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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 16, pages 452–.

Journal URL

http://www.math.washington.edu/~ejpecp/

Local Time Rough Path for Lévy Processes

∗

Chunrong Feng

Department of Mathematics, Shanghai Jiao Tong University, 200240, China

Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK

fcr@sjtu.edu.cn

Huaizhong Zhao

Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK

H.Zhao@lboro.ac.uk

Abstract

In this paper, we will prove that the local time of a Lévy process is a rough path of roughness p a.s. for any 2< p <3 under some condition for the Lévy measure. This is a new class of rough path processes. Then for any function g of finiteq-variation (1≤q <3), we establish the integralR_−∞∞ g(x)d L_tx as a Young integral when 1≤q<2 and a Lyons’ rough path integral when 2≤q<3. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function f if f₋′ exists and is of finiteq-variation when 1≤q < 3, for both continuous semi-martingales and a class of Lévy processes.

Key words:semimartingale local time; geometric rough path; Young integral; rough path inte-gral; Lévy processes.

AMS 2000 Subject Classification:Primary 60H05, 58J99.

Submitted to EJP on October 2, 2009, final version accepted April 15, 2010.

∗_{CF would like to acknowledge the support of National Basic Research Program of China (973 Program No.}

2007CB814903), National Natural Science Foundation of China (No. 70671069 and No. 10971032), the Mathemati-cal Tianyuan Foundation of China (No. 10826090) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, Loughborough University and the London Mathematical Society that enabled her to visit Loughborough University.

1

Introduction

Integration of a stochastic process is a fundamentally important problem in probability theory. Dif-ferent integration theory may result in completely difDif-ferent 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 con-struction of an integral with respect to the local time. There have been many works on the quadratic orp-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 whenn→ ∞, lim

n→∞ P

Dn

(Lxi+1

t −L

xi

t )2 =4

Rb

a L x td x

in probability. Subsequently, the processx →L_tx can be regarded as a semimartingale (with appro-priate 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 Itˆo’s formula have been done, e.g. Mar-cus 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 Itˆo’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 classicalp-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-Itˆo decomposition

X_t = X0+σBt+bt+

Z t+

0 Z

R\{0}

y1{|y|≥1}Np(dsd y)

+ Z t+

0 Z

R\{0}

y1{|y|<1}N˜p(dsd y). (1.1)

Recall that for a general semimartingaleXt, L={Ltx;x ∈R}is defined from the following formula

(Meyer[26]):

Z t

0

g(X_s)d[X,X]c_s = Z ∞

−∞

g(x)L_txd x, (1.2) where[X,X]_.cis 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 ofX with respect to the Lebesgue measure onRi.e.

Z t

0

g(X_s)ds= Z ∞

−∞

g(x)γx_td x, (1.3)

for every Borel function g:R→R+. For the Lévy process (1.1), if σ=6 0, Lx_t andγx_t are the same (up to a multiple of a constant). In caseσ=0 e.g. for a stable process, there is no diffusion part so these two definitions are different. In fact, in this case Lx_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 whenσ6= 0, and (2.2) is satisfied, for any p >2 and

t ≥ 0, the process x 7→ L_tx, is of finite p-variation in the classical sense almost surely. As a direct application, one can define the path integralR_−∞+∞g(x)d L_tx as a Young integral for any g being of boundedq-variation for aq∈[1, 2). But whenq≥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_tx,g(x)), for a deterministic function g being of finite q-variation when q∈[2, 3). This implies establishing the path integralsR_−∞∞ Lx_td_xL_tx andR_−∞∞ g(x)d_xL_tx. 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 processZ.. 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,Z1,Z2)when 2≤q<3. For this, an important step is to computeE(Lxi+1

t −L

xi

t )(L xj+1

t −L

xj

t ),

and obtain the correct order in terms of the incrementsx_i+1−x_i andx_j+1−x_j, especially in disjoint intervals [x_i,x_i+1] and [x_j,x_j+1] when i 6= j. 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,Z1,Z2). Then from Chen’s identity, we define the following two integrals

Z b

a

L_txd L_tx = lim

m(D[a,b])→0

r−1 X

i=0 ((Z2_x

i,xi+1)1,1+Lt(xi)(L

xi+1

t −L

xi

t )) (1.4)

and

Z b

a

g(x)d L_tx = lim

m(D[a,b])→0

r−1 X

i=0 ((Z2_x

i,xi+1)2,1+g(xi)(L

xi+1

t −L

xi

t )). (1.5)

Note that the Riemann sum

r−1 P

i=0

Lxi

t (L xi+1

t −L

xi

t )and r−1

P

i=0

g(x_i)(Lxi+1

t −L

xi

t )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 allω. Here it is essential to add Lévy areas to the Riemann sum. Furthermore, we can prove if a sequence of smooth functions gj → g as j → ∞, then the Riemann integral

Rb

a gj(x)d L x

t converges to the rough path

integralR_abg(x)d L_tx 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 Itˆo’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 boundedq-variation, where 1≤q<3, then P-a.s.

f(Xt) = f(X0) + Z t

0

f₋′(Xs)d Xs−

Z ∞

−∞

f₋′(x)dxLtx

+ X

0≤s≤t

[f(Xs)−f(Xs−)−∆Xsf−′(Xs−)], 0≤t<∞. (1.6)

Here the path integralR_−∞∞ f₋′(x)d_xL_tx is a Lebesgue-Stieltjes integral whenq=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 Itˆo’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 con-tinuous 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 inte-gral 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 semi-martingales and a class of Lévy processes satisfying (3.23) andσ6=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].

2

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 Xt be a one dimensional time

ho-mogeneous Lévy process, and(F_t)_t≥0 be generated by the sample paths X_t, p(·) be a stationary (F_t)-Poisson process on R\ {0}. From the well-known Lévy-Itˆo decomposition theorem, we can writeXt as follows:

X_t := X₀+σB_t+V_t+M˜_t, (2.1) where

V_t := bt+ Z t+

0 Z

R\{0}

y1_{|y|≥1}N_p(dsd y),

˜

M_t := Z t+

0 Z

R\{0}

Here, N_p is the Poisson random measure of p, the compensator of p is of the form Nˆ(dsd y) =

dsn(d y), where n(d y) is the Lévy measure of process X. The compensated random measure ˜

N_p(t,U) =N_p(t,U)−Nˆ_p(t,U)is an(F_t)-martingale. Proposition 2.1. Ifσ6=0, the Lévy measure n satisfies

Z

R\{0}

(|y|2∧1)n(d y)<∞, (2.2)

then the local time La_t of time homogeneous Lévy process X_t given by (1.1) is of bounded p-variation in a for any t≥0, for any p>2, almost surely, i.e.

sup

D(−∞,∞) X

i

|Lai+1

t −L

ai

t |p<∞ a.s.,

where the supremum is taken over all finite partition on R, D(−∞,∞) :={−∞ < a₀ < a₁ < · · ·<

an<∞}.

