Consistent Price Systems in Multiasset Markets

International Journal of Stochastic Analysis Volume 2012, Article ID 687376,14pages doi:10.1155/2012/687376

Research Article

Consistent Price Systems in Multiasset Markets

Florian Maris and Hasanjan Sayit

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

Correspondence should be addressed to Hasanjan Sayit,[email protected] Received 27 May 2012; Accepted 9 July 2012

Academic Editor: Lukasz Stettner

Copyrightq2012 F. Maris and H. 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.

LetXtbe any d-dimensional continuous process that takes values in an open connected domain Oin R^d. In this paper, we give equivalent formulations of the conditional full support CFS property ofXtinO. We use them to show that the CFS property of X inOimplies the existence of a martingale M under an equivalent probability measure such that M lies in the >0 neighborhood of Xt for any given under the supremum norm. The existence of such martingales, which are called consistent price systemsCPSs, 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 withdrisky assets and a risk-free asset which is used as a num´eraire and therefore assumed to be equal to one. We assume that the price processes of the d risky assets are given by anR^d-valued process Y_t Y_t¹, Y_t², . . . , Y_t^d, where Y_tⁱ e^Xit, 1 ≤ i ≤ d, and thed-dimensional processXt X¹_t, X_t², . . . , X_t^dis defined on a filtered probability space Ω,F,F Ft_t∈0,T, Pand 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 strategiesθt θ¹_t, θ²_t, . . . , θ_t^dare given by the difference of two processesL_t L¹_t, L²_t, . . . , L^d_tandM_t M¹_t, M²_t, . . . , M^d_trepresenting respectively the cumulative number of shares purchased and sold up to timet, namely,θt Lt−Mt. We are also required to start and end without any position in the risky assets to and this requirement corresponds toθ₀ θ_T0.

For each such trading strategyθt Lt−Mt, the corresponding wealth process, after taking into account the incurred transaction costs, is given by

V_tθ ^d

i1

_t

0

θ_sⁱdY_sⁱ− d

i1

_t

0

Y_sⁱdVar θⁱ

s, 1.1

where Varθⁱ_sLⁱ_sM_sⁱ is the total variation ofθⁱin0, sfor each 1≤i≤dand >0 is the proportion of the transaction costs. In our model1.1, 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 θt θ¹_t, θ²_t, . . . , θ_t^dof finite variation withθ₀θ_T 0 such that the corresponding wealth process V_tθsatisfiesV_tθ≥ −Cfor some deterministicC >0 and for allt∈0, T.

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

Definition 1.2. We say that the market1, Y_t¹, Y_t², . . . , Y_t^ddoes not admit arbitrage with-sized transaction costs if there is no admissible trading strategyθ_t θ¹_t, θ_t², . . . , θ^d_tsuch that the corresponding value processVtθsatisfies

PVTθ>0>0, PVTθ≥0 1. 1.2

The absence of arbitrage condition excludes trading strategies that enables the investors to have nonnegative payoffwith the possibility of positive payoffwith zero initial investment. The purpose of this note is to study the sufficient conditions onY_t¹, Y_t², . . . , Y_t^d that ensure absence of arbitrage in the market1, Y_t¹, Y_t², . . . , Y_t^d. It is clear that if the stock price process Y_t Y_t¹, Y_t², . . . , Y_t^d 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 pricingsee1that states that martingale price processes do not admit arbitrage in frictionless marketsi.e., 0. In the absence of such martingale measure forY, the existence of a process Yt Y_t¹,Y_t², . . . ,Y_t^dwhich is a martingale under an equivalent measureQand which has the following property:

Y_tⁱ−Y_tⁱ≤Y_tⁱ, fori1,2, . . . , d, ∀t∈0, T, 1.3

also implies absence of arbitrage for the model 1.1. To see this simple fact, observe the following:

VTθ ^d

i1

_T

0

θ_sⁱd

Y_sⁱ−Y_s^{i d}

i1

_T

0

θ_sⁱdY_sⁱ− d

i1

_T

0

Y_sⁱdVar θⁱ

s

^d

i1

_T

0

Y_sⁱ−Y_sⁱ dθ_sⁱ−

_T

0

Y_sⁱdVar θⁱ

s ^d

i1

_T

0

θⁱ_sdY_sⁱ.

