑛−11_{{Δ𝑛−1≠0}}=1−lim_𝑛→∞12^𝑛=1.(2.24) Combining Fatou's lemma with the equation 𝐸(𝐿_𝑛)=𝐸(𝐿)=1, we obtain 𝐸(𝐿∣𝒢_𝑛)=𝐿_𝑛. Also, 𝐸𝑀_𝑛𝐿∣𝒢_𝑛−1==𝐸𝑀𝐸_𝑛−1𝑀_𝑛𝐿∣𝒢_𝑛∣𝒢_𝑛−1=𝐸𝑀_𝑛𝐿∣𝒢_𝑛−1𝐿_𝑛−1=𝐸𝑀_𝑛−1𝐿∣𝒢_𝑛−1.(2.25) Hence, 𝐿 is the density of a measure 𝑄 under which our discrete process 𝑀_𝑛 is a martingale. And since 𝐿>0 (𝐿_𝑛>0 for all n), 𝑄 is equivalent to 𝑃.

Lemma 2.9. Under the measure 𝑄 of Lemma 2.8 the process 𝑀^𝑖𝑛 is uniformly integrable for each 1≤𝑖≤𝑑. In particular, 𝐸_𝑄(sup_𝑛≥0|𝑀^𝑖𝑛|)<∞ for each 𝑖=1,2,…,𝑑.

Proof. For any 1≤𝑖≤𝑑, set 𝑀^𝑖∗=sup_𝑛≥0|𝑀^𝑖𝑛| and observe that on {Δ_𝑘≠0,Δ_𝑘+1=0} we have 𝑀^𝑖∗≤|𝑋^𝑖0|+𝑘𝜖. Observing that 𝑄(Δ_𝑘≠0)=𝑄(Δ_𝑘≠0∣Δ_𝑘−1≠0)⋯𝑄(Δ₁≠0∣Δ₀≠0)𝑄(Δ₀≠0) and that 𝑄(Δ_𝑘≠0∣Δ_𝑘−1≠0)=1/2 we obtain the following: 𝐸_𝑄𝑀^𝑖∗=^∞_𝑘=0𝐸_𝑄𝑀^𝑖∗1_{{Δ𝑘≠0}∩{Δ𝑘+1=0}}≤^∞_𝑘=0||𝑋^𝑖0||+𝑘𝜖𝑄Δ_𝑘≠0,Δ_𝑘+1=0<∞.(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 𝑋_𝑡 be an adapted continuous process with state space 𝒪 and 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑). Then, 𝑋_𝑡 is weak 𝑓-sticky if and only if it is 𝑓-sticky.