Proof:Letχ(θ)be the exponent of Lévy processX, i.e.

EeiθXt _=e−tχ(θ)_,

where

χ(θ) =−i bθ+σ2θ2−

Z

[eiθy −1−iθy1_{|y|<1}]n(d y). (2.3)

Recall from[2],

φ(x)2= 1 π

Z

(1−cosθx)Re( 1

1+χ(θ))dθ Ifσ2>0, it is easy to check that

Re( 1

1+χ(θ))≤c(σ,θ)(1+θ 2₎−1_,

andφ(x)≤c|x|12. So using Theorem 0.2 in[2], one obtains that ifρ(x) = (x l o g(1

x))

1 2

sup

|b−a|≤1

2

|La_t −L_tb| ≤cρ(|b−a|)(sup

x

L_tx)12. (2.4)

Now we use Proposition 4.1.1 in[23](i=1,γ >p−1), for any finite partition{a_l}of[a,b]

sup

D

X

l

|Lal+1

t −L

al

t |p≤c(p,γ)

∞ X

n=1

nγ

2n X

k=1

|La

n k

t −L

an k−1

t |p,

where

a_kn=a+ k

2n(b−a), k=0, 1,· · ·, 2 n_.

The key point here is that the right hand side does not depend on the partition D. Using (2.4), we know that

∞ X

n=1

nγ

2n

X

k=1

|La

n k

t −L

an_k−1

t |p ≤ c

∞ X

n=1

nγ(ρ(b−a 2n ))

p

2

≤ ˜c

∞ X

n=1

nγ(b−a 2n )

p

2−1−ǫ<∞,

asp>2, whereǫ < p₂−1. Therefore for any interval[a,b]⊂R

sup

D

X

l

|Lal+1

t −L

al

t |p<∞a.s. (2.5)

But we knowLa_t has a compact support[−K,K]ina. So for the partitionD:=D−K,K ={−K=a0<

a₁<· · ·<a_r=K}, we obtain

sup

D

X

l

|Lai+1

t −L

ai

t |p<∞a.s. ⋄

The p-variation (p > 2) result of the local time enables one to use Young’s integration theory to defineR_−∞∞ g(x)d_xL_tx for g being of boundedq-variation when 1≤q <2. This is because in this case, for anyq∈[1, 2), one can always find a constantp>2 such that the condition 1

p+

1

q >1 for

the existence of the Young integral is satisfied. However, whenq≥ 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 pathZ= (1,Z1,Z2)associated toZ_.. Lyons’ integration of rough path provides a way to push the result further. This will be studied in the rest of this paper.

3

The local time rough path

To establish the rough path integral of local time, we need to estimate the p-th moment of the in-crement of local time over a space interval and the covariance of the inin-crements over two nonover-lapping 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,

E

X

0≤s≤t

|f(s,p_s(ω),ω)|

p

≤ c_p m

X

k=0

E

Z t 0

Z

R

|f(s,y,ω)|2kn(d y)ds

p

2k

+c_p

E

Z t

0 Z

R

|f(s,y,ω)|2m+1n(d y)ds

p

2m+1 ,

Proof: From the definition ofN_p, ˜N_p, the Burkholder-Davis-Gundy inequality and Jensen’s inequal-ity, we can have the p-moment estimation:

E

X

0≤s≤t

|f(s,p_s(ω),ω)|

p

= E

Z t+ 0

Z

R\{0}

|f(s,y,ω)|Np(dsd y)

p = E Z t 0 Z R

|f(s,y,ω)|n(d y)ds+ Z t+

0 Z

R\{0}

|f(s,y,ω)| Np(dsd y)−n(d y)ds

p ≤ pE Z t 0 Z R

|f(s,y,ω)|n(d y)ds

p +pE

Z t+ 0

Z

R\{0}

|f(s,y,ω)|_N˜_p(dsd y) p ≤ pE Z t 0 Z R

|f(s,y,ω)|n(d y)ds

p +cpE

Z t 0

Z

R

|f(s,y,ω)|2n(d y)ds

p

2

+cpE

Z t 0

Z

R\{0}

|f(s,y,ω)|2N˜p(dsd y)

p 2 ≤ pE Z t 0 Z R

|f(s,y,ω)|n(d y)ds

p +cpE

Z t 0

Z

R

|f(s,y,ω)|2n(d y)ds

p

2 +· · ·

+c_pE

Z t 0

Z

R

|f(s,y,ω)|2mn(d y)ds

p

2m

+c_p E

Z t 0

Z

R\{0}

|f(s,y,ω)|2mN˜_p(dsd y) 2! p

2m+1

= pE

Z t 0

Z

R

|f(s,y,ω)|n(d y)ds

p +c_pE

Z t 0

Z

R

|f(s,y,ω)|2n(d y)ds

p

2 +· · ·

+c_pE

Z t 0

Z

R

|f(s,y,ω)|2mn(d y)ds

p

2m

+c_p

E Z t 0 Z R

|f(s,y,ω)|2m+1n(d y)ds

p

2m+1 ,

wheremis the smallest integer such that 2m+1≥p. ⋄

Recall the Tanaka formula for the Lévy processX_t (c.f.[1]), we have

La_t = (X_t−a)+−(X₀−a)+−

Z t

0 1_{X

s−>a}d Xs

+ X

0≤s≤t

[(X_s−−a)+−(Xs−a)++1{Xs−>a}∆Xs]. (3.1)

Since

X

0≤s≤t

1{Xs−>a}∆Xs1{|∆Xs|≥1}=

Z t+

0 Z

R\{0}

1{Xs−>a}y1{|y|≥1}Np(dsd y),

the following alternative form is often convenient

where

ϕ_t(a) := (X_t−a)+−(X₀−a)+, I_t(a):= Z t

0 1_{X

s−>a}ds, Bˆ a t :=

Z t

0 1_{X

s−>a}d Bs, K1(t,a) :=

X

0≤s≤t

[(Xs−−a)+−(Xs−a)+]1{|∆Xs|≥1},

K2(t,a) := Z t

0 Z

R\{0}

[(Xs−y−a)+−(Xs−a)+]1{|y|<1}N˜p(d y ds),

K₃(t,a) := Z t

0 Z

R

[(X_s−y−a)+−(X_s−a)++1{Xs−y>a}y]1{|y|<1}n(d y)ds.