1.4

Note that because of1.3, we have_d

i1_T

0Y_sⁱ−Y_sⁱdθⁱ_s−_T

0 Y_sⁱdVarθⁱ_s≤0 a.s. This implies that

V_Tθ≤^d

i1

_T

0

θ_sⁱdY_sⁱ. 1.5

The financial interpretation of 1.5is that trading at price process Yt without transaction costs is always at least as profitable as trading at price processY_twith transaction costs. The martingale property ofYtimplies that trading onYtis arbitrage free, and therefore trading on Y with transaction costs is also arbitrage-free.

The processYtis called consistent price systemsCPSsfor the price processY. The origin of CPSs is due to2and the name consistent price system first appeared in3. In the following, we write down the formal definition of CPSs.

Definition 1.3. Let >0. We say thatY_t Y_t¹,Y_t², . . . ,Y_t^dis an-consistent price system for Yt Y_t¹, Y_t², . . . , Y_t^d, if there exists a measureQ∼Psuch thatYtis a martingale underQ, and

1 1 ≤ Y_tⁱ

Y_tⁱ ≤1, fori1,2, . . . , 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 papers4,5studied CPSs for semimartingale models and the papers 6–11 studied CPSs for non-semi-martingale models. Other papers that studied similar problems include4,8,12–17. Particularly, the recent paper10introduced a general condition, conditional full supportCFS, for price processes and showed that if a continuous processX_t X¹_t, X_t², . . . , X_t^dwith state spaceR^dhas the CFS property, then the exponential processY_t Y_t¹, Y_t², . . . , Y_t^dadmits-CPS for any >0. The proof of this result is based on a clever approximation ofY by a discrete process which is called random walk with retirement see10. In this paper, we consider continuous processes X_t with general state spaceO, whereOis any connected open set inR^d. Unlike the original paper10, 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 processXtwith the state space O has the corresponding CFS property, then for any given > 0 there exists a martingale Mt M¹_t, M_t², . . . , M_t^d, under an equivalent change of measure, such that

M_tⁱ−Xⁱ_t≤ for anyi1,2, . . . , dand anyt∈0, T. 1.7 By an abuse of language we call suchMa -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 2.10is a stronger result than the corresponding result in10that 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 isTheorem 2.6which states that the CFS property of X in any open connected domainOimplies the existence of CPSs. To prove this result, we first prove Lemmas2.7,2.8,2.9,2.10, and2.11. InLemma 2.7, we show that the CFS property implies the necessary properties of a random walk with retirementsee10for the formal definition of random walk with retirement. InLemma 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 paper10. 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 papers10,11. InLemma 2.10, we show the equivalence of the f-stickiness with the weakf-stickiness for each givenf. InLemma 2.11, we show that the CFS property is equivalent to the seemingly weaker linear stickiness property.

2. Main Results

LetX_t X¹_t, X²_t, . . . , X_t^d, t∈0, Tbe ad-dimensional continuous process that takes values in an open connected domainO ⊂R^d. For simplicity of our discussion, we assume that 0∈ O.

We also assume that the processXtis defined on a probability spaceΩ,F, Pand adapted to a filtrationF Ft_t∈0,Tthat satisfies the usual assumptions in this space. LetCu, v,O denote the set of continuous functionsf defined on the intervalu, vand with values inO and, for anyx∈R^d, letCxu, v,Odenote the set of functions inCu, v,Owithfu x.

Definition 2.1. An adapted continuous processX_tsatisfies the CFS property inO, if for any t∈0, Tand for almost allω∈Ω,

Supp Law

Xs_s∈t,T| Ftω

CXtωt, 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 domainO ⊂ R^d. We say thatX_tis linear sticky if for anyα ∈R^d, > 0, and any deterministic 0≤s≤θ≤T,

P

sup

t∈s,θ|Xt−X_s−αt−s|<

| Fs

>0, a.s. 2.2

on the set{Xs∈

t∈0,θ−sO −αt}.

The equivalence of the CFS and the linear stickiness properties will be established in Lemmas2.10and2.11. We also need the following definition.

Definition 2.3. Let Xt be an adapted continuous process that takes values in an open and connected domainO ⊂R^d.

aWe say thatXtisf-sticky forf∈C00, T,R^dif

P

sup

t∈τ,T

Xt−Xτ−ft−τ<

| Fτ

>0, a.s. 2.3

on the set{Xτ ∈

t∈0,T−τO −ft}for any >0 and any stopping timeτ. bWe say thatXtis weakf-sticky forf∈C00, T,R^dif

P

sup

t∈s,T

X_t−X_s−ft−s<

| Fs

>0, a.s. 2.4

on the set {Xs ∈

t∈0,T−sO −ft} for any > 0 and any deterministic time s∈0, T.

