Stochastic Calculus For Symmetric Markov Processes

The Annals of Probability
2008, Vol. 36, No. 3, 931–970
DOI: 10.1214/07-AOP347
© Institute of Mathematical Statistics, 2008
STOCHASTIC CALCULUS FOR SYMMETRIC
MARKOV PROCESSES
BY Z.-Q. CHEN,1 P. J. FITZSIMMONS,2 K. KUWAE3 AND T.-S. ZHANG4
University of Washington, University of California at San Diego,
Kumamoto University and University of Manchester
Dedicated to S. Nakao on the occasion of his 60th birthday
Using time-reversal, we introduce a stochastic integral for zero-energy
additive functionals of symmetric Markov processes, extending earlier work
of S. Nakao. Various properties of such stochastic integrals are discussed and
an Itô formula for Dirichlet processes is obtained.
1. Introduction and framework.
It is well known that stochastic integrals
and Itô’s formula for semimartingales play a central role in modern probability the-
ory. However, there are many important classes of Markov processes that are not
semimartingales. For example, symmetric diffusions on Rd whose inﬁnitesimal
generators are elliptic operators in divergence form L =
d
∂ (a
)
i,j =1 ∂x
ij (x) ∂
i
∂xj
with merely measurable coefﬁcients need not be semimartingales. Even when
X is a Brownian motion in Rd and u ∈ W 1,2(Rd) := {u ∈ L2(Rd; dx) | |∇u| ∈
L2(Rd; dx)}, the process u(Xt ) is not generally a semimartingale. To study such
processes, Fukushima obtained the following substitute for Itô’s formula (see [7]):
for u ∈ W 1,2(Rd),
(1.1)
u(Xt ) = u(X0) + Mu +
t
N u
t
for t ≥ 0,
Px-a.s. for quasi-every x ∈ Rd , where Mu is a square-integrable martingale and
N u is a continuous additive functional of zero energy. The decomposition (1.1)
is called Fukushima’s decomposition and holds for a general symmetric Markov
Received August 2006; revised April 2007.
1Supported in part by NSF Grant DMS-06-00206.
2Supported by a foundation based on the academic cooperation between Yokohama City Univer-
sity and UCSD.
3Supported by a foundation based on the academic cooperation between Yokohama City Univer-
sity and UCSD, and partially supported by a Grant-in-Aid for Scientiﬁc Research (C) No. 16540201
from the Japan Society for the Promotion of Science.
4Supported in part by the British EPSRC.
AMS 2000 subject classiﬁcations. Primary 31C25; secondary 60J57, 60J55, 60H05.
Key words and phrases. Symmetric Markov process, time reversal, stochastic integral, general-
ized Itô formula, additive functional, martingale additive functional, dual additive functional, Revuz
measure, dual predictable projection.
931

932
CHEN, FITZSIMMONS, KUWAE AND ZHANG
process X and for u ∈ F , where (E, F ) is the Dirichlet space for X. In this pa-
per, a stochastic process ξ = {ξt , t ≥ 0} under some σ -ﬁnite measure P is called a
Dirichlet process if ξ has locally ﬁnite quadratic variation under P. The composite
process u(X) is a Dirichlet process under Pm, where m is the Lebesgue measure on
Rd, as it has ﬁnite quadratic variation on compact time intervals. Nakao introduced
a stochastic integral t0 f (Xs) dNus in [14] by using a Riesz representation theorem
in a suitably constructed Hilbert space. Nakao’s stochastic integral played an im-
portant role in the study of lower order perturbation of diffusion processes by Lunt,
Lyons and Zhang [12] and by Fitzsimmons and Kuwae [5]. However, Nakao’s de-
ﬁnition of the stochastic integral t f (X
0
s ) d N u, requiring u to be in the domain of
the Dirichlet form of X and f to be square-integrable with respect to the energy
measure of u, is too restrictive to be useful in the study of lower-order perturbation
for symmetric Markov processes with discontinuous sample paths, such as stable
processes. Such a study requires stochastic integrals for more general integrators
as well as integrands. The purpose of this paper is to present a new way of deﬁning
the stochastic integral for Dirichlet processes associated with a symmetric Markov
process. Our new approach uses only the time-reversal operator for the process
X and is therefore more direct and provides additional insight into stochastic in-
tegration for Dirichlet processes. This approach enables us to deﬁne
(M) [see
(1.5)] for any locally square-integrable martingale additive functional (MAF) M,
subject to some mild conditions. Thus, it not only recovers Nakao’s results in [14],
but also extends them signiﬁcantly. The new stochastic integral allows us to study
various transforms for symmetric Markov processes, a project that is carried out in
a subsequent paper [2].
A more detailed description of the current paper appears below.
Let X = { , F∞, Ft , Xt, θt , ζ, Px, x ∈ E} be an m-symmetric right Markov
process with a Lusin state space E, where m is a σ -ﬁnite measure with full
support on E. Its associated Dirichlet space (E , F ) on L2(E; m) is known to
be quasi-regular (see [13]). By [1], (E , F ) is quasi-homeomorphic to a reg-
ular Dirichlet space on a locally compact separable metric space. Using this
quasi-homeomorphism, there is no loss of generality in assuming that X is an
m-symmetric Hunt process on a locally compact metric space E such that its as-
sociated Dirichlet space (E , F ) is regular on L2(E; m) and that m is a positive
Radon measure with full topological support on E. We assume this throughout the
sequel.
Without loss of generality, we can take
to be the canonical path space
D([0, ∞[ → E ) of right-continuous, left-limited (rcll, for short) functions from
[0, ∞[ to E , for which is a trap [i.e., if ω(t) = , then ω(s) = for all s > t].
For any ω ∈ , we set Xt (ω) := ω(t). Let ζ(ω) := inf{t ≥ 0 | Xt (ω) = } be the
lifetime of X. As usual, F∞ and Ft are the minimal augmented σ -algebras ob-
tained from F 0
∞ := σ {Xs | 0 ≤ s < ∞} and F 0 :=
t
σ {Xs | 0 ≤ s ≤ t}, respectively,
under Px; see the next section for more details. We sometimes use a ﬁltration de-
noted by (Mt ) on ( , M) in order to represent several ﬁltrations, for example,

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
933
(F 0
t ), (F 0
t +) on (
, F 0
∞), (Ft ) on ( , F∞) and others introduced later. We use θt
to denote the shift operator deﬁned by θt (ω)(s) := ω(t + s), t, s ≥ 0. Let ω be the
path starting from
. Then, ω (s) ≡
for all s ∈ [0, ∞[. Note that θζ(ω)(ω) = ω
if ζ (ω) < ∞, {ω } ∈ F 0 ⊂ F 0
0
t
for all t > 0 and Px({ω }) ≤ Px(X0 = ) = 0 for
x ∈ E. For a Borel subset B of E, τB := inf{t > 0 | Xt /
∈ B} (the exit time of B) is
an (Ft )-stopping time. If B is closed, then τB is an (F 0
t +)-stopping time. Also, ζ is
an (F 0
t )-stopping time because {ζ ≤ t } = {Xt =
} ∈ F 0t, t ≥ 0. The transition
semigroup of X, {Pt , t ≥ 0}, is deﬁned by
Pt f (x) := Ex[f (Xt )] = Ex[f (Xt ) : t < ζ ],
t ≥ 0.
Each Pt may be viewed as an operator on L2(E; m); collectively, these operators
form a strongly L2-continuous semigroup of self-adjoint contractions. The Dirich-
let form associated with X is the bilinear form
1
E (u, v) := lim (u − Pt u, v)m
t ↓0 t
deﬁned on the space
F := u ∈ L2(E; m) sup t−1(u − Pt u, u)m < ∞ .
t >0
Here, we use the notation (f, g)m :=
f (x)g(x)m(dx).
E
For the reader’s convenience, we recall the following deﬁnitions from [13] and
[7].
DEFINITION 1.1.
(i) An increasing sequence {Fn}n≥1 of closed subsets of E
is an E -nest (or simply nest) if and only if
n≥1 FF is E
n
1-dense in F , where
E1 = E + (·, ·)L2(E,m) and
FF := {u ∈ F : u = 0 m-a.e. on E \ F
n
n}.
(ii) A subset N ⊂ E is E -polar if and only if there is an E -nest {Fn}n≥1 such
that N ⊂
n≥1(E \ Fn).
(iii) A function f on E is said to be quasi-continuous if there is an E -nest
{Fn}n≥1 such that f |F is continuous on F
n
n for each n ≥ 1; we denote this situation
brieﬂy by writing f ∈ C({Fn}).
(iv) A statement depending on x ∈ A is said to hold quasi-everywhere (q.e. in
abbreviation) on A if there is an E -polar set N ⊂ A such that the statement is true
for every x ∈ A \ N .
(v) A nearly Borel subset N ⊂ E is called properly exceptional if m(N) = 0
and
Px(Xt ∈ E \ N for t ≥ 0 and Xt− ∈ E \ N for t > 0) = 1
for every x ∈ E \ N.

934
CHEN, FITZSIMMONS, KUWAE AND ZHANG
It is known (cf. [7]) that a family {Fn} of closed sets is an E -nest if and only if
Px
lim τF = ζ = 1
for q.e. x ∈ E.
n→∞
n
It is also known that a properly exceptional set is E -polar and that every E -polar
set is contained in a properly exceptional set. Every element u in F admits a
quasi-continuous m-version. We assume throughout this section that functions in
F are always represented by their quasi-continuous m-versions. In the sequel, the
abbreviations CAF, PCAF and MAF stand for “continuous additive functional,”
“positive continuous additive functional” and “martingale additive functional,” re-
spectively; the deﬁnitions of these terms can be found in [7].
◦
Let M and Nc denote, respectively, the space of MAF’s of ﬁnite energy and the
space of continuous additive functionals of zero energy. For u ∈ F , Fukushima’s
decomposition holds:
(1.2)
u(Xt ) − u(X0) = Mu +
t
N u
t
for every t ∈ [0, ∞[,
◦
Px -a.s. for q.e. x ∈ E, where Mu ∈M and Nu ∈ Nc.
A positive continuous additive functional (PCAF) of X (call it A) determines a
measure μ = μA on the Borel subsets of E via the formula
1
t
(1.3)
μ(f ) =↑ lim Em
f (Xs) dAs ,
t ↓0 t
0
in which f : E → [0, ∞] is Borel measurable. Here, ↑ limt↓0 indicates an increas-
ing limit as t ↓ 0. The measure μ is necessarily smooth, in the sense that μ charges
no E -polar set of X and there is an E -nest {Fn} of closed subsets of E such that
μ(Fn) < ∞ for each n ∈ N. Conversely, given a smooth measure μ, with A = Aμ.
In the sequel, we refer to this bijection between smooth measures and PCAF’s as
the Revuz correspondence and to μ as the Revuz measure of Aμ.
If M is a locally square-integrable martingale additive functional (MAF) of X
on the random time interval [ 0, ζ [ , then the process M (the dual predictable
projection of [M]) is a PCAF (Proposition 2.8) and the associated Revuz measure
[as in (1.3)] is denoted by μ M . More generally, if Mu is the martingale part in the
Fukushima decomposition of u ∈ F , then Mu, M is a CAF locally of bounded
variation and we have the associated Revuz measure μ Mu,M , which is locally the
difference of smooth (positive) measures. For u ∈ F , the Revuz measure μ Mu of
Mu will usually be denoted by μ u .
Let (N (x, dy), Ht ) be a Lévy system for X; that is, N(x, dy) is a kernel on
(E , B(E )) and Ht is a PCAF with bounded 1-potential such that for any non-
negative Borel function φ on E × E vanishing on the diagonal and any x ∈ E ,
t
Ex
φ(Xs−, Xs) = Ex
φ(Xs, y)N(Xs, dy) dHs .
s≤t
0
E

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
935
To simplify notation, we will write
N φ(x) :=
φ(x, y)N (x, dy)
E
and
t
(N φ ∗ H )t :=
N φ(Xs) dHs.
0
Let μH be the Revuz measure of the PCAF H . The jumping measure J and the
killing measure κ of X are then given by
J (dx, dy) = 1 N(x, dy)μ
2
H (d x)
and
κ(dx) = N(x, { })μH (dx).
These measures feature in the Beurling–Deny decomposition of E : for f, g ∈ F ,
E (f, g) = E(c)(f, g) +
f (x) − f (y) g(x) − g(y) J (dx, dy)
E×E
+
f (x)g(x)κ(dx),
E
where E (c) is the strongly local part of E .
For u ∈ F , the martingale part Mut in (1.2) can be decomposed as
u,j
Mu =
t
Mu,c
t
+ Mt + Mu,κ
t
for every t ∈ [0, ∞[,
Px-a.s. for q.e. x ∈ E, where Mu,c
t
is the continuous part of the martingale Mu and
u,j
Mt
= lim
u(Xs) − u(Xs−) 1{|u(Xs)−u(Xs−)|>ε}1{s<ζ}
ε↓0 0<s≤t
t
−
u(y) − u(Xs) N(Xs, dy) dHs ,
0
{y∈E:|u(y)−u(Xs)|>ε}
t
Mu,κ
t
=
u(Xs)N(Xs, { }) dHs − u(Xζ−)1{t≥ζ}
0
are the jump and killing parts of Mu, respectively. All three terms in this decom-
◦
position of Mu are elements of M; see Theorem A.3.9 of [7]. The limit in the
expression for Mu,j is in the sense of convergence in the norm of the space of
MAF’s of ﬁnite energy and of convergence in probability under Px for q.e. x ∈ E
(see [7]).
Let N ∗ ⊂
c
Nc denote the class of continuous additive functionals of the form
N u + ·0 g(Xs) ds for some u ∈ F and g ∈ L2(E; m). Nakao [14] constructed a
◦
linear map
from M into N ∗
c in the following way. It is shown in [14] that, for
◦
every Z ∈M, there is a unique w ∈ F such that
(1.4)
E1(w, f ) = 1 μ
2
Mf +Mf,κ ,Z (E)
for every f ∈ F .

