Optional processes with non-exploding realized power variation along stopping times are làglàd

We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely laglad paths. This result is useful for the analysis of some imperfect market models in mathematical finance. In the finance applications variation naturally appears along stopping times and not pathwise. On the other hand, if the power variation were only taken along deterministic points in time, the assertion would obviously be wrong.


Introduction
In financial market models with proportional transaction costs and effective friction trading strategies have to be almost surely of finite variation in order to avoid infinite losses (see Campi and Schachermayer [2]). In models with a "large" trader having a smooth impact on the price process of an illiquid stock, as introduced by Bank and Baum [1] and Çetin, Jarrow, and Protter [3], a trading strategy should be of non-exploding quadratic variation. Besides the interest in its own, the result of this note is of use for the analysis of these models. It guarantees that trading strategies possess limits from the left and from the right, i.e. left and right jumps of the process can be defined and used for the analysis, even if one does not start with càglàd strategies (resp. càdlàg, depending on the precise interpretation of a strategy) from the very beginning. Let the real-valued process X model the number of shares of the illiquid stock the trader plans to hold, T 0 = 0, and T 1 ≤ T 2 ≤ . . . the stopping times at which he rebalances his portfolio. Roughly speaking in these models appears some transaction costs term of the order k=1,...,n (X T k − X T k−1 ) 2 . Thus the realized quadratic variation, naturally arising along stopping times and not pathwise, should be non-exploding when passing to a time-continuous limit. In contrast to the total variation, for the quadratic variation the restriction to stopping times is 1 DOI: 10.1214/ECP.v16-1591 crucial. Namely, it is well-known that for any r ∈ + ∪ {+∞} and for almost all paths B · (ω) of a Brownian motion on [0, t] with t > 0 there exist sequences (τ n ) n∈ of grids τ n = (t n 0 , t n 1 , . . . , t n r (see the footnote on page 192 of Lévy [7] and Freedman [5], Proposition 70 and the arguments given on pages 48 and 49). This means that in a pathwise sense the quadratic variation does not exist and is exploding, but of course if grid points are restricted to stopping times the realized quadratic variation converges to t in probability. On the other hand, if the power variation is only taken along deterministic points in time, a non-exploding variation does obviously not imply that paths possess left and right limits, see Example 3.1 for an easy counterexample. The reason for this is that for processes having neither left-nor right-continuous paths arbitrary sequences of grids with vanishing mesh do not always capture the entire variation.

Main part
Throughout the note we fix a terminal time T ∈ + and a complete probability space (Ω, , P) equipped with a filtration ( t ) t∈[0,T ] satisfying the usual conditions. The optional σ-field is the σ-field on Ω × [0, T ] that is generated by all adapted processes with càdlàg paths (considered as mappings on Ω × [0, T ]). A stochastic process that is -measurable is called optional. We have that {ω ∈ Ω | X · (ω) is làglàd} ∈ . If X has non-exploding realized power variation of some order p > 0 (in the sense of Definition 2.1), then P({ω ∈ Ω | X · (ω) is làglàd}) = 1.
The proof of Theorem 2.3 uses a section theorem for optional sets. Example 3.2 shows that the assertion of Theorem 2.3 would not hold under the slightly weaker assumption that X is only progressively measurable instead of optional.
The proof of implication "⇐" shows that the equivalence also holds without introducing a maximal distance δ between two neighboring points, but the lemma is needed in the current form.
Proof of Lemma 2.4. "⇒": Assume that x has no limit from the left at t ∈ (0, T ] or this limit lies in {−∞, ∞}. In both cases there exists a sequence (t n ) n∈ strictly increasing to t s.t. (x(t n )) n∈ is no Cauchy sequence. Thus there exists an M > 0 and a subsequence (t n k ) k∈ s.t. |x(t n k )− x(t n k−1 )| p ≥ M for all k ∈ . As |t − t n k | → 0 we can find for any l ≥ 2 and δ > 0 a k 0 s.t. |t n k − t n k−1 | ≤ δ for k = k 0 + 1, . . . , k 0 + l and the finitely many non-vanishing distances are bounded away from zero. In the case of a missing (finite) limit from the right the argument is the same.
By compactness of [0, T ], the sequence possesses a subsequence such that all components converge to some t * ∈ [0, T ]. Either t 2,n < t * for infinitely many n from the subsequence or t 3,n > t * for infinitely many n from the subsequence. By t 2,n −1/n ≤ t 1,n < t 2,n and |x(t 2,n )− x(t 1,n )| p ≥ M , the former would contradict to the existence of the left limit of x at t * . The latter would contradict to the existence of the right limit of x at t * .
Step 2: Let us now prove (2.6) for arbitrary l ∈ . If l ≥ 5 we have by Step 1 that the variation up to some s j with s j ≤ t 4 + η/2 is at most M 2 −p . It remains to show that the variation on the This implies that the mapping is ( ⊗ l ) − l -measurable and we obtain that By completeness of , the projection of a set in ⊗ l onto Ω is in (see e.g. Theorem 1.32 combined with Theorem 1.36 of He, Wang, and Yan [6]). This means that A m,l,n,k ∈ and thus by (2.8) {ω ∈ Ω | X · (ω) is làglàd} ∈ .
Step 2: Assume that P({ω ∈ Ω | X · (ω) is not làglàd}) > 0 (2.9) and let p > 0. We have to show that X is not of non-exploding realized power variation of order p in the sense of Definition 2.1.
Optional processes with non-exploding realized power variation along stopping times are làglàd5 The idea of the proof is as follows. We want to construct a sequence of grids ( τ i ) i∈ with | T i j − T i j−1 | ≤ 1 i such that on a set of positive probability the power variation of X along τ i exceeds i for all i ∈ . For this we use a section theorem to construct recursively stopping times that constitute an admissible collection in the sense of Definition 2.5 for approximately all paths. By Lemma 2.4 the paths which are not làglàd possess "enough" power variation. It follows from Lemma 2.6 that the variation also appears along the timepoints of an admissible collection which we have constructed with stopping times. Let us now start with the formal proof. By (2.9), there exists an m ∈ such that P l∈ \{1} n∈ k∈ A m,l,n,k =: r > 0. Given i ∈ , choose n = n(i) = 4i and l = l(i) ≥ 2 large enough such that