Remark 2.4. It is clear that the CFS property ofX inOis equivalent to the weakf-stickiness ofX for allf ∈C₀0, T,O. The linear stickiness ofX_tis seemingly weaker condition than the weakf-stickiness of X_t for all f ∈ C₀0, T,O. However, this is not the case and in Lemma 2.11we will show that linear stickiness is equivalent to weakf-stickiness ofXtfor all f∈C00, T,R^d. This, in turn, implies that the linear stickiness property is equivalent to the CFS property.

Remark 2.5. When a processXtis 0-sticky as inbinDefinition 2.3, we say thatXtis jointly sticky and this property was studied in the recent paper14. Thef-stickiness roughly means that starting from any stopping timeτ on, the processX_thas paths that are as close as one wants to the path ft Xτ. As it was shown in 14, the f-stickiness holds for any f ∈ C₀0, T,R^d for the processB^H_t ¹, B^H_t ², . . . , B^H_t ^d, whereB^H_t ¹, B^H_t ², . . . , B^H_t ^d are independent fractional Brownian motions with respective Hurst parametersH1, H2, . . . , Hd∈0,1. From 10, the f-stickiness also holds for any continuous Markov process with the full support property inC₀0, T,R^dfor anyf ∈C₀0, T,R^d.

The following is the main result of this paper. This result is an extension of the main result in10to processes with more general state space. We use10as a road map in the proof of this result.

Theorem 2.6. LetXt X¹_t, X_t², . . . , X_t^dbe a continuous process that takes values in a connected domainOinR^d. IfXtis linear sticky, thenXtadmits CPSs for all >0.

To show this result, one fix any >0 and define the following increasing sequence of stopping times associated with the processX:

τ₀ 0, τ_n1inf

t≥τn{|Xt−X_τn| ≥_n1} ∧T, ∀n≥0, 2.5

with _n1 : ∧dXτn, ∂O/2. One should mention that the paper10 defined the corres- ponding stopping times in a slightly different way, see the proof of Theorem 1.2 in10.

In addition, for eachn≥1 we define

Δn

Xτn−Xτ_n−1 whenτn< T,

0 otherwise. 2.6

LetGnFτn for everyn≥0. Note that_nis bounded andG_n−1measurable.

In the following, we use the notation SuppΔn| Gn−1to denote the smallest closed set ofR^dthat contains the values of the random variableEΔn| Gn−1with probability one. We useB_rxto denote the open ball inR^d with centerxand radiusr. When the center is 0, we simply writeB_r. We first prove the following lemma.

Lemma 2.7. IfXtisf-sticky inOfor allf∈C00, T, then the process{Δn}in2.6satisfies the following three properties:

i PΔm0,∀m≥n|Δn0 1;

ii SuppΔn| G_n−1 0∪∂B_n almost surely on {Δ_n−1/0}, iii PΔm/0,∀m≥1 0.

2.7

Proof. Propertyiis obvious since{Δn 0} {τn T}andτ_n is increasing. Propertyiii follows from the fact that almost surely each path ofXtis contained in a compact set ofO and therefore min_nnω>0 almost surelyω∈Ω. To prove propertyii, let us assume that Pτn−1 < T>0 and letG_n−1be anyGn−1 measurable set such thatPGn−1∩ {τn−1 < T}>0.

Then, it is clear that there existT < T,y ∈ Oandζ > 0 such thatζ < ∧dy, ∂O/4 and PG_n−1 ∩ {τ_n−1 < T} ∩ {Xτn−1 ∈ B_ζy} > 0, wheredy, ∂O is the distance ofy with the boundary∂OofO. It is also clear that on the setG_n−1∩ {τn−1< T} ∩ {Xτn−1 ∈B_ζy}we have ∧3ζ/2≤n<2ζ.

