Stochastic Calculus for Symmetric Markov Processes

(Dedicated to S. Nakao on the occasion of his 60th birthday)

Z.-Q. Chen^{∗}, P. J. Fitzsimmons^{†}, K. Kuwae^{‡} and T.-S. Zhang^{§}

(Revised Version: April 18, 2007)

Abstract 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.

AMS 2000 Mathematics Subject Classification: Primary 31C25; Secondary 60J57, 60J55, 60H05.

Keywords and phrases: Symmetric Markov process, time reversal, stochastic integral, generalized Itô formula, additive functional, martingale additive functional, dual additive functional, Revuz measure, dual predictable projection.

∗The research of this author is supported in part by NSF Grant DMS-0600206. †The research of this author is supported by a foundation based on the academic cooperation between Yokohama

City University and UCSD

‡The research of this author is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD, and partially supported by a Grant-in-Aid for Scientific Research (C) No. 16540201 from Japan Society for the Promotion of Science.

§The research of this author is supported in part by the British EPSRC

1 Introduction and Framework

It is well-known that stochastic integrals and Itô’s formula for semimartingales play a central role in modern probability theory. However there are many important classes of Markov processes that are not semimartingales. For example, symmetric diffusions on ℝ^{d} whose infinitesimal generators are elliptic operators in divergence form L = ∑_{i,j=1}^{d} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂}{∂x_{j}} ) with merely measurable coefficients need not be semimartingales. Even when X is a Brownian motion in ℝ^{d} and u ∈ W^{1,2}(ℝ^{d}) := {u ∈ L^{2}(ℝ^{d}; dx) | |∇u| ∈ L^{2}(ℝ^{d}; dx)}, the process u(X_{t}) is not a semimartingale in general. To study such processes, Fukushima obtained the following substitute for Itô’s formula (see [8]): for u ∈ W^{1,2}(ℝ^{d}),

u(X_{t}) = u(X_{0}) + M_{t}^{u} + N_{t}^{u}  for t ≥ 0,  (1.1)

P_{x}-a.s. for quasi-every x ∈ ℝ^{d}, where M^{u} 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 process X and for u ∈ F, where (E, F) is the Dirichlet space for X. In this paper, a stochastic process ξ = {ξ_{t}, 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 P_{m}, where m is the Lebesgue measure on ℝ^{d}, as it has finite quadratic variation on compact time intervals. Nakao introduced a stochastic integral ∫_{0}^{t} f(X_{s}) dN_{s}^{u} in [17] by using a Riesz representation theorem in a suitably constructed Hilbert space. Nakao’s stochastic integral played an important role in the study of lower order perturbation of diffusion processes by Lunt, Lyons and Zhang [15] and by Fitzsimmons and Kuwae [6]. However Nakao’s definition of the stochastic integral ∫_{0}^{t} f(X_{s}) dN^{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 thereby more direct and provides additional insight into stochastic integration for Dirichlet processes. This approach enables us to define Λ(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 [17] 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 = {Ω, F_{∞}, F_{t}, X_{t}, θ_{t}, ζ, P_{x}, 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 L^{2}(E; m) is known to be quasi-regular (see [16]). By [1], (E, F) is quasihomeomorphic to a regular 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 associated Dirichlet space (E, F) is regular on L^{2}(E; m) and that m is a positive Radon measure with full topological support on E. This we do throughout the sequel.

Without loss of generality, we can take Ω to be the canonical path space D([0, ∞[→ E_{Δ}) of right-continuous, left-limited (rcll, for short) functions from [0, ∞[ to E_{Δ}, for which Δ is a trap (that is, if ω(t) = Δ then ω(s) = Δ for all s > t). For any ω ∈ Ω, we set X_{t}(ω) := ω(t). Let ζ(ω) := inf{t ≥ 0 | X_{t}(ω) = Δ} be the lifetime of X. As usual, F_{∞} and F_{t} are the minimal augmented σ-algebras obtained from F_{∞}^{0} := σ{X_{s} | 0 ≤ s < ∞} and F_{t}^{0} := σ{X_{s} | 0 ≤ s ≤ t}, respectively, under P_{x}; see the next section for more details. We sometimes use a filtration denoted by (M_{t}) on (Ω, M) in order to represent several filtrations, for example, (F_{t}^{0}), (F_{t+}^{0}) on (Ω, F_{∞}^{0}), (F_{t}) on (Ω, F_{∞}) and others introduced later. We use θ_{t} to denote the shift operator defined by θ_{t}(ω)(s) := ω(t + s), t, s ≥ 0. Let ω_{Δ} be the path starting from Δ. Then ω_{Δ}(s) ≡ Δ for all s ∈ [0, ∞[. Note that θ_{ζ(ω)}(ω) = ω_{Δ} if ζ(ω) < ∞, {ω_{Δ}} ∈ F_{0}^{0} ⊂ F_{t}^{0} for all t > 0 and P_{x}({ω_{Δ}}) ≤ P_{x}(X_{0} = Δ) = 0 for x ∈ E. For a Borel subset B of E, τ_{B} := inf{t > 0 | X_{t} ∉ B} (the exit time of B) is an (F_{t})-stopping time. If B is closed, then τ_{B} is an (F_{t+}^{0})-stopping time. Also, ζ is an (F_{t}^{0})-stopping time because {ζ ≤ t} = {X_{t} = Δ} ∈ F_{t}^{0}, t ≥ 0. The transition

semigroup of X, {P_{t}, t ≥ 0}, is defined by

P_{t}f(x) := E_{x}[f(X_{t})] = E_{x}[f(X_{t}) : t < ζ], t ≥ 0.

Each P_{t} may be viewed as an operator on L^{2}(E; m); collectively these operators form a strongly L^{2}-continuous semigroup of self-adjoint contractions. The Dirichlet form associated with X is the bilinear form

E(u, v) := lim_{t↓0} \frac{1}{t}(u − P_{t}u, v)_{m}

defined on the space

F := {u ∈ L^{2}(E; m) | sup_{t>0} t^{−1}(u − P_{t}u, u)_{m} < ∞} .

Here we use the notation (f, g)_{m} := ∫_{E} f(x)g(x) m(dx).

For the reader’s convenience, we recall the following definitions from [16] and [8].

Definition 1.1 (i) An increasing sequence {F_{n}}_{n≥1} of closed subsets of E is an E-nest (or simply nest) if and only if ∪_{n≥1}F_{F_{n}} is E_{1}-dense in F, where E_{1} = E + ( , )_{L^{2}(E,m)} and

F_{F_{n}} := {u ∈ F : u = 0 m-a.e. on E \ F_{n}}.

(ii) A subset N ⊂ E is E-polar if and only if there is an E-nest {F_{n}}_{n≥1} such that N ⊂ ∩_{n≥1}(E\F_{n}).

(iii) A function f on E is said to be quasi-continuous if there is an E-nest {F_{n}}_{n≥1} such that f|_{F_{n}} is continuous on F_{n} for each n ≥ 1; we denote this situation briefly by writing f ∈ C({F_{n}}).

(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

P_{x} (X_{t} ∈ E_{Δ} \ N for t ≥ 0 and X_{t−} ∈ E_{Δ} \ N for t > 0) = 1 for every x ∈ E \ N.

It is known (cf. [8]) that a family {F_{n}} of closed sets is an E-nest if and only if

P_{x} ( lim_{n→∞} τ_{F_{n}} = ζ) = 1 for q.e. x ∈ E.

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 stands for “continuous additive functional”, “positive continuous additive functional” and “martingale additive functional”, respectively; the definitions of these terms can be found in [8].

Let M^{∘} and N_{c} denote, respectively, the space of MAFs of finite energy and the space of continuous additive functionals of zero energy. For u ∈ F, Fukushima’s decomposition holds:

u(X_{t}) − u(X_{0}) = M_{t}^{u} + N_{t}^{u},  for every t ∈ [0, ∞[,  (1.2)

P_{x}-a.s. for q.e. x ∈ E, where M^{u} ∈M^{∘} and N^{u} ∈ N_{c}.

A positive continuous additive functional (PCAF) of X (call it A) determines a measure μ = μ_{A} on the Borel subsets of E via the formula

μ(f) =↑ lim_{t↓0} \frac{1}{t}E_{m} [∫_{0}^{t} f(X_{s}) dA_{s}] ,  (1.3)

in which f : E → [0, ∞] is Borel measurable. Here ↑ lim_{t↓0} indicates an increasing 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 {F_{n}} of closed subsets of E such that μ(F_{n}) < ∞ for each n ∈ ℕ. Conversely, given a smooth measure μ, there is a unique PCAF A^{μ} such that (1.3) holds with A = A^{μ}. In the sequel we refer to this bijection between smooth measures and PCAFs as the Revuz correspondence, and to μ as the Revuz measure of A^{μ}.

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 M^{u} is the martingale part in the Fukushima decomposition of u ∈ F, then ⟨M^{u}, M⟩ is a CAF locally of bounded variation, and we have the associated Revuz measure μ_{⟨M^{u},M⟩}, which is locally the difference of smooth (positive) measures. For u ∈ F, the Revuz measure μ_{⟨M^{u}⟩} of ⟨M^{u}⟩ will usually be denoted by μ_{⟨u⟩}.

Let (N(x, dy), H_{t}) be a Lévy system for X; that is, N(x, dy) is a kernel on (E_{Δ}, B(E_{Δ})) and H_{t} is a PCAF with bounded 1-potential such that for any nonnegative Borel function ϕ on E_{Δ} × E_{Δ} vanishing on the diagonal and any x ∈ E_{Δ},

E_{x} [∑_{s≤t} ϕ(X_{s−}, X_{s})] = E_{x} [∫_{0}^{t} ∫_{E_{Δ}} ϕ(X_{s}, y)N(X_{s}, dy) dH_{s}] .

To simplify notation, we will write

Nϕ(x) := ∫_{E_{Δ}} ϕ(x, y)N(x, dy)

and

(Nϕ ∗ H)_{t} := ∫_{0}^{t} Nϕ(X_{s}) dH_{s}.

Let μ_{H} be the Revuz measure of the PCAF H. Then the jumping measure J and the killing measure κ of X are given by

J(dx, dy) = \frac{1}{2}N(x, dy)μ_{H}(dx), and κ(dx) = N(x, \{Δ\})μ_{H}(dx).

These measures feature in the Beurling-Deny decomposition of E: for f, g ∈ F,

E(f, g) = E^{(c)}(f, g) + ∫_{E×E}(f(x) − f(y))(g(x) − g(y)) J(dx, dy) + ∫_{E} f(x)g(x) κ(dx),

where E^{(c)} is the strongly local part of E.

For u ∈ F, the martingale part M_{t}^{u} in (1.2) can be decomposed as

M_{t}^{u} = M_{t}^{u,c} + M_{t}^{u,j} + M_{t}^{u,κ} for every t ∈ [0, ∞[,

P_{x}-a.s. for q.e. x ∈ E, where M_{t}^{u,c} is the continuous part of the martingale M^{u}, and

M_{t}^{u,j} = lim_{ε↓0} { ∑_{0<s≤t}(u(X_{s}) − u(X_{s−}))1_{{|u(X_{s})−u(X_{s−})|>ε}}1_{{s<ζ}}

− ∫_{0}^{t} (∫_{{y∈E: |u(y)−u(X_{s})|>ε}}(u(y) − u(X_{s}))N(X_{s}, dy)) dH_{s}} ,
M_{t}^{u,κ} = ∫_{0}^{t} u(X_{s})N(X_{s}, {Δ}) dH_{s} − u(X_{ζ−})1_{{t≥ζ}},

are the jump and killing parts of M^{u}, respectively. All three terms in this decomposition of M^{u} are elements of M^{∘}. See Theorem A.3.9 of [8]. The limit in the expression for M^{u,j} is in the sense of convergence in the norm of the space of MAFs of finite energy, and of convergence in probability under P_{x} for q.e. x ∈ E (see [8]).

Let N_{c}^{∗} ⊂ N_{c} denote the class of continuous additive functionals of the form N^{u} + ∫_{0}^{·} g(X_{s}) ds for some u ∈ F and g ∈ L^{2}(E; m). Nakao [17] constructed a linear map Γ from M^{∘} into N_{c}^{∗} in the following way. It is shown in [17] that, for every Z ∈M^{∘}, there is a unique w ∈ F such that

E_{1}(w, f) = \frac{1}{2}μ_{⟨M^{f} +M^{f,κ}, Z⟩}(E) for every f ∈ F.  (1.4)

This unique w is denoted by γ(Z). The operator Γ is now defined by

Γ(Z)_{t} := N_{t}^{γ(Z)} − ∫_{0}^{t} γ(Z)(X_{s}) ds  for every Z ∈M^{∘} .  (1.5)

Nakao showed that Γ(Z) is characterized by the following equation

lim_{t↓0} \frac{1}{t}E_{g·m} [Γ(Z)_{t}] = −\frac{1}{2}μ_{⟨M^{g}+M^{g,κ}, Z⟩}(E)  for every g ∈ F_{b}.  (1.6)

Here F_{b} := F ∩ L^{∞}(E; m). So in particular we have Γ(M^{u}) = N^{u} for u ∈ F. Nakao [17] then used the operator Γ to define a stochastic integral

∫_{0}^{t} f(X_{s}) dN_{s}^{u} := Γ(f ∗ M^{u})_{t} − \frac{1}{2}⟨M^{f,c} + M^{f,j}, M^{u,c} + M^{u,j}⟩_{t},  (1.7)

where u ∈ F, f ∈ F ∩ L^{2}(E; μ_{⟨u⟩}) and (f ∗ M^{u})_{t} := ∫_{0}^{t} f(X_{s−}) dM_{s}^{u}. If we define

Ñ_{c} := {N ∈ N_{c} | N = N^{u} + A^{μ} for some u ∈ F and some signed smooth measure μ} ,

then we see by (1.5) that ∫_{0}^{·} f(X_{s}) dN_{s}^{u} ∈ Ñ_{c} if u ∈ F and f ∈ F ∩ L^{2}(E; μ_{⟨u⟩}). However, the conditions imposed on the integrand f(X_{t}) 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 give a new way of defining Γ(M) and Nakao’s stochastic integral for zero energy AFs N^{u}.

