Integral and Local Limit Theorems for Level Crossings of Diffusions and the Skorohod Problem

J o u r n a l o f P r o b a b i l i t y Electron. Abstract Using a new technique, based on the regularization of a càdlàg process via the double Skorohod map, we obtain limit theorems for integrated numbers of level crossings of diffusions. The results are related to the recent results on the limit theorems for the truncated variation. We also extend to diffusions the classical result of Kasahara on the " local" limit theorem for the number of crossings of a Wiener process. We establish the correspondence between the truncated variation and the double Skoro-hod map. Additionally, we prove some auxiliary formulas for the Skorohod map with time-dependent boundaries.


Introduction
Let X = (X t , t ≥ 0) be a continuous semimartingale adapted to the filtration F = (F t , t ≥ 0) on a probability space (Ω, F, P) such that the usual conditions hold.The purpose of this study is to establish a connection between the level crossings of X, the local time of X, quadratic variation X of X and its truncated variation, denoted by T V c , defined for c > 0 and T > 0 by the following formula The concept of truncated variation of a stochastic process has been recently introduced by Łochowski in [13,14] and proved relevant for interpreting maximal returns from trading in transaction costs problems.The difference with the total variation is that the truncated variation considers only jumps greater than some constant level c and is always finite for any càdlàg process X.
We will call the result given by (1.4) integral limit theorem.The "functional" version of (1.4) is the following.Let N a (Y, T ) be the number of times that Y crosses (from above or from below -for the precise definition of N a (Y, T ) see Section 3 and Subsection 4.4) the level a on the interval [0; T ].For any twice differentiable function f : R → R such that f is continuous we have where X c,x is the (already mentioned) regularization of X.Moreover, we have a more direct result corresponding to (1.5) which may be expressed in terms of interval crossings by X : (1.6) These results shall be compared with the main result of [18] where it was shown (for a slightly more general family of processes) that for X ε being a smoothed version of X, i.e.
where ψ is a smooth (C ∞ ) kernel with compact support [−1; 1] , one has as ε ↓ 0. k ψ and c ψ are here positive constants depending only on ψ.Notice that the smoothed version X ε approximates X on average with accuracy √ ε, i.e.
and in the view of (1.3), (1.5) and (1.7) give the same order of convergence.Other problems of the same type for level crossings by Gaussian processes are an intensive field of study and a good survey of the results obtained so far is [9].It is worth mentioning that besides the number of interval crossings we consider interval downcrossings and interval upcrossings.For d a c , u a c being the numbers of relevant interval downcrossings and upcrossings respectively by the process X we establish a joint convergence result for quadruples X, and obtain an interesting result (cf.Theorem 4.12 and Theorem 4.11) that e.g.With a weaker condition on Xthat it is a continuous semimartingale and there exists a probability measure Q under which X is a local martingale and P is absolutely continuous with respect to Q, we will obtain a "local" counterpart of (1.4), (1.6).Namely, let n c (X, T ) be the number of times that the reflected process |X| crosses down from c to 0 by time T (this is the same as the number of times that X crosses down from c to 0 and crosses up from −c to 0), then where B is a standard Brownian motion, independent from X, and L is the local time of X at 0. This result is a direct generalisation of the main result of [8], where the same statement was proven for X being a standard Brownian motion.It may be viewed as the "local" counterpart of (1.6) since, by the occupation times formula, T da.Notice that the integrated process R f (a)n a c (X, •) da reveals much stronger concentration than the process n c (X, •) (where the multiplication by √ c is needed for convergence).Again, the result will identify the limit for the whole quadruple Remark 1.3.As far as we know, there is no "local" counterpart of (1.7) in the same sense as the generalisation of Kasahara's result, (1.8), is the local counterpart of (1.4), (1.6).
Let us comment on the organisation of the paper.In the next section we summarize the main results and properties of the truncated variation processes and construct the regularization, X c,x , of the process X via the double Skorohod map on [−c/2; c/2].To prove that this regularization satisfies relevant conditions we will need to establish some additional formulas which are (as far as we know) not available in the literature.Thus we will present the solution of the Skorohod problem in a setting suited to our purposes.Next, in Section 3, we establish an important correspondence between the number of interval crossings by the process X and the number of level crossings by the process X c,x .Finally, in the last section we prove convergence results.