For the convenience in what follows in later part, we denote

J₁(s,a):= (X_s−−a)+−(Xs−a)+, J2(s,a):=1{Xs−>a}∆Xs.

Note we have the following important decompositions that we will use often: for anya_i<a_i+1,

J∗(Xs,Xs−,ai,ai+1) := J₁(s,a_i+1)−J₁(s,a_i)

= −(X_s−−a_i)1_{X

s≤ai}1{ai<Xs−≤ai+1}−(ai+1−ai)1{Xs≤ai}1{Xs−>ai+1} +(X_s−a_i)1{ai<Xs≤ai+1}1{Xs−≤ai}−(ai+1−Xs)1{ai<Xs≤ai+1}1{Xs−>ai+1} +(a_i+1−ai)1{Xs>ai+1}1{Xs−≤ai}+ (ai+1−Xs−)1{Xs>ai+1}1{ai<Xs−≤ai+1}

+(Xs−Xs−)1{ai<Xs≤ai+1}1{ai<Xs−≤ai+1}, (3.3)

and

J∗(X_s,X_s−,a_i,a_i+1)

:= [J₁(s,a_i+1)−J1(s,ai)] + [J2(s,ai+1)−J2(s,ai)]

= −(X_s−a_i)1_{X

s≤ai}1{ai<Xs−≤ai+1}−(ai+1−ai)1{Xs≤ai}1{Xs−>ai+1} +(Xs−ai)1{ai<Xs≤ai+1}1{Xs−≤ai}−(ai+1−Xs)1{ai<Xs≤ai+1}1{Xs−>ai+1} +(a_i+1−a_i)1_{X

s>ai+1}1{Xs−≤ai}+ (ai+1−Xs)1{Xs>ai+1}1{ai<Xs−≤ai+1}. (3.4) 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

Z

R\{0}

(|y|32 ∧1)n(d y)<∞, (3.5)

then for any p≥2, there exists a constant c>0such that

E|Lai+1

t −L

ai

t |p≤c|ai+1−ai|

p

Proof: We will estimate every term in (3.2). First note that the function ϕ_t(a) := (X_t−a)+−

(X0−a)+is Lipschitz continuous inawith Lipschitz constant 2. This implies that for anyp>2 and

a_i<a_i+1,

|ϕ_t(a_i+1)−ϕ_t(a_i)|p≤2p(a_i+1−a_i)p. (3.7)

Secondly, for the second term, by the occupation times formula, Jensen’s inequality and Fubini theorem,

E|It(ai+1)−It(ai)|p =

1

σ2p(ai+1−ai)

p_E( 1

a_i+1−a_i

Z ai+1

ai

L_txd x)p

≤ 1

σ2p(ai+1−ai) p_sup

x

E(L_tx)p. (3.8) We now estimate sup_xE(Lx_t)p. By (3.1) and noting that P

0≤s≤t

[(X_s−−a)+−(Xs−a)++1{Xs−>a}∆Xs]

is a decreasing process int, we have

La_t ≤(X_t−a)+−(X₀−a)+−

Z t

0 1_{X

s−>a}d Xs.

Now using the Burkholder-Davis-Gundy inequality, we have

E(La_t)p ≤ p

h

E|Xt−X0|p+E| Z t

0 σ1_{X

s−>a}d Bs| p

+E|

Z t

0

1{Xs−>a}d Vs| p_+E|

Z t

0

1{Xs−>a}dM˜s| pi

≤ cσptp2+c E(

Z t

0

|d V_s|)p+c_p m+1 X

k=1 –Z t

0 Z

R

|y|2k1{|y|<1}n(d y)ds ™p

2k

≤ c(p,b,σ,t),

wherem>0 is the smallest integer such that 2m+1≥p,c(p,b,σ,t)is a universal constant depend-ing onp,b,σ, andt. By Jensen’s inequality, we also have

E(La_t)2p ≤c(p,b,σ,t). (3.9)

This inequality will be used later. So

E|I_t(a_i+1)−I_t(a_i)|p≤c(p,b,σ,t)(a_i+1−a_i)p. (3.10)

Thirdly, for the termBˆa_t, by the Burkholder-Davis-Gundy inequality and a similar argument in deriv-ing (3.10), we have

E|Bˆai+1

t −Bˆ ai

t |p ≤ cpE

‚Z t

0 1_{a

i<Xs≤ai+1}ds

Œ

p

2

≤ c(t,p,σ)(ai+1−ai)

p

AboutK₁(t,a), it is easy to see that

|K₁(t,a_i+1)−K₁(t,a_i)| ≤2(a_i+1−a_i) X 0≤s≤t

1_{|∆X

s|≥1}.

So

E|K₁(t,a_i+1)−K₁(t,a_i)|p≤C(a_i+1−a_i)p. (3.12) About K2(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

E

Z t

0 Z

R\{0}

|Xs−−ai|1{Xs≤ai}1{ai<Xs−≤ai+1}1{|∆Xs|<1}N˜p(d y ds)

!p

≤ c_p m

X

k=1

E

Z t

0

Z X_s−a_i

Xs−ai+1

|X_s−y−a_i|2k1_{X

s≤ai}1{|y|<1}n(d y)ds

!p

2k

+c_p E

Z t

0

Z X_s−a_i

Xs−ai+1

|X_s− y−a_i|2m+11{Xs≤ai}1{|y|<1}n(d y)ds

! p

2m+1

= c(p,σ)

m

X

k=1

E

Z ai

−∞

L_tx

Z x−ai

x−ai+1

|x−y−ai|2

k

1{|y|<1}n(d y)d x !p

2k

+c(p,σ) E

Z ai

−∞

L_tx

Z x−ai

x−ai+1

|x−y−ai|2

m+1

1{|y|<1}n(d y)d x ! p

2m+1

, (3.13) wheremis the smallest integer such that 2m+1≥p. In the following we will often use the following type of method to estimate integrals with respect to the Lévy measure: let

Q :=

Z 0

ai−ai+1

Z a_i

y+ai

(x− y−a_i)d x1_{|y|<1}n(d y)

= 1

2 Z 0

ai−ai+1

|y|21{|y|<1}n(d y).