936
CHEN, FITZSIMMONS, KUWAE AND ZHANG
This unique w is denoted by γ (Z). The operator
is now deﬁned by
t
γ (Z)
◦
(1.5)
(Z)t := Nt
−
γ (Z)(Xs) ds
for every Z ∈M .
0
Nakao showed that
(Z) is characterized by the following equation
1
(1.6)
lim Eg·m[ (Z)t ] = −1μ Mg+Mg,κ,Z (E)
for every g ∈ Fb.
t ↓0 t
2
Here, Fb := F ∩ L∞(E; m). So, in particular, we have (Mu) = Nu for u ∈ F .
Nakao [14] then used the operator
to deﬁne a stochastic integral
t
(1.7)
f (Xs) dNu :=
s
(f ∗ Mu)t − 1 Mf,c + Mf,j , Mu,c + Mu,j
2
t ,
0
where u ∈ F , f ∈ F ∩ L2(E; μ u ) and (f ∗ Mu)t := t0 f (Xs−) dMus. If we de-
ﬁne
Nc := {N ∈ Nc | N = Nu + Aμ for some u ∈ F
and some signed smooth measure μ},
then we see, by (1.5), that · f (X
∈
0
s ) d N u
s
Nc if u ∈ F and f ∈ F ∩ L2(E; μ u ).
However, the conditions imposed on the integrand f (Xt ) and on the integrator
N u in Nakao’s stochastic integral are too restrictive for certain applications, in
particular the perturbation theory of general symmetric Markov processes, which
requires more general integrators as well as integrands; see [2].
The purpose of this paper is to provide a new way of deﬁning (M) and Nakao’s
stochastic integral for zero-energy AF’s N u.
For a ﬁnite rcll AF Mt , it is known (see [3], Lemma 3.2) that there is a Borel
function ϕ on E × E with ϕ(x, x) = 0 for all x ∈ E so that
(1.8)
Mt − Mt− = ϕ(Xt−, Xt)
for every t ∈ ]0, ζ [, Pm-a.e.
Such a ϕ is uniquely determined up to J -negligible sets. We will call ϕ the jump
function of M. When M = Mu, u ∈ F , the jump function ϕ for Mu can be taken to
be as ϕ(x, y) = u(y) − u(x) for (x, y) ∈ E × E, with u( ) := 0. We have a similar
result for locally square-integrable MAF’s on [ 0, ζ [ [see Deﬁnition 2.5(iii) for
the deﬁnition of a locally square-integrable MAF on [ 0, ζ [ ]. Let M be a locally
square-integrable MAF on [ 0, ζ [ . There then exists a jump function ϕ on E × E
for M satisfying the property (1.8) (see Corollary 2.9). Assume that
t
ϕ21{|ϕ|≤1} + |ϕ|1{|ϕ|>1} (Xs, y)N(Xs, dy) dHs < ∞
0
E
(1.9)
for every t < ζ,

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
937
Px-a.s. for q.e. x ∈ E, where ϕ(x, y) := ϕ(x, y) + ϕ(y, x) for x, y ∈ E. By
Lemma 3.2 below, there is a unique purely discontinuous local MAF K on [ 0, ζ [
with
Kt − Kt− = −ϕ(Xt−, Xt )
for t < ζ, Px-a.s. for q.e. x ∈ E.
Deﬁne Pm-a.e. on [0, ζ [,
(M)t := − 1 M
2
t + Mt ◦ rt + ϕ(Xt , Xt−) + Kt
for t ∈ [0, ζ [,
where rt is the time-reversal operator at time t > 0. Note that since X is symmet-
ric, the measure Pm, when restricted to {t < ζ }, is invariant under rt . This time
reversibility plays an important role in this paper. So,
(M) is clearly well deﬁned
on [ 0, ζ [ under the σ -ﬁnite measure Pm. It will be shown in Theorem 2.18 and
Remark 3.4(ii) below that
(M) is a continuous even AF of X on [ 0, ζ [ admit-
ting m-null set. Note that when M = Mu for some u ∈ F , ϕ(x, y) = u(y) − u(x)
is antisymmetric and so ϕ = 0. Thus, Pm-a.e. on {t < ζ },
(Mu)t := − 1 Mu + Mu ◦ r
2
t
t
t + u(Xt−) − u(Xt ) = N u
t .
The last identity follows by applying the time-reversal operator to both sides
of (1.2) and using the fact that N u ◦
t
rt = Nut Pm-a.e. on [ 0, ζ [ (cf. [4], Theo-
rem 2.1). It then follows for every u ∈ F that
(Mu) = (Mu) on [ 0, ζ [ Pm-a.e.
We will show in Theorem 3.6 below that this holds when Mu is replaced by any
◦
M ∈M. Therefore, under the σ -ﬁnite measure Pm,
is a genuine extension of
Nakao’s map
.
A function f is said to be locally in F (denoted as f ∈ Floc) if there is an
increasing sequence of ﬁnely open Borel sets {Dk, k ≥ 1} with
∞
k=1 Dk = E q.e.
and for every k ≥ 1, there is fk ∈ F such that f = fk m-a.e. on Dk. For two
subsets A, B of E, we denote A = B q.e. if A B := (A \ B) ∪ (B \ A) is E -polar.
By deﬁnition, every f ∈ Floc admits a quasi-continuous m-version, so we may
assume that all f ∈ Floc are quasi-continuous. We then have f = fk q.e. on Dk.
For f ∈ Floc, Mf,c is well deﬁned as a continuous MAF on [ 0, ζ [ of locally ﬁnite
energy. Moreover, for f ∈ Floc and a locally square-integrable MAF M on [ 0, ζ [ ,
t
t → (f ∗ M)t :=
f (Xs−) dMs
0
is a locally square-integrable MAF on [ 0, ζ [ . Here, for a locally square-integrable
MAF M on [ 0, ζ [ , denote by Mc its continuous part, which is also a locally
square-integrable MAF on [ 0, ζ [ (see Theorem 8.23 in [9]).
DEFINITION 1.2 (Stochastic integral).
Suppose that M is a locally square-
integrable MAF on [ 0, ζ [ and that f ∈ Floc. Let ϕ : E ×E → R be a jump function

938
CHEN, FITZSIMMONS, KUWAE AND ZHANG
for M and assume that ϕ satisﬁes condition (1.9). Deﬁne, on [ 0, ζ [ ,
t
f (Xs−) d (M)s
0
:= (f ∗ M)t − 1 Mf,c, Mc
2
t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs ,
0
E
whenever
(f ∗ M) is well deﬁned and the third term in the right-hand side of
(3.10) is absolutely convergent.
The above stochastic integral is well deﬁned on [ 0, ζ [ under the σ -ﬁnite mea-
sure Pm and extends that of Nakao (1.7). [See Remark 3.9(i) and Theorem 3.10
below.] We will show in Theorem 4.7 below that it enjoys a generalized Itô for-
mula.
2. Additive functionals.
In this section, we will prove some facts about ad-
ditive functionals, to be used later. We begin with some details on the completion
of ﬁltrations. Let P (E) be the family of all probability measures on E. For each
ν ∈ P (E), let F ν
∞ (resp., F ν
t ) be the Pν -completion of F 0
∞ (resp., Pν -completion
of F 0
t in F ν
∞) and set F∞ :=
ν∈P (E) F ν
∞ and Ft :=
ν∈P (E) F ν
t . Let F m
∞ (resp.,
F m
t ) be the Pm-completion of F 0
∞ (resp., Pm-completion of F 0
t in F m
∞ ). Although
m may not be a ﬁnite measure on E, we do have F∞ ⊂ F m
∞ , Ft ⊂ F m
t
because
for g ∈ L1(E; m) with 0 < g ≤ 1 on E satisfying gm ∈ P (E), Pgm-negligibility
is the same as Pm-negligibility.
For a ﬁxed ﬁltration (Mt ) on ( , M), we recall the notions of (Mt )-predictabil-
ity, (Mt )-optionality and (Mt )-progressive measurability as follows (see [15] for
more details). On [0, ∞[ × , the (Mt )-predictable [resp., (Mt )-optional] σ -ﬁeld
P (Mt ) [resp., O(Mt )] is deﬁned as the smallest σ -ﬁeld over [0, ∞[ ×
contain-
ing all Pν(M)-evanescent sets for all ν ∈ P (E ) and with respect to which all
Mt -adapted lcrl (left-continuous, right-limited) (resp., rcll) processes are measur-
able. A process φ(s, ω) on [0, ∞[ ×
is said to be (Mt )-progressively measurable
provided [0, t] ×
(s, ω) → φ(s, ω) is B([0, t]) ⊗ Mt -measurable for all t > 0.
It is well known that (Mt )-predictability implies (Mt )-optionality, which in turn
implies (Mt )-progressive measurability.
For [0, ∞]-valued functions S, T on
with S ≤ T , we employ the usual nota-
tion for stochastic intervals; for example,
[ S, T [ := {(t, ω) ∈ [0, ∞[ × | S(ω) ≤ t < T (ω)},
the other species of stochastic intervals being deﬁned analogously. We write
[ S] := [ S, S] for the graph of S. Note that these are all subsets of [0, ∞[× .
If S and T are (Mt )-stopping times, then [ S, T ] , [ S, T [ , . . . and [ S] are (Mt )-
optional (see Theorem 3.16 in [9]).

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
939
DEFINITION 2.1 (AF).
An (Ft )-adapted [resp., (F m
t )-adapted] process A =
(At )t≥0 with values in [−∞, ∞] is said to be an additive functional (AF in short)
(resp., AF admitting m-null set) if there exist a deﬁning set
∈ F∞ and an E-polar
(resp., m-null) set N satisfying the following conditions:
(i) Px( ) = 1 for all x ∈ E \ N ;
(ii) θt
⊂ for all t ≥ 0; in particular, ω ∈ and P ( ) = 1 because ω =
θζ(ω)(ω) for all ω ∈ ;
(iii) for all ω ∈
, A·(ω) is right-continuous with left limits on [0, ζ(ω)[,
A0(ω) = 0, |At (ω)| < ∞ for t < ζ(ω) and At+s(ω) = At (ω) + As(θt ω) for all
t, s ≥ 0;
(iv) for all t ≥ 0, At (ω ) = 0; in particular, under the additivity in (iii),
At (ω) = Aζ(ω)(ω) for all t ≥ ζ(ω) and ω ∈ .
An AF A (admitting m-null set) is called right-continuous with left limits
(rcll AF in brief) if Aζ(ω)− exists for each ω ∈ . An AF A (admitting m-null
set) is said to be ﬁnite [resp., continuous additive functional (CAF in brief)] if
|At(ω)| < ∞, t ∈ [0, ∞[ (resp. t → At(ω) is continuous on [0, ∞[) for each
ω ∈ . A [0, ∞[-valued CAF is called a positive continuous additive functional
(PCAF in short). Two AF’s A and B are called equivalent if there exists a com-
mon deﬁning set
∈ F∞ and an E-polar set N such that At(ω) = Bt(ω) for all
t ∈ [0, ∞[ and ω ∈ . We call A = (At )t≥0 an AF on [ 0, ζ [ or a local AF (ad-
mitting m-null set) if A is (Ft )-adapted and satisﬁes (i), (ii), (iv) and the property
(iii) in which (iii) is modiﬁed so that the additivity condition is required only
for t + s < ζ(ω). The notions of rcll AF, CAF and PCAF on [ 0, ζ [ are deﬁned
similarly. Two AF’s on [ 0, ζ [ , A and B, are called equivalent if there exists a
common deﬁning set
∈ F∞ and an E-polar set N such that At(ω) = Bt(ω) for
all t ∈ [0, ζ [ and ω ∈ .
REMARK 2.2.
Any PCAF A on [ 0, ζ [ can be extended to a PCAF by setting
lim Au(ω),
if t ≥ ζ(ω) > 0,
At (ω) := u↑ζ
0,
if t ≥ ζ(ω) = 0,
for ω ∈
and setting At (ω) ≡ 0 for ω ∈ c. The (Ft )-adaptedness of this ex-
tended A holds as follows: for a ﬁxed T > 0, we know {At ≤ T } ∩ {t < ζ } ∈ Ft .
From this, we have the Fζ -measurability of {Aζ ≤ T }. Indeed, {Aζ ≤ T } =
{
t ∈Q
A
+
t ≤ T , t < ζ } ∈ Fζ as {At ≤ T , t < ζ } ∈ Fζ for any t ≥ 0. Thus, {At ≤
T } ∩ {t ≥ ζ } = {Aζ ≤ T } ∩ {t ≥ ζ } ∈ Ft . Therefore, {At ≤ T } ∈ Ft for any T > 0,
which gives the (Ft )-adaptedness of A. Noting that ζ ◦ θt = ζ − t if t < ζ and
ζ ◦ θt = 0 if t ≥ ζ , we conclude that Aζ = At + Aζ ◦ θt for any t ∈ [0, ∞[ on
.
Consequently, At+s = At + As ◦ θt holds for any t, s ∈ [0, ∞[ on .
The following lemma is a special case of [14], Theorem 2.2.

