Local Time Rough Path for Lévy Processes

In this paper, we will prove that the local time of a Levy process is a rough path of roughness $p$ a.s. for any $2 < p < 3$ under some condition for the Levy measure. This is a new class of rough path processes. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int _{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $f^\prime_{-}$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of Levy processes.


Introduction
Integration of a stochastic process is a fundamentally important problem in probability theory. Different integration theory may result in completely different approach to a problem. Local time is an important and useful stochastic process. The investigation of its variation and integration has attracted attentions of many mathematicians. Similar to the case of the Brownian motion, the variation of the local time of a semimartingale in the spatial variable is also fundamental in the construction of an integral with respect to the local time. There have been many works on the quadratic or p-variations (in the case of stable processes) of local times in the sense of probability. Bouleau and Yor ([3]), Perkins ([28]), first proved that, for the Brownian local time, and a sequence of partitions {D n } of an interval [a, b], with the mesh |D n | → 0 when n → ∞, lim in probability. Subsequently, the process x → L x t can be regarded as a semimartingale (with appropriate filtration). This result allowed one to construct various stochastic integrals of the Brownian local time in the spatial variable. See also Rogers and Walsh [31]. Numerous important extensions on the variations, stochastic integrations of local times and Itô's formula have been done, e.g. Marcus and Rosen [24], [25], Eisenbaum [5], [6], Eisenbaum and Kyprianou [7], Flandoli, Russo and Wolf [11], Föllmer, Protter and Shiryayev [12], Föllmer and Protter [13], Moret and Nualart [27]. In their extensions of Itô's formula, the integrals of the local time are given as stochastic integrals in nature, for example as forward and backward stochastic integrals. In this paper, we study path integration of the local time and prove that the local time is of classical p-variation and also can be considered as a rough path (its meaning will be made precise later). We consider the local time of the Lévy process which is represented by the following Lévy-Itô decomposition X t = X 0 + σB t + bt + Recall that for a general semimartingale X t , L = {L x t ; x ∈ R} is defined from the following formula (Meyer [26]): where [X , X ] c . is the continuous part of the quadratic variation process [X , X ] . . There is a different notion of local time defined as the Radon-Nikodym derivative of the occupation measure of X with respect to the Lebesgue measure on R i.e. for every Borel function g : R → R + . For the Lévy process (1.1), if σ = 0, L x t and γ x t are the same (up to a multiple of a constant). In case σ = 0 e.g. for a stable process, there is no diffusion part so these two definitions are different. In fact, in this case L x t = 0. The increment of γ . t for stable processes was considered by Boylan [4], Getoor and Kesten [14] and Barlow [2], using potential theory approach, in order to establish the continuity of the local time in the space variable. Using Theorem 0.2 in [2], it's easy to prove that when σ = 0, and (2.2) is satisfied, for any p > 2 and t ≥ 0, the process x → L x t , is of finite p-variation in the classical sense almost surely. As a direct application, one can define the path integral +∞ −∞ g(x)d L x t as a Young integral for any g being of bounded q-variation for a q ∈ [1,2). But when q ≥ 2, Young's condition 1 p + 1 q > 1 is broken. The main task of this paper is to construct a geometric rough path over the processes Z(x) = (L x t , g(x)), for a deterministic function g being of finite q-variation when q ∈ [2,3). This implies establishing the path integrals For these two integrals, all classical integration theories such as Riemann, Lebesgue and Young fail to work. To overcome the difficulty, we use the rough path theory pioneered by Lyons, see [21], [22], [23], also [19]. However, our p-variation result of the local time does not automatically make the desired rough path exist or the integral well defined, though it is a crucial step to study first. Actually further hard analysis is needed to establish an iterated path integration theory for Z . . First we introduce a piecewise curve of bounded variation as a generalized Wong-Zakai approximation to the stochastic process Z . . Then we define a smooth rough path by defining the iterated integrals of the piecewise bounded variation process. We need to prove the smooth rough path converges to a geometric rough path Z = (1, Z 1 , Z 2 ) when 2 ≤ q < 3. For this, an important step is to compute E(L and obtain the correct order in terms of the increments One can see the global aspects of Lévy processes are captured in this estimate. Actually, this is a very challenging task. In this analysis, the main difficulty is from dealing with jumps, especially the small jumps of the process. One can also see that to construct the geometric rough path, a slightly stronger condition (3.23) is needed here. Using this key estimate, we can establish the geometric rough path Z = (1, Z 1 , Z 2 ). Then from Chen's identity, we define the following two integrals Note that the Riemann sum themselves may not have limits as the mesh m(D [a,b] ) → 0. At least there are no integration theories, rather than Lyons' rough path theory, to guarantee the convergence of the Riemann sums for almost all ω. Here it is essential to add Lévy areas to the Riemann sum. Furthermore, we can prove if a sequence of smooth functions g j → g as j → ∞, then the Riemann integral b a g j (x)d L x t converges to the rough path integral b a g(x)d L x t defined in (1.5). It is also noted that to establish (1.4), one only needs (3.24). This is true as long as the power of | y| in the condition of Lévy measure is anything less (better) than 3 2 . The main technique here is the Tanaka formula. If we assume the processes is symmetric, one can use estimates of Gaussian processes and Dynkin's isomorphism theorem as our tool. This idea is being developed by Wang ([33]) for symmetric stable processes.
Having established the path integration of local time and the corresponding convergence results, as a simple application, we can easily prove a useful extension of Itô's formula for the Lévy process when the function is less smooth: if f : R → R is an absolutely continuous function and has left derivative f − (x) being left continuous and of bounded q-variation, where 1 ≤ q < 3, then P-a.s. (1.6) Here the path integral Lebesgue-Stieltjes integral when q = 1, a Young integral when 1 < q < 2 and a Lyons' rough path integral when 2 ≤ q < 3 respectively. Needless to say that Tanaka's formula ( [32]) and Meyer's formula ( [26], [34]) are special cases of our formula when q = 1. The investigation of Itô's formula to less smooth functions is crucial and useful in many problems e.g. studying partial differential equations with singularities, the free boundary problem in American options, and certain stochastic differential equations. Time dependent cases for a continuous semimartingale X t were investigated recently by Elworthy, Truman and Zhao ( [8]), Feng and Zhao ( [9]), where two-parameter Lebesgue-Stieltjes integrals and two-parameter Young integral were used respectively. We would like to point out that a two-parameter rough path integration theory, which is important to the study of local times, and some other problems such as SPDEs, still remains open.
A part of the results about the rough path integral of local time for a continuous semimartingale was announced without including proofs in Feng and Zhao [10]. In this paper, we will give a full construction of the local time rough path, and obtain complete results for any continuous semimartingales and a class of Lévy processes satisfying (3.23) and σ = 0. Our proofs are given in the context of Lévy processes. For continuous semimartingales, we believe the reader can see easily that the proof is essentially included in this paper, noticing the idea of decomposing the local time to continuous and discontinuous parts in [9] and the key estimate (8) in [10].