Then, 1_{|y|<1}Q1(x −y −a_i)d x n(d y)is a probability measure on {(x,y): y+a_i ≤ x ≤ a_i,(−1)∧

(ai−ai+1)≤ y≤0}. So by Jensen’s inequality, we have that

E

Z 0

ai−ai+1

Z ai

y+ai

L_tx(ai+1−ai)(x−y−ai)d x1{|y|<1}n(d y) !p

2

≤ Qp2E 1

Q

Z 0

ai−ai+1

Z ai

y+ai

(L_tx)p2(a_i+1−a_i)

p

2(x− y−a_i)d x1_{|y|<1}n(d y)

!

≤ c(σ,t,p)|ai+1−ai|

p

2sup

x

E(L_tx)p2

Z 0

ai−ai+1

|y|21_{|y|<1}n(d y) !p

2

Similarly one can estimate

E

Z a_i−a_i+1

−∞

Z y+a_i+1

y+ai

Lx_td x|y|21{|y|<1}n(d y) !p

2

≤ c(σ,t,p)|ai+1−ai|

p

2sup

x

E(Lx_t)p2

Z ai−ai+1

−∞

|y|21{|y|<1}n(d y) !p

2

. (3.15)

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

E

Z a_i

−∞

L_tx

Z x−a_i

x−ai+1

|x−y−a_i|21{|y|<1}n(d y)d x !p

2

= E

‚ Z ai−ai+1

−∞

Z y+a_i+1

y+ai

L_tx(x−y−a_i)21{|y|<1}d x n(d y)

+ Z 0

ai−ai+1

Z ai

y+ai

Lx_t(x−y−ai)(x−y−ai)1{|y|<1}d x n(d y) Œp

2

≤ E

‚ Z ai−ai+1

−∞

Z y+a_i+1

y+ai

L_txd x|y|21_{|y|<1}n(d y)

+ Z 0

ai−ai+1

Z a_i

y+ai

Lx_t(a_i+1−a_i)(x−y−a_i)d x1_{|y|<1}n(d y) Œp

2

≤ c(σ,t,p)|a_i+1−a_i|2p. (3.16)

For the term when 2≤k≤m, it is easy to see that

E

Z ai

−∞

L_tx

Z x−ai

x−ai+1

|x−y−ai|2

k

1{|y|<1}n(d y)d x !p

2k

≤ E

Z a_i

−∞

L_tx

Z x−a_i

x−ai+1

|x−y−a_i|2k−1|y|2k−11_{|y|<1}n(d y)d x

! p

2k

≤ (a_i+1−ai)

1 2pE

Z a_i

−∞

Lx_t

Z x−a_i

x−ai+1

|y|2k−11{|y|<1}n(d y)d x !p

2k

≤ c(t,p)|a_i+1−ai|

p

2. Actually we can see that R_−∞ai Rx−ai

x−ai+1|y|

2k−1

(3.16). For the last term in (3.13), similarly, we have

E

Z ai

−∞

Lx_t

Z x−ai

x−ai+1

|x−y−ai|2

m+1

1_{|y|<1}n(d y)d x

! p

2m+1

≤ E

Z a_i

−∞

Lx_t

Z x−a_i

x−ai+1

|a_i+1−a_i|2m|y|2m1_{|y|<1}n(d y)d x

! p

2m+1

≤ |a_i+1−ai|

p

2 E

Z a_i

−∞

Lx_t

Z x−a_i

x−ai+1

|y|2m1{|y|<1}n(d y)d x ! p

2m+1

≤ c(t,p)|a_i+1−ai|

p

2.

We can see that the key point is to estimate the term whenk=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

E|K₂(t,a_i+1)−K₂(t,a_i)|p≤c(t,σ,p)|a_i+1−a_i|p2. (3.17)

About K3(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 thep-moment estimate formula and occupation times formula, chang-ing orders of integration and uschang-ing Jensen’s inequality, we have

E

‚Z t

0 Z ∞

−∞

|Xs−ai|1{Xs≤ai}1{ai<Xs−y≤ai+1}1{|y|<1}n(d y)ds

Œp

= 1

σ2pE

Z a_i

−∞

L_tx

Z x−a_i

x−ai+1

|x−a_i|1{|y|<1}n(d y)d x !p

= 1

σ2pE

‚ Z ai−ai+1

−∞

Z y+a_i+1

y+ai

Lx_t(a_i−x)1_{|y|<1}d x n(d y)

+ Z 0

ai−ai+1

Z a_i

y+ai

Lx_t(a_i−x)12(a

i−x)

1 21_{|

y|<1}d x n(d y) Œp

≤ 1

σ2pE

‚ Z ai−ai+1

−∞

Z y+a_i+1

y+ai

Lx_td x|y|1{|y|<1}n(d y)

+ Z 0

ai−ai+1

Z a_i

y+ai

Lx_t(a_i+1−a_i)12(a

i−x)

1 2d x1

{|y|<1}n(d y) Œp

≤ c(σ)|a_i+1−ai|

1 2psup

x

E(Lx_t)p

Z a_i−a_i+1

−∞

|y|321_{|

y|<1}n(d y) !p

+c(σ)|a_i+1−ai|

1 2psup

x

E(L_tx)p Z 0

ai−ai+1

|y|321_{|

y|<1}n(d y) !p

≤ c(t,σ,p)|ai+1−ai|

1

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

E

‚Z t

0 Z ∞

−∞

(a_i+1−ai)1{Xs≤ai}1{Xs−y>ai+1}1{|y|<1}n(d y)ds

Œp

= 1

σ2p(ai+1−ai) p_E

Z a_i

−∞

L_tx

Z x−a_i+1

−∞

1_{|y|<1}n(d y)d x

!p

= c(σ)(a_i+1−a_i)pE

Z a_i−a_i+1

−∞

Z a_i

y+ai+1

L_txd x

!