On the truncated variation and the regularization of the process X via Skorohod's map
In this section, first we summarize the main results and properties of the truncated variation processes obtained by Łochowski in [13,14].We will assume that X = (X t , t ≥ 0) is a càdlàg process adapted to the filtration F = (F t , t ≥ 0) on the probability space (Ω, F, P) such that the usual conditions hold.The truncated variation, given by formula (1.1) is a lower bound for the total variation T V (Y, T ) = T V 0 (Y, T ) of every process Y, uniformly approximating the process X with accuracy c/2, This follows immediately from the fact that X − Y ∞ ≤ c/2 implies for any 0 ≤ s < t the inequality Remark 2.1.Notice that the truncated variation, unlike the total variation T V (X, T ), is always finite.This follows from the fact that every càdlàg function may be uniformly approximated with arbitrary accuracy by step functions, which have finite total variation.
Together with truncated variation, we consider upward and downward truncated variations, defined for c > 0 and T > 0 by the formulas where U T V = U T V 0 , DT V = DT V 0 , are called positive and negative total variations respectively (cf.[3, pages 322-323]), and this follows from inequalities Remark 2.2.We will not need this result in the sequel but it is possible to prove that in fact (cf.[15]): which means that the lower bound (2.1) is indeed the greatest lower bound.Moreover, and there exists a càdlàg process X c with X − X c ∞ ≤ c/2 for which but it may be not adapted to F (see [15,Theorem 4.1 and formula (3.2)]).
In the sequel, for every F 0 measurable random variable x ∈ [−c/2; c/2] we will construct an adapted process X c,x , "c/2"-uniform approximation of X with locally finite variation such that its total variation does not exceed the lower bound (2.1) by c.More precisely, it will satisfy the following conditions (C) X c,x is of finite variation with càdlàg paths and is adapted to the filtration F; (D) for any s ≥ 0, the jumps (if any) at time s of X c,x and X satisfy which follows directly from the equality On the other hand, assuming that the process X c,x is constructed, U T V (X c,x , •) and DT V (X c,x , •) give the Jordan decomposition of X c,x and we have From this and conditions (E), (2.2) and (2.3) we have To construct the appropriate process X c,x we will need a slight generalisation of the double Skorohod map on the interval [−c/2; c/2] (cf.[11]) as well as some alternative formulas for it.Since the construction of this map for time-dependent boundaries (cf.[2]) is almost the same as for constant boundaries, we will present it in the timedependent setting.

The Skorohod problem with time-dependent boundaries and with starting condition
Let D[0; +∞) denote the set of real-valued càdlàg functions.Let also BV + [0; +∞), BV [0; +∞) denote subspaces of D[0; +∞) consisting of nondecreasing functions and functions of bounded variation, respectively.We have The usual Skorohod problem is defined with similar conditions (a) and (b), for some càdlàg functions φ and η, as the Skorohod problem just defined with starting condition.The difference is such that in the former instead of condition (c) it is assumed η(0−) = 0, which determines the starting value of the function φ, φ(0), to equal max {α(0), min {ψ(0), β(0)}} .starting condition is the same as what so called the play operator, encountered in mathematical models of hysteresis (cf.[4]).
The existence and uniqueness of the solution of the Skorohod problem with timedependent boundaries and starting condition, for α, β and x such that and x ∈ [α(0); β(0)], follows easily from already known results (see the proof of Proposition 2.7).However, in the sequel we will need some formulas for the solution of this problem as well as some additional properties which are (as far as we know) not available in the literature.This is why we will present the solution of the problem in the setting suited to our purposes.
Proof.It is easy to see that the solution of our Skorohod problem with starting condition coincides with the solution of the usual Skorohod problem with time-dependent boundaries for the function ψ x = ψ − ψ (0) + x.
In order to prove inequality (2.5) let us notice that from formula (2.4) it follows that for any s / ∈ {T u,k ; T d,k } , k = 0, 1, ..., (2.5) holds, hence let us assume that s ∈ {T u,k ; T d,k } .
We consider three possibilities.
• If s = T u,k+1 , k = 0, 1, ..., then (as already mentioned) −∆η x = ψ x (s) − ψ x (s−) ≥ 0 and, by the definition of T u,k+1 , and, by the definition of T d,k , Remark 2.8.It is possible to prove that the function ψ x has the smallest total variation on the intervals [0 This observation, for constant, symmetric boundaries was proved in [10, Chapt.II, Corollary 1.5] and in full generality in [6, Proposition 6 and Theorem 8], but we will not need this in the sequel.
Proof.By Proposition 2.7 we immediately get that X c,x satisfies conditions (A)-(D).To prove that it satisfies conditions (E) and (F) we assume (without loss of generality) that T u ≤ T d and consider four possibilities.