940
CHEN, FITZSIMMONS, KUWAE AND ZHANG
LEMMA 2.3.
Let A, B be PCAF’s such that for m-a.e. x ∈ E, Ex[At ] =
Ex[Bt ] for all t ≥ 0 and suppose that the Revuz measure μA has ﬁnite total mass.
A is then equivalent to B.
REMARK 2.4.
The above lemma may fail if the condition μA(E) < ∞ is not
satisﬁed. For example, take E = Rd with d ≥ 2 and let X be Brownian motion
on Rd and μA(dx) = |x|−d−1 dx. μA is then a smooth measure and corresponds
to a PCAF A of X. Let Bt = At + t, which is a PCAF of X with Revuz measure
μA(dx) + dx. However,
t
Ex[At ] =
p(s, x, y)|y|−d−1 dy ds = ∞ = Ex[Bt ]
0
Rd
for every x ∈ Rd \ {0}.
Here, p(s, x, y) = (2πt)−d/2 exp(−|x −y|2/(2t)) is the transition density function
of X.
As usual, if T is an (Ft )-stopping time and M a process, then MT is the stopped
process deﬁned by MT :=
t
Mt∧T . Following [9], we give the notion of local mar-
tingales of interval type.
DEFINITION 2.5 (Processes of interval type). Let D be a class of (Ft )-adapted
processes and denote by Dloc its localization (resp., by Df -loc its localization by
a nest of ﬁnely open Borel sets); that is, M ∈ Dloc (resp., M ∈ Df -loc) if and
only if there exists a sequence Mn ∈ D and an increasing sequence of stopping
times Tn with Tn → ∞ (resp., a nest {Gn} of ﬁnely open Borel sets) such that
MTn = (Mn)Tn (resp., Mt = Mnt for t < τG ) for each n. Here, a family {G
n
n}
of ﬁnely open Borel sets is called a nest if Px(limn→∞ τG = ζ ) = 1 for q.e.
n
x ∈ E. (However, see Lemma 3.1.) Clearly, D ⊂ Dloc (resp., D ⊂ Df -loc) and
(Dloc)loc = Dloc [resp., (Df -loc)f -loc = Df -loc]. If D is a subclass of AF’s, then
so is Dloc [for if M ∈ Dloc, then there exist Mn and Tn as above and for each
ω and t, s ≥ 0, there exists n ∈ N with s + t < Tn(ω) and s < Tn(θt ω), hence
Mt+s(ω) = Mt (ω) + Ms(θt ω)], while Df -loc is contained in the class of AF’s on
[ 0, ζ [ .
(i) B ⊂ [0, ∞[ ×
is called a set of interval type if there exists a nonnegative
random variable S such that for each ω ∈ , the section Bω := {t ∈ [0, ∞[|(t, ω) ∈
B} is [0, S(ω)] or [0, S(ω)[ and Bω = ∅.
(ii) Let B be an (Ft )-optional set of interval type. A real-valued stochas-
tic process M on B [i.e., M1B = (Mt (ω)1B(t, ω))t≥0 is a real-valued stochas-
tic process] is said to be in DB if and only if there exists N ∈ D such that
M1B = N1B and is said to be locally in D on B [write M ∈ (Dloc)B ] if and
only if S := DBc is the debut of Bc and there exists an increasing sequence of
(Ft )-stopping times {Sn} with limn→∞ Sn = S and a sequence of Mn ∈ D such

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
941
that Bω ⊂
∞ [
n=1 0, Sn(ω)] Px -a.s. ω ∈
and (M1B)Sn = (Mn1B)Sn for all n ∈ N
and t ≥ 0, Px-a.s. ω ∈
for q.e. x ∈ E. Clearly, DB ⊂ (Dloc)B . Moreover,
DB2 ⊂ DB1 and (Dloc)B2 ⊂ (Dloc)B1 for any pair of (Ft )-optional sets B1, B2
of interval type with B1 ⊂ B2.
(iii) Let B be an (Ft )-optional set of interval type. We set
M1 := {M | M is a ﬁnite rcll AF, Ex[|Mt |] < ∞,
Ex[Mt ] = 0 for E-q.e. x ∈ E and all t ≥ 0}
and speak of an element of (M1)B [resp., (M1 )B ] as being an MAF on B (resp.,
loc
a local MAF on B). Similarly,
M := {M | M is a ﬁnite rcll AF, Ex[M2]
t
< ∞,
Ex[Mt ] = 0 for E-q.e. x ∈ E and all t ≥ 0}
and an element of MB [resp., (Mloc)B ] is a square-integrable MAF on B (resp.,
locally square-integrable MAF on B). We further set
Mc : = {M ∈ M | M is a CAF},
Md : = {M ∈ M | M is a purely discontinuous AF}
and an element of (Mc )B [resp., (Md )B ] is called a locally square-integrable
loc
loc
continuous MAF on B (resp., locally square-integrable purely discontinuous MAF
on B). For M ∈ (Mloc)B , M admits a unique decomposition M = Mc + Md with
Mc ∈ (Mc )B and Md ∈ (Md )B (see Theorem 8.23 in [9]). In these deﬁnitions,
loc
loc
we omit the usage “on B” when B = [0, ∞[× .
For a [0, ∞]-valued function R on
and A ⊂ , RA := R · 1A + (+∞) · 1Ac
is called the restriction of R on A. Clearly, R ≤ RA.
REMARK 2.6.
When B = [ 0, R[ for a given (Ft )-stopping time R, there
is another notion of “locally in D on B,” obtained by replacing (M1B)Sn =
(Mn1B)Sn with MSn1B = (Mn)Sn1B in our deﬁnition; this is a weaker notion than
ours because t → 1B(t, ω) is decreasing and 1B(t, ω)1B(s, ω) = 1B(t, ω) for s ≤ t
and ω ∈ . This weaker notion is described in [15].
◦
DEFINITION 2.7 (MAF locally of ﬁnite energy).
Recall that M is the totality
of MAF’s of ﬁnite energy, that is,
◦
1
M:= M ∈ M e(M) := lim
Em[M2]
t
< ∞ .
t ↓0 2t
◦
◦
We say that an AF M on [ 0, ζ [ is locally in M (and write M ∈Mf -loc) if there ex-
◦
ists a sequence {Mn} in M and a nest {Gn} of ﬁnely open Borel sets such that

942
CHEN, FITZSIMMONS, KUWAE AND ZHANG
Mt = Mnt for t < τG for each n ∈ N. In case X is a diffusion process with
n
no killing inside E, we can deﬁne the predictable quadratic variation M for
◦
M ∈Mf -loc as follows. First, note that Mn
=
t ∧τ
Mm
for n < m because of
Gn
t ∧τGn
the continuity of Mn. Owing to the uniqueness of Doob–Meyer decomposition,
we see that Mn t∧τ
= Mm
. The predictable quadratic variation M of
Gn
t ∧τGn
◦
M ∈Mf -loc as a PCAF is well deﬁned by setting M t = Mn t , t < τG , n ∈ N,
n
with Remark 2.2 and by choosing an appropriate deﬁning set and E -polar set
◦
of M , where Mn ∈M and {Gn} is a nest of ﬁnely open Borel sets such that
Mt = Mnt, t < τG .
n
◦
PROPOSITION 2.8.
(Mloc)[ 0,ζ[ ⊂ Mf -loc. More precisely, for each M ∈
◦
(Mloc)[ 0,ζ[ , there exists a nest {Gk} of ﬁnely open Borel sets such that 1G ∗M ∈M
k
for each k ∈ N and the predictable quadratic variation process M can be con-
structed as a PCAF.
PROOF.
Let M ∈ (Mloc)[ 0,ζ[ . There then exists an increasing sequence {Tn}
of stopping times with limn→∞ Tn = ζ (Px-a.s. ω ∈
for q.e. x ∈ E) and
Mn ∈ Mloc such that Mt∧T 1
1
n
[0,ζ[(t ∧ Tn) = Mn
t ∧T
[0,ζ[(t ∧ Tn) holds for all t ≥ 0
n
Px-a.s. for q.e. x ∈ E. We may assume that it holds for all ω ∈
by changing
the sample space. Note that [0, ζ(ω)[ ⊂
∞ [
n=1 0, Tn(ω)] for all ω ∈
. Hence,
Mm
1
1
t ∧T
[0,ζ[(t ∧ Tn) = Mn
[0,ζ[(t ∧ Tn) for n < m. As noted in Deﬁnition 2.5,
n
t ∧Tn
we see that M is an AF on [ 0, ζ [ . Owing to the uniqueness of the Doob–Meyer de-
composition for semimartingales on [ 0, ζ [ (see [9]), we have Mm t∧T 1
n
[0,ζ[(t ∧
Tn) = Mn t∧T 1
n
[0,ζ[(t ∧ Tn) for n < m. Thus, we have Mm t = Mn t for t < Tn
and n < m. The predictable quadratic variation M of M is therefore well deﬁned
by setting M t := Mn t for t < Tn. Setting M t := M ζ := lims↑ζ M s for all
t ≥ ζ , we obtain a PCAF because of Remark 2.2. Let μ M be the Revuz measure
corresponding to M and {Fk} an E -nest of closed sets such that μ M (Fk) < ∞
for each k, and let Gk be the ﬁne interior of Fk. {Gk} is then a nest. In view of the
proofs of Theorem 5.6.1 and Lemma 5.6.2 in [7], the stochastic integral 1G ∗ M
k
is of ﬁnite energy with e(1G ∗ M) = 1 μ
k
2
M (Gk ) and its predictable quadratic
variation 1G ∗ M is a PCAF. Let μ
k
k (resp., μk ) be the Revuz measure corre-
◦
sponding to 1G ∗ M (resp., 1
∗ M, M ). By Lemma 5.6.2 in [7], for M M
k
Gk
i ∈
and fi ∈ L2(E; dμ M ) (i = 1, 2), we have f
= dμ
,
i
1f2dμ M1,M2
f1∗M1,f2∗M2
hence
t
0 (f1f2)(Xs ) d M1, M2 s = f1 ∗ M1, f2 ∗ M2 t . From this, we see that
μk, f 2 = μk, f 2 = 1G μ
k
M , f 2
for any f ∈ L2(E; μ M ); consequently,
we have μk = μk = 1G μ
∗ M
k
M
by μ M (Gk) < ∞. This yields 1Gk
t =
1G ∗ M, M
1
(X
∗ M
k
t =
t
0
Gk
s ) d M s for t < ζ , hence
M − 1Gk
t = 0 for
◦
t < τG . Therefore, M
∗ M)
and 1
∗ M ∈M.
k
t = (1Gk
t for t < τGk
Gk

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
943
COROLLARY 2.9.
Let M be a locally square-integrable MAF on [ 0, ζ [ ,
that is, M ∈ (Mloc)[ 0,ζ[ . There then exists a Borel function ϕ on E × E with
ϕ(x, x) = 0 for all x ∈ E such that
Mt − Mt− = ϕ(Xt−, Xt)
for every t ∈]0, ζ [, Pm-a.e.
PROOF.
By the proof of Proposition 2.8, there exists an E -nest {Fk} such that
◦
for each k ∈ N Mk := 1F ∗ M ∈M and M
. Let ϕ
k
t = M k
t , t < τFk
k be the jump
function corresponding to Mk. We then have ϕk(Xt−, Xt ) = ϕ (Xt−, Xt ), t < τF ,
k
Pm-a.e., for k < . From this, we see that ϕk = ϕ J -a.e. on Fk × Fk. We construct
a Borel function ϕ on E × E in the following manner. We set F0 := ∅, ϕ(x, y) :=
ϕk(x, y) for (x, y) ∈ Fk × Fk \ (Fk−1 × Fk−1), k ∈ N, ϕ(x, y) := 0 if (x, y) ∈
E × E \ ( ∞
k=1 Fk ×
∞
k=1 Fk). ϕ then satisﬁes ϕ(x, x) = 0 for x ∈ E. We also
have ϕ = ϕk J -a.e. on Fk × Fk. Consequently, ϕ(Xt−, Xt ) = ϕk(Xt−, Xt), t <
τF , P
, P
k
m-a.e. This means that Mt − Mt− = ϕ(Xt−, Xt ), t < τFk
m-a.e. Therefore,
Mt − Mt− = ϕ(Xt−, Xt ), 0 < t < ζ Pm-a.e.
We recall the deﬁnition of the shift operator θs and the time-reversal operator rt
on the path space
. For each s ≥ 0, the shift operator θs is deﬁned by θsω(t) :=
ω(t + s) for t ∈ [0, ∞[. Given a path ω ∈ {t < ζ }, the operator rt is deﬁned by
(2.1)
rt (ω)(s) := ω (t − s)− ,
if 0 ≤ s ≤ t,
ω(0),
if s ≥ t.
Here, for r > 0, ω(r−) := lims↑r ω(s) is the left limit at r and we use the con-
vention that ω(0−) := ω(0). For a path ω ∈ {t ≥ ζ }, we set rt (ω) := ω . We note
that
lim rt (ω)(s) = ω(t−) = rt (ω)(0) and
s↓0
(2.2)
lim rt (ω)(s) = ω(0) = rt (ω)(t).
s↑t
A key consequence of the m-symmetry assumption on the Hunt process X is that
the measure Pm, when restricted to {t < ζ }, is invariant under the time-reversal
operator rt .
Clearly for t, s > 0, θs :
→ is F m
t +s /F m
t -measurable. The following lemma
deals with the measurability issue of the time-reversal operator rt .
LEMMA 2.10.
For each t > 0, rt :
→
is F 0
t /F 0
∞-measurable and
F m
t /F m
t -measurable.
PROOF.
Let Fi ∈ B(E ) and si ∈ [0, ∞[, i = 1, 2, . . . , n, with s1 < s2 <
· · · < sk ≤ t < sk+1 < · · · < sn for some k ∈ {1, 2, . . . , n}. Then,
r−1
t
( n
(F
◦ r
i=1 X−1
s
i )) =
n
t )−1(Fi ) is equal to
k
i
i=1(Xsi
i=1({X(t−si )− ∈ Fi , t <