1_{|y|<1}n(d y) !p

≤ c(σ)|a_i+1−a_i|12psup

x

E(Lx_t)p

Z a_i−a_i+1

−∞

|y|321_{|y|<1}n(d y)

!p

≤ c(t,σ,p)|a_i+1−ai|

1 2p. 3) For the third term, we have

E

‚Z t

0 Z ∞

−∞

|Xs−ai|1{ai<Xs≤ai+1}1{Xs−y≤ai}1{|y|<1}n(d y)ds

Œp

= 1

σ2pE

Z a_i+1

ai

L_tx

Z ∞

x−ai

|x−a_i|1{|y|<1}n(d y)d x !p

≤ 1

σ2pE

‚ Z ai+1−ai

0

Z a_i+y

ai

L_tx(a_i+1−ai)

1 2(x−a

i)

1 2d x1_{|

y|<1}n(d y)

+ Z ∞

ai+1−ai

Z a_i+1

ai

Lx_td x|y|1{|y|<1}n(d y) Œp

(3.19)

≤ c(σ,p)|a_i+1−ai|

1 2psup

x

E(Lx_t)p ‚Z ∞

0

|y|321_{|

y|<1}n(d y) Œp

≤ c(t,σ,p)|a_i+1−a_i|12p.

The fourth, fifth and the last terms are symmetric to the third, second and first terms respectively. In summary, we have

E|K₃(t,a_i+1)−K₃(t,a_i)|p≤c(t,σ,p)|a_i+1−a_i|12p. (3.20)

So we proved the result. ⋄

Remark 3.1. From (3.18) and (3.19), we can see easily that if we require the following slightly stronger condition on the Lévy measure

Z

R\{0}

for aξ∈(0,1₂], then for any p≥1,

E|K3(t,ai+1)−K3(t,ai)|p≤c(t,σ,p)|ai+1−ai|(

1

2+ξ)p. (3.22)

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 theθ-variation distance.

Now, define Z_x := (Lx_t,g(x)) as a process of variable x in R2. Here g is of bounded q-variation, 2≤q<3. Then it’s easy to know thatZ_x is of boundedˆq-variation in x, whereˆq=q, ifq>2, and ˆ

q>2 can be taken any number whenq=2. We assume a sightly stronger condition than (3.5) for the Lévy measure: there exists a constantǫ >0 such that

Z

R\{0}

(|y|1+

1

q−(3−q)ǫ_{∧1)n(d y)<∞. (3.23)}

We will prove with this condition, the desired geometric rough pathZ= (1,Z1,Z2)is well defined. We need to point out that in the following when we consider the control function and the conver-gence of the first level path, condition (3.5) is still adequate. But we need (3.23) in the converconver-gence of the second level path. Denoteδ=1

q−(3−q)ǫ. Note whenq=2,δ=

1

2−ǫ. So condition (3.23) becomes: there existsǫ >0 such that

Z

R\{0}

(|y|32−ǫ∧1)n(d y)<∞. (3.24)

Later in this section, we will see under this condition, the integralR_−∞∞ Lx_td L_tx can be well-defined as a rough path integral. Also note inf

2≤q<3δ(q,ǫ) = 1

3. So under the condition Z

R\{0}

(|y|43 ∧1)n(d y)<∞, (3.25)

(3.23) is satisfied for any 2 ≤ q < 3. In this case, our results imply that we can construct the geometric rough path for anygbeing of finiteq-variation, where 2≤q<3 can be arbitrary. Recall theθ-variation metricdθ onC0,θ(∆,T([θ])(R2))defined in[23],

d_θ(Z,Y) = max

1≤i≤[θ]di,θ(Z

i_,Yi_{) = max}

1≤i≤[θ]supD

X

l

|Zi_x

l−1,xl −Y

i xl−1,xl|

θ

i

!i θ .

Assume condition (3.5) through to Proposition 3.1. Let[x′,x′′]be any interval inR. From the proof of Theorem 2.1, for anyp≥2, we know there exists a constantc>0 such that

E|Lb_t −La_t|p≤c|b−a|p2, (3.26) i.e. Lx_t satisfies H¨older condition in [23] with exponent 1₂. First we consider the case when g is continuous. Recall in [23], a control w is a continuous super-additive function on∆ := {(a,b) :

x′≤a<b≤x′′}with values in[0,∞)such thatw(a,a) =0. Therefore

If g(x)is of boundedq-variation, we can find a control ws.t.

|g(b)−g(a)|q≤w(a,b),

for any(a,b)∈∆:={(a,b): x′≤a< b≤ x′′}. It is obvious thatw1(a,b):=w(a,b) + (b−a) is also a control of g. Seth= 1_q, it is trivial to see for anyθ >q(sohθ >1) we have,

|g(b)−g(a)|θ ≤w₁(a,b)hθ, f or an y(a,b)∈∆. (3.27)

Considering (3.26), we can see Zx satisfies, for suchh= 1_q, and anyθ >q, there exists a constantc

such that

E|Z_b−Z_a|θ ≤cw₁(a,b)hθ, f or an y (a,b)∈∆. (3.28)

For anym∈N, define a continuous and bounded variation pathZ(m)by

Z(m)_x :=Zxm l−1+

w1(x)−w1(x_lm−1)

w₁(x_lm)−w₁(x_lm₋₁)∆

m

l Z, (3.29)

if x_lm₋₁ ≤ x < x_lm, forl =1,· · ·, 2m, and∆m_l Z = Z_xm l −Zx

m

l−1. Here Dm :={x

′ _{= x}m

0 < x

m

1 <· · ·<

x₂mm=x′′}is a partition of[x′,x′′]such that

w₁(x_lm)−w₁(xm_l−1) = 1 2mw1(x

′_,x′′_),

wherew1(x):=w1(x′,x). It is obvious thatx_lm−x_lm−1≤ 1

2mw1(x′,x′′)and by the superadditivity of the control functionw1,

w₁(x_lm₋₁,xm_l )≤w₁(x_lm)−w₁(xm_l−1) = 1 2mw1(x

′_,x′′_).