The variation of local time
We see soon that the variation of local time follows immediately from Barlow's celebrated result of modulus of continuity of local time (Theorem 0.2, [2]). Let X t be a one dimensional time homogeneous Lévy process, and ( t ) t≥0 be generated by the sample paths X t , p(·) be a stationary ( t )-Poisson process on R \ {0}. From the well-known Lévy-Itô decomposition theorem, we can write X t as follows: where Here, N p is the Poisson random measure of p, the compensator of p is of the formN (dsd y) = dsn(d y), where n(d y) is the Lévy measure of process X . The compensated random measurẽ where the supremum is taken over all finite partition on R, D(−∞, ∞) := {−∞ < a 0 < a 1 < · · · < a n < ∞}.
The key point here is that the right hand side does not depend on the partition D. Using (2.4), we know that But we know L a t has a compact support [−K, K] in a. So for the partition D : The p-variation (p > 2) result of the local time enables one to use Young's integration theory to define ∞ −∞ g(x)d x L x t for g being of bounded q-variation when 1 ≤ q < 2. This is because in this case, for any q ∈ [1, 2), one can always find a constant p > 2 such that the condition 1 p + 1 q > 1 for the existence of the Young integral is satisfied. However, when q ≥ 2, Young integral is no longer well defined. We have to use a new integration theory. In the following, we only consider the case that 2 ≤ q < 3. We obtain the existence of the geometric rough path Z = (1, Z 1 , Z 2 ) associated to Z . . Lyons' integration of rough path provides a way to push the result further. This will be studied in the rest of this paper.