944
CHEN, FITZSIMMONS, KUWAE AND ZHANG
ζ } ∪ { ∈ Fi, t ≥ ζ }) ∩
n
i=k+1({X0 ∈ Fi , t < ζ } ∪ {
∈ Fi, t ≥ ζ}) ∈ F 0t. Next,
we show the F m
t /F m
t -measurability of rt . Take C ∈ F m
t . There then exist D ∈ F 0
t
and N ∈ F 0
∞ such that C
D ⊂ N and Pm(N) = 0. Since Pm({ω }) = 0, by delet-
ing {ω } = {ω ∈
| ζ(ω) = 0} ∈ F 0 ⊂ F 0
0
t , we may assume that ω
/
∈ C ∪D ∪N.
Then, r−1
t
(C)
r−1
t
(D) ⊂ r−1
t
(N ), r−1
t
(D), r−1
t
(N ) ∈ F 0
t
and Pm(r−1
t
(N )) =
Pm(r−1
t
(N ) ∩ {t < ζ }) + 1N (ω )Pm(t ≥ ζ ) = Pm(N ∩ {t < ζ }) = 0.
DEFINITION 2.11.
For any t > 0, we say that two sample paths ω and ω are
t -equivalent if ω(s) = ω (s) for all s ∈ [0, t]. We say that two sample paths ω and
ω are pre-t -equivalent if ω(s) = ω (s) for all s ∈ [0, t[.
For an rcll AF At adapted to (F 0
t )t ≥0, At (ω) = At (ω ) if ω and ω
are
t -equivalent At−(ω) = At−(ω ) if ω and ω are pre-t-equivalent. These conclu-
sions may fail to hold if the measurability conditions are not satisﬁed. We need the
following notion.
DEFINITION 2.12 (PrAF).
A process A = (At )t≥0 with values in R :=
[−∞, ∞] is said to be a progressively additive functional (PrAF in short) (resp.,
PrAF admitting m-null set) if A is (Ft )-adapted [resp., (F m
t )-adapted] and there
exist deﬁning sets
∈ F∞, t ∈ Ft [resp., ∈ F m
∞ ,
t ∈ F m
t ] for each t > 0 and
an E -polar (resp., m-null) set N satisfying the following conditions:
(i) Px( ) = 1 for all x ∈ E \ N ,
⊂ t ⊂ s for every t > s > 0 and
=
t >0
t ;
(ii) θt
⊂ for all t ≥ 0 and θt−s( t) ⊂ s for all s ∈ ]0, t[, and, in particular,
ω ∈
⊂ t and P ( ) = P ( t) = 1 under (i);
(iii) for all ω ∈ t , A(ω) is deﬁned on [0, t[, is right continuous on [0, t ∧ ζ(ω)[
and has left limit on ]0, t] ∩ ]0, ζ(ω)[ such that A0(ω) = 0, |As(ω)| < ∞ for s ∈
[0, t ∧ ζ(ω)[ and Ap+q(ω) = Ap(ω) + Aq(θpω) for all p, q ≥ 0 with p + q < t;
(iv) for all t ≥ 0, At (ω ) = 0;
(v) for any t > 0 and pre-t -equivalent paths ω, ω ∈
, ω ∈ t implies that
ω ∈ t , As(ω) = As(ω ) for any s ∈ [0, t[ and As−(ω) = As−(ω ) for any s ∈
]0, t].
Furthermore, A is called an rcll PrAF (or an rcll PrAF admitting m-null set) if, for
each t > 0 and ω ∈ t , s → As(ω) is right-continuous on [0, t[ and has left-hand
limits on ]0, t] and a PrAF (or a PrAF admitting m-null set) is said to be ﬁnite
(resp., continuous) if |As(ω)| < ∞ for all s ∈ [0, t[ (resp., continuous on [0, t[) for
every ω ∈ t .
We say that an AF A on [ 0, ζ [ (resp., AF A on [ 0, ζ [ admitting m-null set) is a
PrAF on [ 0, ζ [ (resp., PrAF on [ 0, ζ [ admitting m-null set) if A is (Ft )-adapted
[resp., (F m
t )-adapted] and there exist
∈ F∞, t ∈ Ft (resp., ∈ F m
∞ , t ∈ F m
t )
for each t > 0 and an E -polar (resp., m-null) set N such that (i ), (ii), (iii ), (iv) and

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
945
(v ) hold—(i ): Px( ) = 1 for all x ∈ E \ N ,
⊂ t for all t > 0,
= t>0 t
and
t ∩ {t < ζ } ⊂
s ∩ {s < ζ } for s < t ; (iii ): for each ω ∈
t ∩ {t < ζ }, the
same conclusion as in (iii) holds; (v ): for any t > 0 and pre-t -equivalent paths
ω, ω ∈
∩ {t < ζ}, the same conclusion as in (v) holds.
The notion of rcll PrAF on [ 0, ζ [ (or rcll PrAF admitting m-null set) is simi-
larly deﬁned.
REMARK 2.13.
(i) Our notion of PrAF is different from what is found in
Walsh [16].
(ii) Every PrAF (resp., PrAF on [ 0, ζ [ ) is an AF (resp., AF on [ 0, ζ [ ).
(iii) The MAF Mu and the CAF N u of 0-energy appearing in Fukushima’s
decomposition (1.2) can be regarded as ﬁnite rcll PrAF’s in view of the proof of
Theorem 5.2.2 in [7]. In this case, the deﬁning sets for Mu as PrAF are given by
:= {ω ∈ | Mun
s−(ω) converges uniformly on ]0, t ] for ∀t ≥ 0
for some subsequence nk} ∈ F∞,
t := {ω ∈
| Mun
s−(ω) converges uniformly on ]0, t ]
for some subsequence nk} ∈ Ft
for every t > 0, where Mun
t
:= un(Xt) − un(X0) − t0(un(Xs) − fn(Xs))ds with
fn := n(u − nRn+1u) and un := R1fn = nRn+1u. Hence, an MAF of stochastic
integral type t0 g(Xs−) dMus [g, u ∈ F with g ∈ L2(E; μ u )] can be regarded as
a ﬁnite rcll PrAF. Consequently, any MAF of ﬁnite energy can also be regarded
as an rcll PrAF, in view of the assertion of Lemma 5.6.3 in [7] and Lemma 2.14
below.
◦
(iv) Every M ∈Mf -loc can be regarded as a PrAF on [ 0, ζ [ , hence every M ∈
[
M 0,ζ [ is also. Since every local martingale can be written as the sum of a local
loc
martingale with bounded jumps (and hence a locally square-integrable martingale)
and a local martingale of ﬁnite variation, we conclude that every local MAF is a
PrAF.
LEMMA 2.14.
Let (An) be a sequence of ﬁnite rcll PrAF’s with deﬁning sets
n ∈ F∞ and n ∈
t
Ft . For each t > 0, set
n
t :=
ω ∈
t
An converges uniformly on [0, t[ ∈ Ft
n∈N
and
:= ω ∈
n
An converges uniformly on [0, t[ for every t ∈ [0, ∞[ ∈ F∞.
n∈N

946
CHEN, FITZSIMMONS, KUWAE AND ZHANG
Suppose that there exists an E -polar set N such that Px( ) = 1 for x ∈ E \ N . If
we deﬁne At := limn→∞ Ant on , then A is a ﬁnite rcll PrAF with its deﬁning sets
,
t .
PROOF.
We only show that for any t > 0 and pre-t -equivalent paths ω, ω ,
ω ∈
t implies that ω ∈
t . Suppose that ω ∈
t and ω is pre-t -equivalent
to ω . It easy to see that ω ∈
n
n∈N
t . We then see the uniform convergence of
Ans−(ω ) = Ans−(ω) for s ∈ ]0, t]. Therefore, ω ∈ t, As(ω ) = As(ω) for s ∈ [0, t[
and As−(ω ) = As−(ω) for s ∈ ]0, t].
Recall that {θt , t > 0} denotes the time-shift operators on the path space for the
process X.
LEMMA 2.15.
For t, s > 0:
(i) θt rt+sω is s-equivalent to rsω if t + s < ζ(ω) or s ≥ ζ(ω);
(ii) rt θsω is pre-t-equivalent to rt+sω and, moreover, if ω is continuous at s,
then rt θsω is t-equivalent to rt+sω.
PROOF.
(i) We may assume that t + s < ζ(ω). For v ∈ [0, s],
θt rt+sω(v) = ω (s − v)− = rsω(v)
and so θt rt+sω is s-equivalent to rsω.
(ii) Note that t + s < ζ(ω) is equivalent to t < ζ(θsω). It follows from the
deﬁnition, if t + s < ζ(ω), that
(2.3)
(rt θsω)(v) = ω (t + s − v)− ,
if 0 ≤ v < t,
ω(s),
if v = t,
while rt+sω(v) = ω((t + s − v)−) for 0 ≤ v ≤ t. Hence, typically, rt θsω is only
pre-t -equivalent to rt+sω.
Fix t > 0. Set H t :=
:=
s
Ft for s ∈ [0, t] and Hts
Fs for s ∈ ]t, ∞[. (Hts)s≥0 is
then a ﬁltration over ( , F∞) and Fs ⊂ Hts for all s ≥ 0.
LEMMA 2.16.
The following assertions hold for any ﬁxed t > 0:
(i) if we let ϕ be a Borel function on E × E and set X0− := X0, then
[0, ∞[ ×
(s, ω) → 1[ 0,ζ[ (s, ω)1 (ω)ϕ(X
t
s−(ω), Xs (ω)) is (H ts )-optional for
any t ∈ Ft ;
(ii) if we let A be an rcll PrAF with deﬁning sets
∈ F∞, t ∈ Ft and we set
A0−(ω) := 0 and Ats(ω) := 1 (ω)(1
t
[0,t](s)As(ω) + 1]t,∞[(s)At (ω)) for ω ∈
,
then [0, ∞[ ×
(s, ω) → 1[ 0,ζ[ (s, ω)(Ats(ω) − Ats−(ω)) is (Hts)-optional.

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
947
PROOF.
(i) Note that 1[ 0,ζ[ is (Hts )-predictable. The assertion is clear if ϕ =
f ⊗ g for bounded Borel functions f, g on E . The monotone class theorem for
functions gives us the desired result.
(ii) Since At is (H ts )-adapted and rcll on
and At− is (Hts)-adapted and lcrl on
, (s, ω) → As(ω) is (Hts )-optional and (s, ω) → Ats−(ω) is (Hts)-predictable.
Consequently, (s, ω) → Ats(ω) − Ats−(ω) is (Hts)-optional.
By Lemma 3.2 of [3], for a ﬁnite rcll AF A = (At )t≥0, there is a Borel function
ϕ : E × E → R with ϕ(x, x) = 0 for all x ∈ E such that
(2.4)
At − At− = ϕ(Xt−, Xt )
for every t ∈ ]0, ζ [, Pm-a.e.
Moreover, if ϕ is another such function, then J (ϕ = ϕ) = 0. As before, we refer
[
to such a function
0,ζ [
ϕ as a jump function for A. Recall that if M ∈ M
, then
loc
there exists a jump function ϕ (unique in the above sense) so that Mt − Mt− =
ϕ(Xt−, Xt ) for t ∈ ]0, ζ [, Pm-a.e.
LEMMA 2.17.
Let A be a ﬁnite rcll PrAF with deﬁning sets { , t , t ≥ 0}.
There then exists a real-valued Borel function ϕ on E × E with ϕ(x, x) = 0 for
x ∈ E such that A with deﬁning sets
:= {ω ∈ | As(ω) − As−(ω) = ϕ(Xs−(ω), Xs(ω)) for s ∈ ]0, ζ(ω)[},
t := {ω ∈
t | As (ω) − As−(ω) = ϕ(Xs−(ω), Xs (ω)) for s ∈ ]0, t [ ∩ ]0, ζ (ω)[}
is again an rcll PrAF admitting m-null set. The analogous assertion holds for
PrAF’s on [ 0, ζ [ and, in particular, for elements of (Mloc)[ 0,ζ[ .
PROOF.
Let ϕ : E × E → R be a Borel function vanishing on the diagonal
and deﬁne
,
t in terms of ϕ, as above. Clearly,
= t>0 t, t ⊂ s for
s < t . Moreover, we see that θt
⊂ for t ≥ 0 and θt−s( t) ⊂ s for s < t. For
two pre-t -equivalent paths ω, ω , we see that ω ∈ t implies that ω ∈ t .
By the previous lemma,
:= (s, ω) | 1[ 0,ζ[ (s, ω)1 (ω) At
t
s (ω) − Ats−(ω) − ϕ(Xs−(ω), Xs (ω)) = 0
is (H ts )-progressively measurable for any ﬁxed t > 0 and the debut of
is
D (ω) := inf s ≥ 0 | 1[ 0,ζ[ (s, ω)1 (ω) At
t
s (ω) − Ats−(ω)
− ϕ(Xs−(ω), Xs(ω)) = 0 ,
which is an (H ts )-stopping time by (A5.1) in [15]. In particular,
ω ∈
| 1[ 0,ζ[ (s, ω)1 (ω) A
t
s (ω) − As−(ω) − ϕ(Xs−(ω), Xs (ω)) = 0
for s ∈ [0, t[
= {ω ∈ | t < D (ω)} ∈ Ht =
t
Ft .

948
CHEN, FITZSIMMONS, KUWAE AND ZHANG
Hence,
{ω ∈ t | As(ω) − As−(ω) − ϕ(Xs−(ω), Xs(ω)) = 0 for s ∈ ]0, t[ ∩ ]0, ζ(ω)[}
= {ω ∈ t | As(ω) − As−(ω) − ϕ(Xs−(ω), Xs(ω)) = 0
for s ∈ [0, t[ ∩ [0, ζ(ω)[}
= ω ∈ t | 1[ 0,ζ[ (s, ω) As(ω) − As−(ω) − ϕ(Xs−(ω), Xs(ω)) = 0
for s ∈ [0, t[
∈ Ft.
Therefore,
t ∈ Ft and
∈ F∞. The proof for PrAF’s on [ 0, ζ[ is similar, so we
omit it.
The following theorem is a key to our extension of Nakao’s operator . Its proof
is complicated by measurability issues, but the idea behind it is fairly transparent.
We will use the convention X0−(ω) := X0(ω).
THEOREM 2.18 (Dual PrAF).
Let A be a ﬁnite rcll PrAF on [ 0, ζ [ with deﬁn-
ing sets
,
t admitting m-null set. Suppose that there is a Borel function ϕ on
E × E with ϕ(x, x) = 0 for x ∈ E such that ϕ(Xs−(ω), Xs(ω)) = As(ω)−As−(ω)
for all s ∈]0, t[∩]0, ζ [ and all ω ∈ t . Set
At (ω) := At (rt (ω)) + ϕ(Xt (ω), Xt−(ω))
(2.5)
for t ∈ [0, ζ(ω)[ and At (ω) := 0 for t ∈ [ζ(ω), ∞[.
A is then an rcll PrAF on [ 0, ζ [ admitting m-null set such that
At = At− ◦ rt + ϕ(Xt , Xt−) and At − At− = ϕ(Xt , Xt−)
for all t ∈ ]0, ζ [, Pm-a.e.
PROOF.
Let
∈ F∞, t ∈ F m
t , t > 0 be the deﬁning sets of A admitting m-
null set. We easily see r−1
t
( t ) ∩ {t < ζ } ⊂ r−1
s
( s) ∩ {s < ζ } for s ∈ ]0, t[ by use
of Lemma 2.15(i) and θt−s t ⊂ s.
Set
t := r −1
t
( t ) for t > 0 and
:= t>0 t. We then see that
=
t >0,t ∈Q
t by use of r −1
t
( t ) ∩ {t ≥ ζ } = {t ≥ ζ } and the monotonicity of
r−1
t
( t ) ∩ {t < ζ }. Indeed, we have
⊂ t>0,t∈Q t ⊂ ( s ∩ {s < ζ}) ∪ {t ≥ ζ}
for any 0 < s < t with t ∈ Q. Taking the intersection over t ∈ ]s, ∞[ ∩ Q, we have
⊂ t>0,t∈Q t ⊂ s for all s > 0, which yields the assertion.
We prove θt
⊂
for each t ≥ 0, in particular, θt
⊂ θs
and, equiva-
lently, θ −1
⊂
s
θ −1
t
if s ∈ [0, t]. Suppose that ω ∈ . Then, rt+sω ∈ t+s. If
t + s < ζ(ω), then rt+sω ∈ s, otherwise rt+sω = ω ∈ s. Hence, we have