Now, by the definition of
) , for some k = 0, 1, 2, ... The proof follows similarly as in the previous case.

Interval down-and upcrossings of the process X and level crossings by the regularization X c,x
Now for a càdlàg process X t , t ≥ 0, (not necessarily starting at 0) and c > 0 let us consider the number of downcrossings of X from above the level c to the level 0 before time T. We define it in the following way Definition 3.1.For c > 0 set σ c 0 = 0 and for n = 0, 1, ...
The number of downcrossings of X from above the level c to the level 0 before time T is defined as We will prove that it is almost the same as the number of crossings the level c/2 from above on the interval [0; T ] by the regularization X c,x .Here, for a càdlàg process Y t , t ≥ 0, we define the number of crossings the level c/2 from above on the interval [0; T ] in the following way.Definition 3.2.For c > 0 let u c 0 = 0 and for n = 0, 1, ... we set We define the number of crossings the level c/2 from above on the interval [0; T ] by Y as e c (Y, T ) = max {n : u c n ≤ T } .

Now we have
Lemma 3.3.Let X t , t ≥ 0, be a càdlàg process adapted to the filtration F = (F t , t ≥ 0) on the probability space (Ω, F, P) .For c > 0 and a F 0 -measurable random variable x ∈ [−c/2; c/2] consider the regularization X c,x t , t ≥ 0, defined in Proposition 2.9.For any T > 0 we have Moreover, if x ≡ c/2 we get exact equality, i.e.
Proof.We will use the stopping times introduced in Definition 3.1 and Definition 3.2.To ∞ ≤ c/2 and by X σ c n ≤ 0, we get X c,x σ c n ≤ c/2.Thus, on the interval τ c n−1 ; σ c n we have at least one crossing the level c/2 from above by X c,x and we obtain d c (X, T ) ≤ e c (X c,x , T ) .
To prove the upper bound we notice that the process X c,x does not change its value as long as . By the construction of X c,x , for every ω the interval [0; T ] may be split into a finite sum of disjoint intervals, such that on each of them X c,x (ω) is monotonic.Thus e c (X c,x , T ) is a.s.finite.If e c (X c,x , T ) ∈ {0, 1} the inequality e c (X c,x , T ) ≤ d c (X, T ) + 1 is obvious.Hence assume that e c (X c,x , T ) ≥ 2 and for some n = 2, 3, ...
have at least one downcrossing of X from above the level c to the level 0 before time T and we obtain d c (X, T ) ≥ e c (X c,x , T ) − 1.
To prove the exact equality d c (X, T ) = e c (X c,x , T ) when x ≡ c/2 it is enough to see that if e c (X c,x , T ) ≥ 1 and we consider such Similarly we consider upcrossings from below the level −c to the level 0, u c (X, T ) , and crossings the level −c/2 from below, g c (Y, T ) .Note, that their numbers may be easily calculated as numbers of downcrossings or crossings from above, respectively, of the processes −X, −Y.Naturally, we have EJP 19 (2014), paper 10.Lemma 3.4.Let c, x, X, and X c,x be as in Lemma 3.3.For any T > 0 we have Moreover, if x ≡ −c/2 we get exact equality, i.e. u c (X, T ) = g c (X c,x , T ) .
In this subsection we will assume that X t , t ≥ 0, is a continuous semimartingale (not necessarily starting from 0).Notice that for T > 0 and any a ∈ R, u c (X − a − c, T ) is equal to the number of times that X upcrosses from below the level a to the level a + c before time T. Assume moreover that the bicontinuous version of the local time L of X exists.By [12, page 18, Theorem II.2.4] we have that for 0 uniformly in t and a ∈ R as c ↓ 0.
as c ↓ 0, in C ([0; T ] , R).Now let us consider X c,−c/2 , i.e. regularization of X defined in Proposition 2.9 with x ≡ −c/2.Notice that by condition (D), X c,−c/2 is also continuous and that for T > 0 and any a ∈ R, g c X c,−c/2 − a − c/2, T is equal to the number of times that X c,−c/2 crosses the level a from below on the interval [0; T ].By the extended version of the Banach-Vitali Indicatrix Theorem (cf.[3, page 328], see also Remark 4.10) for t > 0 we have c for any continuous semimartingale X (the second estimate on page 20 in [12]).Thus, applying [12, Theorem III.3.3(a)] and the Lebesgue dominated convergence we get that (4.2) and hence Theorem 4.2 hold for any continuous semimartingale X.