For a finite rcll AF M_{t}, 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

M_{t} − M_{t−} = φ(X_{t−}, X_{t})  for every t ∈]0, ζ[, P_{m}-a.e.  (1.8)

Such a φ is uniquely determined up to J-negligible sets. We will call φ the jump function of M. When M = M^{u}, u ∈ F, the jump function φ for M^{u} can be taken to be as φ(x, y) = u(y) − u(x) for (x, y) ∈ E × E, with u(Δ) := 0. We have a similar result for locally square-integrable MAFs on [[0, ζ[[ (see Definition 2.5(iii) for the definition of locally square-integrable MAF on [[0, ζ[[. Let M be a locally square-integrable MAF on [[0, ζ[[. Then there exists a jump function φ on E × E for M satisfying the property (1.8) (see Corollary 2.9). Assume

∫_{0}^{t} ∫_{E} (φ^{2}1_{{|φ̂|≤1}} + |φ̂|1_{{|φ̂|>1}}) (X_{s}, y)N(X_{s}, dy) dH_{s} < ∞  for every t < ζ,  (1.9)

P_{x}-a.s. for q.e. x ∈ E, where φ̂(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

K_{t} − K_{t−} = −φ̂(X_{t−}, X_{t}) for t < ζ, P_{x}-a.s. for q.e. x ∈ E,

Define P_{m}-a.e. on [0, ζ[,

Λ(M)_{t} := −\frac{1}{2} (M_{t} + M_{t} ∘ r_{t} + φ(X_{t}, X_{t−}) + K_{t}) for t ∈ [0, ζ[,

where r_{t} is the time-reversal operator at time t > 0. Note that since X is symmetric, the measure P_{m}, when restricted to {t < ζ}, is invariant under r_{t}. This time reversibility plays an important role in this paper. So Λ(M) is clearly well-defined on [[0, ζ[[ under the σ-finite measure P_{m}. It will be shown in Theorem 2.18 and Remark 3.4(ii) below that Λ(M) is a continuous even AF of X on [[0, ζ[[ admitting m-null set. Note that when M = M^{u} for some u ∈ F, φ(x, y) = u(y) − u(x) is antisymmetric and so φ̂ = 0. Thus P_{m}-a.e. on {t < ζ},

Λ(M^{u})_{t} := −\frac{1}{2} (M_{t}^{u} + M_{t}^{u} ∘ r_{t} + u(X_{t−}) − u(X_{t})) = N_{t}^{u}

The last identity follows by applying the time-reversal operator to both sides of (1.2) and using the fact that N_{t}^{u} ∘ r_{t} = N_{t}^{u} P_{m}-a.e. on [[0, ζ[[ (cf. [5, Theorem 2.1]). It follows then for every u ∈ F, Λ(M^{u}) = Γ(M^{u}) on [[0, ζ[[ P_{m}-a.e. We will show in Theorem 3.6 below that this holds when M^{u} is replaced by any M ∈M^{∘}. Therefore, under the σ-finite measure P_{m}, Λ is a genuine extension of Nakao’s map Γ.

A function f is said to be locally in F (denoted as f ∈ F_{loc}) if there is an increasing sequence of finely open Borel sets {D_{k}, k ≥ 1} with ∪_{k=1}^{∞}D_{k} = E q.e., and for every k ≥ 1 there is f_{k} ∈ F such that f = f_{k} m-a.e. on D_{k}. 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 ∈ F_{loc} admits a quasi-continuous m-version, so we may assume all f ∈ F_{loc} are quasi-continuous. Then we have f = f_{k} q.e. on D_{k}. For f ∈ F_{loc}, M^{f,c} is well defined as a continuous MAF on [[0, ζ[[ of locally finite energy. Moreover, for f ∈ F_{loc} and a locally square-integrable MAF M on [[0, ζ[[,

t ↦ (f ∗ M)_{t} := ∫_{0}^{t} f(X_{s−}) dM_{s}

is a locally square-integrable MAF on [[0, ζ[[. Here for a locally square-integrable MAF M on [[0, ζ[[ denote by M^{c} its continuous part, which is also a locally square-integrable MAF on [[0, ζ[[ (see Theorem 8.23 in [10]).

Definition 1.2 (Stochastic integral) Suppose that M is a locally square-integrable MAF on [[0, ζ[[ and f ∈ F_{loc}. Let φ : E × E → ℝ be a jump function for M, and assume that φ satisfies condition (1.9). Define on [[0, ζ[[

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s}
:= Λ(f ∗ M)_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s},

whenever Λ(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 measure P_{m} 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 formula.

2 Additive functionals

In this section, we will prove some facts about additive 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_{t}^{ν}) be the P_{ν}-completion of F_{∞}^{0} (resp. P_{ν}-completion of F_{t}^{0} in F_{∞}^{ν}) and set F_{∞} := ⋂_{ν∈P(E)} F_{∞}^{ν} and F_{t} := ⋂_{ν∈P(E)} F_{t}^{ν}. Let F_{∞}^{m} (resp. F_{t}^{m}) be the P_{m}-completion of F_{∞}^{0} (resp. P_{m}-completion of F_{t}^{0} in F_{∞}^{m}). Although m may not be a finite measure on E, we do have F_{∞} ⊂ F_{∞}^{m}, F_{t} ⊂ F_{t}^{m}, because for g ∈ L^{1}(E; m) with 0 < g ≤ 1 on E satisfying gm ∈ P(E), P_{gm}-negligibility is the same as P_{m}-negligibility.

For a fixed filtration (M_{t}) on (Ω, M), we recall the notions of (M_{t})-predictability, (M_{t})- optionality and (M_{t})-progressive measurability as follows (see [18] for more details): On [0, ∞[×Ω, the (M_{t})-predictable (resp. (M_{t})-optional) σ-field P(M_{t}) (resp. O(M_{t})) is defined as the smallest σ-field over [0, ∞[×Ω containing all P_{ν}(M)-evanescent sets for all ν ∈ P(E_{Δ}) and with respect to which all M_{t}-adapted lcrl (left-continuous, right-limited) (resp. rcll) processes are measurable. A process ϕ(s, ω) on [0, ∞[×Ω is said to be (M_{t})-progressively measurable provided [0, t] × Ω ∋ (s, ω) ↦ ϕ(s, ω) is B([0, t])⊗M_{t}-measurable for all t > 0. It is well-known that (M_{t})-predictability implies (M_{t})-optionality, which in turn implies (M_{t})-progressive measurability.

For [0, ∞]-valued functions S, T on Ω with S ≤ T, we employ the usual notation for stochastic intervals; for example,

[[S, T [[ := {(t, ω) ∈ [0, ∞[×Ω | 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, ∞[×Ω. If S and T are (M_{t})-stopping times, then [[S, T]], [[S, T [[,⋯, and [[S]] are (M_{t})-optional (see Theorem 3.16 in [10]).

Definition 2.1 (AF) An (F_{t})-adapted (resp. (F_{t}^{m})-adapted) process A = (A_{t})_{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 Ξ ∈ F_{∞} and an E-polar (resp. m-null) set N satisfying the following conditions;

(i) P_{x}(Ξ) = 1 for all x ∈ E \ N,

(ii) θ_{t}Ξ ⊂ Ξ for all t ≥ 0; in particular, ω_{Δ} ∈ Ξ and P_{Δ}(Ξ) = 1, because of ω_{Δ} = θ_{ζ(ω)}(ω) for all ω ∈ Ξ, (iii) for all ω ∈ Ξ, A_{·}(ω) is right continuous with left hand limits on [0, ζ(ω)[, A_{0}(ω) = 0, |A_{t}(ω)| <

∞ for t < ζ(ω) and A_{t+s}(ω) = A_{t}(ω) + A_{s}(θ_{t}ω) for all t, s ≥ 0,

(iv) for all t ≥ 0, A_{t}(ω_{Δ}) = 0; in particular, under the additivity in (iii), A_{t}(ω) = A_{ζ(ω)}(ω) for all t ≥ ζ(ω) and ω ∈ Ξ.

An AF A (admitting m-null set) is called right-continuous with left limits (rcll AF in brief) if A_{ζ(ω)−} exists for each ω ∈ Ξ. An AF A (admitting m-null set) is said to be finite (resp. continuous additive functional (CAF in brief)) if |A_{t}(ω)| < ∞, t ∈ [0, ∞[ (resp. t ↦ A_{t}(ω) is continuous on [0, ∞[) for each ω ∈ Ξ. A [0, ∞[-valued CAF is called a positive continuous additive functional (PCAF in short). Two AFs A and B are called equivalent if there exists a common defining set Ξ ∈ F_{∞} and an E-polar set N such that A_{t}(ω) = B_{t}(ω) for all t ∈ [0, ∞[ and ω ∈ Ξ. We call A = (A_{t})_{t≥0} an AF on [[0, ζ[[ or a local AF (admitting m-null set) if A is (F_{t})-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 similarly defined. Two AFs on [[0, ζ[[, A and B, are called equivalent if there exists a common defining set Ξ ∈ F_{∞} and an E-polar set N such that A_{t}(ω) = B_{t}(ω) for all t ∈ [0, ζ[ and ω ∈ Ξ.

Remark 2.2 Any PCAF A on [[0, ζ[[ can be extended to a PCAF by setting

A_{t}(ω) := \begin{cases}lim_{u↑ζ} A_{u}(ω), & if t ≥ ζ(ω) > 0, \\ 0, & if t ≥ ζ(ω) = 0 \end{cases}

for ω ∈ Ξ and setting A_{t}(ω) ≡ 0 for ω ∈ Ξ^{c}. The (F_{t})-adaptedness of this extended A holds as follows: for a fixed T > 0, we know {A_{t} ≤ T } ∩ {t < ζ} ∈ F_{t}. From this, we have the F_{ζ}- measurability of {A_{ζ} ≤ T }. Indeed, {A_{ζ} ≤ T } = ⋂_{t∈ℚ_{+}}{A_{t} ≤ T, t < ζ} ∈ F_{ζ} as {A_{t} ≤ T, t < ζ} ∈ F_{ζ} for any t ≥ 0. Thus {A_{t} ≤ T } ∩ {t ≥ ζ} = {A_{ζ} ≤ T } ∩ {t ≥ ζ} ∈ F_{t}. Therefore, {A_{t} ≤ T } ∈ F_{t} for any T > 0, which gives the (F_{t})-adaptedness of A. Noting ζ ∘ θ_{t} = ζ − t if t < ζ and ζ ∘ θ_{t} = 0 if t ≥ ζ, we conclude that A_{ζ} = A_{t} +A_{ζ} ∘θ_{t} for any t ∈ [0, ∞[ on Ξ. Consequently, A_{t+s} = A_{t} +A_{s} ∘θ_{t} holds for any t, s ∈ [0, ∞[ on Ξ. □

The following lemma is a special case of [17, Theorem 2.2].

Lemma 2.3 Let A, B be PCAFs such that for m-a.e. x ∈ E, E_{x}[A_{t}] = E_{x}[B_{t}] for all t ≥ 0, and suppose that the Revuz measure μ_{A} has finite total mass. Then A is equivalent to B.

Remark 2.4 The above lemma may fail if the condition μ_{A}(E) < ∞ is not satisfied. For example, take E = ℝ^{d} with d ≥ 2, and let X be Brownian motion on ℝ^{d} and μ_{A}(dx) = |x|^{−d−1}dx. Then μ_{A} is a smooth measure and it corresponds to a PCAF A of X. Let B_{t} = A_{t} + t, which is a PCAF of X with Revuz measure μ_{A}(dx) + dx. However

E_{x}[A_{t}] = ∫_{0}^{t} (∫_{ℝ^{d}} p(s, x, y)|y|^{−d−1}dy) ds = ∞ = E_{x}[B_{t}] for every x ∈ ℝ^{d} \ {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 (F_{t})-stopping time and M a process, then M^{T} is the stopped process defined by M_{t}^{T} := M_{t∧T}. Following [10], we give the notion of local martingales of interval type:

Definition 2.5 (Processes of interval type) Let D be a class of (F_{t})-adapted processes and denote by D_{loc} its localization (resp. by D_{f-loc} its localization by a nest of finely open Borel sets); that is, M ∈ D_{loc} (resp. M ∈ D_{f-loc}) if and only if there exists a sequence M^{n} ∈ D and an increasing sequence of stopping times T_{n} with T_{n} → ∞ (resp. a nest {G_{n}} of finely open Borel sets) such that M^{T_{n}} = (M^{n})^{T_{n}} (resp. M_{t} = M_{t}^{n} for t < τ_{G_{n}}) for each n. Here a family {G_{n}} of finely open Borel sets is called a nest if P_{x}(lim_{n→∞} τ_{G_{n}} = ζ) = 1 for q.e. x ∈ E. (But see Lemma 3.1.) Clearly, D ⊂ D_{loc} (resp. D ⊂ D_{f-loc}) and (D_{loc})_{loc} = D_{loc} (resp. (D_{f-loc})_{f-loc} = D_{f-loc}). If D is a subclass of AFs, then so is D_{loc} (for if M ∈ D_{loc}, then there exists M^{n} and T_{n} as above, and for each ω and t, s ≥ 0, there exists n ∈ ℕ with s+t < T_{n}(ω) and s < T_{n}(θ_{t}ω), hence M_{t+s}(ω) = M_{t}(ω)+M_{s}(θ_{t}ω)), while D_{f-loc} is contained in the class of AFs on [[0, ζ[[.

(i) B ⊂ [0, ∞[×Ω is called a set of interval type if there exists a non-negative random variable S such that for each ω ∈ Ω the section B_{ω} := {t ∈ [0, ∞[ | (t, ω) ∈ B} is [0, S(ω)] or [0, S(ω)[ and B_{ω} ≠ ∅.

(ii) Let B be an (F_{t})-optional set of interval type. A real-valued stochastic process M on B (that is, M1_{B} = (M_{t}(ω)1_{B}(t, ω))_{t≥0} is a real-valued stochastic process) is said to be in D^{B} if and only if there exists N ∈ D such that M1_{B} = N1_{B}, and is said to be locally in D on B (write M ∈ (D_{loc})^{B}) if and only if S := D_{B^{c}} is the debut of B^{c} and there exists an increasing sequence of (F_{t})-stopping times {S_{n}} with lim_{n→∞} S_{n} = S and a sequence of M^{n} ∈ D such that B_{ω} ⊂ ⋃_{n=1}^{∞}[0, S_{n}(ω)] P_{x}-a.s. ω ∈ Ω and (M1_{B})^{S_{n}} = (M^{n}1_{B})^{S_{n}} for all n ∈ ℕ and t ≥ 0, P_{x}-a.s. ω ∈ Ω for q.e. x ∈ E. Clearly, D^{B} ⊂ (D_{loc})^{B}. Moreover, D^{B_{2}} ⊂ D^{B_{1}} and (D_{loc})^{B_{2}} ⊂ (D_{loc})^{B_{1}} for any pair of (F_{t})-optional sets B_{1}, B_{2} of interval type with B_{1} ⊂ B_{2}.

(iii) Let B be an (F_{t})-optional set of interval type. We set

M^{1} := {M | M is a finite rcll AF, E_{x}[|M_{t}|] < ∞, E_{x}[M_{t}] = 0 for E-q.e. x ∈ E and all t ≥ 0},

and speak of an element of (M^{1})^{B} (resp. (M_{loc}^{1})^{B}) as being an MAF on B (resp. a local MAF on B). Similarly,

M := {M | M is a finite rcll AF, E_{x}[M_{t}^{2}] < ∞, E_{x}[M_{t}] = 0 for E-q.e. x ∈ E and all t ≥ 0},

and an element of M^{B} (resp. (M_{loc})^{B}) is a square-integrable MAF on B (resp. locally squareintegrable MAF on B). We further set

M^{c} : = {M ∈ M | M is a CAF},
M^{d} : = {M ∈ M | M is a purely discontinuous AF},

and an element of (M_{loc}^{c})^{B} (resp. (M_{loc}^{d})^{B}) is called a locally square-integrable continuous MAF on B (resp. locally square-integrable purely discontinuous MAF on B). For M ∈ (M_{loc})^{B}, M admits a unique decomposition M = M^{c} + M^{d} with M^{c} ∈ (M_{loc}^{c})^{B} and M^{d} ∈ (M_{loc}^{d})^{B} (see Theorem 8.23 in [10]). In these definitions, we omit the usage “on B” when B = [0, ∞[×Ω.

For a [0, ∞]-valued function R on Ω and A ⊂ Ω, R_{A} := R · 1_{A} + (+∞) · 1_{A^{c}} is called the restriction of R on A. Clearly, R ≤ R_{A}.