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
949
rsθt ω ∈ s, by Lemma 2.15(ii). Therefore, rsθt ω ∈ s for all s > 0, which im-
plies that θt ω ∈ .
Next, we prove θt−s( t ) ⊂ s for s ∈ ]0, t[. Take ω ∈ t . Then, rsθt−sω is
pre-s-equivalent to rt ω ∈ t ⊂ s, by Lemma 2.15(ii) and hence rsθt−sω ∈ s.
Therefore, θt−sω ∈ s for all s ∈ ]0, t[.
From
t ⊂ F m
t , we obtain
t ∈ F m
t , by Lemma 2.10. Since (
t )c =
r−1
t
(( t )c) = r−1
t
(( t )c) ∩ {t < ζ } holds by noting ω
∈ t, we have
Pm(( t )c) = Pm(( )c) = 0.
By (2.2), v → rs(ω)(v) is continuous at v = s. Hence, on
t ∩ {t < ζ }, we
have ϕ(Xs−, Xs) ◦ rs = ϕ(Xs, Xs−) ◦ rs = 0, in particular, As ◦ rs = As− ◦ rs for
s ∈ ]0, t[.
The remainder of the proof is devoted to showing that A is an rcll PrAF on [0, ζ [
with deﬁning sets
,
t such that on
t ∩ {t < ζ }, As = As− ◦ rs + ϕ(Xs , Xs−),
s ∈ ]0, t[. First, note that for ω ∈ , |At (ω)| < ∞ for any t ∈ ]0, ζ(ω)[ because,
by taking T ∈]t, ζ(ω)[, rT ω ∈ T implies that rt ω ∈ t , hence |At−(rt ω)| < ∞.
Moreover, for ω ∈ t ∩ {t < ζ }, we see that rsω ∈ s ∩ {s < ζ } and |As−(rsω)| <
∞ for all 0 < s < t.
For two pre-t -equivalent paths ω, ω ∈
∩ {t < ζ} with t > 0, we show that
ω ∈ t implies ω ∈ t and As(ω) = As(ω ) for s ∈ [0, t[. Recall that ω ∈ t ∩
{t < ζ } ⊂ s ∩{s < ζ } for s ∈ [0, t] and note that ω and ω are s-equivalent for any
s ∈ [0, t[. On the other hand, s < ζ(ω) is equivalent to s < ζ(ω ) for any s ∈ [0, t[.
We then see that rsω ∈ s is s-equivalent to rsω for any s ∈ [0, t], which implies
that rsω ∈ s for any ]0, t] and As−(rsω) = As−(rsω ) for any s ∈ [0, t].
Fix t > 0. On
t ∩{t < ζ } and for any p, q > 0 with p + q < t , by Lemma 2.15,
Ap+q = A(p+q)− ◦ rp+q + ϕ Xp+q, X(p+q)−
= (Ap + Aq− ◦ θp) ◦ rp+q + ϕ Xp+q, X(p+q)−
= Ap ◦ rp+q + Aq− ◦ θp ◦ rp+q + ϕ Xp+q, X(p+q)−
= Ap− ◦ rp+q + ϕ(Xp−, Xp) ◦ rp+q + Aq− ◦ rq + ϕ Xp+q, X(p+q)−
= Ap− ◦ rp ◦ θq + ϕ(Xq, Xq−) + Aq − ϕ(Xq, Xq−)
+ ϕ Xp+q, X(p+q)−
= Ap − ϕ(Xp, Xp−) ◦ θq + Aq + ϕ Xp+q, X(p+q)−
= Ap ◦ θq + Aq.
On
t ∩ {t < ζ }, again by Lemma 2.15 and (2.2), for any s > 0 and u ∈ ]0, s[,
As − As−u = Au ◦ θs−u
= Au− ◦ ru + ϕ(Xu, Xu−) ◦ θs−u
= Au− ◦ ru ◦ θs−u + ϕ(Xs, Xs−)
= Au− ◦ rs + ϕ(Xs, Xs−).

950
CHEN, FITZSIMMONS, KUWAE AND ZHANG
So,
lim(As − As−u) = ϕ(Xs, Xs−).
u↓0
This shows that A has left limit at s ∈ ]0, t[ and As − As− = ϕ(Xs, Xs−).
To show the right continuity of A on
t ∩ {t < ζ } at any s ∈ ]0, t [, note that for
any u ∈ ]0, t − s[, by Lemma 2.15 and (2.2),
As+u − As = Au ◦ θs
= Au− ◦ ru + ϕ(Xu, Xu−) ◦ θs
= Au− ◦ ru ◦ θs + ϕ Xs+u, X(s+u)−
= Au− ◦ rs+u + ϕ Xs+u, X(s+u)− .
Since (Av −Av−)◦rs+v = ϕ(Xv−, Xv)◦rs+v = ϕ(Xs, Xs−), while, by Lemma 2.15
and (2.2),
(Av − Av−) ◦ rs+v = lim(Av − Av−u) ◦ rs+v
u↓0
= lim Au ◦ θv−u ◦ rs+v
u↓0
= lim Au− ◦ rv+u + ϕ(Xs, Xs−),
u↓0
we conclude that
lim Au− ◦ rs+u = 0.
u↓0
On the other hand, for any s ≥ 0,
lim ϕ Xs+u, X(s+u)− = lim ϕ X(v−u)−, Xv−u ◦ rs+v
u↓0
u↓0
= lim Av−u − A(v−u)− ◦ rs+v
u↓0
= (Av− − Av−) ◦ rs+v = 0.
Hence, we have, for s > 0,
lim(As+u − As) = 0.
u↓0
In other words, A is right-continuous at any s ∈ ]0, t[ on t ∩ {t < ζ }. We also see
that
lim
(As+u − As) = 0.
u<s,s↓0,u↓0
We can thus deﬁne the limit A0(ω) := lims↓0 As(ω) for ω ∈ t ∩ {t < ζ } for any
t > 0. We also see that A0(ω) = lims↓0 As−(ω) for ω ∈ t ∩ {t < ζ } for any t > 0
because lims↓0 ϕ(Xs, Xs−) = 0. Next, we prove that A0(ω) = 0 for ω ∈ t ∩ {t <

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
951
ζ } for any t > 0. Take ω ∈ t ∩ {t < ζ } for some ﬁxed t > 0. It sufﬁces to show
that limu↓0 As−u(θuω) = As(ω) for s ∈ [0, t[. Owing to Lemma 2.15(ii), we have
As−u(θuω) = A(s−u)−(rs−uθuω) + ϕ(Xs(ω), Xs−(ω))
= A(s−u)−(rsω) + ϕ(Xs(ω), Xs−(ω))
= As−u(rsω) − ϕ(Xu(ω), Xu−(ω)) + ϕ(Xs(ω), Xs−(ω))
= As−u(rsω) − Au(ω) + Au−(ω) + ϕ(Xs(ω), Xs−(ω))
→ As−(rsω) + ϕ(Xs(ω), Xs−(ω))
as u ↓ 0
= As(ω).
The F m
t -measurability of At is clear from (2.5). This proves the theorem.
3. Stochastic integral for Dirichlet processes.
The following fact will be
used repeatedly in this section. Since a Hunt process is quasi-left continuous, for
each ﬁxed t > 0, we have Xt− = Xt , Px-a.s. for every x ∈ E.
Before embarking on the deﬁnition of our stochastic integral, we prepare the
following lemma for later use.
LEMMA 3.1.
The following assertions hold.
(i) Let {Gn} be an increasing sequence of ﬁnely open Borel sets. The following
are then equivalent:
(a) {Gn} is a nest, that is, Px(limn→∞ σE\G ∧ ζ = ζ ) = 1 for q.e. x ∈ E;
n
(b) E =
∞
n=1 Gn q.e.;
(c) Px(limn→∞ σE\G = ∞) = 1 for m-a.e. x ∈ E;
n
(d) Px(limn→∞ σE\G = ∞) = 1 for q.e. x ∈ E.
n
In particular, for an increasing sequence {Fn} of closed sets, {Fn} is an E -nest if
and only if Px(limn→∞ σE\F = ∞) = 1 for m-a.e. x ∈ E.
n
(ii) For a function f on E, f ∈ Floc if and only if there exist an E -nest {Fk} of
closed sets and {fk | k ∈ N} ⊂ Fb such that f = fk q.e. on Fk.
PROOF.
(i) For the implications (i)(a) ⇐⇒ (i)(b), see Theorem 4.6 in [11]. The
implication (i)(d) ⇒ (i)(a) is clear. Next, we show (i)(b) ⇒ (i)(c). Since each
Gn is ﬁnely open, it is quasi-open by Theorem 4.6.1(i) in [7]. So, there exists a
common nest {A } of closed sets such that (E \ Gn) ∩ A is closed for all n, ∈ N.
Set σ := limn→∞ σE\G . We then have that for all n ∈ N, X
∈ E \ G
n
σE\Gn
n Px -
a.s. on {σ < ∞} for q.e. x ∈ E. We have Px(lim →∞ σE\A = ∞) = 1 q.e. x ∈ E.
Since σ (ω) < ∞ and lim →∞ σE\A (ω) = ∞ together imply σ (ω) < σE\A (ω)
0
for some 0 = 0(ω) ∈ N, we have that there exists 0 ∈ N such that σE\G <
n

952
CHEN, FITZSIMMONS, KUWAE AND ZHANG
σE\A for all n > ≥ 0, Px-a.s. on {σ < ∞} for q.e. x ∈ E. This means that
Px(σ < ∞) ≤ Px lim {Xσ
∈ (E \ G
E\G
n) ∩ A for all n > , σ < ∞}
→∞
n
≤ lim Px Xσ
∈ (E \ G ) ∩ A for all n > , σ < ∞
E\G
→∞
n
≤ lim Px Xσ ∈ (E \ G ) ∩ A , σ < ∞
→∞
≤ lim Px(Xσ ∈ E \ G , σ < ∞)
→∞
∞
= Px Xσ ∈ E \
G , σ < ∞ = 0
=1
for m-a.e. x ∈ E because of the E -polarity of E \
∞=1 G , where we use the
quasi-left continuity of X up to ∞ and the closedness of (E \ G ) ∩ A . The
implication (i)(c) ⇐⇒ (i)(d) follows from the fact that x → Px(σ < ∞) is the limit
of a decreasing sequence of excessive functions and Lemma 4.1.7 in [7].
(i) The “if” part is clear by (i) because τF = τ , where G
k
Gk
k is the ﬁne interior
of Fk. We only prove the “only if” part. Take f ∈ Floc. There then exist {fk | k ∈
N} ⊂ F and an increasing sequence {Gk} of ﬁnely open sets with E = ∞
k=1 Gk
q.e. such that f = fk m-a.e. on Gk. We may take fk ∈ Fb for each k ∈ N by
replacing fk with (−k) ∨ fk ∧ k and Gk with Gk ∩ {|f | < k}. Note that f and
E
fk are quasi-continuous, so f = fk q.e. on Gk. Taking an E -quasi-closure Gk of
E
Gk, we have f = fk q.e. on Gk (see [10] for the deﬁnition of E -quasi-closure).
E
Let {An} be a common E -nest of closed sets such that for each k, n ∈ N, Gk ∩ An
E
E
is closed. Set Fk := Gk ∩ Ak. By (i), {Gk} is a nest, hence Gk is a nest of q.e.
E
ﬁnely closed sets because of τG ≤ τ
as a ﬁnely closed
k
E . Here, we recognize G
G
k
k
Borel sets by deleting an E -polar set. Since {An} is a nest of closed sets, {Fk} is
also, that is, Pm(limk→∞ τF = ζ ) = 0. Therefore, {F
k
k } is an E -nest of closed sets.
We easily see that for each k ∈ N, f = fk q.e. on Fk.
Recall that any locally square-integrable MAF M on [ 0, ζ [ admits a jump func-
tion ϕ on E × E with ϕ(x, x) = 0 for x ∈ E such that
Mt = ϕ(Xt−, Xt) for
◦
t ∈ ]0, ζ [, Pm-a.e. When M ∈M, we can strengthen this statement by replacing
]0, ζ[ with ]0, ∞[ in view of Fukushima’s decomposition and the combination of
Theorem 5.2.1 and Lemma 5.6.3 in [7].
LEMMA 3.2.
Let φ be a Borel function on E × E satisfying φ(x, x) = 0
for all x ∈ E .
(i) Suppose that
N 1E×E(|φ|2 ∧ |φ|) μH ∈ S.

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
953
There then exists a unique, purely discontinuous local MAF K on [ 0, ζ [ [i.e.,
K ∈ (M1 )[ 0,ζ[ ] such that K
loc
t − Kt− = φ(Xt−, Xt ) for all t ∈ [0, ζ [, Px -a.s. for
q.e. x ∈ E.
(ii) If
N 1E×E (|φ|2 ∧ |φ|) μH ∈ S,
then K can be taken to be a local MAF (i.e., K ∈ M1 ) and K
loc
t − Kt− =
φ(Xt−, Xt) for all t ∈ [0, ∞[, Px-a.s. for q.e. x ∈ E.
PROOF.
The proof of (ii) is similar to that of (i), so we only prove (i). By mar-
tingale theory (see, e.g. [9]), the hypothesis implies that the compensated process
K(2)
t
:=
φ(Xs−, Xs)1{|φ(Xs−,Xs)|>1}1{s<ζ}
0<s≤t
t
−
N (Xs, dy)φ(Xs, y)1{|φ(Xs,y)|>1} dHs
0
E
is a local MAF of X of ﬁnite variation on [ 0, ζ [ and
K(1)
t
:= lim
φ(Xs−, Xs)1{ε<|φ(Xs−,Xs)|≤1}1{s<ζ}
ε→0 0<s≤t
t
−
N (Xs, dy)φ(Xs, y)1{ε<|φ(Xs,y)|≤1} dHs
0
E
is a purely discontinuous locally square-integrable MAF of X on [ 0, ζ [ . Thus,
K := K(1) + K(2) is a purely discontinuous MAF on [ 0, ζ [ with jump function φ.
The uniqueness is clear from martingale theory.
DEFINITION 3.3.
Let M be a local MAF on [ 0, ζ [ [i.e., M ∈ (M1 )[ 0,ζ[ ]
loc
with jump function ϕ. Assume that for q.e. x ∈ E, Px-a.s.
t
ϕ21{|ϕ|≤1} + |ϕ|1{|ϕ|>1} (Xs, y)N(Xs, dy) dHs < ∞
0
E
(3.1)
for every t < ζ,
where ϕ(x, y) := ϕ(x, y) + ϕ(y, x). Deﬁne, Pm-a.e. on [ 0, ζ [ ,
(3.2)
(M)t := − 1 M
2
t + Mt ◦ rt + ϕ(Xt , Xt−) + Kt
for t ∈ [0, ζ [,
where Kt is the purely discontinuous local MAF on [ 0, ζ [ with
(3.3)
Kt − Kt− = −ϕ(Xt−, Xt)
for every t < ζ, Px-a.s.
for q.e. x ∈ E.