The local limit theorem -generalisation of Kasahara's result on CLT for number of interval crossings
Let T > 0 be given and fixed.In this subsection we will work with a continuous semimartingale X satisfying the following conditions.
(i) There exists a probability measure Q, under which X is a local martingale; (ii) the measure P is absolutely continuous with respect to Q.
Theorem 4.5.Fix T > 0. Let X be as above and d c (X, t) and u c (X, t) , t ∈ [0; T ] , be numbers of times that the process X downcrosses from above and upcrosses from EJP 19 (2014), paper 10.
below the intervals [0; c] and [−c; 0] till time t respectively.Moreover, let L be the local time of X at 0. We have that where B 1 , B 2 , are independent standard Brownian motions, which are also independent from X. γ l should be replaced by γ l/2 .
The immediate consequence of the obtained result is the following generalisation of the main result of [8].
Corollary 4.7.Let n c (X, t) = d c (X, t) + u c (X, t) be the number of downcrossings the interval [0; c] by the process |X| till time t.Then where B is a standard Brownian motion, independent from X.
and because X does not vanish on [τ n ; σ n+1 ) , L σn+1∧t = L τn∧t .As a result and similarly where η and η are two predictable processes and random variables r 1 (c, t), r 2 (c, t) belong to the interval [0; c] .
We define With this notation we have By the usual localization argument, we may assume that there exists some M > 0 such that Q a.s.sup t∈[0;T ] |X t | < M.
For 0 ≤ s < t, by the Burkholder inequality, the definition of K c and occupation times formula, for some constant A 1 Now, combining (4.11) and (4.12), for Similarly, (X u − X s ) .
By (4.13)-(4.14)and Chebyschev's inequality, with the aid of Aldous' criterion we have that the family of measures Q c , c ∈ (0; 1) , is weakly relatively compact (cf.[7, Theorem VI.4.5] and notice that by the Lebesgue dominated convergence theorem and continuity of X we have lim Let Q * be any limit of Q c , c ∈ (0; 1) , and let x, k, k, l denote the coordinate process in Therefore, by the Knight representation theorem for continuous local martingales we have that for a two-dimensional standard Brownian motion where "= d " denotes the equality in distributions (in C [0; T ] , R 4 ).Notice that the assumption in the Knight therorem that Q a.s.K c ∧ Kc → +∞ may be omitted, since we consider the Brownian motion B 1 , B 2 on the interval [0; L T /2] only (see remark below [19, Theorem V.1.9]).Since Q * is unique we get the desired weak convergence.
The stable convergence follows e.g. from the fact that the inequalities (4.17)-(4.21),(4.13)-(4.14),have their counterparts when we restrict to any subset F ∈ F with Q(F ) > 0, for example and one may again apply the Knight theorem to obtain the desired weak convergence on F. Now the stable convergence follows from [7, Sect.VIII, Proposition 5.33,(iv)].
To obtain the stable convergence under the measure P one may notice that for any ε ∈ (0; 1) there exists F ε ∈ F with P(F ε ) > 1 − ε such that for some M ε < +∞, dP/dQ < M ε and P a.s.sup 0≤t≤T |X s | ≤ M ε on F ε .Now notice that the inequalities (4.17)-(4.21),(4.13)-(4.14)have their counterparts under measure P when we restrict to the subset F ε , for example |X s | .