Remark 2.6 When B = [[0, R[[ for a given (F_{t})-stopping time R, there is another notion of “locally in D on B”, obtained by replacing (M1_{B})^{S_{n}} = (M^{n}1_{B})^{S_{n}} with M^{S_{n}}1_{B} = (M^{n})^{S_{n}}1_{B} in our definition; this is a weaker notion than ours, because t ↦ 1_{B}(t, ω) is decreasing and 1_{B}(t, ω)1_{B}(s, ω) = 1_{B}(t, ω) for s ≤ t and ω ∈ Ω. This weaker notion is described in [18]. □

Definition 2.7 (MAF locally of finite energy) Recall that M^{∘} is the totality of MAFs of finite energy, that is,

M^{∘}:= \{M ∈ M | e(M) := lim_{t↓0} \frac{1}{2t}E_{m}[M_{t}^{2}] < ∞\} .

We say that an AF M on [[0, ζ[[ is locally in M^{∘} (and write M ∈M^{∘}_{f-loc})) if there exists a sequence {M^{n}} in M^{∘} and a nest {G_{n}} of finely open Borel sets such that M_{t} = M_{t}^{n} for t < τ_{G_{n}} for each n ∈ ℕ. In case X is a diffusion process with no killing inside E, we can define the predictable quadratic variation ⟨M⟩ for M ∈M^{∘}_{f-loc} as follows: Note that M_{t∧τ_{Gn}}^{n} = M_{t∧τ_{Gn}}^{m} for n < m because of the continuity of M^{n}. Owing to the uniqueness of Doob-Meyer decomposition, we see ⟨M^{n}⟩_{t∧τ_{Gn}} = ⟨M^{m}⟩_{t∧τ_{Gn}}. The predictable quadratic variation ⟨M⟩ of M ∈M^{∘}_{f-loc} as a PCAF is well-defined by setting ⟨M⟩_{t} = ⟨M^{n}⟩_{t}, t < τ_{G_{n}}, n ∈ ℕ, with Remark 2.2 and by choosing an appropriate defining set and E-polar set of ⟨M⟩, where M^{n} ∈M^{∘} and {G_{n}} is a nest of finely open Borel sets such that M_{t} = M_{t}^{n}, t < τ_{G_{n}}.

Proposition 2.8 (M_{loc})^{[[0,ζ[[} ⊂M^{∘}_{f-loc}. More precisely, for each M ∈ (M_{loc})^{[[0,ζ[[}, there exists a nest {G_{k}} of finely open Borel sets such that 1_{G_{k}} ∗ M ∈M^{∘} for each k ∈ ℕ, and the predictable quadratic variation process ⟨M⟩ can be constructed as a PCAF.

Proof. Let M ∈ (M_{loc})^{[[0,ζ[[}. Then there exists an increasing sequence {T_{n}} of stopping times with lim_{n→∞} T_{n} = ζ, (P_{x}-a.s. ω ∈ Ω for q.e. x ∈ E) and M^{n} ∈ M_{loc} such that M_{t∧T_{n}}1_{[0,ζ[}(t ∧ T_{n}) = M_{t∧T_{n}}^{n}1_{[0,ζ[}(t ∧ T_{n}) holds for all t ≥ 0 P_{x}-a.s. for q.e. x ∈ E. We may assume that it holds for all ω ∈ Ω by changing the sample space. Note that [0, ζ(ω)[⊂ ⋃_{n=1}^{∞}[0, T_{n}(ω)] for all ω ∈ Ω. Hence, M_{t∧T_{n}}^{m}1_{[0,ζ[}(t ∧ T_{n}) = M_{t∧T_{n}}^{n}1_{[0,ζ[}(t ∧ T_{n}) for n < m. As noted in Definition 2.5, we see that M is an AF on [[0, ζ[[. Owing to the uniqueness of the Doob-Meyer decomposition for semimartingales on [[0, ζ[[ (see [10]), we have ⟨M^{m}⟩_{t∧T_{n}}1_{[0,ζ[}(t ∧ T_{n}) = ⟨M^{n}⟩_{t∧T_{n}}1_{[0,ζ[}(t ∧ T_{n}) for n < m. Thus, we have ⟨M^{m}⟩_{t} = ⟨M^{n}⟩_{t} for t < T_{n} and n < m. The predictable quadratic variation ⟨M⟩ of M is therefore well defined by setting ⟨M⟩_{t} := ⟨M^{n}⟩_{t} for t < T_{n}. Setting ⟨M⟩_{t} := ⟨M⟩_{ζ} := lim_{s↑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 {F_{k}} an E-nest of closed sets such that μ_{⟨M⟩}(F_{k}) < ∞ for each k, and let G_{k} be the fine interior of F_{k}. Then {G_{k}} is a nest. In view of the proofs of Theorem 5.6.1 and Lemma 5.6.2 in [8], the stochastic integral 1_{G_{k}} ∗M is of finite energy with e(1_{G_{k}} ∗M) = \frac{1}{2}μ_{⟨M⟩}(G_{k}), and its predictable quadratic variation ⟨1_{G_{k}} ∗M⟩ is a PCAF. Let μ_{k} (resp. μ_{k}) be the Revuz measure corresponding to ⟨1_{G_{k}} ∗M⟩ (resp. ⟨1_{G_{k}} ∗M, M⟩). By Lemma 5.6.2 in [8], for M_{i} ∈M^{∘} and f_{i} ∈ L^{2}(E; μ_{⟨M_{i}⟩}) (i = 1, 2), we have f_{1}f_{2}μ_{⟨M_{1},M_{2}⟩} = μ_{⟨f_{1}∗M_{1},f_{2}∗M_{2}⟩}, hence ∫_{0}^{t}(f_{1}f_{2})(X_{s}) d⟨M_{1}, M_{2}⟩_{s} = ⟨f_{1} ∗ M_{1}, f_{2} ∗ M_{2}⟩_{t}. From this, we see ⟨μ_{k}, f^{2}⟩ = ⟨μ_{k}, f^{2}⟩ = ⟨1_{G_{k}}μ_{⟨M⟩}, f^{2}⟩ for any f ∈ L^{2}(E; μ_{⟨M⟩}), consequently we have μ_{k} = μ_{k} = 1_{G_{k}}μ_{⟨M⟩} by μ_{⟨M⟩}(G_{k}) < ∞. This yields ⟨1_{G_{k}} ∗M⟩_{t} = ⟨1_{G_{k}} ∗M, M⟩_{t} = ∫_{0}^{t} 1_{G_{k}}(X_{s}) d⟨M⟩_{s} for t < ζ, hence ⟨M − 1_{G_{k}} ∗ M⟩_{t} = 0 for t < τ_{G_{k}}. Therefore, M_{t} = (1_{G_{k}} ∗ M)_{t} for t < τ_{G_{k}} and

1_{G_{k}} ∗ M ∈M^{∘}. □

Corollary 2.9 Let M be a locally square-integrable MAF on [[0, ζ[[ that is, M ∈ (M_{loc})^{[[0,ζ[[}. Then there exists a Borel function φ on E × E with φ(x, x) = 0 for all x ∈ E such that

M_{t} − M_{t−} = φ(X_{t−}, X_{t}) for every t ∈]0, ζ[, P_{m}-a.e.

Proof. By the proof of Proposition 2.8, there exists an E-nest {F_{k}} such that for each k ∈ ℕ M^{k} := 1_{F_{k}} ∗ M ∈M^{∘} and M_{t} = M_{t}^{k}, t < τ_{F_{k}}. Let φ_{k} be the jump function corresponding to M^{k}. Then we have φ_{k}(X_{t−}, X_{t}) = φ_{ℓ}(X_{t−}, X_{t}), t < τ_{F_{k}} P_{m}-a.e. for k < ℓ. From this, we see φ_{k} = φ_{ℓ} J-a.e. on F_{k} × F_{k}. We construct a Borel function φ on E × E in the following manner. We set F_{0} := ∅, φ(x, y) := φ_{k}(x, y) for (x, y) ∈ F_{k} × F_{k} \ (F_{k−1} × F_{k−1}), k ∈ ℕ, φ(x, y) := 0 if (x, y) ∈ E × E \ (⋃_{k=1}^{∞} F_{k} × ⋃_{k=1}^{∞} F_{k}). Then φ satisfies φ(x, x) = 0 for x ∈ E. We also have φ = φ_{k} J-a.e. on F_{k} × F_{k}. Consequently, φ(X_{t−}, X_{t}) = φ_{k}(X_{t−}, X_{t}), t < τ_{F_{k}} P_{m}-a.e. This means that M_{t} − M_{t−} = φ(X_{t−}, X_{t}), t < τ_{F_{k}} P_{m}-a.e. Therefore M_{t} − M_{t−} = φ(X_{t−}, X_{t}), 0 < t < ζ P_{m}-a.e. □

We recall the definition of the shift operator θ_{s} and the time-reversal operator r_{t} on the path space Ω. For each s ≥ 0, the shift operator θ_{s} is defined by θ_{s}ω(t) := ω(t + s) for t ∈ [0, ∞[. Given a path ω ∈ {t < ζ}, the operator r_{t} is defined by

r_{t}(ω)(s) := \begin{cases}ω((t − s)−), & if 0 ≤ s ≤ t, \\ ω(0), & if s ≥ t. \end{cases}  (2.1)

Here for r > 0, ω(r−) := lim_{s↑r} ω(s) is the left limit at r, and we use the convention that ω(0−) := ω(0). For a path ω ∈ {t ≥ ζ}, we set r_{t}(ω) := ω_{Δ}. We note that

lim_{s↓0} r_{t}(ω)(s) = ω(t−) = r_{t}(ω)(0) and lim_{s↑t} r_{t}(ω)(s) = ω(0) = r_{t}(ω)(t).  (2.2)

A key consequence of the m-symmetry assumption on the Hunt process X is that the measure P_{m}, when restricted to {t < ζ}, is invariant under the time-reversal operator r_{t}.

Clearly for t, s > 0, θ_{s} : Ω → Ω is F_{t+s}^{m}/F_{t}^{m}-measurable. The following lemma deals with the measurability issue of the time-reversal operator r_{t}.

Lemma 2.10 For each t > 0, r_{t} : Ω → Ω is F_{t}^{0}/F_{∞}^{0}-measurable and F_{t}^{m}/F_{t}^{m}-measurable.

Proof. Let F_{i} ∈ B(E_{Δ}) and s_{i} ∈ [0, ∞[, i = 1, 2, ⋯, n with s_{1} < s_{2} < ⋯ < s_{k} ≤ t < s_{k+1} < ⋯ < s_{n} for some k ∈ {1, 2, ⋯, n}. Then r_{t}^{−1}(⋂_{i=1}^{n} X_{s_{i}}^{−1}(F_{i})) = ⋂_{i=1}^{n}(X_{s_{i}} ∘ r_{t})^{−1}(F_{i}) is equal to ⋂_{i=1}^{k}({X_{(t−s_{i})−} ∈ F_{i}, t < ζ} ∪ {Δ ∈ F_{i}, t ≥ ζ}) ∩ ⋂_{i=k+1}^{n}({X_{0} ∈ F_{i}, t < ζ} ∪ {Δ ∈ F_{i}, t ≥ ζ}) ∈ F_{t}^{0}. Next we show the F_{t}^{m}/F_{t}^{m}-measurability of r_{t}. Take C ∈ F_{t}^{m}. Then there exists D ∈ F_{t}^{0} and N ∈ F_{∞}^{0} such that C △ D ⊂ N and P_{m}(N) = 0. Since P_{m}({ω_{Δ}}) = 0, by deleting {ω_{Δ}} = {ω ∈ Ω | ζ(ω) = 0} ∈ F_{0}^{0} ⊂ F_{t}^{0}, we may assume ω_{Δ} ∉ C ∪ D ∪ N. Then, r_{t}^{−1}(C) △ r_{t}^{−1}(D) ⊂ r_{t}^{−1}(N), r_{t}^{−1}(D), r_{t}^{−1}(N) ∈ F_{t}^{0} and P_{m}(r_{t}^{−1}(N)) = P_{m}(r_{t}^{−1}(N) ∩ {t < ζ}) + 1_{N}(ω_{Δ})P_{m}(t ≥ ζ) = P_{m}(N ∩ {t < ζ}) = 0. □

Definition 2.11 For any t > 0, we say two sample paths ω and ω′ are t-equivalent if ω(s) = ω′(s) for all s ∈ [0, t]. We say two sample paths ω and ω′ are pre-t-equivalent if ω(s) = ω′(s) for all s ∈ [0, t[.

For an rcll AF A_{t} adapted to (F_{t}^{0})_{t≥0}, A_{t}(ω) = A_{t}(ω′) if ω and ω′ are t-equivalent and A_{t−}(ω) = A_{t−}(ω′) if ω and ω′ are pre-t-equivalent. These conclusions may fail to hold if the measurability conditions are not satisfied. We need the following notion:

Definition 2.12 (PrAF) A process A = (A_{t})_{t≥0} with values in \overline{ℝ} := [−∞, ∞] is said to be a progressively additive functional (PrAF in short) (resp. PrAF admitting m-null set) if A is (F_{t})- adapted (resp. (F_{t}^{m})-adapted) and there exist defining sets Ξ ∈ F_{∞}, Ξ_{t} ∈ F_{t} (resp. Ξ ∈ F_{∞}^{m}, Ξ_{t} ∈ F_{t}^{m}) for each t > 0 and an E-polar (resp. m-null) set N satisfying the following condition;

(i) P_{x}(Ξ) = 1 for all x ∈ E \ N, and Ξ ⊂ Ξ_{t} ⊂ Ξ_{s} for every t > s > 0, and Ξ = ⋂_{t>0} Ξ_{t},

(ii) θ_{t}Ξ ⊂ Ξ for all t ≥ 0 and θ_{t−s}(Ξ_{t}) ⊂ Ξ_{s} for all s ∈]0, t[; in particular, ω_{Δ} ∈ Ξ ⊂ Ξ_{t} and P_{Δ}(Ξ) = P_{Δ}(Ξ_{t}) = 1 under (i),

(iii) for all ω ∈ Ξ_{t}, A_{·}(ω) is defined on [0, t[ and it is right continuous on [0, t ∧ ζ(ω)[ and has left limit on ]0, t]∩]0, ζ(ω)[ such that A_{0}(ω) = 0, |A_{s}(ω)| < ∞ for s ∈ [0, t ∧ ζ(ω)[ and A_{p+q}(ω) = A_{p}(ω) + A_{q}(θ_{p}ω) for all p, q ≥ 0 with p + q < t,

(iv) for all t ≥ 0, A_{t}(ω_{Δ}) = 0,

(v) for any t > 0 and pre-t-equivalent paths ω, ω′ ∈ Ω, ω ∈ Ξ_{t} implies ω′ ∈ Ξ_{t}, A_{s}(ω) = A_{s}(ω′) for any s ∈ [0, t[ and A_{s−}(ω) = A_{s−}(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 ∈ Ξ_{t}, s ↦ A_{s}(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 |A_{s}(ω)| < ∞, ∀s ∈ [0, t[ (resp. continuous on [0, t[) for every ω ∈ Ξ_{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 (F_{t})-adapted (resp. (F_{t}^{m})-adapted), and there exist Ξ ∈ F_{∞}, Ξ_{t} ∈ F_{t} (resp. Ξ ∈ F_{∞}^{m}, Ξ_{t} ∈ F_{t}^{m}) for each t > 0 and an E-polar (resp. m-null) set N such that (i’), (ii), (iii’), (iv) and (v’) hold: (i’): P_{x}(Ξ) = 1 for all x ∈ E \ N, Ξ ⊂ Ξ_{t} for all t > 0, Ξ = ⋂_{t>0} Ξ_{t}, and Ξ_{t} ∩ {t < ζ} ⊂ Ξ_{s} ∩ {s < ζ} for s < t. (iii’): For each ω ∈ Ξ_{t} ∩ {t < ζ}, the same conclusion as in (iii) holds. (v’): For any t > 0 and pre-t-equivalent paths ω, 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 similarly defined.