First we show thatPΔn 0 | G_n−1 > 0 a.s. on{Δ_n−1/0}. To see this, define the following stopping time:

τ

τ_n−1 on G_n−1∩ {τ_n−1< T} ∩

X_τn−1∈B_ζ y

,

T otherwise. 2.8

The 0-stickiness ofXimplies that

P

sup

t∈τ,T|Xt−X_τ|< ζ, τ < T

>0. 2.9

But{sup_t∈τ,T|Xt−Xτ|< ζ, τ < T} ⊂ {Δn0}and sinceG_n−1was an arbitraryG_n−1measurable set withPG_n−1∩ {τ_n−1< T}>0, we have

PΔn0| G_n−1>0 a.s. on{Δ_n−1/0}. 2.10

Next we show that∂Bnω ⊂ SuppΔn | Gn−1ωalmost surely on{Δn−1/0}. Too see this, take anyx∈∂B₁, 0< < ζand define

ft

⎧⎪

⎨

⎪⎩ 6ζt

T−Tx if 0≤t≤ T−T 2 ,

3ζx otherwise,

2.11

and note that

P

⎛

⎝τ < T, Xτ ∈

t∈τ,T

O −ft−τ⎞

⎠>0. 2.12

Byf-stickiness ofX, we obtain

P

sup

t∈τ,T

X_t−X_τ−ft−τ< , τ < T

>0, 2.13

or equivalentlyPA∩G_n−1>0, where

A

sup

t∈τn−1,T

X_t−X_τn−1−ft−τ_n−1<

∩

τ_n−1 < T

∩

⎧⎨

⎩X_τn−1 ∈

t∈τn−1,T

O −ft−τ_n−1⎫

⎬

⎭.

2.14

We claim thatA⊂ {Δn⊂B₂nx}. Indeed, ifω∈A, we get Xτ_n−1T−T/2ω−Xτ_n−1ω

≥ f

#T−T 2

$−

X_{τn−1T−T/2}ω−X_τn−1ω−f

#T−T 2

$

≥3ζ−> _nω.

2.15

Hence{τn< T}onA. Also, forω∈Awe have 0dXτn−X_τn−1, ∂B_n≥d

fτn−τ_n−1, ∂Bn

−X_τn−X_τn−1−ft−τ_n−1

>fτn−τ_n−1−nx −,

|Xτn −X_τn−1−_nx| ≤fτn−τ_n−1−_nxX_τn−X_τn−1−ft−τ_n−1

<fτn−τ_n−1−nx <2.

2.16

So for allω∈A,dΔn, nx<2. Since this is true for any small,x∈B1and any arbitrary G_n−1 measurable set withPG_n−1∩ {τ_n−1 < T} > 0, we conclude that∂B_nω ⊂ SuppΔn | Gn−1ωalmost surely on{Δn−1/0}. Note that the other direction∂B_nω∪ {0} ⊃SuppΔn| Gn−1ωis clear from the definition ofΔn.

Now, definen,Δn,n ≥0 as above and letMn X0_n

i1Δi,n ≥0. TheR^d-valued process M_n : M¹_n, M²_n, . . . , M^d_nwill be used to construct CPSs for X_t. Next, we prove a lemma that shows that all ofM_nⁱ, 1≤i≤dare in fact uniformly integrable martingales under an equivalent change of measure. The proof of this lemma uses Lemma 3.1 of10as a road mapsee also Proposition 2.2.14 of18.

Lemma 2.8. There exists a measureQequivalent toP under which theR^d-valued discrete process {Mn,Gn}^∞_n0is a martingale.

Proof. For anyn≥0, letμ_nbe the regular conditional probability ofΔnwith respect toG_n−1 and let Ωn {ω ∈ Ω | SuppΔn | Gn−1ω 0 ∪∂B_n}. Let k be any strictly increasing convex function defined onRwith values in0,∞such thatkt tfor everyt≥1. Define G_n:Ωn×R^d → R^das follows:

Gnω, α

R^dkα·xxdμnω,·. 2.17

Obviously for each n, Gn·, a is Gn−1 measurable and convex with respect to α. As a consequence, for any fixedω∈Ωn, ImGnω,·, the image of the functionGnω,·is convex.

We first prove that for everyn≥1 andω∈Ωn:

|α| → ∞lim Gnω, α· α

|α| ∞. 2.18

By the way of contrary, assume that this is not true, for somen≥1 andω ∈Ωn. Then, there exists a sequenceαm_m≥1with|αm| → ∞such thatGnω, αm·αm/|αm|is bounded above.