Integral limit theorem -CLT for integrated number of interval crossings
In this subsection we will assume that X is the unique strong solution of the following s.d.e., driven by a standard Brownian motion W, with Lipschitz µ, σ and σ > 0. For X satisfying (4.22) we have a more precise result than Theorem 4.2, namely (cf.[16, Theorem 5]) where B is another standard Brownian motion, independent from W.
Remark 4.8.The just cited Theorem 5 from [16] describes convergence of a different vector than the vector appearing in (4.23) and only for diffusions starting from 0, but (4.23) follows simply from the Mapping Theorem (cf.[1, Sect.2]) and from the fact that we may set an arbitrary starting value for X, x 0 , and then consider the diffusion X − x 0 , which has the same values of U T V c , DT V c , T V c and • as X.The stable convergence follows from [16, Remark 6], see also [7, Chapt.VIII, Proposition 5.33(ii),(iii)].
From (4.23) we shall obtain a convergence result concerning the integrated number of down(up-)crossings.
For a càdlàg Y t , t ≥ 0, let us denote Theorem 4.9.Fix T > 0. For X satisfying (4.22) we have where B is another standard Brownian motion, independent from W.
Proof.By Lemma 3.3 and the extended Banach-Vitali Indicatrix Theorem ([3, page 328]) we have where by (2.3) and condition (F), the random variable r d (c, t) defined by equality (4.25) belongs to where r u (c, t) ∈ [0; c].Finally, by the just obtained equalities (4.25), (4.26) and by Re- It is interesting to compare the formulas just obtained for the integrated number of crossings with that obtained in the Subsection 4.2 generalisation of Kasahara's result.By the occupation times formula Thus we get that the integrated with respect to a process d a c (X, •)−L a /(2c) reveals much stronger concentration than the same process for a given a.Moreover, for numbers of (up-)crossings we get EJP 19 (2014), paper 10.

Functional integral limit theorems
In this subsection we assume that X is the same process as in Subsection 4.3.Let us fix T > 0 and let f ∈ C 2 , where C 2 is the class of functions with continuous second derivative.The aim of this subsection is to obtain a "functional" limit theorem of the form This is a considerable generalisation of the theorem obtained in the previous section and this result can not be obtained (as far as we know) by a straightforward application of [16,Theorem 5], like in the previous subsection.The convergence result (4.28) and its relevant version for the quadruple will be obtained via establishing the convergence result for the quadruple of integrals involving level crossings of the regularised processes X c,x , as in Proposition 2.9, where for each c > 0, Numbers U a (X c,x , t) , D a (X c,x , t) and N a (X c,x , t) in (4.29) denote the number of level crossings and for any càdlàg process Y t , t ≥ 0, are defined as i.e.U a (Y, t) is the number of times that the process Y crosses the level a from below and D a (Y, t) is the number of times that the process Y crosses the level a from above the interval [0; t].
The main tool we will use will be the "functional" version of the Banach-Vitali Indicatrix Theorem, from which follows that for any continuous function f : R → R and t > 0 Further, we also have two other equalities analogous to (4.30): EJP 19 (2014), paper 10.
Remark 4.10.Since these results are not easily found in the literature, for the reader's convenience we present a simple proof of (4.31) and (4.30).
Proof.Let us notice that for every ω ∈ Ω the continuous function ψ 0 otherwise.
From this we have Now let X c,x , c > 0, be a sequence of regularizations of X as in Proposition 2.9 with x = x(c) such that |x| ≤ c/2.We have Theorem 4.11.For any f ∈ C 2 the following convergence holds The analog of Theorem 4.11 for Q c 1 is Theorem 4.12.The same convergence as for the quadruple Q c 2 holds for the quadruple First, we will prove the following two-dimensional version of Theorem 4.11.
Theorem 4.13.For any f ∈ C 2 the following convergence holds and  notice that it implies that The total variation of the drift part of X is globally bounded: and, since f is continuous, By (4.30) it may be split into two parts: Step 1. Proof that S c 1 → 0 a.s. as c ↓ 0 in C ([0; T ] , R) .By Taylor's formula, for any real x and δ, f where the remainder r, defined for δ = 0 by the equality (4.43), may be represented as for some θ ∈ (x; x + δ).For δ = 0 we may simply set r(x, 0) = 1 2 f (x).
Let us consider the stopping times: υ 0 = 0 and To simplify the notation we will omit x in the superscript X c,x .By (4.43) we get To bound the first term in (4.46), notice that by (4.35)-(4.38) A small difficulty may arise from the fact that the number of summands in (4.46) may be arbitrary large, but since X is a.s.continuous, we may always restrict to a subset of Θ ⊂ Ω with probability arbitrary close to 1 and such that for X ∈ Θ the number of summands in (4.46) is bounded by some fixed integer.Let us also notice that the EJP 19 (2014), paper 10.
To bound the second term in (4.46) notice that by the mean value theorem, for where Z is a random number belonging to the interval [X s ∧ X υ k ; X s ∨ X υ k ].Hence, by (4.39), (4.42) and the definition of stopping times υ k , k = 0, 1, . . ., Thus, for any ε > 0, by the choice of sufficiently large K, the absolute value of the second term in (4.46) may be bounded by ε as c ↓ 0.
Step 2. Convergence in probability of S c 2 .By integration by parts formula (recall that T V (X c , [0; t]) − X t /c is a process with locally finite total variation), we have and, since X c is continuous and has locally finite total variation, where Thus, in view of (4.51) (and (4.39) -(4.41)), it is enough to consider Again, by integration by parts Finally, We prove that the second term in (4.55), i.e.
vanishes as c → 0. First, using the stopping times υ k , k = 0, 1, . . ., defined in Step 1, we decompose To bound the second term in (4.56) we use the mean value theorem and similarly as in Step 1, for EJP 19 (2014), paper 10. Hence (4.57) Now we will prove that for every k = 0, 1, . . ., as c ↓ 0. First, let us introduce the "natural clock" for the process X, i.e. we define γ (t) := inf {s : X s > t} .
By this time-change we obtain the process Xt := X γ(t) satisfying the following s.d.e.(cf. [17, Theorem 8.5.7]): where W is another Brownian motion.Defining Xc t := X c γ(t) and stopping times υk , k = 0, 1, . . .analogous to the times υ k , k = 0, 1, . . ., i.e. υk = 0 and for k = 1, 2, . . ., υk = inf t > υk−1 : Xt ∈ I i for some i = 1, 2, . . ., K such that Xυ k−1 / ∈ I i , we obtain the equality Moreover, we may assume that the process Xt − x 0 is a standard Brownian motion on the set sup 0≤t≤T Xt < M under some measure Q, such that the measure P is absolute continuous with respect to Q.
More precisely, take a standard Brownian motion B on some filtered probability space Ω, G, G = (G t ) t≥0 , Q with probability measure Q, such that usual conditions hold, define v (M ) = inf {t ≥ 0 : |B t + x 0 | ≥ M } and the measure P, absolutely continuous with respect to Q, with the Radon-Nikodym derivative Now we will prove that for k = 0, 1, . . ., as c ↓ 0, where → Q denotes convergence in probability with respect to the measure Q, B c = B c,x (with x = x (c) ∈ [−c/2; c/2] being a G 0 -measurable random variable) is the regularization of B, defined similarly as the regularization X c,x in Proposition 2.9; w 0 = 0 and for k = 1, 2, . . ., (notice that for fixed k = 0, 1, . . ., Ew k+1 < +∞).Now, denoting by U c,k the quadratic variation of U c,k , by (4.66) and (4.64) we have that is the regularization of Xs − x 0 with the same x as the regularization X c = X c,x of X, for k = 0, 1, . . .Hence, by (4.59) we have Step 3. Approximation of S c 4 with step processes and establishing its weak convergence.Let us recall the stopping times defined in (4.45) and define the càdlàg process