Remark 2.13 (i) Our notion of PrAF is different from what is found in Walsh [19].

(ii) Every PrAF (resp. PrAF on [[0, ζ[[) is an AF (resp. AF on [[0, ζ[[).

(iii) The MAF M^{u} and the CAF N^{u} of 0-energy appearing in Fukushima’s decomposition (1.2) can be regarded as finite rcll PrAFs in view of the proof of Theorem 5.2.2 in [8]. In this case, the defining sets for M^{u} as PrAF are given by

Ξ := {ω ∈ Ω | M_{s−}^{u_{n}}(ω) converges uniformly on ]0, t] for ∀t ≥ 0
for some subsequence n_{k}} ∈ F_{∞}
Ξ_{t} := {ω ∈ Ω | M_{s−}^{u_{n}}(ω) converges uniformly on ]0, t]
for some subsequence n_{k}} ∈ F_{t}

for every t > 0, where M_{t}^{u_{n}} := u_{n}(X_{t}) − u_{n}(X_{0}) − ∫_{0}^{t}(u_{n}(X_{s}) − f_{n}(X_{s})) ds with f_{n} := n(u − nR_{n+1}u) and u_{n} := R_{1}f_{n} = nR_{n+1}u. Hence a MAF of stochastic integral type ∫_{0}^{t} g(X_{s−}) dM_{s}^{u} (g, u ∈ F with g ∈ L^{2}(E; μ_{⟨u⟩})) can be regarded as a finite rcll PrAF. Consequently, any MAF of finite energy also can be regarded as an rcll PrAF, in view of the assertion of Lemma 5.6.3 in [8] and Lemma 2.14 below.

(iv) Every M ∈M^{∘}_{f-loc} can be regarded as a PrAF on [[0, ζ[[, hence every M ∈ M_{loc}^{[[0,ζ[[} is so. Since every local martingale can be written as the sum of a local 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 (A^{n}) be a sequence of finite rcll PrAFs with defining sets Ξ^{n} ∈ F_{∞}, Ξ_{t}^{n} ∈ F_{t}. For each t > 0, set

Ξ_{t} := {ω ∈ ⋂_{n∈ℕ}Ξ_{t}^{n} | A^{n} converges uniformly on [0, t[} ∈ F_{t}

and

Ξ := {ω ∈ ⋂_{n∈ℕ}Ξ^{n} | A^{n} converges uniformly on [0, t[ for every t ∈ [0, ∞[} ∈ F_{∞}.

Suppose that there exists an E-polar set N such that P_{x}(Ξ) = 1 for x ∈ E \ N. If we define A_{t} := \underline{lim}_{n→∞} A_{t}^{n} on Ω, then A is a finite rcll PrAF with its defining sets Ξ, Ξ_{t}.

Proof. We only show that for any t > 0 and pre-t-equivalent paths ω, ω′, ω ∈ Ξ_{t} implies ω′ ∈ Ξ_{t}. Suppose that ω ∈ Ξ_{t} and ω is pre-t-equivalent to ω′. It easy to see that ω′ ∈ ⋂_{n∈ℕ} Ξ_{t}^{n}. We then see the uniform convergence of A_{s−}^{n}(ω′) = A_{s−}^{n}(ω) for s ∈]0, t]. Therefore ω′ ∈ Ξ_{t}, A_{s}(ω′) = A_{s}(ω) for s ∈ [0, t[ and A_{s−}(ω′) = A_{s−}(ω) for s ∈]0, t]. □

Recall that {θ_{t}, t > 0} denotes the time shift operators on the path space for the process X.

Lemma 2.15 For t, s > 0,

(i) θ_{t}r_{t+s}ω is s-equivalent to r_{s}ω if t + s < ζ(ω) or s ≥ ζ(ω);

(ii) r_{t}θ_{s}ω is pre-t-equivalent to r_{t+s}ω. Moreover, if ω is continuous at s, then r_{t}θ_{s}ω is t-equivalent to r_{t+s}ω.

Proof. (i): We may assume t + s < ζ(ω). For v ∈ [0, s],

θ_{t}r_{t+s}ω(v) = ω((s − v)−) = r_{s}ω(v)

and so θ_{t}r_{t+s}ω is s-equivalent to r_{s}ω.

(ii): Note that t + s < ζ(ω) is equivalent to t < ζ(θ_{s}ω). It follows from the definition, if t + s < ζ(ω),

(r_{t}θ_{s}ω)(v) = \begin{cases}ω((t + s − v)−), & if 0 ≤ v < t, \\ ω(s), & if v = t, \end{cases}  (2.3)

while r_{t+s}ω(v) = ω((t + s − v)−) for 0 ≤ v ≤ t. Hence typically r_{t}θ_{s}ω is only pre-t-equivalent to r_{t+s}ω. □

Fix t > 0. Set H_{s}^{t} := F_{t} for s ∈ [0, t]; and H_{s}^{t} := F_{s} for s ∈]t, ∞[. Then (H_{s}^{t})_{s≥0} is a filtration over (Ω, F_{∞}), and F_{s} ⊂ H_{s}^{t} for all s ≥ 0.

Lemma 2.16 The following assertions hold for any fixed t > 0:

(i) Let φ be a Borel function on E × E and set X_{0−} := X_{0}. Then [0, ∞[×Ω ∋ (s, ω) ↦ 1_{[[0,ζ[[}(s, ω)1_{Γ_{t}}(ω)φ(X_{s−}(ω), X_{s}(ω)) is (H_{s}^{t})-optional for any Γ_{t} ∈ F_{t}.

(ii) Let A be an rcll PrAF with defining sets Ξ ∈ F_{∞}, Ξ_{t} ∈ F_{t}. If we set A_{0−}(ω) := 0 and A_{s}^{t}(ω) := 1_{Ξ_{t}}(ω)(1_{[0,t]}(s)A_{s}(ω) + 1_{]t,∞[}(s)A_{t}(ω)) for ω ∈ Ω, then [0, ∞[×Ω ∋ (s, ω) ↦ 1_{[[0,ζ[[}(s, ω)(A_{s}^{t}(ω) − A_{s−}^{t}(ω)) is (H_{s}^{t})-optional.

Proof. (i): Note that 1_{[[0,ζ[[} is (H_{s}^{t})-predictable. The assertion is clear if φ = f ⊗ g for bounded Borel functions f, g on E_{Δ}. The monotone class theorem for functions tells us the desired result.

(ii): Since A^{t} is (H_{s}^{t})-adapted and rcll on Ω and A_{−}^{t} is (H_{s}^{t})-adapted and lcrl on Ω, (s, ω) ↦ A_{s}(ω) is (H_{s}^{t})-optional and (s, ω) ↦ A_{s−}^{t}(ω) is (H_{s}^{t})-predictable. Consequently, (s, ω) ↦ A_{s}^{t}(ω) − A_{s−}^{t}(ω) is (H_{s}^{t})-optional. □

By Lemma 3.2 of [3], for a finite rcll AF A = (A_{t})_{t≥0}, there is a Borel function φ : E × E → ℝ with φ(x, x) = 0 for all x ∈ E such that

A_{t} − A_{t−} = φ(X_{t−}, X_{t}),  for every t ∈]0, ζ[, P_{m}-a.e.  (2.4)

Moreover, if φ̃ is another such function, then J(φ ≠ φ̃) = 0. As before, we refer to such a function φ as a jump function for A. Recall that if M ∈ M_{loc}^{[[0,ζ[[}, then there exists a jump function φ (unique in the above sense) so that M_{t} − M_{t−} = φ(X_{t−}, X_{t}) for t ∈]0, ζ[, P_{m}-a.e.

Lemma 2.17 Let A be a finite rcll PrAF with defining sets {Ξ, Ξ_{t}, t ≥ 0}. Then there exists a real valued Borel function φ on E_{Δ} × E_{Δ} with φ(x, x) = 0 for x ∈ E_{Δ} such that A with defining sets

Ξ̃ := {ω ∈ Ξ | A_{s}(ω) − A_{s−}(ω) = φ(X_{s−}(ω), X_{s}(ω)) for s ∈]0, ζ(ω)[ },
Ξ̃_{t} := {ω ∈ Ξ_{t} | A_{s}(ω) − A_{s−}(ω) = φ(X_{s−}(ω), X_{s}(ω)) for s ∈]0, t[∩]0, ζ(ω)[ },

is again an rcll PrAF admitting m-null set. The analogous assertion holds for PrAFs on [[0, ζ[[, and in particular for elements of (M_{loc})^{[[0,ζ[[}

Proof. Let φ : E_{Δ} × E_{Δ} → ℝ be a Borel function vanishing on the diagonal and define Ξ̃, Ξ̃_{t} in terms of φ as above. Clearly, Ξ̃ = ⋂_{t>0} Ξ̃_{t}, Ξ̃_{t} ⊂ Ξ̃_{s} for s < t. Moreover, we see that θ_{t}Ξ̃ ⊂ Ξ̃ for t ≥ 0, θ_{t−s}(Ξ̃_{t}) ⊂ Ξ̃_{s} for s < t. For two pre-t-equivalent paths ω, ω′, we see that ω ∈ Ξ̃_{t} implies ω′ ∈ Ξ̃_{t}.

By the previous lemma,

Γ := {(s, ω) | 1_{[[0,ζ[[}(s, ω)1_{Ξ_{t}}(ω)(A_{s}^{t}(ω) − A_{s−}^{t}(ω) − φ(X_{s−}(ω), X_{s}(ω))) ≠ 0}

is (H_{s}^{t})-progressively measurable for any fixed t > 0 and the debut of Γ is

D_{Γ}(ω) := inf{s ≥ 0 | 1_{[[0,ζ[[}(s, ω)1_{Ξ_{t}}(ω)(A_{s}^{t}(ω) − A_{s−}^{t}(ω) − φ(X_{s−}(ω), X_{s}(ω))) ≠ 0},

which is an (H_{s}^{t})-stopping time by (A5.1) in [18]. In particular,

{ω ∈ Ω | 1_{[[0,ζ[[}(s, ω)1_{Ξ_{t}}(ω)(A_{s}(ω) − A_{s−}(ω) − φ(X_{s−}(ω), X_{s}(ω))) = 0 for s ∈ [0, t[}
= {ω ∈ Ω | t < D_{Γ}(ω)} ∈ H_{t}^{t} = F_{t}.

Hence,

{ω ∈ Ξ_{t} | A_{s}(ω) − A_{s−}(ω) − φ(X_{s−}(ω), X_{s}(ω)) = 0 for s ∈]0, t[∩]0, ζ(ω)[}
= {ω ∈ Ξ_{t} | A_{s}(ω) − A_{s−}(ω) − φ(X_{s−}(ω), X_{s}(ω)) = 0 for s ∈ [0, t[∩[0, ζ(ω)[}
= {ω ∈ Ξ_{t} | 1_{[[0,ζ[[}(s, ω)(A_{s}(ω) − A_{s−}(ω) − φ(X_{s−}(ω), X_{s}(ω))) = 0 for s ∈ [0, t[}
∈ F_{t}.

Therefore, Ξ̃_{t} ∈ F_{t} and Ξ̃ ∈ F_{∞}. The proof for PrAFs on [[0, ζ[[ is similar, so we omit it.

The following theorem is a key to our extension of Nakao’s operator Γ. Its proof is complicated by measurability issues but the idea behind it is fairly transparent. We will use the convention X_{0−}(ω) := X_{0}(ω).

Theorem 2.18 (Dual PrAF) Let A be a finite rcll PrAF on [[0, ζ[[ with defining sets Ξ, Ξ_{t} admitting m-null set. Suppose there is a Borel function φ on E × E with φ(x, x) = 0 for x ∈ E such that φ(X_{s−}(ω), X_{s}(ω)) = A_{s}(ω) − A_{s−}(ω), ∀s ∈]0, t[∩]0, ζ[ and all ω ∈ Ξ_{t}. Set

Â_{t}(ω) := A_{t}(r_{t}(ω)) + φ(X_{t}(ω), X_{t−}(ω)) for t ∈ [0, ζ(ω)[ and Â_{t}(ω) := 0 for t ∈ [ζ(ω), ∞[. (2.5)

Then Â is an rcll PrAF on [[0, ζ[[ admitting m-null set such that

Â_{t} = A_{t−} ∘ r_{t} + φ(X_{t}, X_{t−})  and  Â_{t} − Â_{t−} = φ(X_{t}, X_{t−})

for all t ∈]0, ζ[, P_{m}-a.e.

Proof. Let Ξ ∈ F_{∞}, Ξ_{t} ∈ F_{t}^{m}, t > 0 be the defining sets of A admitting m-null set. We easily see r_{t}^{−1}(Ξ_{t}) ∩ {t < ζ} ⊂ r_{s}^{−1}(Ξ_{s}) ∩ {s < ζ} for s ∈]0, t[ by use of Lemma 2.15(i) and θ_{t−s}Ξ_{t} ⊂ Ξ_{s}.

Set Ξ̂_{t} := r_{t}^{−1}(Ξ_{t}) for t > 0 and Ξ̂ := ⋂_{t>0} Ξ̂_{t}. Then, we see Ξ̂ = ⋂_{t>0,t∈ℚ} Ξ̂_{t} by use of r_{t}^{−1}(Ξ_{t}) ∩ {t ≥ ζ} = {t ≥ ζ} and the monotonicity of r_{t}^{−1}(Ξ_{t}) ∩ {t < ζ}. Indeed, we have Ξ̂ ⊂ ⋂_{t>0,t∈ℚ} Ξ̂_{t} ⊂ (Ξ̂_{s} ∩ {s < ζ}) ∪ {t ≥ ζ} for any 0 < s < t with t ∈ ℚ. Taking the intersection over t ∈]s, ∞[∩ℚ, we have Ξ̂ ⊂ ⋂_{t>0,t∈ℚ} Ξ_{t} ⊂ Ξ̂_{s} for all s > 0, which yields the assertion.

We prove θ_{t}Ξ̂ ⊂ Ξ̂ for each t ≥ 0, in particular, θ_{t}Ξ̂ ⊂ θ_{s}Ξ̂, equivalently θ_{s}^{−1}Ξ̂ ⊂ θ_{t}^{−1}Ξ̂ if s ∈ [0, t]. Suppose ω ∈ Ξ̂. Then r_{t+s}ω ∈ Ξ_{t+s}. If t + s < ζ(ω), then r_{t+s}ω ∈ Ξ_{s}, otherwise r_{t+s}ω = ω_{Δ} ∈ Ξ_{s}. Hence we have r_{s}θ_{t}ω ∈ Ξ_{s} by Lemma 2.15(ii). Therefore r_{s}θ_{t}ω ∈ Ξ_{s} for all s > 0, which implies θ_{t}ω ∈ Ξ̂.

Next we prove θ_{t−s}(Ξ̂_{t}) ⊂ Ξ̂_{s} for s ∈]0, t[. Take ω ∈ Ξ̂_{t}. Then r_{s}θ_{t−s}ω is pre-s-equivalent to r_{t}ω ∈ Ξ_{t} ⊂ Ξ_{s} by Lemma 2.15(ii), hence r_{s}θ_{t−s}ω ∈ Ξ_{s}. Therefore θ_{t−s}ω ∈ Ξ̂_{s} for all s ∈]0, t[.

