Mathematisches Institut
Justus-Liebig-Universität Giessen

Functional Itō-Calculus for Superprocesses
and the Historical Martingale Representation

Dissertation

Submitted by

Christian Mandler

in fulfillment of the requirements for the degree of
“Doctor rerum naturalium” (Dr. rer. nat.)

Supervisor: Prof. Dr. Ludger Overbeck

March 2021



Abstract

We derive an Itō-formula for the Dawson-Watanabe superprocess, a well-known class of
measure-valued processes, extending the classical Itō-formula with respect to two aspects.
Firstly, we extend the state-space of the underlying process (X(t))t∈[0,T ] to an infinite-
dimensional one – the space of finite measures. Secondly, we extend the formula to functionals
F (t,Xt) depending on the entire stopped paths Xt = (X(s ∧ t))s∈[0,T ], t ∈ [0, T ]. This later
extension is usually called functional Itō-formula.

Given the filtration (Ft)t∈[0,T ] generated by an underlying superprocess, we show that by
extending the functional derivative used in the functional Itō-formula we obtain the integrand
in the martingale representation formula for square-integrable (Ft)t-martingales. This result
is finally extended to square-integrable historical martingales. These are (Ht)t-martingales,
where (Ht)t∈[τ,T ] is the filtration generated by a historical Brownian motion, an enriched
version of a Dawson-Watanabe superprocess.

Kurzfassung

Wir leiten die funktionale Itō-Formel für eine bestimmte Klasse maßwertiger Prozesse, die
Dawson-Watabe Superprozesse, her. Diese erweitert die klassiche Itō-Formel wie folgt: Während
die klassiche Itō-Formel für Rd-wertige Prozesse gilt, betrachten wird Prozesse (X(t))t∈[0,T ]
mit einem unendlichdimensionalen Zustandsraum. Zudem betrachten wir Funktionale von
Xt = (X(s ∧ t))s∈[0,T ], dem zur Zeit t gestoppten Pfad von X, anstelle von Funktionen von
X(t), dem Zustand von X zur Zeit t.

Weiter zeigen wir, dass, gegeben der von einem Superprozess erzeugten Filtration (Ft)t∈[0,T ],
die funktionale Ableitung, welche in der funktionalen Itō-Formel auftaucht, genutzt wer-
den kann, um den Integranden in der Martingaldarstellung für quadratintegrierbare (Ft)t-
Martingale zu bestimmen. Zum Schluss erweitern wir dieses Resultat auf quadratintegrier-
bare historische Martingale. Dies sind (Ht)t-Martingale, wobei die Filtration (Ht)t∈[τ,T ] von
einer historischen Brownschen Bewegung erzeugt wird.

I





Contents

List of Figures V

A Remark on Notations VII

Introduction IX

1 Preliminaries 1
1.1 Superprocesses and the Historical Brownian Motion . . . . . . . . . . . . . . 2

1.1.1 Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Historical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Functional Itō-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 The Approach by Levental et al. . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 The Approach by Cont and Fournié . . . . . . . . . . . . . . . . . . . 11

1.3 Martingale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Martingale Measures and Integration with respect to

Martingale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 The Martingale Measure of the B(A, c)-Superprocess . . . . . . . . . . 17
1.3.3 Extending the Class of Integrands . . . . . . . . . . . . . . . . . . . . 19

2 Functional Itō-Calculus for Superprocesses 25
2.1 The Itō-Formula for Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The Functional Itō-Formula for Superprocesses . . . . . . . . . . . . . . . . . 35
2.3 Extension to the Locally Compact Case . . . . . . . . . . . . . . . . . . . . . 45

3 Martingale Representation 47
3.1 The Representation Formula for Square-Integrable (Ft)t-Martingales . . . . . 48
3.2 The Representation Formula for Square-Integrable (Ht)t-Martingales . . . . . 56
3.3 Comparison to the Results by Evans and Perkins . . . . . . . . . . . . . . . . 68

4 Outlook 71

Bibliography 76

Index of Notations 77

III





List of Figures

1.1 Illustration of different definitions of stopped paths. . . . . . . . . . . . . . . 10
1.2 Illustration of the two paths playing a role in the definition of functional deriva-

tives by Levental and co-authors. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Illustration of the two paths playing a role in the definition of functional deriva-

tives by Cont and Fournié. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

V





A Remark on Notations

While we try to introduce most of the notations used throughout this monograph when they
first appear, we have included some basic notations at the outset of this monograph to provide
for an easier read. In addition, an index of all notations, regardless of initial introduction,
appears after the bibliography.