Xt
where K is the same number which was used in the definition of the intervals I i , i = 1, 2, ..., K. From (4.69), (4.39) and the mean value theorem it follows that where C 2 is a universal constant and V denotes finite variation part of X.
By (4.72) we also have the same convergence for the differences, Since Xs is a step process, and hence, by (4.73) we also have Let π be the Prohorov metric (cf.[1, Sect.6]) defined on the space of probability laws on C ([0; T ] , R) .Reasoning similarly as above (but instead of convergence in probability considering the Prohorov metric), by the weak convergence result for T V c (X, •)− X /c (i.e.[16,Theorem 5]) and the fact that T V c (X, •) and T V (X c , •) differ by no more than c, we get Finally, notice that by (4.69) Since K may be chosen arbitrary large and δ (K) → 0 as K ↑ +∞, we get Step 4. Convergence.By Steps 1-3 and the fact that the convergence in the defined Prohorov metric is equivalent to weak convergence in C ([0; T ] , R) , c.f. [1, Theorem 6.8], we obtain the weak convergence of To finish the proof let us notice that instead of considering the processes S c i , i = 1, . . ., 5 alone, we could consider pairs (X, S c i ) .By [16, Theorem 5 and Remark 6] we have stable convergence of the pair (X, T V c (X, •) − X /c) and thus we obtain the desired stable convergence with respect to the σ-field generated by X (as described in Remark   Using integration by parts we get that  Now, using 2 c (X c s − X s ) dX c s = −dT V (X c , s) and again integration by parts we get which, in view of (4.54) converges in probability as c ↓ 0 to the same limit as To finalize the proof we need to notice that From this we get R f (a) U a (X c , t) da =  The convergence of the quadruple Q 1 holds by the Mapping Theorem, since n a c (X, t) = u a c (X, t) + d a c (X, t) .remarks about connections between so called play operator and the Skorohod Problem and Prof. Leszek Słomiński for his remark about the uniqueness and the existence of the solution of the Skorohod problem with starting condition.Acknowledgements are also directed to the anonymous referee for a very careful review of this paper, which helped to fix many mistakes and considerably improved the exposition.This paper was written while the second author was staying at the African Institute for Mathematical Sciences, the warm hospitality of AIMS is gratefully acknowledged.