From Ξ_{t} ⊂ F_{t}^{m}, we get Ξ̂_{t} ∈ F_{t}^{m} by Lemma 2.10. Since (Ξ̂_{t})^{c} = r_{t}^{−1}((Ξ_{t})^{c}) = r_{t}^{−1}((Ξ_{t})^{c})∩{t < ζ} holds by noting ω_{Δ} ∈ Ξ_{t}, we have P_{m}((Ξ̂_{t})^{c}) = P_{m}((Ξ̂)^{c}) = 0.

By (2.2), v ↦ r_{s}(ω)(v) is continuous at v = s. Hence, on Ξ̂_{t}∩{t < ζ}, we have φ(X_{s−}, X_{s})∘r_{s} = φ(X_{s}, X_{s−}) ∘ r_{s} = 0, in particular, A_{s} ∘ r_{s} = A_{s−} ∘ r_{s} for s ∈]0, t[.

The remainder of the proof is devoted to showing that Â is an rcll PrAF on [0, ζ[ with defining sets Ξ̂, Ξ̂_{t} such that on Ξ̂_{t} ∩{t < ζ}, Â_{s} = A_{s−} ∘r_{s} +φ(X_{s}, X_{s−}), s ∈]0, t[. First note that for ω ∈ Ξ̂, |Â_{t}(ω)| < ∞ for any t ∈]0, ζ(ω)[, because by taking T ∈]t, ζ(ω)[, r_{T} ω ∈ Ξ_{T} implies r_{t}ω ∈ Ξ_{t}, hence |A_{t−}(r_{t}ω)| < ∞. Moreover, for ω ∈ Ξ̂_{t} ∩ {t < ζ}, we see r_{s}ω ∈ Ξ_{s} ∩ {s < ζ} and |A_{s−}(r_{s}ω)| < ∞ for all 0 < s < t.

For two pre-t-equivalent paths ω, ω′ ∈ Ω ∩ {t < ζ} with t > 0, we show ω ∈ Ξ̂_{t} implies ω′ ∈ Ξ̂_{t} and Â_{s}(ω) = Â_{s}(ω′) for s ∈ [0, t[. Recall ω ∈ Ξ̂_{t} ∩ {t < ζ} ⊂ Ξ̂_{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[. Then we see r_{s}ω ∈ Ξ_{s} is s-equivalent to r_{s}ω′ for any s ∈ [0, t], which implies r_{s}ω′ ∈ Ξ_{s} for any ]0, t] and A_{s−}(r_{s}ω) = A_{s−}(r_{s}ω′) for any s ∈ [0, t].

Fix t > 0. On Ξ̂_{t} ∩ {t < ζ} and for any p, q > 0 with p + q < t, by Lemma 2.15,

Â_{p+q} = A_{(p+q)−} ∘ r_{p+q} + φ(X_{p+q}, X_{(p+q)−})
= (A_{p} + A_{q−} ∘ θ_{p}) ∘ r_{p+q} + φ(X_{p+q}, X_{(p+q)−})
= A_{p} ∘ r_{p+q} + A_{q−} ∘ θ_{p} ∘ r_{p+q} + φ(X_{p+q}, X_{(p+q)−})
= (A_{p−} ∘ r_{p+q} + φ(X_{p−}, X_{p}) ∘ r_{p+q}) + A_{q−} ∘ r_{q} + φ(X_{p+q}, X_{(p+q)−})
= (A_{p−} ∘ r_{p} ∘ θ_{q} + φ(X_{q}, X_{q−})) + (Â_{q} − φ(X_{q}, X_{q−})) + φ(X_{p+q}, X_{(p+q)−})
= (Â_{p} − φ(X_{p}, X_{p−})) ∘ θ_{q} + Â_{q} + φ(X_{p+q}, X_{(p+q)−})
= Â_{p} ∘ θ_{q} + Â_{q}.

On Ξ̂_{t} ∩ {t < ζ}, again by Lemma 2.15 and (2.2), for any s > 0 and u ∈]0, s[,

Â_{s} − Â_{s−u} = Â_{u} ∘ θ_{s−u}
= (A_{u−} ∘ r_{u} + φ(X_{u}, X_{u−})) ∘ θ_{s−u}
= A_{u−} ∘ r_{u} ∘ θ_{s−u} + φ(X_{s}, X_{s−})
= A_{u−} ∘ r_{s} + φ(X_{s}, X_{s−}).

So

lim_{u↓0}(Â_{s} − Â_{s−u}) = φ(X_{s}, X_{s−}).

This shows that Â has left limit at s ∈]0, t[ and Â_{s} − Â_{s−} = φ(X_{s}, X_{s−}).

To show the right continuity of Â on Ξ̂_{t} ∩ {t < ζ} at any s ∈]0, t[, note for any u ∈]0, t − s[, by Lemma 2.15 and (2.2),

Â_{s+u} − Â_{s} = Â_{u} ∘ θ_{s}
= (A_{u−} ∘ r_{u} + φ(X_{u}, X_{u−})) ∘ θ_{s}
= A_{u−} ∘ r_{u} ∘ θ_{s} + φ(X_{s+u}, X_{(s+u)−})
= A_{u−} ∘ r_{s+u} + φ(X_{s+u}, X_{(s+u)−}).

Since (A_{v} − A_{v−}) ∘ r_{s+v} = φ(X_{v−}, X_{v}) ∘ r_{s+v} = φ(X_{s}, X_{s−}), while by Lemma 2.15 and (2.2),

(A_{v} − A_{v−}) ∘ r_{s+v} = lim_{u↓0}(A_{v} − A_{v−u}) ∘ r_{s+v} = lim_{u↓0} A_{u} ∘ θ_{v−u} ∘ r_{s+v} = lim_{u↓0} A_{u−} ∘ r_{v+u} + φ(X_{s}, X_{s−}).

we conclude that

lim_{u↓0} A_{u−} ∘ r_{s+u} = 0.

On the other hand, for any s ≥ 0

lim_{u↓0} φ(X_{s+u}, X_{(s+u)−}) = lim_{u↓0} φ(X_{(v−u)−}, X_{v−u}) ∘ r_{s+v} = lim_{u↓0}(A_{v−u} − A_{(v−u)−}) ∘ r_{s+v}
= (A_{v−} − A_{v−}) ∘ r_{s+v} = 0.

Hence we have for s > 0

lim_{u↓0}(Â_{s+u} − Â_{s}) = 0.

In other words, Â is right continuous at any s ∈]0, t[ on Ξ̂_{t} ∩ {t < ζ}. We also see

lim_{u<s,s↓0,u↓0}(Â_{s+u} − Â_{s}) = 0.

Thus we can define the limit Â_{0}(ω) := lim_{s↓0} Â_{s}(ω) for ω ∈ Ξ̂_{t} ∩ {t < ζ} for any t > 0. We also see Â_{0}(ω) = lim_{s↓0} Â_{s−}(ω) for ω ∈ Ξ̂_{t} ∩ {t < ζ} for any t > 0, because lim_{s↓0} φ(X_{s}, X_{s−}) = 0. Next we prove Â_{0}(ω) = 0 for ω ∈ Ξ̂_{t} ∩ {t < ζ} for any t > 0. Take ω ∈ Ξ̂_{t} ∩ {t < ζ} for some fixed t > 0. It suffices to show that lim_{u↓0} Â_{s−u}(θ_{u}ω) = Â_{s}(ω) for s ∈ [0, t[. Owing to Lemma 2.15(ii), we have

Â_{s−u}(θ_{u}ω) = A_{(s−u)−}(r_{s−u}θ_{u}ω) + φ(X_{s}(ω), X_{s−}(ω))
= A_{(s−u)−}(r_{s}ω) + φ(X_{s}(ω), X_{s−}(ω))
= A_{s−u}(r_{s}ω) − φ(X_{u}(ω), X_{u−}(ω)) + φ(X_{s}(ω), X_{s−}(ω))
= A_{s−u}(r_{s}ω) − Â_{u}(ω) + Â_{u−}(ω) + φ(X_{s}(ω), X_{s−}(ω))
→ A_{s−}(r_{s}ω) + φ(X_{s}(ω), X_{s−}(ω)) as u ↓ 0
= Â_{s}(ω).

The F_{t}^{m}-measurability of Â_{t} 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 X_{t−} = X_{t}, P_{x}-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 {G_{n}} be an increasing sequence of finely open Borel sets. Then the following are equivalent.

(a) {G_{n}} is a nest, that is, P_{x}(lim_{n→∞} σ_{E\G_{n}} ∧ ζ = ζ) = 1 for q.e. x ∈ E.

(b) E = ⋃_{n=1}^{∞} G_{n} q.e.

(c) P_{x}(lim_{n→∞} σ_{E\G_{n}} = ∞) = 1 for m-a.e. x ∈ E.

(d) P_{x}(lim_{n→∞} σ_{E\G_{n}} = ∞) = 1 for q.e. x ∈ E.

In particular, for an increasing sequence {F_{n}} of closed sets, {F_{n}} is an E-nest if and only if P_{x}(lim_{n→∞} σ_{E\F_{n}} = ∞) = 1 for m-a.e. x ∈ E.

(ii) For a function f on E, f ∈ F_{loc} if and only if there exist an E-nest {F_{k}} of closed sets and {f_{k} | k ∈ ℕ} ⊂ F_{b} such that f = f_{k} q.e. on F_{k}.

Proof. (i): For the implications (ia)⟺(ib), see Theorem 4.6 in [14]. The implication (id)⟹(ia) is clear. Next we show (ib)⟹(ic). Since each G_{n} is finely open, it is quasi-open by Theorem 4.6.1(i) in [8]. So there exists a common nest {A_{ℓ}} of closed sets such that (E \ G_{n}) ∩ A_{ℓ} is closed for all n, ℓ ∈ ℕ. Set σ := lim_{n→∞} σ_{E\G_{n}}. We then have that for all n ∈ ℕ X_{σ_{E\Gn}} ∈ E \ G_{n} P_{x}-a.s. on {σ < ∞} for q.e. x ∈ E. We have P_{x}(lim_{ℓ→∞} σ_{E\A_{ℓ}} = ∞) = 1 q.e. x ∈ E. Since σ(ω) < ∞ and lim_{ℓ→∞} σ_{E\A_{ℓ}}(ω) = ∞ together imply σ(ω) < σ_{E\A_{ℓ_{0}}} (ω) for some ℓ_{0} = ℓ_{0}(ω) ∈ ℕ, we have that there exists ℓ_{0} ∈ ℕ such that σ_{E\G_{n}} < σ_{E\A_{ℓ}} for all n > ℓ ≥ ℓ_{0}, P_{x}-a.s. on {σ < ∞} for q.e. x ∈ E. This means

P_{x}(σ < ∞) ≤ P_{x} ( \underline{lim}_{ℓ→∞}\{X_{σ_{E\Gn}} ∈ (E \ G_{n}) ∩ A_{ℓ} for all n > ℓ, σ < ∞\})

≤ \underline{lim}_{ℓ→∞} P_{x}(X_{σ_{E\Gn}} ∈ (E \ G_{ℓ}) ∩ A_{ℓ} for all n > ℓ, σ < ∞)

≤ \underline{lim}_{ℓ→∞} P_{x}(X_{σ} ∈ (E \ G_{ℓ}) ∩ A_{ℓ}, σ < ∞)

≤ \underline{lim}_{ℓ→∞} P_{x}(X_{σ} ∈ E \ G_{ℓ}, σ < ∞)

= P_{x} (X_{σ} ∈ E \ ⋃_{ℓ=1}^{∞} G_{ℓ}, σ < ∞) = 0

for m-a.e. x ∈ E, because of the E-polarity of E \ ⋃_{ℓ=1}^{∞} G_{ℓ}, where we use the quasi-left continuity of X up to ∞ and the closedness of (E \ G_{ℓ}) ∩ A_{ℓ}. The implication (ic)⟺(id) follows from the fact that x ↦ P_{x}(σ < ∞) is the limit of a decreasing sequence of excessive functions and Lemma 4.1.7 in [8].

(ii): The “if”part is clear by (i) because τ_{F_{k}} = τ_{G_{k}}, where G_{k} is the fine interior of F_{k}. We only prove the “only if”part. Take f ∈ F_{loc}. Then there exist {f_{k} | k ∈ ℕ} ⊂ F and an increasing sequence {G_{k}} of finely open sets with E = ⋃_{k=1}^{∞} G_{k} q.e. such that f = f_{k} m-a.e. on G_{k}. We may take f_{k} ∈ F_{b} for each k ∈ ℕ, by replacing f_{k} with (−k) ∨ f_{k} ∧ k, and G_{k} with G_{k} ∩ {|f| < k}. Note that f and f_{k} are quasi-continuous, so f = f_{k} q.e. on G_{k}. Taking an E-quasi-closure \overline{G_{k}}^{E} of G_{k}, we have f = f_{k} q.e. on \overline{G_{k}}^{E} (see [13] for the definition of E-quasi-closure). Let \{A_{n}\} be a common E-nest of closed sets such that for each k, n ∈ ℕ, \overline{G_{k}}^{E} ∩ A_{n} is closed. Set F_{k} := \overline{G_{k}}^{E} ∩ A_{k}. By (i), \{G_{k}\} is a nest, hence \overline{G_{k}}^{E} is a nest of q.e. finely closed sets, because of τ_{G_{k}} ≤ τ\overline{G_{k}}_{E}. Here we recognize \overline{G_{k}}^{E} as a finely closed Borel sets by deleting an E-polar set. Since \{A_{n}\} is a nest of closed sets, {F_{k}} is so, that is, P_{m}(lim_{k→∞} τ_{F_{k}} ≠ ζ) = 0. Therefore {F_{k}} is an E-nest of closed sets. We easily see that for each k ∈ ℕ, f = f_{k} q.e. on F_{k}. □

Recall that any locally square-integrable MAF M on [[0, ζ[[ admits a jump function φ on E × E with φ(x, x) = 0 for x ∈ E such that ΔM_{t} = φ(X_{t−}, X_{t}) for t ∈]0, ζ[ P_{m}-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 [8].

Lemma 3.2 Let ϕ be a Borel function on E_{Δ} × E_{Δ} satisfying ϕ(x, x) = 0 for all x ∈ E_{Δ}.

(i) Suppose that

N(1_{E×E}(|ϕ|^{2} ∧ |ϕ|))μ_{H} ∈ S.

Then there exists a unique purely discontinuous local MAF K on [[0, ζ[[ (i.e. K ∈ (M_{loc}^{1})^{[[0,ζ[[}) such that K_{t} − K_{t−} = ϕ(X_{t−}, X_{t}) for all t ∈ [0, ζ[, P_{x}-a.s. for q.e. x ∈ E.

(ii) If

N(1_{E×E_{Δ}}(|ϕ|^{2} ∧ |ϕ|))μ_{H} ∈ S,

then K can be taken to be a local MAF (i.e. K ∈ M_{loc}^{1}) and K_{t} − K_{t−} = ϕ(X_{t−}, X_{t}) for all t ∈ [0, ∞[, P_{x}-a.s. for q.e. x ∈ E.