As usual, the set of natural numbers is denoted by N. We write d ∈ N0 if we allow d to be
a natural number or zero. The space of real numbers is denoted by R and, for d ∈ N, its
d-dimensional counterpart is denoted by Rd.

Further, we write B(·) for a Borel-σ-algebra, that is B([0, T ]), B(R) and B(Rd) stand for the
Borel-σ-algebra on [0, T ], R and Rd, respectively.

For a majority of this monograph, we consider an abstract metric space E and denote its
Borel-σ-algebra by E . We write δy, y ∈ E, for the Dirac measure on (E, E), which is given by

δy(x) =
{

1, if x = y,
0, if x 6= y,

for all x ∈ E. In addition, for any B ⊂ E, the indicator function 1B is defined for all x ∈ E
by

1B(x) =
{

1, if x ∈ B,
0, if x /∈ B.

Now, let f : Rd → R be a continuous function. The partial derivative of f at x ∈ Rd in
direction i, i = 1, . . . , d, is given by

∂if(x) = lim
ε→0

f(x+ εei)− f(x)
ε

if the limit exists, where ei denotes the d-dimensional basis vector with a 1 in the i-th coor-
dinate and all other entries being 0’s. The second order partial derivative of f in directions

VII



i and j, i, j = 1, . . . , d, denoted by ∂ij , is defined iteratively and the Laplacian ∆ of f is
defined as

∆f =
d∑
i=1

∂iif.

If f can be interpreted as a function of time and space, i.e.

f : R× Rd 3 (s, x) 7→ f(s, x) ∈ R,

the partial derivative in direction of the first coordinate is denoted by ∂s.

Finally, let T be an interval on the real line and (Ω,F , (Ft)t∈T ,P) a filtered probability
space. Then, we denote by ([X]t)t∈T the quadratic variation process of a continuous (Ft)t-
local martingale (X(t))t∈T . The quadratic variation process is the unique continuous and
adapted process such that

(X2(t)− [X]t)t∈T
is a (Ft)t-local martingale. Analogously, we write ([X,Y ]t)t∈T for the quadratic covariation
process of two continuous (Ft)t-local martingales X and Y , which is the unique continuous
and adapted process such that

(X(t)Y (t)− [X,Y ]t)t∈T

is a (Ft)t-local martingale.

VIII



Introduction

In order to introduce superprocesses, it is reasonable to start with a simple branching diffusion
process on Rn. Thus, consider a number N(0) ∈ N of particles moving around independently
in Rn. At independent times, the particles die and leave behind a random number of descen-
dants that behave analogously. The number of descendants is determined by independent
draws from a common probability distribution on N0. If we denote by N(t) the number of
particles alive at time t and denote by Z1(t), . . . , ZN(t)(t) their locations in Rn at time t, the
process

X(t) =
N(t)∑
i=1

wδZi(t),

where δx denotes the Dirac measure with unit mass at x ∈ Rd and w > 0 is some weight,
takes values in the space of finite measures on Rn. Watanabe ([Watanabe, 1968]) was the
first to show that

“when we change the scale of time and mass in an appropriate way, [the process
X converges] to a continuous random motion on the space of mass distributions
on Rn.”

The resulting limit process is a superprocess. To pay homage to Watanabe’s findings as well as
to Dawson’s work on these processes in the following years (e.g. [Dawson, 1977] and [Dawson
and Hochberg, 1979]), superprocesses are also often called Dawson-Watanabe superprocesses
(see [Etheridge, 2000]).

Since the introduction of superprocesses, it has been shown that superprocesses arise as a
scaling limit of numerous so-called branching particle systems (see e.g. [Dynkin, 1991a]) in-
cluding contact processes (see e.g. [Durrett and Perkins, 1999]) and other interacting particle
systems (see e.g. [Cox et al., 2000] or [Durrett et al., 2005]).

Superprocesses have attracted particular interest in stochastic analysis due to their connec-
tion to non-linear (partial) differential equations (see e.g. [Dynkin, 1991b], [Dynkin, 1992],
[Dynkin, 1993] or [Le Gall, 1999]). Chapter 8 in [Etheridge, 2000] provides an extended

IX



overview of the research on this relation, which allows for a fruitful interplay between stochas-
tic analysis and the traditional analysis of partial differential equations and, more recently,
has led to some applications of the theory of superprocesses in mathematical finance as in
[Guyon and Henry-Labordere, 2013] and [Schied, 2013].