We can assume thatαm/|αm|converges to someαthis is a bounded sequence and therefore has a convergent subsequence. We have

G_nω, αm· α_m

|αm|≥

αm·x≤0kαm·xα_m·x

|αm| dμ_nω,·

αm·x/|αm|>nω/4kαm·xαm·x

|αm| dμnω,·

≥ −k0

2α·x>nω

αm·x²

|αm| dμnω,·,

2.19

for big enoughm. Therefore, we can conclude that

2α·x>nωαm·x²/|αm|·1/|am|dμnω,· converges to 0 as|am| → ∞, which will imply after passing to the limit that μnω,{2α· x > n} and this is a contradiction. From this it follows easily that 0 ∈ ImGnω,·. If 0 /∈ ImGnω,·, then using the geometric form of Hahn Banach theorem, there exists

a unit vector β ∈ R^d such that

R^dkα·xβxdμnω,· < 0 for every α ∈ R^d. Therefore, lim sup_{t→ ∞}

R^dktβ·xβxdμnω,·≤0. But

R^dk tβ·x

βxdμ_nω,· G_n ω, tβ

· tβ

tβ, 2.20 and so it contradicts2.18.

Next, we want to show that ImGnω,·is closed. Leta∈ ImGnω,·, so there exists a sequenceαm_m≥1such thatGnω, αm → a. But then|αm|is unbounded, and therefore this contradicts2.18. So based on the continuity ofG_nω,·,a∈ImGnω,·.

Therefore, we conclude that for anyn ≥ 1 andω ∈ Ωn, there exists anα_nω ∈ R^d, unique, as a consequence of the strict monotonicity ofk, such that Gnω, αnω 0. Gn

being continuous with respect toαandG_n−1measurable with respect toω, it follows thatα_n isGn−1measurable. We extendα_nwith 1 outsideΩnand define:

Z_n kαn·Δn1_{Δn/0}

2E

kαn·Δn1{Δn/0}| Gn−1 1_{Δn0}

2PΔn0| G_n−1. 2.21 It is easy to check thatZnsatisfies

EZn| G_n−1 1,

EZnΔn| Gn−1 0. 2.22

LetL_n%_n

i1Z_iandLlim_{n→ ∞}L_n. Note that this limit exists almost surely sinceL_n1L_n a.s. on{Δn0}and{Δn0} Ω. From2.22, we get

ELn| Gn−1 L_n−1

ELnMn| G_n−1 L_n−1M_n−1, 2.23

which shows that Ln_n≥1 and MnL_nn≥1 are martingales under P. We thus get ELn

EZ1 1, and Fatou’s lemma givesEL≤1. We will show thatEL 1. We have

EL E

#

nlim→ ∞L1_{Δn0}

$ lim

n→ ∞E

L1_{Δn0}

lim

n→ ∞E

Ln1_{Δn0}

1− lim

n→ ∞E

L_n1_{Δn/0}

1− lim

n→ ∞E E

L_n1_{Δn/0}| G_n−1 1−1

2 lim

n→ ∞E

L_n−11_{Δn−1/0}

1− lim

n→ ∞

#1 2

$_n 1.

2.24

Combining Fatou’s lemma with the equationELn EL 1, we obtainEL | Gn Ln. Also,

EMnL| G_n−1 EEMnL| Gn| G_n−1 EMnL| G_n−1

M_n−1L_n−1EMn−1L| Gn−1. 2.25

Hence,Lis the density of a measureQunder which our discrete processMnis a martingale.

And sinceL >0Ln>0 for all n,Qis equivalent toP.

Lemma 2.9. Under the measureQ ofLemma 2.8 the processMⁱ_n is uniformly integrable for each 1≤i≤d. In particular,EQsup_n≥0|Mⁱ_n|<∞for eachi1,2, . . . , d.

Proof. For any 1≤i≤d, setMⁱ_∗sup_n≥0|Mⁱ_n|and observe that on{Δk/0,Δ_k1 0}we have Mⁱ_∗≤ |X₀ⁱ|k. Observing thatQΔk/0 QΔk/0|Δk−1/0· · ·QΔ1/0|Δ0/0QΔ0/0 and thatQΔk/0|Δ_k−1/0 1/2 we obtain the following:

E_Q M_∗ⁱ

^∞

k0

E_Q

Mⁱ_∗1_{{Δk/0}∩{Δk10}}

≤^∞

k0

Xⁱ₀k

Q{Δk/0,Δk10}<∞.