Proof. The proof of (ii) is similar to that of (i), so we only prove (i). By martingale theory (see, e.g. [10]), the hypothesis implies that the compensated process

K_{t}^{(2)} := ∑_{0<s≤t} ϕ(X_{s−}, X_{s}) 1_{{|ϕ(X_{s−},X_{s})|>1}}1_{{s<ζ}} − ∫_{0}^{t} ∫_{E} N(X_{s}, dy)ϕ(X_{s}, y) 1_{{|ϕ(X_{s},y)|>1}} dH_{s}

is a local MAF of X of finite variation on [[0, ζ[[ and

K_{t}^{(1)} := lim_{ε→0} ( ∑_{0<s≤t} ϕ(X_{s−}, X_{s}) 1_{{ε<|ϕ(X_{s−},X_{s})|≤1}} 1_{{s<ζ}}−∫_{0}^{t} ∫_{E} N(X_{s}, dy)ϕ(X_{s}, y) 1_{{ε<|ϕ(X_{s},y)|≤1}} dH_{s})

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 the martingale theory. □

Definition 3.3 Let M be a local MAF on [[0, ζ[[ (that is, M ∈ (M_{loc}^{1})^{[[0,ζ[[}) with jump function φ. Assume that for q.e. x ∈ E, P_{x}-a.s.

∫_{0}^{t} ∫_{E} (φ^{2}1_{{|φ̂|≤1}} + |φ̂|1_{{|φ̂|>1}}) (X_{s}, y)N(X_{s}, dy)dH_{s} < ∞  for every t < ζ,  (3.1)

where φ̂(x, y) := φ(x, y) + φ(y, x). Define, P_{m}-a.e. on [[0, ζ[[,

Λ(M)_{t} := −\frac{1}{2} (M_{t} + M_{t} ∘ r_{t} + φ(X_{t}, X_{t−}) + K_{t}) for t ∈ [0, ζ[,  (3.2)

where K_{t} is the purely discontinuous local MAF on [[0, ζ[[ with

K_{t} − K_{t−} = −φ̂(X_{t−}, X_{t}) for every t < ζ, P_{x}-a.s.  (3.3)

for q.e. x ∈ E.

Remark 3.4 (i) The condition (3.1) is nothing but N(1_{E×E}(|φ̂|^{2} ∧ |φ̂|))μ_{H} ∈ S. In particular, condition (3.1) is satisfied by the jump function of any element of (M_{loc})^{[[0,ζ[[}.

(ii) It follows from Remark 2.13(iv) and Theorem 2.18 that Λ(M) is a continuous 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 B_{t} := M_{t} ∘ r_{t} + φ(X_{t}, X_{t−}) is a rcll PrAF, with left-limit process B_{t−} = B_{t} −φ(X_{t}, X_{t−}). It follows that Λ(M) is rcll on [0, ζ[ and that Λ(M)_{t−} = Λ(M)_{t} for all t ∈]0, ζ[, P_{m}-a.e.) Note that −K_{t} := ∑_{s≤t} φ̂(X_{s−}, X_{s})1_{{s<ζ}} − ∫_{0}^{t} ∫_{E} φ̂(X_{s}, y)N(X_{s}, dy)dH_{s}, t < ζ, satisfies K_{t} = K_{t} ∘ r_{t} P_{m}-a.e. on {t < ζ} for fixed t > 0. In view of Theorem 2.18, it is then clear from the definition that Λ is a linear operator that maps local MAFs on [[0, ζ[[ satisfying condition (3.1) into even CAFs on [[0, ζ[[ admitting m-null set.

(iii) If {M^{n}, n ≥ 1} is a sequence of MAFs having finite energy and converging in probability to M, then it is easy to see that M_{t}^{n} ∘ r_{t}, φ^{n}(X_{t−}, X_{t}) = M_{t}^{n} − M_{t−}^{n} and φ^{n}(X_{t}, X_{t−}) converge to M_{t} ∘ r_{t}, φ(X_{t−}, X_{t}) = M_{t} − M_{t−} and φ(X_{t}, X_{t−}) in probability, respectively, under P_{m}. Hence we have Λ(M^{n})_{t} converges to Λ(M)_{t} in measure for each t > 0.

(iv) For u ∈ F,

Λ(M^{u})_{t} = −\frac{1}{2} (M_{t}^{u} + M_{t}^{u} ∘ r_{t} + u(X_{t−}) − u(X_{t})) = N_{t}^{u}

P_{m}-a.e. on {t < ζ} for each fixed t ≥ 0. The first equality above is just the definition of Λ(M^{u}) while the second follows by applying r_{t} to both sides of (1.2), because X_{t} = X_{t−} and N^{u} ∘ r_{t} = N_{t}^{u}, P_{m}-a.e. on {t < ζ}. (The last property is proved in [5, Theorem 2.1] when X is a diffusion, but the same proof works for general symmetric Markov process X.) Since both Λ(M^{u})_{t} and N_{t}^{u} are continuous in t, we even have, P_{m}-a.e.,

Λ(M^{u})_{t} = N_{t}^{u} for all t < ζ.
□

We are going to show that Λ(M) defined above coincides on [0, ζ[ with Γ(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 Z_{t} ∘ r_{t} = Z_{t} (resp. Z_{t} ∘ r_{t} = −Z_{t}) P_{m}-a.e. on {t < ζ} for each t > 0. For a rcll process Z with Z_{0} = 0 and T > 0, we define

R_{T} Z_{t} := (R_{T} Z)_{t} := Z_{T −} − Z_{(T −t)−} for 0 ≤ t ≤ T,

with the convention Z_{0−} = Z_{0} = 0. Note that R_{T} Z_{t} so defined is an rcll process in t ∈ [0, T].

Lemma 3.5 Suppose that Z is an rcll PrAF. Then P_{m}-a.e. on {T < ζ},

R_{T} Z_{t} = \begin{cases}Z_{t} ∘ r_{T} , & if Z is even \\ −Z_{t} ∘ r_{T} , & if Z is odd \end{cases} for every t ∈ [0, T].  (3.4)

Proof. Let Z be an rcll PrAF and let T > 0. By Lemma 2.15,

Z_{t} ∘ r_{T} = (Z_{T} − Z_{T −t} ∘ θ_{t}) ∘ r_{T} = Z_{T} ∘ r_{T} − Z_{T −t} ∘ r_{T −t} for all t < T.  (3.5)

When Z is even,

Z_{t} ∘ r_{T} = Z_{T} − Z_{T −t} = Z_{T −} − Z_{(T −t)−} = R_{T} Z_{t}

P_{m}-a.e. on {T < ζ} for each fixed 0 ≤ t < T. Since both sides are right continuous in t ∈ [0, T [, we have P_{m}-a.e. R_{T} Z_{t} = Z_{t} ∘ r_{T} for every t ∈ [0, T]. When Z is an odd AF of Z, (3.4) can be proved similarly. □

Theorem 3.6 For an MAF M of finite energy, Λ(M) defined above coincides on [[0, ζ[[ with Γ(M) defined in (1.5) P_{m}-a.e..

Proof. For u ∈ F and 0 < t < T, since N^{u} is an even CAF, by Lemma 3.5,

(M_{t}^{u} + 2N_{t}^{u}) ∘ r_{T} = (u(X_{t}) − u(X_{0}) + N_{t}^{u}) ∘ r_{T}
= u(X_{(T −t)−}) − u(X_{T −}) + N_{T −}^{u} − N_{(T −t)−}^{u}
= M_{(T −t)−}^{u} − M_{T −}^{u}
= −R_{T} M_{t}^{u}.

Since both (M_{t}^{u} + 2N_{t}^{u}) ∘ r_{T} and R_{T} M_{t}^{u} are right continuous in t, we have P_{m}-a.e. on {T < ζ},

R_{T} M_{t}^{u} = − (M_{t}^{u} + 2N_{t}^{u}) ∘ r_{T} for every t ∈ [0, T].  (3.6)

For u ∈ D(L) ⊂ F and v ∈ F_{b}, define M_{t} = ∫_{0}^{t} v(X_{s−})dM_{s}^{u}, which is a MAF of finite energy. Note that, since u ∈ D(L), N_{t}^{u} = ∫_{0}^{t} Lu(X_{s})ds is a continuous process of finite variation. For each fixed 0 < t < T and n ≥ 1, define t_{i} = it/n and s_{i} = T − t + t_{i}. Using the standard Riemann-sum approximation of the Itô integral and of the covariance process [M^{v}, M^{u}], we have P_{m}-a.e. on {T < ζ}

M_{T} − M_{T −t} + [M^{v}, M^{u}]_{T} − [M^{v}, M^{u}]_{T −t}
= lim_{n→∞} (∑_{i=0}^{n−1} v(X_{s_{i}}) (M_{s_{i+1}}^{u} − M_{s_{i}}^{u}) + (M_{s_{i+1}}^{v} − M_{s_{i}}^{v}) (M_{s_{i+1}}^{u} − M_{s_{i}}^{u}))

= lim_{n→∞} (∑_{i=0}^{n−1} v(X_{s_{i+1}}) (M_{s_{i+1}}^{u} − M_{s_{i}}^{u}) − (N_{s_{i+1}}^{v} − N_{s_{i}}^{v}) (M_{s_{i+1}}^{u} − M_{s_{i}}^{u}))

= lim_{n→∞} ∑_{i=0}^{n−1} v(X_{s_{i+1}}) (M_{s_{i+1}}^{u} − M_{s_{i}}^{u})

= lim_{n→∞} ∑_{i=0}^{n−1} v(X_{T −t+t_{i}}) (R_{T} M_{t−t_{i}}^{u} − R_{T} M_{t−t_{i+1}}^{u})

= lim_{n→∞} (∑_{i=0}^{n−1} v(X_{t−t_{i+1}}) (M_{t−t_{i+1}}^{u} − M_{t−t_{i}}^{u} + 2N_{t−t_{i+1}}^{u} − 2N_{t−t_{i}}^{u})) ∘ r_{T}
= − (∫_{0}^{t} v(X_{s−})d(M_{s}^{u} + 2N_{s}^{u})) ∘ r_{T} ,

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. Note also that X_{t} = X_{t−} P_{m}-a.e. for each fixed t > 0. So we have for each fixed t < T, P_{m}-a.e on {T < ζ},

R_{T} M_{t} + R_{T} [M^{v}, M^{u}]_{t} = − (∫_{0}^{t} v(X_{s−})d(M_{s}^{u} + 2N_{s}^{u})) ∘ r_{T} .

Since both sides are right continuous in t ∈ [0, T], we have P_{m}-a.e. on {T < ζ},

R_{T} M_{t} + R_{T} [M^{v}, M^{u}]_{t} = − (∫_{0}^{t} v(X_{s−}) d(M_{s}^{u} + 2N_{s}^{u})) ∘ r_{T} for every t ∈ [0, T].  (3.7)

By [17, Theorem 3.1] and (1.7),

∫_{0}^{t} v(X_{s−}) dN_{s}^{u} = ∫_{0}^{t} v(X_{s}) dN_{s}^{u} = Γ(M)_{t} − \frac{1}{2}⟨M^{v,c} + M^{v,j}, M^{u,c} + M^{u,j}⟩_{t}.

It follows that P_{m}-a.e. on {T < ζ},

R_{T} M_{t} + R_{T} [M^{v}, M^{u}]_{t}
= − (M_{t} + 2Γ(M)_{t} − ⟨M^{v,c} + M^{v,j}, M^{u,c} + M^{u,j}⟩_{t}) ∘ r_{T}
= − (M_{t} + 2Γ(M)_{t} − ⟨M^{v,c}, M^{u,c}⟩_{t} − ⟨M^{v,j}, M^{u,j}⟩_{t}) ∘ r_{T} for all t ≤ T.

Recall that

[M^{v}, M^{u}]_{t} = ⟨M^{v,c}, M^{u,c}⟩_{t} + ∑_{s≤t}(M_{s}^{v} − M_{s−}^{v})(M_{s}^{u} − M_{s−}^{u})
= ⟨M^{v,c}, M^{u,c}⟩_{t} + ∑_{s≤t}(v(X_{s}) − v(X_{s−}))(u(X_{s}) − u(X_{s−})).

Taking t = T and noting that both Γ(M) and ⟨M^{v,c}, M^{u,c}⟩ are continuous even AFs, we have from above that P_{m}-a.e. on {t < ζ},

Γ(M)_{t} = −\frac{1}{2} (M_{t} + M_{t} ∘ r_{t} + v(X_{t})(u(X_{t−}) − u(X_{t})) + K_{t}) ,  (3.8)

where

K_{t} = ∑_{s≤t}(v(X_{s}) − v(X_{s−}))(u(X_{s}) − u(X_{s−})) − ⟨M^{v,j}, M^{u,j}⟩_{t}

is the purely discontinuous MAF with K_{t} − K_{t−} = (v(X_{t}) − v(X_{t−}))(u(X_{t}) − u(X_{t−})). Notice that the right side of (3.8) is right continuous on [0, ζ[, P_{m}-a.e.; cf. Remark 3.4(ii). Also observe that M_{t} − M_{t−} = φ(X_{t−}, X_{t}), where φ(x, y) = v(x)(u(y) − u(x)), and that

K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}).

This shows that Γ(M)_{t} = Λ(M)_{t} P_{m}-a.e. on {t < ζ} for each fixed t ≥ 0. Since both processes are continuous in t ∈ [0, ζ[, we have P_{m}-a.e.

Γ(M) = Λ(M) on [0, ζ[

for an MAF M of the form M_{t} = ∫_{0}^{t} v(X_{s−}) dM_{s}^{u} with u ∈ D(L) and v ∈ F_{b}. By Lemma 5.4.5 in [7], such MAFs form a dense subset in the space of MAFs having finite energy. Thus by Lemma 3.1 in Nakao [17] and Remark 3.4(iii) we have for a general MAF M of finite energy, Γ(M)_{t} = Λ(M)_{t} P_{m}-a.e. on {t < ζ} for every fixed t ≥ 0. Since both processes are continuous in t ∈ [0, ζ[, it follows that Γ(M) = Λ(M) on [[0, ζ[[ P_{m}-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,

lim_{n→∞} ∑_{ℓ=0}^{n−1} (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n})^{2} = 0,  (3.9)

where the convergence is in P_{gm}-measure on {t < ζ} for any g ∈ L^{1}(E; m) with 0 < g ≤ 1 m-a.e.

Proof. By (1.5) and Theorem 3.6, (3.9) clearly holds when M is a MAF of finite energy. For a locally square-integrable MAF M on [[0, ζ[[, there is an E-nest {F_{k}} of closed sets such that 1_{F_{k}} ∗ M ∈M^{∘} for each k ≥ 1 in view of the proof of Proposition 2.8 and so (3.9) holds with 1_{F_{k}} ∗ M in place of M. For each fixed k ≥ 1,

Λ(M)_{t} = Λ(1_{F_{k}} ∗ M)_{t} − \frac{1}{2}K_{t}^{k} P_{m}-a.e. on [0, τ_{F_{k}}[,

where K_{t}^{k} is a purely discontinuous local MAF on [[0, ζ[[ with

K_{t}^{k} − K_{t−}^{k} = 1_{F_{k}^{c}}(X_{t−})φ(X_{t−}, X_{t}) + 1_{F_{k}^{c}}(X_{t})φ(X_{t}, X_{t−}) for t < ζ.

Since 1_{F_{k}} ∗ M ∈M^{∘}, we have

∫_{E} N(1_{F_{k}×E}φ^{2}) dμ_{H} = ∫_{E} N(1_{E×F_{k}}φ^{2}) dμ_{H} < ∞.