More typical applications of superprocesses can be found in population genetics. These are
often driven by the close link between superprocesses and a second class of measure-valued pro-
cesses, the so-called Fleming-Viot processes (see e.g. [Etheridge and March, 1991], [Perkins,
1992] or [Ethier and Krone, 1995]). Note that, in the literature, Fleming-Viot processes are of-
ten also referred to as superprocesses. To avoid confusion, we only refer to Dawson-Watanabe
superprocesses as superprocesses.

Thorough introductions to superprocesses in general can be found in [Dawson, 1993] and
[Perkins, 2002]. In this work, we focus on a particular subclass of superprocesses, the so-called
B(A, c)-superprocesses, which are also intensively studied in [Dawson, 1993] and introduced
in Section 1.1.1 of this monograph. Processes in this subclass can be characterized as limits
of branching diffusion processes with a specific motion and branching mechanism and come
with two favorable properties that are essential for the proofs of our result: Firstly, B(A, c)-
superprocesses have continuous sample paths and, secondly, they give rise to a continuous
orthogonal martingale measure in the sense of [Walsh, 1986] (see Section 1.3.2).

As measure-valued Markov processes, superprocesses take values in infinite-dimensional spaces.
Therefore, a fundamental tool in stochastic analysis, the traditional Itō-formula, is not directly
applicable to functions of such processes. The first main result presented in this monograph
is the Itō-formula for functions of B(A, c)-superprocesses. Dawson ([Dawson, 1978]) was the
first to prove an Itō-formula for measure-valued processes, but his result is limited to what
we call finitely based functions and compares to Theorem 2.4 in this monograph. Our result
(Theorem 2.9), which is obtained by rewriting a result in [Jacka and Tribe, 2003], extends
the Itō-formula to a much wider class of functions.

Given a Wiener process X, the traditional Itō-formula states that, for suitable functions
f : R→ R, the value of f at X(t), the value of X at time t, is given by

f(X(t)) = f(X(0)) +
∫ t

0
f ′(X(s))dX(s) + 1

2

∫ t

0
f ′′(X(s))d[X]s.

In many application, however, it is necessary to consider functionals of the whole path of X
up to time t, given by Xt = {X(t ∧ s) : s ∈ [0, T ]}, instead of functions of the value of X at
time t. In [Dupire, 2009], Dupire writes

“[...] in many cases, uncertainty affects the current situation not only through
the current state of the process but through its whole history. For instance, the
quality of a harvest does not only depend on the current temperature, but also on
the whole pattern of past temperatures; the price of a path dependent option may
depend on the whole history of the underlying price; [...]”.

Motivated by this fact, Dupire develops the so-called functional Itō-calculus for R-valued Itō-
processes in [Dupire, 2009]. For suitable functionals f , the functional Itō-formula is given

X



by

f(Xt) = f(X0) +
∫ t

0
∆sf(Xs)ds+

∫ t

0
∆xf(Xs)dX(s) + 1

2

∫ t

0
∆xxf(Xs)d[X]s,

where ∆ denotes the so-called functional derivatives. We skip over details like the domain of
the functionals or the definition of the functional derivatives at this point and deal with them
in Section 1.2. Also in this section, we introduce two different approaches to formalize and
extend Dupire’s result. The first of the two is due to Levental and co-authors ([Levental et al.,
2013]). The second one is by Cont and Fournié, who address the topic in a series of publi-
cations ([Cont and Fournié, 2010], [Cont and Fournié, 2013], [Cont, 2016]). In addition to
deriving the functional Itō-formula, Cont and Fournié use the functional derivatives to derive
the martingale representation formula for square-integrable (σ(X(s) : s ≤ t))t-martingales.

The martingale representation formula expresses a martingale as the sum of its expectation
and a stochastic integral term. While the integrator of the stochastic integral is given by the
underlying setting, obtaining the integrand requires more work. The traditional approach to
obtain the integrand is the so-called Clark-Ocone-Haussmann formula ([Clark, 1970], [Clark,
1971], [Haussmann, 1978], [Haussmann, 1979], [Karatzas et al., 1991], [Ocone, 1984]), in which
the integrand is obtained using Malliavin calculus.

The derivation of the functional Itō-formula for functionals of superprocesses is our second
main result (Theorem 2.14). The proof of this result follows the approach by Cont and
Fournié and, as in [Cont and Fournié, 2013] and [Cont, 2016], we also show that we can work
with the functional derivative used in the functional Itō-formula to obtain our third main re-
sult, the martingale representation formula for square-integrable (Ft)t-martingales (Theorem
3.10), where (Ft)t∈[0,T ] is the natural filtration of the considered superprocess.