The corresponding smooth rough pathZ(m)is built by taking its iterated path integrals, i.e. for any (a,b)∈∆,

Z(m)_aj_,b= Z

a<x1<···<xj<b

d Z(m)_x1⊗ · · · ⊗d Z(m)_xj. (3.30)

In the following, we will prove{Z(m)}_m∈Nconverges to a geometric rough pathZin theθ-variation topology when 2≤q<3. We callZthe 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 boundedˆq-variation. Here we divide[x′,x′′]by equally partitioning the range of w1. 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 inθ-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 pathZ(m)1_a,b. The method is similar to Chapter 4 in[23]for Brownian motion. Similar to Proposition 4.2.1 in[23], we can prove that for alln∈N,m7→

2n

P

k=1

|Z(m)1_xn k−1,x

n k

|θ

is increasing and form≥n,

Z(m)1_xn k−1,xkn

=Z(m+1)1_xn k−1,xnk

=Z_xn k −Zx

n

k−1. (3.31)

LetZ1_a,b=Z_b−Z_a. Then (3.28) impliesE|Z1_a,b|θ ≤cw₁(a,b)hθ. For such points{x_kn}, k=1,· · ·, 2n,

n=1, 2,· · ·, defined above we still have the inequality in Proposition 4.1.1 in[23],

Esup

D

X

l

|Z1_x

l−1,xl|

θ _{≤ C(θ,γ)E} ∞ X

n=1

nγ

2n

X

k=1

|Z1_xn k−1,x

n k

|θ

≤ C1 ∞ X

n=1

nγ( 1 2n)

hθ−1_w

1(x′,x′′)hθ, (3.32)

for constantC1=C(θ,γ)c. Sincehθ−1>0, the series on the right-hand side of (3.32) is conver-gent, so that sup_DP_l|Z1_x

l−1,xl|

θ _{<∞almost surely. This shows thatZ}1 _{has finiteθ-variation almost} surely. Moreover, for anyγ > θ −1, there exists a constantC₁(θ,γ,c)>0 such that

Esup

m

sup

D

X

l

|Z(m)1_x

l−1,xl|

θ _{≤ C(θ,γ)Esup} m

∞ X

n=1

nγ

2n

X

k=1

|Z(m)1_xn k−1,x

n k

|θ

≤ C(θ,γ)E

∞ X

n=1

nγ

2n

X

k=1

|Z1_xn k−1,x

n k

|θ

≤ C1(θ,γ,c) ∞ X

n=1

nγ( 1 2n)

hθ−1_w

1(x′,x′′)hθ (3.33) < ∞.

So

sup

m

sup

D

X

l

|Z(m)1_x

l−1,xl|

θ _{<∞ a.s. (3.34)}

This means thatZ(m)1

a,bhave finiteθ-variation uniformly inm. And furthermore, from (3.31) and

some standard arguments,

E

∞ X

n=1

nγ

2n

X

k=1

|Z(m)1_xn k−1,x

n k

−Z1_xn k−1,x

n k

|θ ≤C( 1 2m)

hθ−1

2 , (3.35)

inequality and (3.35),

E

∞ X

m=1 sup

D

X

l

|Z(m)1_x

l−1,xl−Z

1

xl−1,xl|

θ 1 θ ≤ E ∞ X

m=1 ∞ X

n=1

nγ

2n

X

k=1

|Z(m)1_xn k−1,x

n k

−Z1_xn k−1,x

n k

|θ

!1_θ

≤

∞ X

m=1

E

∞ X

n=1

nγ

2n X

k=1

|Z(m)1_xn k−1,x

n k

−Z1_xn k−1,x

n k |θ !1 θ ≤ C ∞ X

m=1 ( 1

2m)

hθ−1 2θ

< ∞, (3.36)

forhθ >1. So we obtain

Theorem 3.1. Let L_tx be the local time of the time homogeneous Lévy process Xt given by (1.1), and g be a continuous function of bounded q-variation. Assume q≥1,σ6=0and the Lévy measure n(d y)

satisfies (3.5). Then for anyθ >q, the continuous process Z_x = (L_tx,g(x))satisfying (3.28), we have

∞ X

m=1 sup

D

X

l

|Z(m)1_x

l−1,xl−Z

1

xl−1,xl|

θ

1 θ

<∞a.s. (3.37)

In particular,(Z(m)1_a,b)converges to(Z1_a,b)in theθ-variation distance a.s. for any(a,b)∈∆.

We next consider the second level pathZ(m)2_a,b. As in[23], we can also see that ifm≤n,

Z(m)2_xn k−1,x

n k

=22(m−n)−1(∆m_l Z)⊗2, (3.38) wherel is chosen such that xm_l−1≤ x_kn₋₁<x_kn≤ x_lm; ifm>n,

Z(m)2_xn k−1,x

n k

= 1 2∆

n kZ⊗∆

n kZ+

1 2

2m−nk

X

r,l=2m−n(k−1)+1

r<l

(∆m_r Z⊗∆m_l Z−∆m_l Z⊗∆m_r Z),

so

Z(m+1)2_xn k−1,x

n k

−Z(m)2_xn k−1,x

n k

= 1

2

2m−nk

X

l=2m−n_(k−1)+1

(∆m_2l+₋1₁Z⊗∆₂m_l+1Z−∆m_2l+1Z⊗∆₂m_l+₋1₁Z), (3.39)

k=1,· · ·, 2n. Similar to the proof of Proposition 4.3.3 in[23], we have

Proposition 3.1. Assume g is a continuous function of finite q-variation with a real number q ≥2, and the Lévy measure satisfies (3.5). Letθ >q. Then for m≤n,

2n

X

k=1

E|Z(m+1)2_xn k−1,x

n

k−Z(m)

2

xn_k−1,xn_k|

θ 2 ≤C(

1 2n+m)

θh−1

2 , (3.40)

where C depends onθ, h(:= 1

q), w1(x

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 ofK2(t,ai+1)−K2(t,ai)andK2(t,aj+1)−

K2(t,aj), and (3.22) for the termK3, forξ=

q−2

2q + (3−q)ǫ.

Lemma 3.3. Assume the Lévy measure satisfies (3.21) with0< ξ≤ 1

6, then for any a0 <a1<· · ·<

a_m,

E K2(t,ai+1)−K2(t,ai)

K₂(t,a_j+1)−K₂(t,a_j)

≤

¨

c(t,σ)(a_i+1−a_i), when 0≤i= j≤m,

c(t,σ)[(a_i+1−ai)1+2ξ+ (aj+1−aj)1+2ξ], when 0≤i6= j≤m.