954
CHEN, FITZSIMMONS, KUWAE AND ZHANG
REMARK 3.4. (i) The condition (3.1) is nothing but N (1E×E(|ϕ|2 ∧|ϕ|))μH ∈
S. In particular, condition (3.1) is satisﬁed by the jump function of any element of
(Mloc)[ 0,ζ[ .
(ii) It follows from Remark 2.13(iv) and Theorem 2.18 that
(M) is a con-
tinuous PrAF admitting m-null set on [ 0, ζ [ . (This is because Remark 2.13(iv)
and Theorem 2.18 imply that the process deﬁned on [0, ζ [ by Bt := Mt ◦ rt +
ϕ(Xt , Xt−) is an rcll PrAF, with left-limit process Bt− = Bt − ϕ(Xt , Xt−). It fol-
lows that
(M) is rcll on [0, ζ [ and that
(M)t− = (M)t for all t ∈ ]0, ζ [, Pm-
a.e.) Note that −Kt :=
s≤t ϕ(Xs−, Xs )1{s<ζ } −
t
0
ϕ(X
E
s , y)N (Xs , dy) d Hs ,
t < ζ , satisﬁes Kt = Kt ◦ rt Pm-a.e. on {t < ζ } for ﬁxed t > 0. In view of Theo-
rem 2.18, it is then clear from the deﬁnition that
is a linear operator that maps
local MAF’s on [ 0, ζ [ satisfying condition (3.1) into even CAF’s on [ 0, ζ [ ad-
mitting m-null set.
(iii) If {Mn, n ≥ 1} is a sequence of MAF’s having ﬁnite energy and converging
in probability to M, then it is easy to see that Mn ◦
−
t
rt , ϕn(Xt−, Xt ) = Mnt
Mn
t −
and ϕn(Xt , Xt−) converge to Mt ◦ rt , ϕ(Xt−, Xt ) = Mt − Mt− and ϕ(Xt , Xt−)
in probability, respectively, under Pm. Hence, we have that
(Mn)t converges to
(M)t in measure for each t > 0.
(iv) For u ∈ F ,
(Mu)t = − 1 Mu + Mu ◦ r
2
t
t
t + u(Xt−) − u(Xt ) = N u
t ,
Pm-a.e. on {t < ζ }, for each ﬁxed t ≥ 0. The ﬁrst equality above is just the def-
inition of
(Mu), while the second follows by applying rt to both sides of (1.2)
because Xt = Xt− and Nu ◦ rt = Nut, Pm-a.e. on {t < ζ }. (The last property is
proved in [4], Theorem 2.1, when X is a diffusion, but the same proof works for
general symmetric Markov process X.) Since both
(Mu)t and Nu
t are continuous
in t , we even have, Pm-a.e.,
(Mu)t = Nut
for all t < ζ.
We are going to show that
(M) deﬁned above coincides on [0, ζ [ with (M)
deﬁned in (1.5) by Nakao when M is an MAF of ﬁnite energy. An AF Z is called
even (resp., odd) if and only if Zt ◦ rt = Zt (resp., Zt ◦ rt = −Zt ), Pm-a.e. on
{t < ζ } for each t > 0. For an rcll process Z with Z0 = 0 and T > 0, we deﬁne
RT Zt := (RT Z)t := ZT − − Z(T −t)−
for 0 ≤ t ≤ T ,
with the convention Z0− = Z0 = 0. Note that RT Zt so deﬁned is an rcll process in
t ∈ [0, T ].
LEMMA 3.5.
Suppose that Z is an rcll PrAF. Then, Pm-a.e. on {T < ζ },
(3.4)
RT Zt = Zt ◦ rT ,
if Z is even,
−
for every t ∈ [0, T ].
Zt ◦ rT ,
if Z is odd,

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
955
PROOF.
Let Z be an rcll PrAF and let T > 0. By Lemma 2.15,
Zt ◦ rT = (ZT − ZT −t ◦ θt ) ◦ rT = ZT ◦ rT − ZT −t ◦ rT −t
(3.5)
for all t < T .
When Z is even,
Zt ◦ rT = ZT − ZT −t = ZT − − Z(T −t)− = RT Zt ,
Pm-a.e. on {T < ζ } for each ﬁxed 0 ≤ t < T . Since both sides are right-continuous
in t ∈ [0, T [, we have, Pm-a.e., RT Zt = Zt ◦ rT for every t ∈ [0, T ]. When Z is an
odd AF of Z, (3.4) can be proven similarly.
THEOREM 3.6.
For an MAF M of ﬁnite energy,
(M) deﬁned above coin-
cides on [ 0, ζ [ with (M) deﬁned in (1.5), Pm-a.e.
PROOF.
For u ∈ F and 0 < t < T , since Nu is an even CAF, by Lemma 3.5,
(Mu +
◦
t
2N u
t ) ◦ rT = u(Xt ) − u(X0) + N u
t
rT
= u X(T −t)− − u(XT −) + NuT− − Nu(T−t)−
= Mu(T −t)− − MuT−
= −RT Mut.
Since both (Mu +
t
2N u
t ) ◦ rT and RT M u
t are right-continuous in t , we have, Pm-a.e.
on {T < ζ },
(3.6)
RT Mu = −
+
t
(Mu
t
2N u
t ) ◦ rT
for every t ∈ [0, T ].
For u ∈ D(L) ⊂ F and v ∈ Fb, deﬁne Mt = t0 v(Xs−) dMus, which is an MAF
of ﬁnite energy. Note that, since u ∈ D(L), Nu = t
t
0 Lu(Xs ) d s is a continuous
process of ﬁnite variation. For each ﬁxed 0 < t < T and n ≥ 1, deﬁne ti = it/n
and si = T − t + ti. Using the standard Riemann sum approximation of the Itô
integral and of the covariance process [Mv, Mu], we have, Pm-a.e. on {T < ζ },
MT − MT −t + [Mv, Mu]T − [Mv, Mu]T −t
n−1
= lim
v(Xs )(Mu
− Mu) + (Mv − Mv )(Mu − Mu)
n→∞
i
si+1
si
si+1
si
si+1
si
i=0
n−1
= lim
v(Xs
)(Mu
− Mu) − (Nv − Nv )(Mu − Mu)
n→∞
i+1
si+1
si
si+1
si
si+1
si
i=0
n−1
= lim
v(Xs
)(Mu
− Mu)
n→∞
i+1
si+1
si
i=0

956
CHEN, FITZSIMMONS, KUWAE AND ZHANG
n−1
= lim
v(XT −t+t )(RT Mu − RT Mu
)
n→∞
i
t −ti
t −ti+1
i=0
n−1
= lim
v(Xt−t )(Mu
− Mu + 2Nu
− 2Nu ) ◦ rT
n→∞
i+1
t −ti+1
t −ti
t −ti+1
t −ti
i=0
t
= −
v(Xs−) d(Mu +
s
2N u
s )
◦ rT ,
0
where in the third equality, we used the fact that N u has zero energy, while in the
second to the last equality, we used (3.6). Note that the stochastic integral involving
N u in the last equality is just the Lebesgue–Stieltjes integral since N u is of ﬁnite
variation. Also, note that Xt = Xt−, Pm-a.e., for each ﬁxed t > 0. So, we have, for
each ﬁxed t < T , Pm-a.e. on {T < ζ },
t
RT Mt + RT [Mv, Mu]t = −
v(Xs−) d(Mu +
s
2N u
s )
◦ rT .
0
Since both sides are right-continuous in t ∈ [0, T ], we have, Pm-a.e. on {T < ζ },
t
RT Mt + RT [Mv, Mu]t = −
v(Xs−)d(Mu +
s
2N u
s )
◦ rT
0
(3.7)
for every t ∈ [0, T ].
By [14], Theorem 3.1 and (1.7),
t
t
v(Xs−) dNu =
=
s
v(Xs) dNu
s
(M)t − 1 Mv,c + Mv,j , Mu,c + Mu,j
2
t .
0
0
It follows that Pm-a.e. on {T < ζ },
RT Mt + RT [Mv, Mu]t
= − Mt + 2 (M)t − Mv,c + Mv,j, Mu,c + Mu,j t ◦ rT
= − Mt + 2 (M)t − Mv,c, Mu,c t − Mv,j, Mu,j t ◦ rT
for all t ≤ T .
Recall that
[Mv, Mu]t = Mv,c, Mu,c t +
(Mv −
−
s
Mvs−)(Mus Mus−)
s≤t
= Mv,c, Mu,c t +
v(Xs) − v(Xs−) u(Xs) − u(Xs−) .
s≤t
Taking t = T and noting that both (M) and Mv,c, Mu,c are continuous even
AF’s, we have, from above, that, Pm-a.e. on {t < ζ },
(3.8)
(M)t = − 1 M
2
t + Mt ◦ rt + v(Xt ) u(Xt−) − u(Xt ) + Kt ,

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
957
where
Kt =
v(Xs) − v(Xs−) u(Xs) − u(Xs−) − Mv,j , Mu,j t
s≤t
is the purely discontinuous MAF with Kt − Kt− = (v(Xt ) − v(Xt−))(u(Xt ) −
u(Xt−)). Note that the right-hand side of (3.8) is right-continuous on [0, ζ [,
Pm-a.e. [cf. Remark 3.4(ii)]. Also, observe that Mt − Mt− = ϕ(Xt−, Xt ), where
ϕ(x, y) = v(x)(u(y) − u(x)), and that
Kt − Kt− = −ϕ(Xt−, Xt ) − ϕ(Xt , Xt−).
This shows that
(M)t = (M)t , Pm-a.e. on {t < ζ } for each ﬁxed t ≥ 0. Since
both processes are continuous in t ∈ [0, ζ [, we have, Pm-a.e.,
(M) = (M)
on [0, ζ [
for an MAF M of the form Mt = t v(X
0
s−) dM u
s with u ∈ D (L) and v ∈ Fb .
By Lemma 5.4.5 in [6], such MAF’s form a dense subset in the space of MAF’s
having ﬁnite energy. Thus, by Lemma 3.1 in Nakao [14] and Remark 3.4(iii), we
have, for a general MAF M of ﬁnite energy,
(M)t = (M)t Pm-a.e. on {t < ζ }
for every ﬁxed t ≥ 0. Since both processes are continuous in t ∈ [0, ζ [, it follows
that
(M) = (M) on [ 0, ζ [ , Pm-a.e.
THEOREM 3.7.
Let M be a locally square-integrable MAF on [ 0, ζ [ with
jump function ϕ. Suppose that ϕ satisﬁes condition (3.1). Then, for every t > 0,
n−1
(3.9)
lim
(M)
2
( +1)t/n −
(M) t/n
= 0,
n→∞ =0
where the convergence is in Pgm-measure on {t < ζ } for any g ∈ L1(E; m) with
0 < g ≤ 1 m-a.e.
PROOF.
By (1.5) and Theorem 3.6, (3.9) clearly holds when M is an MAF of
ﬁnite energy. For a locally square-integrable MAF M on [ 0, ζ [ , there is an E -nest
◦
{Fk} of closed sets such that 1F ∗ M ∈M for each k ≥ 1 in view of the proof of
k
Proposition 2.8 and so (3.9) holds with 1F ∗ M in place of M. For each ﬁxed
k
k ≥ 1,
(M)t = (1F ∗ M)
Kk
[,
k
t − 1
2
t ,
Pm-a.e. on [0, τFk
where Kkt is a purely discontinuous local MAF on [ 0, ζ [ with
Kk −
t
Kkt− = 1Fc(Xt−)ϕ(Xt−, Xt) + 1Fc(Xt)ϕ(Xt, Xt−)
for t < ζ.
k
k
◦
Since 1F ∗ M ∈M, we have
k
N (1F
N (1
ϕ2) dμ
k ×E ϕ2) d μH =
E×Fk
H < ∞.
E
E

958
CHEN, FITZSIMMONS, KUWAE AND ZHANG
Consequently, by Lemma 3.2, we have the existence of a purely discontinuous
local MAF on [ 0, ζ [ with jumps given by 1F (X
(X
k
t −)ϕ(Xt−, Xt ) + 1Fk
t )ϕ(Xt ,
Xt−), t < ζ . So, we obtain the existence of such Kkt. Since the square bracket
of Kk is given by
s≤t 1F c (Xs−)ϕ2(Xs−, Xs ) + 1F c (Xs )ϕ2(Xs , Xs−) and it van-
k
k
ishes at t < τF , we have, for each ﬁxed t > 0,
k
n−1
2
lim
(M)( +1)t/n − (M) t/n = 0
in Pgm-measure on {t < τF }.
n→∞
k
=0
Passing to the limit as k ↑ ∞ establishes (3.9).
We are now in a position to deﬁne stochastic integrals against
(M) as integra-
tor. Note that, for f ∈ Floc, Mf,c is well deﬁned as a continuous MAF on [ 0, ζ [ of
locally ﬁnite energy (see Theorem 8.2 in [9]). Moreover, for f ∈ Floc and a locally
square-integrable MAF M on [ 0, ζ [ ,
t
t → (f ∗ M)t :=
f (Xs−) dMs
0
is a locally square-integrable MAF on [ 0, ζ [ .
DEFINITION 3.8 (Stochastic integral).
Suppose that M is a locally square-
integrable MAF on [ 0, ζ [ and f ∈ Floc. Let ϕ : E × E → R be a jump function
for M and assume that ϕ satisﬁes condition (3.1). Deﬁne, Pm-a.e. on [ 0, ζ [ ,
t
f (Xs−) d (M)s
0
(3.10)
:= (f ∗ M)t − 1 Mf,c, Mc
2
t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs ,
0
E
whenever
(f ∗ M) is well deﬁned and the third term in the right-hand side of
(3.10) is absolutely convergent.
REMARK 3.9.
(i) Under the above condition, the stochastic integral is clearly
well deﬁned on [ 0, ζ [ under Pm and is a PrAF of X admitting m-null set.
(ii) Here are some sufﬁcient conditions for every term on the right-hand side
of (3.10) to be well deﬁned. In addition to the conditions in Deﬁnition 3.8, we
assume that, Pm-a.e.,
t
2
(3.11)
f (Xs) − f (y) N(Xs, dy) dHs < ∞
for every t < ζ
0
E
and that
t
(3.12)
ϕ(y, Xs)2N(Xs, dy) dHs < ∞
for every t < ζ.
0
E

