On the Existence of Consistent Price Systems

In markets with proportional transaction costs, a consistent price system (CPS) plays the role of a martingale measure in both hedging and absence of arbitrage problems, as highlighted by the recent results of Guasoni, Rásonyi, and Schachermayer (see [3,Theorem 1.3] and [4,Theorem 1.11]). Therefore it is crucial to study the existence of CPSs. Recall that a strictly positive adapted stochastic process (Y t ) t∈[0,T ] defined on a filtered probability space (Ω, F, F = (F t ) t∈[0,T ] , P ) that satisfies the usual conditions (i.e., the filtration F is right continuous, and F 0 contains all of the P null sets of F) admits an -CPS for > 0 if there exists an equivalent probability measure P ∼ P and a (F, P )-martingale ( Ỹt ) t∈[0,T ] such that

Originally, the concept of CPS is due to Jouini and Kallal [5]. See [8] for further details.

In [3], Guasoni, Rásonyi, and Schachermayer introduced an important condition, conditional full support (CFS), for continuous stochastic processes and showed that CFS implies the existence of CPSs. (See equation (11), below, for the definition of CFS.) They proved that fractional Brownian motion (fBm) and certain continuous Markov processes possess the CFS property. Motivated by this result, in the subsequent papers [1,2,6,7] several other processes were shown to possess the CFS property.

In Section 2 of this note, we give weaker sufficient conditions for the existence of CPSs. As an application of these results, in Section 3, we study the existence of the CPSs for transformed processes of the form e f (X) , where f : R → R is a continuous function and X is a continuous process with CFS. Moreover, based on these results, we construct examples of processes f (X) that do not have CFS, and yet admit a CPSs.

Let us first recall the definition of random walk with retirement, introduced in [3]. To this end, let (Ω, G, G = (G n ) n≥0 , P ) be a discrete-time filtered probability space such that G 0 = {∅, Ω} and

where > 0 and (R n ) n≥1 is a G-adapted process in {-1, 0, 1} with the following properties:

Any random walk with retirement (Z n ) n≥0 admits an equivalent probability measure Q ∼ P , under which it is a uniformly integrable martingale [3,Lemma 2.6]. This fact will be used in our argument, below.

To state our main results, let (X t ) t∈[0,T ] be a continuous process adapted to the filtration F.

Moreover, for any h ∈ (0, T ), δ > 0, c > 0, and any stopping time τ with values in [0, T -h), let

The event F 0 (τ, h, δ, c) is indeed independent of h and c, but we add these arguments for consistency with F -1 (τ, h, δ, c) and F 1 (τ, h, δ, c). Roughly speaking, these three events correspond to X staying in a tube, moving down, and moving up, respectively, after the stopping time τ -see Figure 1 for an illustration.

Theorem 1. Let (X t ) t∈[0,T ] be a continuous process adapted to filtration F. If there exists 0 > 0 such that for any h ∈ (0, T ) and stopping time τ with values in [0, T -h), and j ∈ {-1, 0, 1},

(2) P F j X (τ, h, log(1 + 0 ), log(1 + 0 )) F τ > 0 a.s., then (Y t ) t∈[0,T ] := (e Xt ) t∈[0,T ] admits an -CPS with = (1 + 0 ) 3 -1.

The events F 0 X (τ, h, δ, c), F 1 X (τ, h, δ, c), and F -1 X (τ, h, δ, c) in (1).

Proof. As in proof of Theorem 1.2 of [3], we set up a CPS for Y using a random walk with retirement associated with Y . We divide the proof into three steps.

Step 1. Define

and

Step 2. We will check that Z satisfies the conditions of a random walk with retirement on the filtered probability space (Ω, G, (G n ) n≥0 , P ), with G = ∨ n≥0 G n . To show this, we need to check (R1)-(R3) in Definition 1. Clearly, condition (R1) is satisfied, and (R3) is a consequence of the continuity of X. Therefore, we only need to check that

for all n ≥ 1. This is equivalent to showing that for any A ∈ F τ n-1 with

and P (A) > 0,

. By the assumption of the theorem, we have

with P (B) > 0, and therefore, the events

have positive probability, which, in turn, implies P ({R n = j} ∩ B) > 0 for any j ∈ {-1, 0, 1}. Since B ⊂ A, the result follows.

Step 3. Since (Z n ) is a random walk with retirement, thanks to Lemma 2.6 of [3], there exists an equivalent probability measure

Yt , and

Therefore Zt is an -CPS for Y t , with = (1 + 0 ) 3 -1.

Remark 1. If X is adapted to a sub-filtration F of F and (2) holds with respect to F for 0 > 0, then it also holds with respect to the smaller filtration F for 0 .

The condition (2) in Theorem 1 needs, of course, to be checked for a very wide class of stopping times. Depending on the process X, direct verification of (2) might be a difficult task. To overcome this difficulty, we establish the following variant of Theorem 1 with a sufficient condition that involves only deterministic times. Theorem 2. Let (X t ) t∈[0,T ] be a continuous process adapted to filtration F. If there exists γ > 0 such that for any h ∈ (0, T ), t ∈ [0, T -h), δ ∈ (0, γ), c ∈ (0, γ), and j ∈ {-1, 0, 1}, (9) P F j X (t, h, δ, c) F t > 0 a.s., then (Y t ) t

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