(3.41)

Proof: When i= j, (3.41) follows from (3.17) directly. Here we only need Lévy measure satisfies (2.2). Now we consider the case when i 6= j. Without losing generality, we assume i < j. From (3.3), it is easy to see that

E K₂(t,a_i+1)−K₂(t,a_i)

· K₂(t,a_j+1)−K₂(t,a_j)

= E



 Z t+

0 Z

R

J∗(Xs,Xs−y,ai,ai+1)N˜p(d y ds)·

Z t+

0 Z

R

J∗(Xs,Xs−y,aj,aj+1)N˜p(d y ds)





= E

Z t

0 Z

R

J∗(Xs,Xs−y,ai,ai+1)J∗(Xs,Xs− y,aj,aj+1)n(d y)ds

= A₁+A₂+A₃+A₄, where

A1 := Z t

0 Z

R

h

−(a_i+1−ai)(aj−Xs+y)

i

1{Xs≤ai}1{aj<Xs−y≤aj+1}1{|y|<1}n(d y)ds

+ Z t

0 Z

R

h

(a_i+1−a_i)(a_j+1−a_j) i

1_{X

s≤ai}1{Xs−y>aj+1}1{|y|<1}n(d y)ds,

A2 := Z t

0 Z

R

h

−(ai+1−Xs)(aj−Xs+y)

i 1_{a

i<Xs≤ai+1}1{aj<Xs−y≤aj+1}1{|y|<1}n(d y)ds

+ Z t

0 Z

R

(a_i+1−Xs)(aj+1−aj)1{ai<Xs≤ai+1}1{Xs−y>aj+1}1{|y|<1}n(d y)ds,

A₃ := Z t

0 Z

R

h

(a_i+1−a_i)(X_s−a_j) i

1_{X

s−y≤ai}1{aj<Xs≤aj+1}1{|y|<1}n(d y)ds

+ Z t

0 Z

R

(ai+1−ai)(aj+1−aj)1{Xs−y≤ai}1{Xs>aj+1}1{|y|<1}n(d y)ds,

A₄ := Z t

0 Z

R

h

−(X_s−y−a_i+1)(Xs−aj)1{ai<Xs−y≤ai+1}1{aj<Xs≤aj+1}1{|y|<1}n(d y)ds

+ Z t

0 Z

R

h

−(X_s−y−a_i+1)(a_j+1−a_j)i1_{a

i<Xs−y≤ai+1}1{Xs>aj+1}

i

the sense of the uniform topology, and also in the sense of theθ-variation topology. As for(Z_j)2_a,b, we can easily see that

d_2,θ((Z_j)2,Z2)≤d_2,θ((Z_j)2,(Z_j(m))2) +d_2,θ((Z_j(m))2,Z(m)2) +d_2,θ(Z(m)2,Z2). (4.1) From Theorem 3.2, we know that d_2,θ(Z(m)2,Z2)→0 as m→ ∞. Moreover, it is not difficult to see from the proofs of Propositions 3.1, 3.2, and Theorem 3.2,d_2,θ((Z_j)2,(Z_j(m))2)→0 asm→ ∞ uniformly in j. So for any givenǫ >0, there exists anm0such that whenm≥m0,d2,θ(Z(m)2,Z2)<

ǫ

3,d2,θ((Zj) 2_,(Z

j(m))2)< ǫ₃ for all j. In particular,d2,θ(Z(m0)2,Z2)< ₃ǫ,d2,θ((Zj)2,(Zj(m0))2)< ǫ₃

for allj. It’s easy to prove for suchm₀,d_2,θ((Z_j(m₀))2,Z(m₀)2)< ǫ₃ for sufficiently largej. Replacing m by m0 in (4.1), we can get d2,θ((Zj)2,Z2) < ǫ for sufficiently large j. Then by (3.48) and the definition ofR_abgj(x)d Ltx, we know that

Rb

a gj(x)d L x t →

Rb

a g(x)d L x

t as j → ∞. Similarly, we can see from the last section, when we considerZt(m) = (1,Zt(m)1,Zt(m)2),d2,θ((Zt)2,(Zt(m))2)→0, as m → ∞ uniformly in t ∈ [0,T], for any T > 0. Therefore we can also conclude that Z2_t is continuous in t in the d_2,θ topology. Note now that the local time L_tx has a compact support in x a.s. So it is easy to see from taking[x′,x′′]covering the support of L_tx that the above construction of the integrals and the convergence can work for the integrals onR. Therefore we have

Proposition 4.1. Let Z_j(x):= (L_tx,g_j(x)), Z(x):= (L_tx,g(x)), where g_j(·), g(·)are continuous and of bounded q-variation uniformly in j,2≤q<3, and the Lévy measure n(d y)satisfies (3.23),σ6=0. Assume g_j(x)→g(x)as j → ∞ for all x∈R. Then as j→ ∞,Z_j(·)→Z(·)a.s. in the θ-variation distance. In particular, as j→ ∞,R_−∞∞ g_j(x)d Lx_t →R_−∞∞ g(x)d Lx_ta.s.Similarly, Z_t(·)is continuous in t in theθ-topology. In particular,R_−∞∞ g(x)d Lx_t is a continuous function of t a.s.

Now for anygbeing continuous and of boundedq-variation (2≤q<3), define gj(x) =

Z ∞

−∞

kj(x−y)g(y)d y, (4.2)

wherekj is the mollifier given by kj(x) =

(

e

1

(j x−1)2−1_{, ifx ∈(0,}2

j),

0, otherwise.

Here c is a constant such thatR₀2kj(x)d x =1. It is well known that gj is a smooth function and g_j(x) → g(x) as j → ∞ for each x. So the integral R_−∞∞ g_j(x)d Lx_t is a Riemann integral for the smooth function gj(x). Moreover, Proposition 4.1 guarantees that

R∞

−∞gj(x)d L x t →

R∞

−∞g(x)d L x t a.s.

In the following, we will show that Proposition 4.1 is true for gbeing of boundedq-variation (2≤ q<3) without assuminggbeing continuous. Note that a function with boundedq-variation (q≥1) may have at most countable discontinuities. Using the method in [35], we will define the rough path integralRx

′′

x′ Lx_td g(x). Here we assumeg(x)is c`adl`ag inx.

First we can define a map

τδ(·):[x′,x′′]→[x′,x′′+δ

∞

X

n=1

in the following way:

τδ(x) =x+δ

∞

X

n=1

|j(xn)|q1{xn≤x}(x),

where j(x_i):=G(x_i)−G(x_i−),{x_i}∞_i=1are the discontinuous points ofG inside[x′,x′′],δ >0. The mapτ_δ(·):[x′,x′′]→[x′,τ_δ(x′′)]extends the space interval into the one where we can define the continuous pathGδ(y)from a c`adl`ag pathGby:

G_δ(y) = ¨

G(x) if y=τ_δ(x),

G(xn−) + (y−τδ(xn−))j(xn)δ−1|j(xn)|−q if y∈[τδ(xn−),τδ(xn)).

(4.3) TakeG to be g and Lt, we can define gδ and Lt,δ respectively. As Ltx is continuous, we can easily see that L_t,δ(y):=L_t,δ(τ_δ(x)) =L_tx.

