Consistent Price Systems in Multiasset Markets

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 (cid:2) CFS (cid:3) 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 (cid:2) > 0 neighborhood of X t for any given (cid:2) under the supremum norm. The existence of such martingales, which are called consistent price systems (cid:2) CPSs (cid:3) , 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. (cid:2) 2008 (cid:3) , where the CFS property is introduced and shown su ﬃ cient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (cid:2) 2008 (cid:3) , to processes with more general state space.


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 1 t , Y 2 t , . . ., Y d t , where Y i t e X i t , 1 ≤ i ≤ d, and the d-dimensional process X t X 1 t , X 2 t , . . ., X d t 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 1 t , θ 2 t , . . ., θ d t are given by the difference of two processes L t L 1 t , L 2 t , . . ., L d t and M t M 1 t , M 2 t , . . ., M d t 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 θ 0 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 where Var θ i s L i s M i s is the total variation of θ i in 0, s for each 1 ≤ i ≤ d and > 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 1 t , θ 2 t , . . ., θ d t of finite variation with θ 0 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.
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 1 t , Y 3 also implies absence of arbitrage for the model 1.1 .To see this simple fact, observe the following:

1.4
International Journal of Stochastic Analysis 3 Note that because of 1.3 , we have The financial interpretation of 1.
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 1 t , X By an abuse of language we call such M a -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.

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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.

Main Results
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 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 X t 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 , > 0, and any deterministic 0 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
on the set {X τ ∈ t∈ 0,T −τ O − f t } for any > 0 and any stopping time τ.
b We say that X t is weak f-sticky on the set {X s ∈ t∈ 0,T −s O − f t } for any > 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 ∈ C 0 0, T , O .The linear stickiness of X t is seemingly weaker condition than the weak f-stickiness of X t for all f ∈ C 0 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 ∈ C 0 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 ∈ C 0 0, T , R d for the process B H 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 1 t , X 2 t , . . ., X d t 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 > 0.
To show this result, one fix any > 0 and define the following increasing sequence of stopping times associated with the process X: with n 1 : ∧ d X τ n , ∂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 .

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In addition, for each n ≥ 1 we define Let G n F τ n for every n ≥ 0. Note that 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 ∈ C 0 0, T , then the process {Δ n } in 2.6 satisfies the following three properties:

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 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: The 0-stickiness of X implies that 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 Next we show that ∂B n ω ⊂ Supp Δ n | G n−1 ω almost surely on {Δ n−1 / 0}.Too see this, take any x ∈ ∂B 1 , 0 < < ζ and define and note that By f-stickiness of X, we obtain

2.14
We claim that A ⊂ {Δ n ⊂ B 2 n x }.Indeed, if ω ∈ A, we get

2.15
Hence {τ n < T} on A. Also, for ω ∈ A we have

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So for all ω ∈ A, d Δ n , n x < 2 .Since this is true for any small , x ∈ B 1 and any arbitrary Now, define n , Δ n , n ≥ 0 as above and let M n X 0 . ., M d n will be used to construct CPSs for X t .Next, we prove a lemma that shows that all of M i n , 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 Proof.For any n ≥ 0, let μ n be the regular conditional probability of Δ n with respect to G n−1 and let 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 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 :

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 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 for big enough m.Therefore, we can conclude that 2α • converges to 0 as |a m | → ∞, which will imply after passing to the limit that μ n ω, {2α • x > 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 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 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:

2.21
It is easy to check that Z n satisfies

2.22
Let L n n i 1 Z 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

2.23
which shows that L n n≥1 and M n L n n≥1 are martingales under P .We thus get E L n E Z 1 1, and Fatou's lemma gives E L ≤ 1.We will show that E L 1.We have

2.24
Combining Fatou's lemma with the equation

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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 i n is uniformly integrable for each 1/2 we obtain the following:

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 ∈ C 0 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 ∈ C 0 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 > 0 such that

2.29
Since P A > 0 and A q∈I A 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 Let C q * A q * ∩ {sup t∈ q * ,T |X t − X q * − f t − q * | < /3}.Then we claim that which contradicts 2.27 .Indeed, if ω ∈ C q * , then for t ∈ τ, q * we have 32 by the definition of A q * and the choice of δ.We will show also that |X t − X τ − f t − τ | < on C q * whenever t ∈ q * , T : < 3 3 3 .

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 ∈ C 0 0, T , R d .
Proof.We only need to show that linear stickiness implies the weak f-stickiness for each 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 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 B 0 {Z ≥ h} has positive probability.Note that B 0 ∈ F s and B 0 ⊂ A ∩ B. In the following, we show that Let min 0 , h and set t 0 0, and define

2.42
International Journal of Stochastic Analysis n 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 < ∧ d y, ∂O /4 and P G n−1 ∩ {τ n−1 < T } ∩ {X τ n−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 τ n−1 ∈ B ζ y } we have ∧ 3ζ/2 ≤ n < 2ζ.
, 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., 0 .In the absence of such martingale measure for Y , the existence of a process Y t t is a martingale under a measure Q that is equivalent to the original measure P 5is 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.

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
Brownian motions with respective Hurst parameters H 1 , H 2 , . . ., H d ∈ 0, 1 .From 10 , the f-stickiness also holds for any continuous Markov process with the full support property in C 0 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

13
By the definition of B k and the above relation, it is easy see thatB k 1 ⊃ B k ∩ sup t∈ s t k ,s t k 1 |X t − X s t k − α k t − s t k | < 2 N−k 1 .2.43On B k we haved X s t k , ∂O ≥ d X s g s t k − s , ∂O − d X s g s t k − s , X s t k > 2 − 2 N−k 1 ≥ 4 , |X s t k − X s t k α k t − s t k | |α k t − s t k | g t − g t k ≤ 4 , .. ., N − 1 and for all t ∈ s t k , s t k 1 .From this, we conclude thatB k ⊂ {X s t k ∈ t∈ s t k ,s t k 1 O − α k t − s t k }.Now,from the linear stickiness and the fact that P B 0 > 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 measureQ ∼ P such that M i n , G n n≥0 is a uniformly integrable martingale for each 1 ≤ i ≤ d.Let M i ∞ lim n → ∞ M i n .For each t ∈ 0, T , set M i | F t on the set {τ n−1 ≤ t ≤ τ n } for all n ≥ 0.Thus the following equation holds:M i t − X i t 1 {τ n−1 ≤t≤τ n } E Q M i n − X i t 1 {τ n−1 ≤t≤τ n } | F t , n ≥ 1. takes values in − , on the set {τ n−1 ≤ t ≤ τ n }.Therefore, we have −3 ≤ | M i t − X i t | ≤ 3 on the set {τ n−1 ≤ t ≤ τ n }.Since ∞ n 1 {τ n−1 ≤ t ≤ τ n } Ω,we conclude that be a sequence of independent fractional Brownian motions with respective Hurst parameters H 1 , H 2 , . . ., 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 f 1 B H 1 admits CPSs for each > 0. This can be easily seen from the CFS property of the process B H 1 t , B H 2 t , . . ., B H d t which was shown in 14 and the fact that the map f x f 1 x , f 2 x , . . ., f d x is a homomorphism from R d to a 1 , b 1 × a 2 , b 2 × • • • × a n , b n .