2.26

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

Lemma 2.10. LetX_tbe an adapted continuous process with state spaceOandf ∈ C₀0, T,R^d. Then,Xtis weakf-sticky if and only if it isf-sticky.

Proof. Let us show first that for anyf ∈C₀0, T,R^dweakf-stickiness impliesf-stickiness.

Suppose for a contradiction that X_t is weakf-sticky but not f-sticky. Then there exists a stopping timeτwithPτ < T>0, and an >0 such that

P

⎛

⎝τ < T, X_τ ∈

t∈0,T−τ

O −ft⎞

⎠>0,

P

sup

t∈τ,T

Xt−Xτ−ft−τ< , τ < T

0.

2.27

Sincef ∈C00, T,R^d, there exists aδ > 0 such that for allt,s ∈0, T,|t−s|< δimplies

|ft−fs|< /3. In addition, we can findt1,t2 ∈0, T, 0< t2−t1< δ, and 0< ζ≤/3 such that

P

⎛

⎝t1≤τ < t2, Xτ∈

t∈τ,T

Oζ−ft−τ⎞

⎠>0, 2.28

whereOζ{x∈ O/dx, ∂O> ζ}.

For eachq∈I Q∩t1, t₂, letA_q:A∩ {t1≤τ < q} ∩ {sup_t∈τ,q|Xt−X_τ|< ζ}, where

A:

⎧⎨

⎩t₁≤τ < t₂, X_τ ∈

t∈τ,T

Oζ−ft−τ⎫

⎬

⎭. 2.29

SincePA>0 andA&

q∈IAq, there exists aq^∗∈Isuch thatPAq^∗>0. Note thatAq^∗∈ Fq^∗

andA_q∗⊂

t∈0,T−q^∗O −ft. Hence, sinceX_tis weakf-sticky, we obtain

P

⎛

⎝A_q^∗∩

⎧⎨

⎩ sup

t∈^q∗,T

X_t−X_q^∗−f

t−q^∗<

3

⎫⎬

⎭

⎞

⎠>0. 2.30

LetCq^∗Aq^∗∩ {sup_t∈q∗,T|Xt−Xq^∗−ft−q^∗|< /3}. Then we claim that

Cq^∗⊂

sup

t∈τ,T

Xt−Xτ−ft−τ<

∩ {τ < T}, 2.31

which contradicts2.27. Indeed, ifω∈Cq^∗, then fort∈τ, q^∗we have Xt−Xτ−ft−τ<|Xt−Xτ|ft−τ<

3

3 < , 2.32

by the definition ofA_q^∗and the choice ofδ. We will show also that|Xt−X_τ−ft−τ|< on Cq^∗whenevert∈q^∗, T:

Xt−Xτ−ft−τ≤Xq^∗−Xτ−ft−τ f

t−q^∗Xt−Xq^∗−f t−q^∗

≤X_τ−X_q^∗f t−q^∗

−ft−τX_t−X_q^∗−f t−q^∗

<

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. LetX_tbe a continuous adapted process with state spaceO. ThenX_tis linear sticky if and only ifXtisf-sticky for allf∈C00, T,R^d.

Proof. We only need to show that linear stickiness implies the weak f-stickiness for each f∈C₀0, T,R^d. Fix anyf∈C₀0, T,R^d, s∈0, T,₀>0. We need to show that

P

sup

t∈s,T

Xt−Xs−ft−s< 0

| Fs

>0, a.s. 2.34

on the setB:{Xs ∈

t∈0,T−sO −ft}. To do this, for anyA∈ FswithPA∩B>0, we need to show that

P

A∩B∩

sup

t∈s,T

X_t−X_s−ft−s< ₀

>0. 2.35

DefineZω inf_r∈0,T−sdXsω fr, ∂Ofor anyω ∈A∩B. From the definition ofB, it is clear thatZ >0 a.s. onA∩B. Leth > 0 be a constant such that the setB₀ {Z ≥ h}has positive probability. Note thatB₀∈ FsandB₀⊂A∩B. In the following, we show that