4
Limit theorems for truncated variations and interval down-and upcrossings of continuous semimartingales and diffusions 4.1 Strong laws of large numbers for
us notice that by the definition of continuous processes K c and Kc , they have disjoint intervals where they are non-constant, hence K c , Kc ≡ 0.
32) and (4.33) we get (4.31).Similar arguments work for DT V. Finally, (4.30) follows from the fact that N a = U a + D a and dT V = dU T V + dDT V.

( 4 .
34) EJP 19 (2014), paper 10.Proof.(of Theorem 4.13.)We start with the following properties of the Skorohod map, which are simply the translation of condition (b) in the definition of the Skorohod problem in Subsection 2.1:

EJP 19 (
2014), paper 10.Page 21/33 ejp.ejpecp.orgIntegral and local limit theorems for level crossings Now we fix positive integer K and split the interval [−M ; M ] into finite sum of intervals

Remark 4 . 14 .
Due to localisation this s.d.e.might be satisfied only on the time interval [0; τM ] , where τM := U ∧ inf t : Xt ≥ M and U := T sup x∈[−M ;M ] σ 2 (x) , but this is sufficient for our purposes.(By (4.40), U is sufficiently large for the inequality X T ≤ U to hold on sup 0≤t≤T |X t | < M.)

For
ψ = B define stopping times U c u,k and U c d,k analogously as stopping times T u,k and T d,k in Subsection 2.1, with α ≡ −c/2, β ≡ c/2.It is easy to see that by symmetry of standard Brownian, for l = 1, 2, . . ., under the measure Q, [U c u,l ;U c d,l ] 2 c (B s − B c s ) ds = d − [U c d,l ;U c u,l+1 ] 2 c (B s − B c s ) ds .

k 2 cc 1 +1
.65) and similar equalities hold for intervals U c d,l ; U c u,l+1 .The integral vw+1 w (B s − B c s ) ds, k = 0, 1, . . ., consists, except maybe two marginal integrals , of the sum of the integrals of the form [U c u,l ;U c d,l ] 2 c (B s − B c s ) ds + [U c d,l ;U c u,l+1 ] 2 c (B s − B c s ) ds.

Define
U c,k t , t ≥ 0, in the following way: U c,k t = 0 for t ≤ U c u,l c 0 +1 and for t > U c u− B c s ) ds, where l c 1 (t) := sup l : U c u,l ≤ t ∧ (l c 1 + 1) .By (4.63) and renewal structure of B s − B c s , U c,k is a martingale under Q (for l = 1, 2, . . .consecutive integrals [U c u,l ;U c u,l+1 ] 2 c (B s − B c s ) ds, are independent).By (4.65), similar equalities for intervals U c d,l ; U c u,l+1 , independence of consecutive intervals U c u,l ; U c u,l+1 , l = 1, 2, . . ., and Wald's identity we have

vw+1 w k 2 c
(B s − B c s ) ds, k = 0, 1, . . ., by two marginal integrals as c ↓ 0) and apply Burkholder's inequality.The same holds for B under the (absolutely continuous with respect to Q) measure P. Thus, by this and by (easy to prove) equality Xs − x 0 − Xs − x 0 c = Xs − Xc s , where Xs − x 0 c = Xs − x 0 c,x

0 f (X c s ) dX c s − • 0 f• 0 f
the fact that X − X c ∞ ≤ c/2, the Taylor expansion and similar reasoning as in Step 1 of the proof of Theorem 4.13 we get that• 0 {f (X c s ) − f (X s )} dX c s − • 0 f (X s ) (X c s − X s ) dX c s → 0 a.s.(4.83)as c ↓ 0. Finally, from (4.79), (4.83) and (4.82) we get that• (X s ) (X c s − X s ) dX c s − (X s ) d X s → P • 0 f (X s ) dX s .(4.84)