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
959
The ﬁrst and third terms on the right-hand side of (3.10) are then well deﬁned. This
is because N (1E×E|ϕ|)μH ∈ S implies that N(1E×E|f ϕ|)μH ∈ S, and
f (x)ϕ(x, y) + f (y)ϕ(y, x) = f (x) ˆ
ϕ(x, y) + f (y) − f (x) ϕ(y, x),
so
(f ∗ M) is well deﬁned on [0, ζ [ in view of the condition (3.1) for f ∗ M,
(3.11) and (3.12). Condition (3.11) is satisﬁed when f is a bounded function in
Floc or f ∈ F . This is because, when f ∈ F , the left-hand side of (3.11) is just
Mf,j t . When f is a bounded function in Floc, by Lemma 3.1(i), there exist a
nest {Fn | n ∈ N} of closed sets and a sequence of functions {fn | n ∈ N} ⊂ Fb
such that f = fn q.e. on Fn for every n ≥ 1. Note that for each n ≥ 1, Mfn,d is a
square-integrable, purely discontinuous martingale and
f
f
M n,d
n,d
t
− Mt− = fn(Xt) − fn(Xt−).
So, t →
s≤t (fn(Xs ) − fn(Xs−))2 is Px -integrable for q.e. x ∈ E. Since f is
bounded, we have, for each n ≥ 1, that
2
t →
f (Xs) − f (Xs−)
s≤t∧τFn
=
2
2
f (Xs) − f (Xs−) + f (Xt∧τ ) − f (X
F
−)
n
t ∧τFn
s<t ∧τFn
=
2
2
fn(Xs) − fn(Xs−) + f (Xt∧τ ) − f (X
F
−)
n
t ∧τFn
s<t ∧τFn
is an increasing process and is Px -integrable for each ﬁxed t ≥ 0 for q.e. x ∈ E.
Similarly, At :=
s≤t (f (Xs ) − f (Xs−))2 is locally integrable in the sense of De-
ﬁnition 5.18 in [9]. Indeed, for a stopping time Tn := inf{t > 0 | At > n}, AT =
n
AT
) − f (X
n− + (f (XTn
Tn−))2 is bounded, hence Px -integrable for q.e. x ∈ E.
Note that the dual predictable projection of At is nothing but t0
(f (X
E
s ) −
f (y))2N (Xs, dy) dHs. The dual predictable projection of
s≤t∧τ
(f (X
F
s ) −
n
t ∧τ
f (X
Fn
s−))2 is then given by
(f (X
0
E
s ) − f (y))2N (Xs , dy) d Hs from Corol-
lary 5.24 in [9], which is Px-integrable for q.e. x ∈ E. This implies that (3.11)
holds for every t < τF . Therefore, (3.11) holds for every t < ζ .
n
Condition (3.12) is satisﬁed when Md is Pm-square-integrable. Indeed,
Em
ϕ2(Xs, Xs−) : t < ζ = Em [Md]t ◦ rt : t < ζ
s≤t
= Em [Md]t : t < ζ < ∞.
Corollary 4.5 in [8] then tells us that
1
1
lim
Em
ϕ2(Xs, Xs−) : t < ζ = lim Em
ϕ2(Xs, Xs−) ,
t →0 t
t →0 t
s≤t
s≤t

960
CHEN, FITZSIMMONS, KUWAE AND ZHANG
which implies that
t
Em
ϕ(y, Xs)2N(Xs, dy) dHs < ∞
0
E
for all t > 0 by way of its subadditivity. Hence, we obtain (3.12).
(iii) Suppose that f ∈ F . Let Kt be a purely discontinuous local MAF on
[ 0, ζ[ with Kt − Kt− = −ϕ(Xt−, Xt) − ϕ(Xt, Xt−) on ]0, ζ[. Then,
t
Mf,j , Mj + K t = −
f (y) − f (Xs) ϕ(y, Xs)N(Xs, dy) dHs.
0
E
In this case, (3.10) can be rewritten as
t
(3.13)
f (Xs−) d (M)s = (f ∗ M)t − 1 Mf,c + Mf,j , Mc + Mj + K
2
t
0
on [0, ζ [. So, when M = Mu for some u ∈ F and f ∈ F ∩ L2(E; μ u ),
t
0 f (Xs−) d
(M)s on [0, ζ [ is just the t0 f (Xs) ◦ d (M)s deﬁned by (1.7). This
shows that the stochastic integral given in Deﬁnition 3.8 extends Nakao’s deﬁni-
tion (1.7) of stochastic integral ﬁrst introduced in [14].
THEOREM 3.10.
The stochastic integral in (3.10) is well deﬁned. That is, if M
and M are two locally square-integrable MAF’s on [ 0, ζ [ such that all conditions
in Deﬁnition 3.3 for M and M are satisﬁed and
(M) ≡ (M) on [ 0, ζ [ , then
for every f ∈ Floc for which t f (X
f (X
0
s−)d
(M)s and t0
s−) d
(M)s are well
deﬁned, we have, Pm-a.e.,
t
t
f (Xs−) d (M)s =
f (Xs−) d (M)s
on [0, ζ [.
0
0
PROOF.
It is equivalent to show that
t
f (Xs−) d (M − M)s = 0
on [0, ζ [.
0
By taking M to be M − M, we may and will assume that M = 0. Moreover,
a localization argument allows us to assume that f is bounded. Let ϕ : E × E → R
be a jump function for M. Let Kt be the purely discontinuous local MAF on [ 0, ζ [
with
Kt − Kt− = −ϕ(Xt−, Xt ) − ϕ(Xt , Xt−)
for t < ζ.
Since
(M) ≡ 0, we have
(3.14)
Mt + Mt ◦ rt + ϕ(Xt , Xt−) + Kt = 0
on [0, ζ [.
Thus, by (3.5) and (3.14), on {T < ζ },
Mt ◦ rT = MT ◦ rT − MT −t ◦ rT −t
(3.15)
= −MT − KT + MT −t + KT −t
− ϕ(XT , XT −) + ϕ XT −t, X(T −t)−

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
961
for every t ∈ [0, T ]. Using the standard Riemann sum approximation and (3.15),
we have, for f ∈ F ,
(f ∗ M)t ◦ rt + f (Xt )ϕ(Xt , Xt−)
= −(f ∗ M)t − (f ∗ K)t − [Mf , M + K]t
= −(f ∗ M)t − (f ∗ K)t − Mf,c, Mc t
+
f (Xs) − f (Xs−) ϕ(Xs, Xs−)
s≤t
Pm-a.e. on {t < ζ } for each ﬁxed t ≥ 0. Consequently, we have, for f ∈ Floc,
Pm-a.e. for all t ∈ [0, ζ [,
(f ∗ M)t ◦ rt + f (Xt )ϕ(Xt , Xt−)
(3.16)
= −(f ∗ M)t − (f ∗ K)t − Mf,c, Mc t
+
f (Xs) − f (Xs−) ϕ(Xs, Xs−),
s≤t
since both sides are right-continuous in t ∈ [0, ζ [. Let K be the purely discontinu-
ous local MAF on [ 0, ζ [ with
Kt − Kt− = −f (Xt−)ϕ(Xt−, Xt ) − f (Xt )ϕ(Xt , Xt−)
for all t ∈ [0, ζ [.
Then, for f ∈ Floc, we have, by (3.16),
(f ∗ M)t = − 1 (f ∗ M)
2
t + (f ∗ M) ◦ rt + f (Xt )ϕ(Xt , Xt−) + Kt
t
= 1
f (X
2
s−) dKs + M f,c, M c t
0
−
f (Xs) − f (Xs−) ϕ(Xs, Xs−) − Kt .
s≤t
Thus,
t
f (Xs−) d (M)s
0
= (f ∗ M)t − 1 Mf,c, Mc
2
t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs
0
E
t
= 1
f (X
f (X
K
2
s−) dKs − 12
s ) − f (Xs−) ϕ(Xs , Xs−) − 12 t
0
s≤t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs .
0
E

962
CHEN, FITZSIMMONS, KUWAE AND ZHANG
Note that
Kt = −
f (Xs−)ϕ(Xs−, Xs) + f (Xs)ϕ(Xs, Xs−)
s≤t
(3.17)
t
+
f (Xs)ϕ(Xs, y) + f (y)ϕ(y, Xs) N(Xs, dy) dHs
0
E
and that
(3.18)
Kt = lim −
ϕ1{|ϕ|>ε} (Xs−, Xs) + N(ϕ1{|ϕ|>ε}) ∗ H
,
t
ε→0
s≤t
where ϕ(x, y) := ϕ(x, y) + ϕ(y, x). It follows that
t
f (Xs−) d (M)s = 0
for all t < ζ,
0
Pm-a.e.
REMARK 3.11.
The above proof actually shows that if
(M) =
(M) on
[0, T ] ∩ [0, ζ [, then, Pm-a.e.,
t
t
f (Xs−) d (M)s =
f (Xs−) d (M)s
on [0, T ] ∩ [0, ζ [.
0
0
4. Further study of the stochastic integral.
THEOREM 4.1.
Suppose that f ∈ Floc and that M is a locally square-
integrable MAF on [ 0, ζ [ satisfying (3.1) such that
(M) is a continuous
process A of ﬁnite variation on [ 0, ζ [ . Assume that the stochastic integral t →
t
0 f (Xs−) d
(M)s is well deﬁned. Then, Pm-a.e.,
t
t
f (Xs−) d (M)s =
f (Xs) dAs
on [0, ζ [,
0
0
where the integral on the right-hand side is the Lebesgue–Stieltjes integral.
PROOF.
Let ϕ : E × E → R be a Borel function with ϕ(x, x) = 0 for x ∈ E
such that ϕ(Xt−, Xt ) = Mt − Mt− for t ∈ [0, ζ [, Pm-a.e. Let Kt be the purely
discontinuous local MAF on [ 0, ζ [ with
Kt − Kt− = −ϕ(Xt−, Xt ) − ϕ(Xt , Xt−)
for t ∈ ]0, ζ [.
Since
(M) = A on [0, ζ [,
Mt ◦ rt + ϕ(Xt , Xt−) = −Mt − Kt − 2At
for all t ∈ [0, ζ [.
Thus, by (3.5), for every T > t > 0, on {T < ζ },
Mt ◦ rT = −MT − KT − 2AT + MT −t + KT −t
(4.1)
+ 2AT −t − ϕ(XT , XT −) + ϕ XT −t, X(T −t)− .

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
963
Now, ﬁx f ∈ Floc; as before, we may assume, without loss of generality, that f is
bounded. Using the standard Riemann sum approximation, we obtain, on {t < ζ },
(f ∗ M)t ◦ rt + f (Xt )ϕ(Xt , Xt−)
= −(f ∗ M)t − (f ∗ K)t − 2(f ∗ A)t − [Mf , M + K + 2A]t
= −(f ∗ M)t − (f ∗ K)t − 2(f ∗ A)t − Mf,c, Mc t
+
f (Xs) − f (Xs−) ϕ(Xs, Xs−).
s≤t
Consequently, we have, Pm-a.e. for all t ∈ [0, ζ [,
(f ∗ M)t ◦ rt + f (Xt )ϕ(Xt , Xt−)
(4.2)
= −(f ∗ M)t − (f ∗ K)t − 2(f ∗ A)t − Mf,c, Mc t
+
f (Xs) − f (Xs−) ϕ(Xs, Xs−)
s≤t
since both sides are right-continuous in t ∈ [0, ζ [. Let K be the purely discontinu-
ous local MAF on [ 0, ζ [ with
Kt − Kt− = −f (Xt−)ϕ(Xt−, Xt ) − f (Xt )ϕ(Xt , Xt−)
for all t ∈ [0, ζ [.
Then, by (4.2),
(f ∗ M)t = − 1 (f ∗ M)
2
t + (f ∗ M) ◦ rt + f (Xt )ϕ(Xt , Xt−) + Kt
t
t
= 1
f (X
f (X
2
s−) dKs + 2
s−) dAs + M f,c, M c t
0
0
−
f (Xs) − f (Xs−) ϕ(Xs, Xs−) − Kt .
s≤t
Thus,
t
f (Xs−) d (M)s
0
= (f ∗ M)t − 1 Mf,c, Mc
2
t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs
0
E
t
t
= 1
f (X
f (X
2
s−) dKs +
s−) dAs
0
0
− 1
f (X
K
2
s ) − f (Xs−) ϕ(Xs , Xs−) − 12 t
s≤t
t
+ 1
f (y) − f (X
2
s ) ϕ(y, Xs )N (Xs , dy) d Hs .
0
E