Consequently, by Lemma 3.2, we have the existence of purely discontinuous local MAF on [[0, ζ[[ with jumps given by 1_{F_{k}}(X_{t−})φ(X_{t−}, X_{t})+1_{F_{k}}(X_{t})φ(X_{t}, X_{t−}), t < ζ. So we obtain the existence of such K_{t}^{k}. Since the square bracket of K^{k} is given by ∑_{s≤t} 1_{F_{k}^{c}}(X_{s−})φ^{2}(X_{s−}, X_{s}) + 1_{F_{k}^{c}}(X_{s})φ^{2}(X_{s}, X_{s−}) and it vanishes at t < τ_{F_{k}}, we have for each fixed t > 0,

lim_{n→∞} ∑_{ℓ=0}^{n−1} (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n})^{2} = 0 in P_{gm}-measure on {t < τ_{F_{k}}}.

Passing to the limit as k ↑ ∞ establishes (3.9). □

We are now in a position to define stochastic integrals against Λ(M) as integrator. Note that for f ∈ F_{loc}, M^{f,c} is well defined as a continuous MAF on [[0, ζ[[ of locally finite energy (see Theorem 8.2 in [10]). Moreover, for f ∈ F_{loc} and a locally square-integrable MAF M on [[0, ζ[[,

t ↦ (f ∗ M)_{t} := ∫_{0}^{t} f(X_{s−}) dM_{s}

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 ∈ F_{loc}. Let φ : E × E → ℝ be a jump function for M, and assume that φ satisfies condition (3.1). Define P_{m}-a.e. on [[0, ζ[[,

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s}
:= Λ(f ∗ M)_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}, (3.10)

whenever Λ(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 P_{m} 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 P_{m}-a.e.

∫_{0}^{t} ∫_{E}(f(X_{s}) − f(y))^{2}N(X_{s}, dy) dH_{s} < ∞ for every t < ζ  (3.11)

and that

∫_{0}^{t} ∫_{E} φ(y, X_{s})^{2}N(X_{s}, dy) dH_{s} < ∞ for every t < ζ.  (3.12)

Then the first and third terms on the right side of (3.10) are well defined. This is because N(1_{E×E}|φ̂|)μ_{H} ∈ S implies N(1_{E×E}|fφ̂|)μ_{H} ∈ S, and

f(x)φ(x, y) + f(y)φ(y, x) = f(x)φ̂(x, y) + (f(y) − f(x))φ(y, x),

so Λ(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 F_{loc} or f ∈ F. This is because when f ∈ F, the left hand side of (3.11) is just ⟨M^{f,j}⟩_{t}. When f is a bounded function in F_{loc}, by Lemma 3.1(ii), there exist a nest {F_{n} | n ∈ ℕ} of closed sets and a sequence of functions {f_{n} | n ∈ ℕ} ⊂ F_{b} such that f = f_{n} q.e. on F_{n} for every n ≥ 1. Note for each n ≥ 1, M^{f_{n},d} is a square-integrable purely discontinuous martingale and

M_{t}^{f_{n},d} − M_{t−}^{f_{n},d} = f_{n}(X_{t}) − f_{n}(X_{t−}).

So t ↦ ∑_{s≤t} (f_{n}(X_{s}) − f_{n}(X_{s−}))^{2} is P_{x}-integrable for q.e. x ∈ E. Since f is bounded, we have for each n ≥ 1 that

t ↦ ∑_{s≤t∧τ_{Fn}} (f(X_{s}) − f(X_{s−}))^{2}
= ∑_{s<t∧τ_{Fn}} (f(X_{s}) − f(X_{s−}))^{2} + (f(X_{t∧τ_{Fn}}) − f(X_{t∧τ_{Fn}−}))^{2}
= ∑_{s<t∧τ_{Fn}} (f_{n}(X_{s}) − f_{n}(X_{s−}))^{2} + (f(X_{t∧τ_{Fn}}) − f(X_{t∧τ_{Fn}−}))^{2}

is an increasing process and is P_{x}-integrable for each fixed t ≥ 0 for q.e. x ∈ E. Similarly, A_{t} := ∑_{s≤t} (f(X_{s}) − f(X_{s−}))^{2} is locally integrable in the sense of Definition 5.18 in [10]. Indeed, for a stopping time T_{n} := inf{t > 0 | A_{t} > n}, A_{T_{n}} = A_{T_{n}−} + (f(X_{T_{n}}) − f(X_{T_{n}−}))^{2} is bounded, hence P_{x}-integrable for q.e. x ∈ E. Note that the dual predictable projection of A_{t} is nothing but ∫_{0}^{t} ∫_{E_{Δ}}(f(X_{s}) − f(y))^{2}N(X_{s}, dy) dH_{s}. Then the dual predictable projection of ∑_{s≤t∧τ_{Fn}} (f(X_{s}) − f(X_{s−}))^{2} is given by ∫_{0}^{t∧τ_{Fn}} ∫_{E_{Δ}}(f(X_{s}) − f(y))^{2}N(X_{s}, dy) dH_{s} from Corollary 5.24 in [10], which is P_{x}-integrable for q.e. x ∈ E. This implies that (3.11) holds for every t < τ_{F_{n}}. Therefore (3.11) holds for every t < ζ.

Condition (3.12) is satisfied when M^{d} is P_{m}-square-integrable. Indeed,

E_{m}[∑_{s≤t} φ^{2}(X_{s}, X_{s−}) : t < ζ] = E_{m}[[M^{d}]_{t} ∘ r_{t} : t < ζ]
= E_{m}[[M^{d}]_{t} : t < ζ] < ∞.

Then Corollary 4.5 in [9] tells us that

lim_{t→0} \frac{1}{t}E_{m}[∑_{s≤t} φ^{2}(X_{s}, X_{s−}) : t < ζ] = lim_{t→0} \frac{1}{t}E_{m}[∑_{s≤t} φ^{2}(X_{s}, X_{s−})],

which implies

E_{m}[∫_{0}^{t} ∫_{E} φ(y, X_{s})^{2}N(X_{s}, dy) dH_{s}] < ∞

for all t > 0 by way of its subadditivity. Hence we get (3.12).

(iii) Suppose that f ∈ F. Let K_{t} be a purely discontinuous local MAF on [[0, ζ[[ with K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}) on ]0, ζ[. Then

⟨M^{f,j}, M^{j} + K⟩_{t} = − ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}.

In this case, (3.10) can be rewritten as

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = Λ(f ∗ M)_{t} − \frac{1}{2}⟨M^{f,c} + M^{f,j}, M^{c} + M^{j} + K⟩_{t}  (3.13)

on [0, ζ[. So when M = M^{u} for some u ∈ F and f ∈ F ∩ L^{2}(E; μ_{⟨u⟩}), ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} on [0, ζ[ is just the ∫_{0}^{t} f(X_{s}) ∘ dΓ(M)_{s} defined by (1.7). This shows that the stochastic integral given in Definition 3.8 extends Nakao’s definition (1.7) of stochastic integral first introduced in [17]. □

Theorem 3.10 The stochastic integral in (3.10) is well defined. That is, if M and M̃ are two locally square-integrable MAFs on [[0, ζ[[ such that all conditions in Definition 3.3 for M and M̃ are satisfied and Λ(M) ≡ Λ(M̃) on [[0, ζ[[, then for every f ∈ F_{loc} for which ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} and ∫_{0}^{t} f(X_{s−}) dΛ(M̃)_{s} are well defined, we have P_{m}-a.e.

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f(X_{s−}) dΛ(M̃)_{s} on [0, ζ[.

Proof. It is equivalent to show that

∫_{0}^{t} f(X_{s−}) dΛ(M − M̃)_{s} = 0 on [0, ζ[.

By taking M to be M −M̃, we may and will assume that M̃ = 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 K_{t} be the purely discontinuous local MAF on [[0, ζ[[ with

K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}) for t < ζ.

Since Λ(M) ≡ 0, we have

M_{t} + M_{t} ∘ r_{t} + φ(X_{t}, X_{t−}) + K_{t} = 0 on [0, ζ[  (3.14)

Thus by (3.5) and (3.14), on {T < ζ},

M_{t} ∘ r_{T} = M_{T} ∘ r_{T} − M_{T −t} ∘ r_{T −t}
= −M_{T} − K_{T} + M_{T −t} + K_{T −t} − φ(X_{T} , X_{T −}) + φ(X_{T −t}, X_{(T −t)−})  (3.15)

for every t ∈ [0, T]. Using the standard Riemann-sum approximation and (3.15), we have for f ∈ F,

(f ∗ M)_{t} ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−})
= −(f ∗ M)_{t} − (f ∗ K)_{t} − [M^{f}, M + K]_{t}
= −(f ∗ M)_{t} − (f ∗ K)_{t} − ⟨M^{f,c}, M^{c}⟩_{t} + ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−})

P_{m}-a.e. on {t < ζ} for each fixed t ≥ 0. Consequently we have for f ∈ F_{loc}, P_{m}-a.e. for all t ∈ [0, ζ[,

(f ∗ M)_{t} ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−})
= −(f ∗ M)_{t} − (f ∗ K)_{t} − ⟨M^{f,c}, M^{c}⟩_{t} + ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}), (3.16)

since both sides are right continuous in t ∈ [0, ζ[. Let K̃ be the purely discontinuous local MAF on [[0, ζ[[ with

K̃_{t} − K̃_{t−} = −f(X_{t−})φ(X_{t−}, X_{t}) − f(X_{t})φ(X_{t}, X_{t−}) for all t ∈ [0, ζ[.

Then for f ∈ F_{loc}, we have by (3.16),

Λ(f ∗ M)_{t} = −\frac{1}{2} ((f ∗ M)_{t} + (f ∗ M) ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−}) + K̃_{t})
= \frac{1}{2} (∫_{0}^{t} f(X_{s−}) dK_{s} + ⟨M^{f,c}, M^{c}⟩_{t} − ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}) − K̃_{t}) .

Thus

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s}
= Λ(f ∗ M)_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}
= \frac{1}{2} ∫_{0}^{t} f(X_{s−}) dK_{s} − \frac{1}{2} ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}) − \frac{1}{2}K̃_{t}

+\frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}.

Note that

K̃_{t} = − ∑_{s≤t} (f(X_{s−})φ(X_{s−}, X_{s}) + f(X_{s})φ(X_{s}, X_{s−}))
+ ∫_{0}^{t} ∫_{E} (f(X_{s})φ(X_{s}, y) + f(y)φ(y, X_{s})) N(X_{s}, dy) dH_{s}  (3.17)

and that

K_{t} = lim_{ε→0} (− ∑_{s≤t}(φ̂1_{{|φ̂|>ε}})(X_{s−}, X_{s}) + (N(φ̂1_{{|φ̂|>ε}}) ∗ H)_{t}) ,  (3.18)

where φ̂(x, y) := φ(x, y) + φ(y, x). It follows that

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = 0 for all t < ζ

P_{m}-a.e. □

Remark 3.11 The above proof actually shows that if Λ(M) = Λ(M̃) on [0, T]∩[0, ζ[, then P_{m}-a.e.,

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f(X_{s−}) dΛ(M̃)_{s} on [0, T] ∩ [0, ζ[.
□

4 Further study of the stochastic integral

Theorem 4.1 Suppose that f ∈ F_{loc} and that M is a locally square-integrable MAF on [[0, ζ[[ satisfying (3.1) such that Λ(M) is a continuous process A of finite variation on [[0, ζ[[. Assume that the stochastic integral t ↦ ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} is well defined. Then P_{m}-a.e.,

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f(X_{s}) dA_{s} on [0, ζ[,

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(X_{t−}, X_{t}) = M_{t} − M_{t−} for t ∈ [0, ζ[, P_{m}-a.e. Let K_{t} be the purely discontinuous local MAF on [[0, ζ[[ with

K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}) for t ∈]0, ζ[.

Since Λ(M) = A on [0, ζ[,

M_{t} ∘ r_{t} + φ(X_{t}, X_{t−}) = −M_{t} − K_{t} − 2A_{t} for all t ∈ [0, ζ[.

Thus by (3.5), for every T > t > 0, on {T < ζ},

M_{t} ∘ r_{T} = −M_{T} − K_{T} − 2A_{T} + M_{T −t} + K_{T −t} + 2A_{T −t} − φ(X_{T} , X_{T −}) + φ(X_{T −t}, X_{(T −t)−}). (4.1)

Now fix f ∈ F_{loc}; 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} ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−})
= −(f ∗ M)_{t} − (f ∗ K)_{t} − 2(f ∗ A)_{t} − [M^{f}, M + K + 2A]_{t}
= −(f ∗ M)_{t} − (f ∗ K)_{t} − 2(f ∗ A)_{t} − ⟨M^{f,c}, M^{c}⟩_{t} + ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}).

Consequently, we have, P_{m}-a.e. for all t ∈ [0, ζ[,

(f ∗ M)_{t} ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−})  (4.2)
= −(f ∗ M)_{t} − (f ∗ K)_{t} − 2(f ∗ A)_{t} − ⟨M^{f,c}, M^{c}⟩_{t} + ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}),

since both sides are right continuous in t ∈ [0, ζ[. Let K̃ be the purely discontinuous local MAF on [[0, ζ[[ with

K̃_{t} − K̃_{t−} = −f(X_{t−})φ(X_{t−}, X_{t}) − f(X_{t})φ(X_{t}, X_{t−}) for all t ∈ [0, ζ[.

Then by (4.2),

Λ(f ∗ M)_{t} = −\frac{1}{2} ((f ∗ M)_{t} + (f ∗ M) ∘ r_{t} + f(X_{t})φ(X_{t}, X_{t−}) + K̃_{t})
= \frac{1}{2}( ∫_{0}^{t} f(X_{s−}) dK_{s} + 2 ∫_{0}^{t} f(X_{s−}) dA_{s} + ⟨M^{f,c}, M^{c}⟩_{t}

− ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}) − K̃_{t}).

Thus

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s}
= Λ(f ∗ M)_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}
= \frac{1}{2} ∫_{0}^{t} f(X_{s−}) dK_{s} + ∫_{0}^{t} f(X_{s−}) dA_{s} − \frac{1}{2} ∑_{s≤t}(f(X_{s}) − f(X_{s−}))φ(X_{s}, X_{s−}) − \frac{1}{2}K̃_{t}

+\frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}.

It now follows from (3.17)-(3.18) that P_{m}-a.e.,

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f(X_{s−}) dA_{s} for all t ∈ [0, ζ[.

This proves the theorem. □

Note that if f, g ∈ F_{loc}, then fg ∈ F_{loc}. Theorem 4.2 Let f, g ∈ F_{loc} and let M be a locally square-integrable MAF on [[0, ζ[[ satisfying (3.1). Then P_{m}-a.e.,

∫_{0}^{t} g(X_{s−}) d (∫_{0}^{s} f(X_{r−}) dΛ(M)_{r}) = ∫_{0}^{t} f(X_{s−})g(X_{s−}) dΛ(M)_{s} for every t < ζ,  (4.3)

whenever all 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 P_{m}-a.e.,

φ(X_{t−}, X_{t}) = M_{t} − M_{t−} for all t ∈]0, ζ[.