P

B₀∩

sup

t∈s,T

X_t−X_s−ft−s< ₀

>0. 2.36

Letmin0, hand sett₀0, and define t_kinf'

t≥t_k−1:ft−ftk−1≥ 4

(∧T−s, 2.37

fork ≥1. LetNbe the smallest positive integer such thatt_N T−s. For eachk ≥1, define gtontk−1, t_k to be equal to the linear function that connects the two pointsftk−1and ftk. We can assume that

gt ftk−1 α_k−1t, ontk−1, tk, 2.38

for some constant vectorα_k−1∈R^dfor eachk≥1. It is clear that sup

t∈0,T−s

ft−gt≤

2. 2.39

Because of2.39, to show2.36we only need to show

P

B₀∩

sup

t∈s,T

X_t−X_s−gt−s<

2

>0. 2.40

For eachk0,1,2, . . . , N−1, let B_k1B_k∩

sup

t∈stk,stk1

X_t−X_s−gt−s<

2^N−k

. 2.41

Note thatB_N is contained in the event in2.40. Therefore, it is sufficient to prove thatB_N has positive probability. Whent ∈ st_k, st_k1, we have gt−s ftk α_kt−s ftk αktkαkt−stk gstk−s αkt−stk. Therefore, we have the following relation:

sup

t∈stk,stk1

X_t−X_s−gt−s<

2^N−k

⊃ )

X_stk −Xs−gstk−s<

2^N−k1

*

∩

sup

t∈stk,stk1|Xt−X_stk−α_kt−st_k|<

2^N−k1

.

2.42

By the definition ofBkand the above relation, it is easy see that

B_k1⊃Bk∩

sup

t∈stk,stk1|Xt−X_stk−αkt−stk|<

2^N−k1

. 2.43

OnB_kwe have

dXstk, ∂O≥d

Xsgstk−s, ∂O

−d

Xsgstk−s, Xstk

>

2 −

2^N−k1 ≥ 4,

|X_stk−X_stkα_kt−st_k||αkt−st_k|gt−gtk≤ 4,

2.44

for each k 0,1, . . . , N −1 and for allt ∈ st_k, s t_k1. From this, we conclude that Bk ⊂ {Xstk ∈

t∈stk,stk1O −αkt−stk}. Now, from the linear stickiness and the fact thatPB0>0 we concludePBN>0. This completes the proof.

Proof ofTheorem 2.6. By Lemmas2.8and2.9, there exists an equivalent probability measure Q ∼ P such that M_nⁱ,Gn_n≥0 is a uniformly integrable martingale for each 1 ≤ i ≤ d. Let Mⁱ_∞lim_{n→ ∞}Mⁱ_n. For eacht∈0, T, setM+ⁱ_tE_QM_∞ⁱ | Ft. Observe thatM+ⁱ_τn E_QMⁱ_∞| Fτn Mⁱ_n, andM+ⁱ_tEQM+_τⁱ_n | Fton the set{τ_n−1≤t≤τn}for alln≥0. Thus the following equation holds:

M+ⁱ_t−X_tⁱ

1_{{τn−1≤t≤τn}}EQ

Mⁱ_n−Xⁱ_t

1_{{τn−1≤t≤τn}}| Ft

, n≥1. 2.45

We writeMⁱ_n−Xⁱ_t Mⁱ_n−Xⁱ_τn X_τⁱ_n−1 −X_tⁱ Xⁱ_τn −Xⁱ_τn−1. Note that each ofM_nⁱ −X_τⁱ_n, X_τⁱ_n−1−X_tⁱ, andX_τⁱ_n−Xⁱ_τn−1takes values in−, on the set{τn−1≤t≤τ_n}. Therefore, we have

−3≤ |M+ⁱ_t−X_tⁱ| ≤3on the set{τ_n−1 ≤ t≤τ_n}. Since&_∞

n1{τ_n−1 ≤t ≤τ_n} Ω, we conclude that

−3≤ +Mⁱ_t−Xⁱ_t≤3. 2.46

Since >0 is arbitrary, the claim follows.

Example 2.12. Let B_t^H1, B_t^H2, . . . , B_t^Hd be a sequence of independent fractional Brownian motions with respective Hurst parametersH₁, H₂, . . . , H_d ∈ 0,1. Letf_i : R → ai, b_ibe a homeomorphism for eachi 1,2, . . . , d, whereai, b_iis an open interval in the real line.

Then the new processf1B_t^H1, f2B^H_t ², . . . , fdB^H_t ^dadmits CPSs for each > 0. This can be easily seen from the CFS property of the processB^H_t ¹, B^H_t ², . . . , B^H_t ^dwhich was shown in 14and the fact that the mapfx f1x, f2x, . . . , fdxis a homomorphism fromR^dto a1, b₁×a2, b₂× · · · ×an, b_n.

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