The local time rough path
To establish the rough path integral of local time, we need to estimate the p-th moment of the increment of local time over a space interval and the covariance of the increments over two nonoverlapping intervals. First, we give a general p-moment estimate formula. This will be used in later proofs.

Lemma 3.1.
We have the p-moment estimate formula: for any p ≥ 1, , for a constant c p > 0. Here m is the smallest integer such that 2 m+1 ≥ p.
Proof: From the definition of N p ,Ñ p , the Burkholder-Davis-Gundy inequality and Jensen's inequality, we can have the p-moment estimation: where m is the smallest integer such that 2 m+1 ≥ p.

Recall the Tanaka formula for the Lévy process
the following alternative form is often convenient For the convenience in what follows in later part, we denote Note we have the following important decompositions that we will use often: for any a i < a i+1 , and We will use the above decomposition and probabilistic tools to prove the following lemma. The proof includes many technically rather hard calculations. However, they are key to get the desired estimates and crucial in our analysis.

Lemma 3.2. Assume the Lévy measure n(d y) satisfies
then for any p ≥ 2, there exists a constant c > 0 such that Proof: We will estimate every term in (3.2). First note that the function ϕ t (a) := (X t − a) + − (X 0 − a) + is Lipschitz continuous in a with Lipschitz constant 2. This implies that for any p > 2 and a i < a i+1 , Secondly, for the second term, by the occupation times formula, Jensen's inequality and Fubini theorem, is a decreasing process in t, we have Now using the Burkholder-Davis-Gundy inequality, we have where m > 0 is the smallest integer such that 2 m+1 ≥ p, c(p, b, σ, t) is a universal constant depending on p, b, σ, and t. By Jensen's inequality, we also have This inequality will be used later. So Thirdly, for the termB a t , by the Burkholder-Davis-Gundy inequality and a similar argument in deriving (3.10), we have (3.11) About K 1 (t, a), it is easy to see that About K 2 (t, a), with the decomposition (3.3), we will estimate the sum of each term for jumps |∆X s | < 1. There are seven such terms.
For the first term in (3.3), by the p-moment estimate formula and occupation times formula, we have , (3.13) where m is the smallest integer such that 2 m+1 ≥ p. In the following we will often use the following type of method to estimate integrals with respect to the Lévy measure: let . (3.14) Similarly one can estimate Then we can estimate each term in (3.13). When k = 1, we change the orders of the integration and use Jensen's inequality to have For the term when 2 ≤ k ≤ m, it is easy to see that Actually we can see that ∞ by using the same method as in (3.16). For the last term in (3.13), similarly, we have We can see that the key point is to estimate the term when k = 1 because the higher order term can always be dealt by the above method easily. We can use the similar method to deal with other terms and derive that (3.17) About K 3 (t, a), with the decomposition (3.4), we will estimate the sum of each term for jumps |∆X s | < 1. There are six such terms.
For the first term in (3.4), by the p-moment estimate formula and occupation times formula, changing orders of integration and using Jensen's inequality, we have We can use the similar method to deal with other terms. In the following, we will only sketch the estimate without giving great details.
2) For the second term, we have 3) For the third term, we have The fourth, fifth and the last terms are symmetric to the third, second and first terms respectively. In summary, we have So we proved the result.