Proof. Let us show first that for any 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑) weak 𝑓-stickiness implies 𝑓-stickiness. Suppose for a contradiction that 𝑋_𝑡 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 𝑃𝑃su𝜏p_{[]𝑡∈𝜏,𝑇}<𝑇,𝑋_𝜏∈_{𝑡∈[0,𝑇−𝜏]}(𝒪−𝑓(𝑡))>0,||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||<𝜖,𝜏<𝑇=0.(2.27) Since 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑), 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 𝑃𝑡₁≤𝜏<𝑡₂,𝑋_𝜏∈_{𝑡∈[𝜏,𝑇]}𝒪_𝜁−𝑓(𝑡−𝜏)>0,(2.28) where 𝒪_𝜁={𝑥∈𝒪/𝑑(𝑥,𝜕𝒪)>𝜁}.
For each 𝑞∈𝐼=ℚ∩[𝑡₁,𝑡₂), let 𝐴_𝑞∶=𝐴∩{𝑡₁≤𝜏<𝑞}∩{sup_{𝑡∈[𝜏,𝑞]}|𝑋_𝑡−𝑋_𝜏|<𝜁}, where 𝐴∶=𝑡₁≤𝜏<𝑡₂,𝑋_𝜏∈_{𝑡∈[𝜏,𝑇]}𝒪_𝜁−𝑓(𝑡−𝜏).(2.29) Since 𝑃(𝐴)>0 and 𝐴=⋃_𝑞∈𝐼𝐴_𝑞, there exists a 𝑞^∗∈𝐼 such that 𝑃(𝐴_𝑞∗)>0. Note that 𝐴_𝑞∗∈ℱ_𝑞∗ and 𝐴_𝑞∗⊂⋂_{𝑡∈[0,𝑇−𝑞∗]}(𝒪−𝑓(𝑡)). Hence, since 𝑋_𝑡 is weak 𝑓-sticky, we obtain 𝑃𝐴_𝑞∗∩sup_{[𝑞𝑡∈∗],𝑇}||𝑋_𝑡−𝑋_𝑞∗−𝑓𝑡−𝑞^∗||<𝜖3>0.(2.30) Let 𝐶_𝑞∗=𝐴_𝑞∗∩{sup_{𝑡∈[𝑞∗,𝑇]}|𝑋_𝑡−𝑋_𝑞∗−𝑓(𝑡−𝑞^∗)|<𝜖/3}. Then we claim that 𝐶_𝑞∗⊂sup_{[]𝑡∈𝜏,𝑇}||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||<𝜖∩{𝜏<𝑇},(2.31) which contradicts (2.27). Indeed, if 𝜔∈𝐶_𝑞∗, then for 𝑡∈[𝜏,𝑞^∗] we have ||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||<||𝑋_𝑡−𝑋_𝜏||+||𝑓(𝑡−𝜏)||<𝜖3+3𝜖<𝜖,(2.32) by the definition of 𝐴_𝑞∗ and the choice of 𝛿. We will show also that |𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)|<𝜖 on 𝐶_𝑞∗ whenever 𝑡∈[𝑞^∗,𝑇]: ||𝑋_𝑡−𝑋_𝜏−𝑓(𝑡−𝜏)||≤≤||||𝑋𝑋_𝑞∗𝜏−𝑋_𝜏−𝑓(𝑡−𝜏)+𝑓𝑡−𝑞^∗||+||𝑋_𝑡−𝑋_𝑞∗−𝑓𝑡−𝑞^∗||<𝜖−𝑋_𝑞∗||+||𝑓𝑡−𝑞^∗−𝑓(𝑡−𝜏)||+||𝑋_𝑡−𝑋_𝑞∗−𝑓𝑡−𝑞^∗||3+3𝜖+3𝜖=𝜖.(2.33) Thus, weak 𝑓-stickiness implies 𝑓-stickiness. Since the opposite direction is obvious, the proposition is proved.

Lemma 2.11. Let 𝑋_𝑡 be a continuous adapted process with state space 𝒪. Then 𝑋_𝑡 is linear sticky if and only if 𝑋_𝑡 is 𝑓-sticky for all 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑).