Theorem 4.1. Let g(x)be a cadl`` ag path with bounded q-variation (2≤q<3), and the Lévy measure n(d y)satisfies (3.23),σ6=0. Then

Z x′′

x′

Lx_td g(x) =

Z τ_δ(x′′)

x′

L_t,δ(y)d g_δ(y). (4.4)

Proof:First it is easy to see that the integralRτδ(x

′′₎

x′ Lt,δ(y)d gδ(y)is a rough path integral that can be defined by the method of last section. Now note that at any discontinuous point xr,

Z xr xr−

Lx_td g(x) =Lt(xr)(g(xr)−g(xr−)) and

X

r

((Z_δ)2_τ

δ(xr−),τδ(xr))2,1 =

X

r

Z τ_δ(x_r) τδ(xr−)

(L_t,δ(y)−L_t,δ(τ_δ(x_r−)))d g_δ(y) =0, whereZ_δ(y):= (L_t,δ(y),g_δ(y)). Thus

X

r Lxr

t (g(xr)−g(xr−))

= X

r

Lt,δ(τδ(xr−))(gδ(τδ(xr))−gδ(τδ(xr−)))

= X

r

h

Lt,δ(τδ(xr−))(gδ(τδ(xr))−gδ(τδ(xr−))) + ((Zδ)2τδ(xr−),τδ(xr))2,1

i

< ∞, so

Z τ_δ(x_r) τδ(xr−)

L_t,δ(y)d g_δ(y) = L_t,δ(τ_δ(x_r−))(g_δ(τ_δ(x_r))−g_δ(τ_δ(x_r−))) = L_t(x_r)(g(x_r)−g(x_r−)).

Thus

Z x_r

xr−

Lx_td g(x) =

Z τ_δ(x_r) τδ(xr−)

L_t,δ(y)d g_δ(y). Now define g(x) =˜g(x) +h(x), whereh(x) = P

xr≤x

(g(x_r)−g(xr−)). Then ˜gis the continuous part ofgandhis the jump part of g. Moreover, ˜gsatisfies theq-variation condition. SoRx

′′

x′ Lt(x)d˜g(x) can be well defined as in the last section. Forh, we can definehδ by takingG=hin (4.3). So the

integralR_xx′”Lx_tdh(x)can be well defined by the followings:

Z τδ(x′′)

x′

Lt,δ(y)dhδ(y) = X

r

Z τδ(xr)

τδ(xr−)

Lt,δ(y)dhδ(y) = X

r

Lt(xr)(h(xr)−h(xr−))

= X

r

L_t(x_r)(g(x_r)−g(xr−)) =

X

r

Z x_r

xr−

L_txdh(x) = Z x′′

x′

Lx_tdh(x). Therefore

Z x′′

x′

L_txd g(x) = Z x′′

x′

Lx_td˜g(x) + Z x′′

x′

Lx_tdh(x)

=

Z τδ(x′′)

x′

Lt,δ(y)d˜gδ(y) +

Z τδ(x′′)

x′

Lt,δ(y)dhδ(y)

=

Z τδ(x′′)

x′

L_t,δ(y)d g_δ(y).

⋄ Similarly to Proposition 4.1, we have

Proposition 4.2. Under the conditions of Proposition 4.1, we have as j → ∞, R_−∞∞ L_txd g_j(x)

→R_−∞∞ L_txd g(x)a.s. for such g with bounded q-variation (2≤q<3).

Proof:DefineF_j(x):= (g_j−g)(x), soF_j(x)→0 as j→ ∞, for allx. It’s easy to see thatF_j,δ(x)→0 as j→ ∞, for all x. From the above theorem and Proposition 4.1, we have

Z x′′

x′

Lx_td(g_j−g)(x) = Z x′′

x′

L_txd F_j(x) =

Z τ_δ(x′′)

x′

L_t,δd 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 Ito’s formula (c.f.ˆ [29]), we have

f_n(X_t) = f_n(X₀) + Z t

0

where

An_t = 1

2

Z t 0

f_n′′(Xs−)d[X,X]sc+

X

0≤s≤t

[fn(Xs)−fn(Xs−)− fn′(Xs−)∆Xs]. (4.6) From the occupation times formula, the definition of the integral of local time and the convergence results of the integrals, we have

lim n→∞

1 2

Z t 0

f_n′′(X_s−)d[X,X]c_s= lim n→∞

1 2

Z ∞

−∞

L_txd f_n′(x) =−1 2

Z ∞

−∞

f₋′(x)d_xLx_t,

and the rough path integral R_−∞∞ f₋′(x)d_xL_tx is continuous in t from Proposition 4.1. For the con-vergence of jump part in (4.6), we can conclude from the proof of Theorem 3 in Eisenbaum and Kyprianou[7], if the following assumption:

Condition (A): R_{|y|<1}|f(x + y)− f(x)− f₋′(x)y|n(d y) is well defined and locally bounded in x, holds, then

lim n→∞

X

0≤s≤t

[f_n(X_s)−f_n(X_s−)−f_n′(X_s−)∆X_s] = X 0≤s≤t

[f(X_s)−f(X_s−)− ∇−f(X_s−)∆X_s],

in L2(d P). Therefore we have:

Theorem 4.2. Let f : R → R be an absolutely continuous function and have left derivative f₋′(x)

being left continuous and locally bounded, f₋′(x) be of bounded q-variation, where1≤ q< 3. Then for X = (X_t)t≥0, a time homogeneous Lévy process with σ 6= 0 and Lévy measure n(d y) satisfying

Condition (A), and (3.5) when1≤q<2, (3.23) when2≤q<3, we have P-a.s. f(X_t) = f(X₀) +

Z t 0

f₋′(X_s)d X_s−1 2

Z ∞

−∞

f₋′(x)d_xL_tx

+ X

0≤s≤t

[f(X_s)−f(Xs−)− f−′(Xs−)∆Xs], 0≤t<∞. (4.7)

Here the integralR_−∞∞ f₋′(x)d_xL_tx is a Lebesgue-Stieltjes integral when q = 1, a Young integral when 1 < q < 2 and a Lyons’ rough path integral when 2≤ q < 3 respectively. In particular, under the condition (3.25), (4.7) holds for any2≤q<3.

Acknowledgement

It is our great pleasure to thank K.D. Elworthy, W.V. Li, T. Lyons, Z.M. Ma, S.G. Peng, Z.M. Qian for stimulating conversations. We are grateful to the referee for his/her very valuable comments.

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