21)
for a ξ ∈ (0, 1 2 ], then for any p ≥ 1, This estimate will be used in the construction of the geometric rough path where (3.20) is not adequate. In particular, this plays an essential role in obtaining (3.42) and (3.47), from which one can calculate (Z(m) 2 ) m∈N is a Cauchy sequence in the θ -variation distance.
) as a process of variable x in R 2 . Here g is of bounded q-variation, 2 ≤ q < 3. Then it's easy to know that Z x is of boundedq-variation in x, whereq = q, if q > 2, and q > 2 can be taken any number when q = 2. We assume a sightly stronger condition than (3.5) for the Lévy measure: there exists a constant > 0 such that (3.23) We will prove with this condition, the desired geometric rough path Z = (1, Z 1 , Z 2 ) is well defined.
We need to point out that in the following when we consider the control function and the convergence of the first level path, condition (3.5) is still adequate. But we need (3.23) in the convergence of the second level path. Denote δ = 1 q − (3 − q) . Note when q = 2, δ = 1 2 − . So condition (3.23) becomes: there exists > 0 such that Later in this section, we will see under this condition, the integral (3.23) is satisfied for any 2 ≤ q < 3. In this case, our results imply that we can construct the geometric rough path for any g being of finite q-variation, where 2 ≤ q < 3 can be arbitrary.
Recall the θ -variation metric d θ on C 0,θ (∆, T ([θ ]) (R 2 )) defined in [23], Assume condition (3.5) through to Proposition 3.1. Let [x , x ] be any interval in R. From the proof of Theorem 2.1, for any p ≥ 2, we know there exists a constant c > 0 such that i.e. L x t satisfies Hölder condition in [23] with exponent 1 2 . First we consider the case when g is continuous. Recall in [23], a control w is a continuous super-additive function on ∆ := {(a, b) : w(a, b) + w(b, c) ≤ w(a, c), f or an y (a, b), (b, c) ∈ ∆.
If g(x) is of bounded q-variation, we can find a control w s.t. ≤ w(a, b), is obvious that w 1 (a, b) := w(a, b) + (b − a) is also a control of g. Set h = 1 q , it is trivial to see for any θ > q (so hθ > 1) we have,

|g(b) − g(a)| q
Considering (3.26), we can see Z x satisfies, for such h = 1 q , and any θ > q, there exists a constant c such that where w 1 (x) := w 1 (x , x). It is obvious that x m l − x m l−1 ≤ 1 2 m w 1 (x , x ) and by the superadditivity of the control function w 1 , The corresponding smooth rough path Z(m) is built by taking its iterated path integrals, i.e. for any (a, b) ∈ ∆, In the following, we will prove {Z(m)} m∈N converges to a geometric rough path Z in the θ -variation topology when 2 ≤ q < 3. We call Z the canonical geometric rough path associated to Z.

Remark 3.2. The bounded variation process Z(m) x is a generalized Wong-Zakai approximation to the process Z of boundedq-variation.
Here we divide [x , x ] by equally partitioning the range of w 1 . We then use (3.29) to form the piecewise curved approximation to Z. Note here Wong-Zakai's standard piecewise linear approximation does not work immediately.

Remark 3.3. It is noted here that there is no unique way to construct rough path. The approach we present in this paper is one construction that makes the Lévy area convergent. More importantly, we will see later the integral constructed coincides with the Lebesgue-Stieltjes integral when g is of bounded variation. And the convergence theorem of the integral (Propostion 4.2) guarantees that the integral we constructed in this paper is the limit of the Lebesgue-Stieltjes integrals in θ -variation topology. This shows that the rough path integral defined by (3.48) is the correct integral in our formula (4.7).
Let's first look at the first level path Z(m)  1 (a, b) hθ . For such points {x n k }, k = 1, · · · , 2 n , n = 1, 2, · · · , defined above we still have the inequality in Proposition 4.1.1 in [23], Since hθ − 1 > 0, the series on the right-hand side of (3.32) is convergent, so that sup D l |Z 1 x l−1 ,x l | θ < ∞ almost surely. This shows that Z 1 has finite θ -variation almost surely. Moreover, for any γ > θ − 1, there exists a constant This means that Z(m) 1 a,b have finite θ -variation uniformly in m. And furthermore, from (3.31) and some standard arguments, where C depends on θ , h, w 1 (x , x ), and c in (3.28 for hθ > 1. So we obtain Theorem 3.1. Let L x t be the local time of the time homogeneous Lévy process X t given by (1.1), and g be a continuous function of bounded q-variation. Assume q ≥ 1, σ = 0 and the Lévy measure n(d y) satisfies (3.5). Then for any θ > q, the continuous process Z We next consider the second level path Z(m) 2 a,b . As in [23], we can also see that if m ≤ n, E|Z(m + 1) 2 where C depends on θ , h(:= 1 q ), w 1 (x , x ), and c in (3.28).
The main step to establish the geometric rough path integral over Z is the following estimate. Delicate and correct power in (3.42) must be obtained to prove the convergence of the approximated Lévy area. We will use Lemma 3.3 about the correlation of K 2 (t, a i+1 ) − K 2 (t, a i ) and K 2 (t, a j+1 ) − K 2 (t, a j ), and (3.22) for the term K 3 , for ξ =