Proof. We only need to show that linear stickiness implies the weak 𝑓-stickiness for each 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑). Fix any 𝑓∈𝐶₀([0,𝑇],ℝ^𝑑),𝑠∈[0,𝑇], 𝜖₀>0. We need to show that 𝑃sup_{[]𝑡∈𝑠,𝑇}||𝑋_𝑡−𝑋_𝑠−𝑓(𝑡−𝑠)||<𝜖₀∣ℱ^𝑠>0,a.s.(2.34) on the set 𝐵=∶{𝑋_𝑠∈⋂_{𝑡∈[0,𝑇−𝑠]}(𝒪−𝑓(𝑡))}. To do this, for any 𝐴∈ℱ_𝑠 with 𝑃(𝐴∩𝐵)>0, we need to show that 𝑃𝐴∩𝐵∩sup_{[]𝑡∈𝑠,𝑇}||𝑋_𝑡−𝑋_𝑠−𝑓(𝑡−𝑠)||<𝜖₀>0.(2.35) Define 𝑍(𝜔)=inf_{𝑟∈[0,𝑇−𝑠]}𝑑(𝑋_𝑠(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∈ℱ_𝑠 and 𝐵₀⊂𝐴∩𝐵. In the following, we show that 𝑃𝐵₀∩sup_{[]𝑡∈𝑠,𝑇}||𝑋_𝑡−𝑋_𝑠−𝑓(𝑡−𝑠)||<𝜖₀>0.(2.36) Let 𝜖=min(𝜖₀,ℎ) and set 𝑡₀=0, and define 𝑡_𝑘=inf𝑡≥𝑡_𝑘−1∶||𝑓(𝑡)−𝑓𝑡_𝑘−1||≥𝜖4∧(𝑇−𝑠),(2.37) for 𝑘≥1. Let 𝑁 be the smallest positive integer such that 𝑡_𝑁=𝑇−𝑠. For each 𝑘≥1, define 𝑔(𝑡) on [𝑡_𝑘−1,𝑡_𝑘] to be equal to the linear function that connects the two points 𝑓(𝑡_𝑘−1) and 𝑓(𝑡_𝑘). We can assume that 𝑔(𝑡)=𝑓𝑡_𝑘−1+𝛼_𝑘−1𝑡,on𝑡_𝑘−1,𝑡_𝑘,(2.38) for some constant vector 𝛼_𝑘−1∈ℝ^𝑑 for each 𝑘≥1. It is clear that sup_{[]𝑡∈0,𝑇−𝑠}||𝑓(𝑡)−𝑔(𝑡)||≤𝜖2.(2.39) Because of (2.39), to show (2.36) we only need to show 𝑃𝐵₀∩sup_{[]𝑡∈𝑠,𝑇}||𝑋_𝑡−𝑋_𝑠−𝑔(𝑡−𝑠)||<𝜖2>0.(2.40)
For each 𝑘=0,1,2,…,𝑁−1, let 𝐵_𝑘+1=𝐵_𝑘∩sup_{𝑡∈𝑠+𝑡𝑘,𝑠+𝑡𝑘+1}||𝑋_𝑡−𝑋_𝑠−𝑔(𝑡−𝑠)||<𝜖2^𝑁−𝑘.(2.41) Note that 𝐵_𝑁 is contained in the event in (2.40). Therefore, it is sufficient to prove that 𝐵_𝑁 has positive probability. When 𝑡∈[𝑠+𝑡_𝑘,𝑠+𝑡_𝑘+1], we have 𝑔(𝑡−𝑠)=𝑓(𝑡_𝑘)+𝛼_𝑘(𝑡−𝑠)=𝑓(𝑡_𝑘)+𝛼_𝑘𝑡_𝑘+𝛼_𝑘[𝑡−(𝑠+𝑡_𝑘)]=𝑔([(𝑠+𝑡_𝑘)−𝑠])+𝛼_𝑘[𝑡−(𝑠+𝑡_𝑘)]. Therefore, we have the following relation: sup_{𝑡∈𝑠+𝑡𝑘,𝑠+𝑡𝑘+1}||𝑋_𝑡−𝑋_𝑠−𝑔(𝑡−𝑠)||<𝜖⊃𝑋_{𝑠+𝑡𝑘}2^𝑁−𝑘−𝑋_𝑠−𝑔𝑠+𝑡_𝑘−𝑠<𝜖∩sup_{𝑡∈𝑠+𝑡𝑘,𝑠+𝑡𝑘+1}2^{𝑁−𝑘+1}||𝑋_𝑡−𝑋_{𝑠+𝑡𝑘}−𝛼_𝑘𝑡−𝑠+𝑡_𝑘||<𝜖2^{𝑁−𝑘+1}.(2.42) By the definition of 𝐵_𝑘 and the above relation, it is easy see that 𝐵_𝑘+1⊃𝐵_𝑘∩sup_{𝑡∈𝑠+𝑡𝑘,𝑠+𝑡𝑘+1}||𝑋_𝑡−𝑋_{𝑠+𝑡𝑘}−𝛼_𝑘𝑡−𝑠+𝑡_𝑘||<𝜖2^{𝑁−𝑘+1}.(2.43) On 𝐵_𝑘 we have 𝑑𝑋_{𝑠+𝑡𝑘},𝜕𝒪≥𝑑𝑋_𝑠+𝑔𝑠+𝑡_𝑘−𝑠,𝜕𝒪−𝑑𝑋_𝑠+𝑔𝑠+𝑡_𝑘−𝑠,𝑋_{𝑠+𝑡𝑘}>𝜖2−2𝜖^{𝑁−𝑘+1}≥𝜖||𝑋_{𝑠+𝑡𝑘}4,−𝑋_{𝑠+𝑡𝑘}+𝛼_𝑘𝑡−𝑠+𝑡_𝑘||=||𝛼_𝑘𝑡−𝑠+𝑡_𝑘||=||𝑔(𝑡)−𝑔𝑡_𝑘||≤𝜖4,(2.44) for each 𝑘=0,1,…,𝑁−1 and for all 𝑡∈[𝑠+𝑡_𝑘,𝑠+𝑡_𝑘+1]. From this, we conclude that 𝐵_𝑘⊂{𝑋_{𝑠+𝑡𝑘}∈⋂_{𝑡∈[𝑠+𝑡𝑘,𝑠+𝑡𝑘+1]}(𝒪−𝛼_𝑘[𝑡−(𝑠+𝑡_𝑘)]}. Now, from the linear stickiness and the fact that 𝑃(𝐵₀)>0 we conclude 𝑃(𝐵_𝑁)>0. This completes the proof.