964
CHEN, FITZSIMMONS, KUWAE AND ZHANG
It now follows from (3.17)–(3.18) that, Pm-a.e.,
t
t
f (Xs−) d (M)s =
f (Xs−) dAs
for all t ∈ [0, ζ [.
0
0
This proves the theorem.
Note that if f, g ∈ Floc, then fg ∈ Floc.
THEOREM 4.2.
Let f, g ∈ Floc and let M be a locally square-integrable MAF
on [ 0, ζ [ satisfying (3.1). Then Pm-a.e.,
t
s
g(Xs−) d
f (Xr−) d (M)r
0
0
(4.3)
t
=
f (Xs−)g(Xs−) d (M)s
for every t < ζ,
0
whenever all of the integrals involved are well deﬁned.
PROOF.
Let ϕ : E × E → R be a Borel function with ϕ(x, x) = 0 for x ∈ E
such that, Pm-a.e.,
ϕ(Xt−, Xt ) = Mt − Mt−
for all t ∈ ]0, ζ [.
Let Kt and Kt be the purely discontinuous local MAF’s on [ 0, ζ [ with
Kt − Kt− = −ϕ(Xt−, Xt ) − ϕ(Xt , Xt−)
for t ∈ ]0, ζ [
and
Kt − Kt− = −f (Xt−)ϕ(Xt−, Xt) − f (Xt )ϕ(Xt , Xt−)
for t ∈ ]0, ζ [,
respectively. The left-hand side of (4.3) is then equal to
t
t
g(Xs−) d (f ∗ M)s − 1
g(X
2
s−) d M f,c, M c s
0
0
t
+ 1
g(X
2
s ) f (y) − f (Xs ) ϕ(y, Xs )N (Xs , dy) d Hs
0
E
= (fg ∗ M)t − 1 Mg,c, (f ∗ M)c
2
t
t
+ 1
g(y) − g(X
2
s ) f (y)ϕ(y, Xs )N (Xs , dy) d Hs
0
E
t
− 1
g(X
2
s−) d M f,c, M c s
0
t
+ 1
g(X
2
s ) f (y) − f (Xs ) ϕ(y, Xs )N (Xs , dy) d Hs
0
E
= (fg ∗ M)t − 1 Mfg,c, Mc
2
t

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
965
t
+ 1
f (y)g(y) − f (X
2
s )g(Xs ) ϕ(y, Xs )N (Xs , dy) d Hs
0
E
t
=
f (Xs−)g(Xs−) d (M)s.
0
This proves the theorem.
Let J denote the class of stochastic processes that can be written as the sum of
an (Ft )-semimartingale Y and
(M) for a locally square-integrable MAF M on
[ 0, ζ[ satisfying the condition of Deﬁnition 3.3. The last two theorems imply that
the following stochastic integral is well deﬁned for integrators Z ∈ J.
DEFINITION 4.3.
For f ∈ Floc and Z = Y + (M) ∈ J, deﬁne on, [0, ζ [,
t
t
t
f (Xs−) dZs :=
f (Xs−) dYs +
f (Xs−) d (M)s,
0
0
0
whenever the latter stochastic integral is well deﬁned.
To establish Itô’s formula, we need the following result.
THEOREM 4.4.
Let f ∈ Floc and let M be a locally square-integrable MAF
on [ 0, ζ [ such that ·0 f (Xs−) d (M) is well deﬁned on [0, ζ [. Then, for every
t > 0, Pm-a.e. on {t < ζ },
t
n−1
(4.4)
f (Xs−) d (M)s = lim
f (X t/n)
(M)( +1)t/n − (M) t/n .
0
n→∞ =0
Here, the convergence is in measure with respect to Pgm on {t < ζ } for every
g ∈ L1(E; m) with 0 < g ≤ 1 m-a.e.
PROOF.
By (3.5), Ms ◦ rt = Mt ◦ rt − Mt−s ◦ rt−s for all s < t. Let ϕ : E ×
E → R be a Borel function with ϕ(x, x) = 0 for x ∈ E such that ϕ(Xt−, Xt ) =
Mt − Mt− for all t ∈ [0, ζ [. Let K be the purely discontinuous local MAF on
[ 0, ζ[ with
Kt − Kt− = −ϕ(Xt−, Xt) − ϕ(Xt , Xt−)
for t ∈ ]0, ζ [.
Then, for each ﬁxed t > 0, Pm-a.e. on {t < ζ },
n−1
lim
f (X t/n)
(M)( +1)t/n − (M) t/n
n→∞ =0
= −1(f ∗ M)
(f ∗ K)
2
t − 1
2
t
n−1
+ 1 lim
f (X
2
t /n) M( +1)t/n ◦ r( +1)t/n − M t/n ◦ r t/n
n→∞ =0

966
CHEN, FITZSIMMONS, KUWAE AND ZHANG
= −1(f ∗ M)
(f ∗ K)
2
t − 1
2
t
n−1
− 1 lim
f X
2
( +1)t/n M( +1)t/n − M t/n
◦ rt
n→∞
=0
= −1(f ∗ M)
(f ∗ K)
(f ∗ M)
[Mf , M]
2
t − 1
2
t − 1
2
t ◦ rt − 1
2
t ◦ rt
= −1(f ∗ M)
(f ∗ K)
(f ∗ M)
Mf,c, Mc
2
t − 1
2
t − 1
2
t ◦ rt − 1
2
t
− 1
f (X
2
s−) − f (Xs ) ϕ(Xs , Xs−)
s≤t
= (f ∗ M)t + 1K
(f ∗ K)
Mf,c, Mc
2
t − 1
2
t − 1
2
t
− 1
f (X
2
s−) − f (Xs ) ϕ(Xs , Xs−)
s≤t
t
=
f (Xs−) d (M)s,
0
where K in the penultimate equality is the purely discontinuous local MAF on
[ 0, ζ[ with Ks − Ks− = −f (Xs−)ϕ(Xs−, Xs) − f (Xs)ϕ(Xs, Xs−) for s ∈ ]0, ζ[.
REMARK 4.5.
(i) Theorem 4.4 immediately implies Theorems 3.10 and 4.1.
(ii) By (3.9),
t
n−1
(4.5)
f (Xs−) d (M)s = lim
f X( +1)t/n
(M)( +1)t/n − (M) t/n
0
n→∞ =0
holds in Pgm-measure on {t < ζ } for any g ∈ L1(E; m) with 0 < g ≤ 1 m-a.e.
Hence, we could denote this stochastic integral by either
t
0 f (Xs ) d
(M)s or
t
0 f (Xs ) ◦ d
(M)s. Here, t0 f (Xs) ◦ d (M)s is the Fisk–Stratonovich type in-
tegral: for t < ζ
t
f (Xs) ◦ d (M)s
0
(4.6)
n−1
:=
f (X( +1)t/n) + f (X t/n)
lim
(M)( +1)t/n − (M) t/n .
n→∞ =
2
0
(iii) For any f ∈ Floc and Pm-square-integrable MAF M, by way of the
Riemann sum approximation (4.4), we can extend the stochastic integral
t f (X
0
s−) d
(M)s without imposing further conditions. Indeed, let {G } be a
nest of ﬁnely open Borel sets and f ∈ Fb with f = f m-a.e. on G [see
the explanation for the condition (3.11) in Remark 3.9]. By (4.4), we see

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
967
t
0 fn(Xs−) d
(M)s = t0 f (Xs−) d (M)s for t < τG and n < . We can then
n
deﬁne t0 f (Xs−) d (M)s = t0 fn(Xs−) d (M)s for t < τG for each n ∈ N and,
n
◦
consequently, for all t < ζ Pm-a.e. More strongly, for M ∈M and f ∈ Floc, we can
deﬁne t0 f (Xs−) d (M)s for all t ∈ [0, ∞[ Pm-a.e. Indeed, by Remark 3.9(ii), our
◦
stochastic integral fn ∗ (M) for M ∈M agrees with that deﬁned by Nakao [14]
on [0, ζ [ Pm-a.e., while the latter is deﬁned as a CAF of X for all t ≥ 0. This
implies that lims↑ζ (fn ∗ (M))s exists and is ﬁnite Pm-a.e. After we extend our
deﬁnition of stochastic integral fn ∗ (M) beyond [0, ζ [ by
fn ∗ (M) = f
= lim f
for t ≥ ζ,
t
n ∗
(M) ζ
n ∗
(M)
s↑ζ
s
fn ∗ (M) becomes a CAF of X on [0, ∞[ Pm-a.e. With this extension for each
n < , we have t0 f (Xs−) d (M)s = t0 f (Xs−) d (M)s for t < σE\G , P
n
m-a.e.
Owing to Lemma 3.1(i) and the existence of the limit lim
t
t ↑ζ
f (X
0
s−) d
(M)s
Pm-a.e., we obtain the stochastic integral t0 f (Xs−) d (M)s, on [0, ∞[, Pm-a.e.
◦
for any f ∈ Floc and M ∈M, extending the stochastic integral of Nakao [14].
Remark
4.5(iii)
says
that
the
stochastic
integral
f ∗
(M)t :=
t
0 f (Xs−) d
(M)s can be deﬁned for t ∈ [0, ∞[, Pm-a.e., for every f ∈ Floc and
◦
M ∈M. We shall reﬁne this statement from m-almost every starting point x ∈ E
to quasi-every x ∈ E.
◦
LEMMA 4.6.
For f ∈ Floc and M ∈M, the stochastic integral f ∗ (M)t :=
t
0 f (Xs−) d
(M)s can be deﬁned for all t ∈ [0, ∞[, Px-a.s. for q.e. x ∈ E, in
particular, f ∗ (M) is a CAF of X on [0, ∞[.
PROOF.
Since f ∈ Floc, we have {fk | k ∈ N} ⊂ Fb and a nest {Gk | k ∈ N} of
ﬁnely open Borel sets such that f = fk q.e. on Gk. We know that the stochastic
integral fk ∗ (M) is deﬁned Px-a.s. for q.e. x ∈ E. Let
k be the deﬁning set
admitting an E -polar set for the CAF fk ∗ (M) of zero energy and set
∞
:= ω ∈
k
for any k, ∈ N with k < ,
k=1
t
fk(Xs−(ω)) d (M)s(ω)
0
t
=
f (Xs−(ω)) d (M)s(ω) for t < σE\G (ω) .
k
0
Then, Px( c) = 0, m-a.e. x ∈ E. Hence, for each s > 0, Px(θ−1
s
( c)) =
Ps(P·( c))(x) = 0 for q.e. x ∈ E. Setting
:= ∞
k=1
k ∩
s∈Q
θ −1
++ s
( ), we

968
CHEN, FITZSIMMONS, KUWAE AND ZHANG
have Px( ) = 1 for q.e. x ∈ E. For ω ∈
with t < σE\G (ω), we can ﬁnd small
k
s0(= s0(ω)) > 0 such that t + s0 < σE\G (ω). We then see that t < σ
(θ
k
E\Gk
s ω)
for any rational s ∈ ]0, s0[. Hence, for such ω, we have for k < and any rational
s ∈ ]0, s0[
t +s
t +s
fk(Xv−(ω)) d (M)v(ω) =
f (Xv−(ω)) d (M)v(ω).
s
s
Letting s → 0 and noting that ω ∈ k, k ∈ N, we have that for k < , fk ∗ (M)t =
f ∗
(M)t for t < σE\G , P
k
x -a.s. for q.e. x ∈ E. By Lemma 3.1(i), we know
that Px(limk→∞ σE\G = ∞) = 1 for q.e. x ∈ E. Therefore, we obtain that the
k
stochastic integral f ∗
(M) deﬁned as in Remark 4.5(4.5) can be established
Px-a.s. for q.e. x ∈ E. This completes the proof.
THEOREM 4.7 (Generalized Itô formula).
Suppose that
∈ C2(Rd) and u =
(u1, . . . , ud) ∈ F d . Then, for q.e. x ∈ E, Px-a.s. for all t ∈ [0, ∞[,
(u(Xt )) − (u(X0))
d
t
=
∂
(u(Xs−)) duk(Xs)
0 ∂x
k=1
k
d
t
(4.7)
+ 1
∂2
(u(Xs−)) d Mui,c, Muj,c s
2
0 ∂x
i,j =1
i ∂xj
+
(u(Xs)) − (u(Xs−))
s≤t
d
−
∂
(u(Xs−)) uk(Xs) − uk(Xs−) .
∂x
k=1
k
PROOF.
Note that both sides appearing in (4.7) are Px -a.s. deﬁned for q.e. x ∈
E in view of Lemma 4.6. First, we show this Itô formula (4.7) under Pm for a ﬁxed
t ≥ 0. Note that
◦ u ∈ Floc and that
uk(Xt ) = uk(X0) + Muk
t
+ Nuk
t
= uk(X0) + Muk
t
+ (Muk)t.
This version of Itô’s formula follows from Theorems 3.7 and 4.4 by a line of
reasoning similar to that used to prove Itô’s formula for semimartingales (cf. [9]).
Since both sides in (4.7) are right-continuous, (4.7) holds under Pm.
Second, we reﬁne the starting point. Recall that
consists of rcll paths. Let
It (ω) be the difference of the left-hand side and the right-hand side of (4.7). Let
be the intersection of all of the deﬁning sets of AF’s appearing in the for-
mula and {ω ∈
| It(ω) = 0, ∀t ∈ [0, ∞[}. Then, Px( c) = 0, m-a.e. x ∈ E.
Let
be the intersection of the deﬁning sets of AF’s appearing in the formula

STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES
969
and
s∈Q
θ −1
++ s
( ). We then have Px( ) = 1 for q.e. x ∈ E, as in the proof of
Lemma 4.6. Take ω ∈ . For any positive rational s > 0, we then have It (θsω) = 0,
that is,
(u(Xt+s(ω))) − (u(Xs(ω)))
d
t +s
=
∂
(u(Xv−(ω))) duk(Xv(ω))
s
∂x
k=1
k
d
t +s
+ 1
∂2
(u(Xv−(ω))) d Mui,c, Muj,c v(ω)
2
s
∂x
i,j =1
i ∂xj
+
(u(Xv(ω))) − (u(Xv−(ω)))
s<v≤t+s
d
−
∂
(u(Xv−(ω))) uk(Xv(ω)) − uk(Xv−(ω)) .
∂x
k=1
k
Letting s → 0 and using the right-continuity of s → u(Xs) and stochastic integrals,
we have It (ω) = 0. This completes the proof.
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Z.-Q. CHEN
P. J. FITZSIMMONS
DEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF WASHINGTON
UNIVERSITY OF CALIFORNIA AT SAN DIEGO
SEATTLE, WASHINGTON 98195
LA JOLLA, CALIFORNIA 92093-0112
USA
USA
E-MAIL: zchen@math.washington.edu
E-MAIL: pﬁtzsim@ucsd.edu
K. KUWAE
T.-S. ZHANG
DEPARTMENT OF MATHEMATICS
SCHOOL OF MATHEMATICS
FACULTY OF EDUCATION
UNIVERSITY OF MANCHESTER
KUMAMOTO UNIVERSITY
SACKVILLE STREET, MANCHESTER M60 1QD
KUMAMOTO 860-8555
UNITED KINGDOM
JAPAN
E-MAIL: tzhang@maths.manchester.ac.uk
E-MAIL: kuwae@gpo.kumamoto-u.ac.jp

Stochastic Calculus For Symmetric Markov Processes

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