Lemma 3.3.
Assume the Lévy measure satisfies (3.21) with 0 < ξ ≤ 1 6 , then for any a 0 < a 1 < · · · < a m , For convenience, we denote the two integrals in A 1 by A 11 and A 12 respectively. To estimate |EA 1 |, first by the occupation times formula, Fubini theorem, Jensen's inequality, similar as before, we have : In the same way, we can have Therefore, we get Using the same method, we can have the similar estimations for |EA 2 |, |EA 3 | and |EA 4 |. It follows that when i = j, Therefore we proved (3.41). ∈ [2, 3), and the Lévy measure satisfies (3.23). Let q < θ < 3. Then for m > n, we have

Proposition 3.2. Assume g is a continuous function of finite q-variation with a real number q
where C is a generic constant and also depends on θ , h(:= 1 q ), w 1 (x , x ), and c in (3.28).
Proof: For m > n, it is easy to see that The main difficulty is to estimate the following expectation which can be derived from Tanaka's formula: Firstly, from (3.7), (3.10), (3.11), the Cauchy-Schwarz inequality and the quadratic variation of stochastic integrals, we have Here C is a generic constant and may depend on t, b, σ, w 1 (x , x ). Secondly, recall the fact that E(P . M . ) = 0, if P is a process of bounded variation and M is a martingale with mean 0 and at least one of M and P is continuous. Note here K 1 is a process of bounded variation. Recall also that the cross-variation of t 0 1 {a i <X s− ≤a i+1 } d B s and the jump parts such as K 2 (t, a j+1 ) − K 2 (t, a j ) are zero.

Corollary 3.1. Under the same assumption as in Proposition 3.2, we have
as 2 < θ < 3, hθ > 1, where C is a generic constant and also depends on θ , h, w 1 (x , x ), and c. Therefore, we know Secondly from Proposition 3.1 and Proposition 3.2, we know as q < θ < 3, and hθ > 1. So Similar to the proof of Theorem 3.1, we can easily deduce that (Z(m) 2 ) m∈N is a Cauchy sequence in the θ -variation distance. So when m → ∞, it has a limit, denote it by Z 2 . And from the completeness under the θ -variation distance (Lemma 3.3.3 in [23]), Z 2 is also of finite θ -variation. The theorem is proved.

Remark 3.4.
We would like to point out that the above method does not seem to work for two arbitrary functions f of p-variation and g of q-variation (2 < p, q < 3) to define a rough path Z x = ( f (x), g(x)). However the special property (3. As local time L x t has a compact support in x for each ω and t, so we can define integral of local time directly in R. For this, we take [x , x ] covering the support of L x t . From Chen's identity, it's easy to know that for any (a, b) ∈ ∆, ).
In particular, exists. Here (Z 2 ) 2,1 means lower-left element of the 2 × 2 matrix Z 2 . It turns out that exists. Denote this limit by  (3.24). Then the local time L x t is a geometrical rough path of roughness p in x for any t ≥ 0 a.s. for any p > 2, and Moreover, if g is a continuous function with bounded q-variation, 2 ≤ q < 3, and the Lévy measure

Continuity of the rough path integrals and applications to extensions of Itô's formula
In this section we will apply the Young integral and rough path integral of local time defined in sections 2 and 3 to prove a useful extension to Itô's formula. First we consider some convergence result of the rough path integrals.
Similarly to Proposition 4.1, we have Proof: Define F j (x) := (g j − g)(x), so F j (x) → 0 as j → ∞, for all x. It's easy to see that F j,δ (x) → 0 as j → ∞, for all x. From the above theorem and Proposition 4.1, we have x L t,δ d F j,δ ( y) → 0, as j → ∞.
Then the proposition follows easily.
Applying the standard smoothing procedure on f (x), we can get f n (x) which is defined in the same way as g j (x) in (4.2). And by Itô's formula (c.f. [29]), we have f n (X t ) = f n (X 0 ) + t 0 f n (X s )d X s + A n t , 0 ≤ t < ∞. (4.5)