Proof of Theorem 2.6. By Lemmas 2.8 and 2.9, there exists an equivalent probability measure 𝑄∼𝑃 such that (𝑀^𝑖𝑛,𝒢_𝑛)_𝑛≥0 is a uniformly integrable martingale for each 1≤𝑖≤𝑑. Let 𝑀^𝑖∞=lim_𝑛→∞𝑀^𝑖𝑛. For each 𝑡∈[0,𝑇], set 𝑀^𝑖𝑡=𝐸_𝑄[𝑀^𝑖∞∣ℱ_𝑡]. Observe that 𝑀^𝑖𝜏_𝑛=𝐸_𝑄[𝑀^𝑖∞∣ℱ_𝜏𝑛]=𝑀^𝑖𝑛, and 𝑀^𝑖𝑡=𝐸_𝑄[𝑀^𝑖𝜏_𝑛∣ℱ_𝑡] on the set {𝜏_𝑛−1≤𝑡≤𝜏_𝑛} for all 𝑛≥0. Thus the following equation holds: 𝑀^𝑖𝑡−𝑋^𝑖𝑡1^{𝜏_𝑛−1^{≤𝑡≤𝜏}_𝑛^}=𝐸_𝑄𝑀^𝑖𝑛−𝑋^𝑖𝑡1_{{𝜏𝑛−1≤𝑡≤𝜏𝑛}}∣ℱ_𝑡,𝑛≥1.(2.45) We write 𝑀^𝑖𝑛−𝑋^𝑖𝑡=(𝑀^𝑖𝑛−𝑋^𝑖𝜏_𝑛)+(𝑋^𝑖𝜏_𝑛−1−𝑋^𝑖𝑡)+(𝑋^𝑖𝜏_𝑛−𝑋^𝑖𝜏_𝑛−1). Note that each of 𝑀^𝑖𝑛−𝑋^𝑖𝜏_𝑛, 𝑋^𝑖𝜏_𝑛−1−𝑋^𝑖𝑡, and 𝑋^𝑖𝜏_𝑛−𝑋^𝑖𝜏_𝑛−1 takes values in (−𝜖,𝜖) on the set {𝜏_𝑛−1≤𝑡≤𝜏_𝑛}. Therefore, we have −3𝜖≤|𝑀^𝑖𝑡−𝑋^𝑖𝑡|≤3𝜖 on the set {𝜏_𝑛−1≤𝑡≤𝜏_𝑛}. Since ⋃^∞𝑛=1{𝜏_𝑛−1≤𝑡≤𝜏_𝑛}=Ω, we conclude that −3𝜖≤|||𝑀^𝑖𝑡−𝑋^𝑖𝑡|||≤3𝜖.(2.46) Since 𝜖>0 is arbitrary, the claim follows.

Example 2.12. Let 𝐵^𝐻1_𝑡,𝐵^𝐻2_𝑡,…,𝐵^𝐻𝑑_𝑡 be a sequence of independent fractional Brownian motions with respective Hurst parameters 𝐻₁,𝐻₂,…,𝐻_𝑑∈(0,1). Let 𝑓_𝑖∶ℝ→(𝑎_𝑖,𝑏_𝑖) be a homeomorphism for each 𝑖=1,2,…,𝑑, where (𝑎_𝑖,𝑏_𝑖) is an open interval in the real line. Then the new process (𝑓₁(𝐵^𝐻1_𝑡),𝑓₂(𝐵^𝐻2_𝑡),…,𝑓_𝑑(𝐵^𝐻𝑑_𝑡)) admits CPSs for each 𝜖>0. This can be easily seen from the CFS property of the process (𝐵^𝐻1_𝑡,𝐵^𝐻2_𝑡,…,𝐵^𝐻𝑑_𝑡) which was shown in [14] and the fact that the map 𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑑(𝑥)) is a homomorphism from ℝ^𝑑 to (𝑎₁,𝑏₁)×(𝑎₂,𝑏₂)×⋯×(𝑎_𝑛,𝑏_𝑛).