0 f
s ) d X s , which is the Stratonovich integral and we get (4.78).Now we proceed to the proof of Theorem 4.11.Proof.By (4.31), R f (a) U a (X c , t) da = t 0 f (X c s ) dU T V (X c , s) and R f (a) D a (X c , t) da = t (X c s ) dDT V (X c , s) .
c s ) dT V (X c , s) .

f• 0 f 2 • 0 f 0 f2c t 0 f
(a) N a (X c , •) da − 1 c (X s ) d X s which (recall (4.30)) is equal to X, • 0 f (X c s ) dT V (X c , s) − 1 c • 0 f (X s ) d X s .By this, (4.87), (4.88) and by Lemma 4.15 we have the thesis.EJP 19 (2014), paper 10.Page 31/33 ejp.ejpecp.orgIntegral and local limit theorems for level crossingsWe finish with the proof of Theorem 4.12.Proof.By (4.87), Lemma 4.15, (4.30) and Step 1 of the proof of Theorem 4.13, for anysequences of F 0 -measurable r.v.s x 1 = x 1 (c), x 2 = x 2 (c) such that x 1 , x 2 ∈ [−c/2; c/2] we have the convergence R f (a) U a (X c,x1 , •) da − R f (a) U a (X c,x2 , •) da − 1 (X s ) d {T V (X c,x1 , s) − T V (X c,x2 , s)} → P 0 in C ([0; T ],R) as c ↓ 0, and a similar assertion holds for crossings from above.But by (2.1) and condition (E), |T V (X c,x1 , s) − T V (X c,x2 , s)| ≤ c, and integration by parts yields• (X s ) d {T V (X c,x1 , s) − T V (X c,x2 , s)} → P 0 in C ([0; T ], R) as c ↓ 0. Thus the triple T = X, R f (a) U a (X c,x1 , t) da − 1 2c t 0 f (X s ) d X s , R f (a) D a (X c,x2 , t) da − 1 (X s ) d X sconverges stably (as c ↓ 0) to the same limit for any sequences x 1 and x 2 as above.Now, it is enough to notice that by Lemma 3.4 we have R f (a) u a c (X, t) da = R f (a) U a X c,−c/2 , t da and similarly R f (a) d a c (X, t) da = R f (a) D a X c,c/2 , t da.
Research of the second author was supported by the AIMS and by the National Science Centre in Poland under decision no.DEC-2011/01/B/ST1/05089. EJP 19 (2014), paper 10.Page 33/33 ejp.ejpecp.org By the formula (2.4) the function η x = −ψ x is nonincreasing on the intervals [T u,k ; T d,k ) and nondecreasing on the intervals [T d,k ; T u,k+1 ) , k = 0, 1, 2, .... Thus one may define η x d , η x u ∈ BV + [0; +∞) in such a way that dη x 1, . . ., (times T d,i , T u,i are now stopping times, defined for every path separately).Using monotonicity of X c,x on the intervals [T u,i ; T d,i ] and [T d,i ; T u,i+1 ] , and formula (2.4) we calculate For a continuous semimartingale X t , t ≥ 0, such that the bicontinuous version of its local time exists, and , (cf.(2.3) and condition (F)) and the analogous reasoning for crossings from above (analog of (4.2), the Banach-Vitali Indicatrix Theorem and Lemma 3.3), we get EJP 19 (2014), paper 10.Theorem 4.2.
Thus, we have obtained an alternative proof of [16, Theorem 1], but using a stronger condition on X -that the bicontinuous version of its local time exists.But we have Remark 4.3.A careful examination of the proof of [12, Theorem II.2.4] gives for any .11) Now, since X is a local martingale under Q, by the Barlow-Yor inequality [19, Theorem XI.2.4], for some universal constant A 2 we have B t ) σ 4 (B t ) dt .U ∧ v (M )] .It is easy to see that for the solutions of (4.60) uniqueness in law holds on the interval [0; U ∧ v (M )] .Thus, by (4.58) and(4.60) .67) Thus, by (4.57) and (4.67), since M and K may be arbitrary large (first we choose M then K), we get that EJP 19 (2014), paper 10.