(by choosing k(i) large enough the latter can be achieved as
For every i ∈ we want to construct stopping times T i 0 ≤ T i 1 ≤ . . .. Assuming that the stopping time T i j−1 is already specified we define S i j and Γ i j are the "random versions" of u j and B j in (2.2) resp. (2.1). Note that X T i j−1 is T i j−1measurable as X is optional (see e.g. Theorem 3.12 of He, Wang, and Yan [6]). Thus S i j is the debut of an optional set and therefore a wide-sense stopping time (see e.g. Theorem 4.30 of [6]). Consequently, S i j + 1/(2k) is a stopping time and Γ i j an optional set. For technical reasons define Let us comment the construction of the set Γ i j and the stopping time T i j . Fixing an ω the ωsection of Γ i j mimics the admissibility condition from Definition 2.5. Namely, if ω ∈ π Ω (Γ i j ), i.e.
there is a s ∈ [0, T ] with (ω, s) ∈ Γ i j , the next point t of the collection has to satisfy (ω, t) ∈ Γ i j . If ω ∈ π Ω (Γ i j ), then we require that t = (T i j−1 (ω)+3/(4i)+1/(2k))∧ T . In addition, Ω×{T } is taken out of the set Γ i j . This is one way to distinguish the case that T i j (ω) = T occurs as the selection of the stopping time from the optional set fails from the case that T i j (ω) = T and T is the only admissible successor of T i j−1 (ω) in the sense of Definition 2.5. (2.10) guarantees that T i j (ω) = T if T is the only admissible successor of T i j−1 (ω) or if we have that already T i j−1 (ω) = T . Putting together, on all paths without any failing selection we obtain admissible collections of timepoints in the sense of Definition 2.5. In the following, this idea is written down mathematically.

Counterexamples
The following easy example shows that the assertion of Theorem 2.3 would be wrong, if the variation were only considered along deterministic points in time. For simplicity define t for all t ∈ [0, T ] as the completion of the sigma-algebra generated by (ξ n ) n∈ .
Then, X is optional as it can be written as X = sup n∈ inf m∈ X n,m with the càdlàg adapted processes X n,m defined by Adaptedness of X n,m holds as already 0 contains all information about ξ n . The variation of X along deterministic times vanishes with probability one as for any fixed t X t vanishes with probability one. On the other hand, for almost all ω (ξ n (ω)) n∈ is dense in [0, T ]. Thus, with probability one the paths of X are not làglàd.
The following example shows that the assertion of Theorem 2.3 would not hold under the slightly weaker assumption that X is only progressively measurable instead of optional. A process X is called progressively measurable if for any t ∈ i.e. for each (ω, t) ∈ A we have that B t (ω) = 0 and there exists an > 0 s.t. B s (ω) = 0 for all s ∈ (t, t + ). The set A is an example due to Dellacherie and Meyer for a progressive set which is not optional (see [4], page 128). The progressively measurable process X := 1 A is not làglàd as for almost all ω there are both infinitely many timepoints s with (ω, s) ∈ A and infinitely many timepoints s with (ω, s) ∈ A in any right neighbourhood of t = 0. On the other hand, the power variation of X does not explode along stopping times in the sense of Definition 2.1 as we have P(X τ = 0) = 1 for any stopping time τ. The latter follows by the strong Markov property of B w.r.t. ( B t ) t∈[0,T ] . The process (B t+τ − B τ ) t≥0 is a standard Brownian motion and stochastically independent of τ which implies that τ cannot be the starting point of an excursion with positive probability.