Let K_{t} and K̃_{t} be the purely discontinuous local MAFs on [[0, ζ[[ with

K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}) for t ∈]0, ζ[

and

K̃_{t} − K̃_{t−} = −f(X_{t−})φ(X_{t−}, X_{t}) − f(X_{t})φ(X_{t}, X_{t−}) for t ∈]0, ζ[.

respectively. Then the left hand side of (4.3) is equal to

∫_{0}^{t} g(X_{s−}) dΛ(f ∗ M)_{s} − \frac{1}{2} ∫_{0}^{t} g(X_{s−}) d⟨M^{f,c}, M^{c}⟩_{s}

+\frac{1}{2} ∫_{0}^{t} ∫_{E} g(X_{s})(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}

= Λ(fg ∗ M)_{t} − \frac{1}{2}⟨M^{g,c}, (f ∗ M)^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(g(y) − g(X_{s}))f(y)φ(y, X_{s})N(X_{s}, dy) dH_{s}

−\frac{1}{2} ∫_{0}^{t} g(X_{s−}) d⟨M^{f,c}, M^{c}⟩_{s} + \frac{1}{2} ∫_{0}^{t} ∫_{E} g(X_{s})(f(y) − f(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}

= Λ(fg ∗ M)_{t} − \frac{1}{2}⟨M^{fg,c}, M^{c}⟩_{t} + \frac{1}{2} ∫_{0}^{t} ∫_{E}(f(y)g(y) − f(X_{s})g(X_{s}))φ(y, X_{s})N(X_{s}, dy) dH_{s}
= ∫_{0}^{t} f(X_{s−})g(X_{s−}) dΛ(M)_{s}.

This proves the theorem. □

Let J denote the class of stochastic processes that can be written as the sum of an (F_{t})- semimartingale Y and Λ(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 ∈ F_{loc} and Z = Y + Λ(M) ∈ J, define on [0, ζ[

∫_{0}^{t} f(X_{s−}) dZ_{s} := ∫_{0}^{t} f(X_{s−}) dY_{s} + ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s},

whenever the latter stochastic integral is well defined.

To establish Itô’s formula, we need the following result.

Theorem 4.4 Let f ∈ F_{loc} and let M be a locally square-integrable MAF on [[0, ζ[[ such that ∫_{0}^{·} f(X_{s−}) dΛ(M) is well defined on [0, ζ[. Then for every t > 0, P_{m}-a.e. on {t < ζ},

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = lim_{n→∞} ∑_{ℓ=0}^{n−1} f(X_{ℓt/n}) (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n}) .  (4.4)

Here the convergence is in measure with respect to P_{gm} on {t < ζ} for every g ∈ L^{1}(E; m) with 0 < g ≤ 1 m-a.e.

Proof. By (3.5), M_{s} ∘ r_{t} = M_{t} ∘ r_{t} − M_{t−s} ∘ r_{t−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(X_{t−}, X_{t}) = M_{t} − M_{t−} for all t ∈ [0, ζ[. Let K be the purely discontinuous local MAF on [[0, ζ[[ with

K_{t} − K_{t−} = −φ(X_{t−}, X_{t}) − φ(X_{t}, X_{t−}) for t ∈]0, ζ[.

Then for each fixed t > 0, P_{m}-a.e. on {t < ζ}

lim_{n→∞} ∑_{ℓ=0}^{n−1} f(X_{ℓt/n}) (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n})

= −\frac{1}{2}(f ∗ M)_{t} − \frac{1}{2}(f ∗ K)_{t} + \frac{1}{2} lim_{n→∞} ∑_{ℓ=0}^{n−1} f(X_{ℓt/n}) (M_{(ℓ+1)t/n} ∘ r_{(ℓ+1)t/n} − M_{ℓt/n} ∘ r_{ℓt/n})

= −\frac{1}{2}(f ∗ M)_{t} − \frac{1}{2}(f ∗ K)_{t} − \frac{1}{2} lim_{n→∞} [∑_{ℓ=0}^{n−1} f(X_{(ℓ+1)t/n}) (M_{(ℓ+1)t/n} − M_{ℓt/n})] ∘ r_{t}
= −\frac{1}{2}(f ∗ M)_{t} − \frac{1}{2}(f ∗ K)_{t} − \frac{1}{2}(f ∗ M)_{t} ∘ r_{t} − \frac{1}{2}[M^{f}, M]_{t} ∘ r_{t}
= −\frac{1}{2}(f ∗ M)_{t} − \frac{1}{2}(f ∗ K)_{t} − \frac{1}{2}(f ∗ M)_{t} ∘ r_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} − \frac{1}{2} ∑_{s≤t}(f(X_{s−}) − f(X_{s}))φ(X_{s}, X_{s−})
= Λ(f ∗ M)_{t} + \frac{1}{2}K̃_{t} − \frac{1}{2}(f ∗ K)_{t} − \frac{1}{2}⟨M^{f,c}, M^{c}⟩_{t} − \frac{1}{2} ∑_{s≤t}(f(X_{s−}) − f(X_{s}))φ(X_{s}, X_{s−})
= ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s},

where K̃ in the second to the last equality is the purely discontinuous local MAF on [[0, ζ[[ with K̃_{s} − K̃_{s−} = −f(X_{s−})φ(X_{s−}, X_{s}) − f(X_{s})φ(X_{s}, X_{s−}) for s ∈]0, ζ[. □

Remark 4.5 (i) Theorem 4.4 immediately implies Theorems 3.10 and 4.1.

(ii) By (3.9),

∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = lim_{n→∞} ∑_{ℓ=0}^{n−1} f(X_{(ℓ+1)t/n}) (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n})  (4.5)

holds in P_{gm}-measure on {t < ζ} for any g ∈ L^{1}(E; m) with 0 < g ≤ 1 m-a.e. Hence we could denote this stochastic integral by either ∫_{0}^{t} f(X_{s}) dΛ(M)_{s} or ∫_{0}^{t} f(X_{s}) ∘ dΛ(M)_{s}. Here ∫_{0}^{t} f(X_{s}) ∘ dΛ(M)_{s} is the Fisk-Stratonovich type integral: for t < ζ

∫_{0}^{t} f(X_{s}) ∘ dΛ(M)_{s} := lim_{n→∞} ∑_{ℓ=0}^{n−1} \frac{f(X_{(ℓ+1)t/n}) + f(X_{ℓt/n})}{2} (Λ(M)_{(ℓ+1)t/n} − Λ(M)_{ℓt/n}) . (4.6)

(iii) For any f ∈ F_{loc} and P_{m}-square-integrable MAF M, by way of the Riemann-sum approximation (4.4), we can extend the stochastic integral ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} without imposing further conditions. Indeed, let {G_{ℓ}} be a nest of finely open Borel sets and f_{ℓ} ∈ F_{b} with f = f_{ℓ} m-a.e. on G_{ℓ} (see the explanation for the condition (3.11) in Remark 3.9.) By (4.4), we see ∫_{0}^{t} f_{n}(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f_{ℓ}(X_{s−}) dΛ(M)_{s} for t < τ_{G_{n}} and n < ℓ. Then we can define ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f_{n}(X_{s−}) dΛ(M)_{s} for t < τ_{G_{n}} for each n ∈ ℕ, consequently, for all t < ζ P_{m}-a.e. More strongly, for M ∈M^{∘} and f ∈ F_{loc}, we can define ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} for all t ∈ [0, ∞[ P_{m}-a.e. Indeed by Remark 3.9(ii), our stochastic integral f_{n} ∗ Λ(M) for M ∈M^{∘} agrees with that defined by Nakao [17] on [0, ζ[ P_{m}-a.e., while the latter is defined as a CAF of X for all t ≥ 0. This implies that lim_{s↑ζ}(f_{n} ∗ Λ(M))_{s} exists and is finite P_{m}-a.e. After we extend our definition of stochastic integral f_{n} ∗ Λ(M) beyond [0, ζ[ by

(f_{n} ∗ Λ(M))_{t} = (f_{n} ∗ Λ(M))_{ζ} = lim_{s↑ζ}(f_{n} ∗ Λ(M))_{s} for t ≥ ζ,

f_{n} ∗Λ(M) becomes a CAF of X on [0, ∞[ P_{m}-a.e. With this extension for each n < ℓ, we have ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} = ∫_{0}^{t} f_{ℓ}(X_{s−}) dΛ(M)_{s} for t < σ_{E\G_{n}} P_{m}-a.e. Owing to Lemma 3.1(i) and the existence of the limit lim_{t↑ζ} ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} P_{m}-a.e., we obtain the stochastic integral ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s}, on [0, ∞[, P_{m}-a.e. for any f ∈ F_{loc} and M ∈M^{∘}, extending the stochastic integral by Nakao [17]. □

Remark 4.5(iii) says that the stochastic integral f ∗ Λ(M)_{t} := ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} can be defined for t ∈ [0, ∞[ P_{m}-a.e. for every f ∈ F_{loc} 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 ∈ F_{loc} and M ∈M^{∘}, the stochastic integral f ∗ Λ(M)_{t} := ∫_{0}^{t} f(X_{s−}) dΛ(M)_{s} can be defined for all t ∈ [0, ∞[ P_{x}-a.s. for q.e. x ∈ E, in particular, f ∗Λ(M) is a CAF of X on [0, ∞[.

Proof. Since f ∈ F_{loc}, we have {f_{k} | k ∈ ℕ} ⊂ F_{b} and a nest {G_{k} | k ∈ ℕ} of finely open Borel sets such that f = f_{k} q.e. on G_{k}. We know that the stochastic integral f_{k} ∗ Λ(M) is defined P_{x}-a.s. for q.e. x ∈ E. Let Ξ_{k} be the defining set admitting an E-polar set for the CAF f_{k} ∗ Λ(M) of zero energy and set

Ξ :={ω ∈ ⋂_{k=1}^{∞} Ξ_{k} | for any k, ℓ ∈ ℕ with k < ℓ,

∫_{0}^{t} f_{k}(X_{s−}(ω)) dΛ(M)_{s}(ω) = ∫_{0}^{t} f_{ℓ}(X_{s−}(ω)) dΛ(M)_{s}(ω) for t < σ_{E\G_{k}}(ω)}.

Then P_{x}(Ξ^{c}) = 0, m-a.e. x ∈ E. Hence for each s > 0, P_{x}(θ_{s}^{−1}(Ξ^{c})) = P_{s}(P_{·}(Ξ^{c}))(x) = 0 for q.e. x ∈ E. Setting Ξ̂ := ⋂_{k=1}^{∞} Ξ_{k} ∩ ⋂_{s∈ℚ_{++}} θ_{s}^{−1}(Ξ), we have P_{x}(Ξ̂) = 1 for q.e. x ∈ E. For ω ∈ Ξ̂ with t < σ_{E\G_{k}}(ω), we can find small s_{0}(= s_{0}(ω)) > 0 such that t + s_{0} < σ_{E\G_{k}}(ω). Then we see t < σ_{E\G_{k}}(θ_{s}ω) for any rational s ∈]0, s_{0}[. Hence for such ω, we have for k < ℓ and any rational s ∈]0, s_{0}[

∫_{s}^{t+s} f_{k}(X_{v−}(ω)) dΛ(M)_{v}(ω) = ∫_{s}^{t+s} f_{ℓ}(X_{v−}(ω)) dΛ(M)_{v}(ω).

Letting s → 0 and noting ω ∈ Ξ_{k}, k ∈ ℕ, we have that for k < ℓ, f_{k} ∗ Λ(M)_{t} = f_{ℓ} ∗ Λ(M)_{t} for t < σ_{E\G_{k}}, P_{x}-a.s. for q.e. x ∈ E. By Lemma 3.1(i), we know P_{x}(lim_{k→∞} σ_{E\G_{k}} = ∞) = 1 for q.e. x ∈ E. Therefore, we obtain that the stochastic integral f ∗ Λ(M) defined as in Remark 4.5(iii) can be established P_{x}-a.s. for q.e. x ∈ E. This completes the proof. □

Theorem 4.7 (Generalized Itô formula) Suppose that Φ ∈ C^{2}(ℝ^{d}) and u = (u_{1}, ⋯, u_{d}) ∈ F^{d}. Then for q.e. x ∈ E, P_{x}-a.s. for all t ∈ [0, ∞[,

Φ(u(X_{t})) − Φ(u(X_{0}))
= ∑_{k=1}^{d} ∫_{0}^{t} \frac{∂Φ}{∂x_{k}} (u(X_{s−})) du_{k}(X_{s}) + \frac{1}{2} ∑_{i,j=1}^{d} ∫_{0}^{t} \frac{∂^{2}Φ}{∂x_{i}∂x_{j}} (u(X_{s−})) d⟨M^{u_{i},c}, M^{u_{j},c}⟩_{s} (4.7)

+ ∑_{s≤t} (Φ(u(X_{s})) − Φ(u(X_{s−})) − ∑_{k=1}^{d} \frac{∂Φ}{∂x_{k}} (u(X_{s−}))(u_{k}(X_{s}) − u_{k}(X_{s−}))) .

Proof. Note that both sides appeared in (4.7) are P_{x}-a.s. defined for q.e. x ∈ E in view of Lemma 4.6. First we show this Itô formula (4.7) under P_{m} for a fixed t ≥ 0. Note that Φ ∘ u ∈ F_{loc} and that

u_{k}(X_{t}) = u_{k}(X_{0}) + M_{t}^{u_{k}} + N_{t}^{u_{k}} = u_{k}(X_{0}) + M_{t}^{u_{k}} + Λ(M^{u_{k}})_{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. [10]). Since both sides in (4.7) are right continuous, (4.7) holds under P_{m}.

Secondly, we refine the starting point. Recall that Ω consists of rcll paths. Let I_{t}(ω) be the difference of the left hand side and the right hand side in (4.7). Let Ξ be the intersection of all the defining sets of AFs appeared in the formula and {ω ∈ Ω | I_{t}(ω) = 0,^{∀}t ∈ [0, ∞[}. Then P_{x}(Ξ^{c}) = 0, m-a.e. x ∈ E. Let Ξ̂ be the intersection of the defining sets of AFs appeared in the formula and ⋂_{s∈ℚ_{++}} θ_{s}^{−1}(Ξ). Then we have P_{x}(Ξ̂) = 1 for q.e. x ∈ E as in the proof of Lemma 4.6. Take ω ∈ Ξ̂. Then for any positive rational s > 0, we have I_{t}(θ_{s}ω) = 0, that is,

Φ(u(X_{t+s}(ω))) − Φ(u(X_{s}(ω)))

= ∑_{k=1}^{d} ∫_{s}^{t+s} \frac{∂Φ}{∂x_{k}} (u(X_{v−}(ω))) du_{k}(X_{v}(ω)) + \frac{1}{2} ∑_{i,j=1}^{d} ∫_{s}^{t+s} \frac{∂^{2}Φ}{∂x_{i}∂x_{j}} (u(X_{v−}(ω))) d⟨M^{u_{i},c}, M^{u_{j},c}⟩_{v}(ω)

+ ∑_{s<v≤t+s} (Φ(u(X_{v}(ω))) − Φ(u(X_{v−}(ω))) − ∑_{k=1}^{d} \frac{∂Φ}{∂x_{k}} (u(X_{v−}(ω)))(u_{k}(X_{v}(ω)) − u_{k}(X_{v−}(ω)))) .

Letting s → 0 and using the right continuity of s ↦ u(X_{s}) and stochastic integrals, we have I_{t}(ω) = 0. This completes the proof. □

References

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Zhen-Qing Chen:

Department of Mathematics, University of Washington, Seattle, WA 98195, USA.

Email: zchen@math.washington.edu Patrick J. Fitzsimmons:

Department of Mathematics, University of California at San Diego, La Jolla, CA 92093-0112, USA, Email: pfitzsim@ucsd.edu

Kazuhiro Kuwae:

Department of Mathematics, Faculty of Education, Kumamoto University, Kumamoto 860-8555, Japan. Email: kuwae@gpo.kumamoto-u.ac.jp

Tusheng Zhang:

School of Mathematics, University of Manchester, Sackville Street, Manchester M60 1QD, UK.

Email: tzhang@maths.manchester.ac.uk