The martingale representation formula in the context of superprocesses was first studied in
[Evans and Perkins, 1994], [Evans and Perkins, 1995] as well as [Overbeck, 1995]. While the
uniqueness of the representation is proved in [Evans and Perkins, 1994] and [Overbeck, 1995],
Evans and Perkins derive the explicit form of the integrand in [Evans and Perkins, 1995]
using an approach following the ideas of Malliavin calculus. Strictly speaking, in [Evans and
Perkins, 1995] the authors consider historical processes instead of superprocesses but note
that one can obtain the result for superprocesses by projection.

A historical process is a time-inhomogeneous Markov process that can be viewed as an en-
riched version of a superprocess that contains information on genealogy (see [Perkins, 1992]).
In Section 3.2, we consider a specific historical process, the so-called historical Brownian mo-
tion and derive the martingale representation formula for square-integrable (Ht)t-martingales,
where (Ht)t∈[τ,T ] is the natural filtration of the underlying historical Brownian motion. This
representation (Theorem 3.24) is our fourth main result.

The structure of this monograph is as follows. We start by introducing three concepts from
stochastic analysis, namely superprocesses and the historical Brownian motion, the functional
Itō-calculus as well as martingale measures, that underlie all of the presented main results.
This is subject of Chapter 1. In Chapter 2, we derive the Itō-formula as well as the func-
tional Itō-formula for functions, respectively functionals, of superprocesses. The next chapter,

XI



Chapter 3, covers our results on the martingale representation in both scenarios, the one con-
sidering superprocesses as well as the one considering the historical Brownian motion. In
the final chapter, we conclude this monograph by outlining ongoing research and discussing
potential future research question related to our work.

XII



Chapter 1

Preliminaries

In this first chapter we lay the foundation for the remainder of this monograph by introducing
the three central underlying mathematical concepts. The first one of these three concepts is
a class of measure-valued Markov processes called B(A, c)-superprocesses. These processes
arise as a scaling limit of branching diffusion processes and are introduced in the first section
of this chapter. While the results in Chapter 2 hold for any B(A, c)-superprocess as defined
in Definition 1.9, the results in Chapter 3 only hold for a specific B(A, c)-superprocess, the
so-called super-Brownian motion, as well as the so-called historical Brownian motion. While
not a B(A, c)-superprocess, the later one is an enriched version of the earlier one and both
processes are also introduced in the first part of this chapter.

The second concept, the functional Itō-calculus, is a relatively new mathematical concept and
introduced in the second part of this chapter. It is based on the work by Dupire ([Dupire,
2009]), which is why the functional derivatives used in the functional Itō-formula are often
referred to as Dupire derivatives. However, instead of presenting the original approach by
Dupire in more detail, we introduce the approaches by Levental as well as Cont and their
respective co-authors, who formalized Dupire’s original ideas in slightly different ways.

In the final section of this chapter, we introduce the concept of martingale measures. Intro-
duced by Walsh ([Walsh, 1986]), martingale measures are a particular class of measure-valued
martingales that play a crucial role in the formulation of the results in both, Chapter 2 as
well as Chapter 3.

While in the first two sections of this chapter we mostly skip the proofs of the results stated
to keep the introduction brief, we deviate from this principle in the final part. The reason for
this is twofold. The fact that there exists a martingale measure associated with a B(A, c)-
superprocess as introduced in the first part of this chapter is well known (see e.g. Example
7.1.3 in [Dawson, 1993]). However, as we could not find a detailed proof of this fact that we can
refer to, fwe carry out the proof in Section 1.3.2 for the sake of completeness. In addition, we
extend the class of valid integrands for the integral with respect to the martingale measure
associated with a B(A, c)-superprocess in Section 1.3.3. While this result can be proved
following standard arguments, the result seems to be new and is thus proved.

1



2 CHAPTER 1. PRELIMINARIES

1.1 Superprocesses and the Historical Brownian Motion
The stochastic processes underlying almost all results in this monograph are superprocesses,
more precisely B(A, c)-superprocesses. As a brief review of the history of superprocesses as
well as their connections to other fields of research is provided in the introduction, the focus
of this section is on introducing the B(A, c)-superprocess, stating some of its properties and
explaining how one can obtain the super-Brownian motion by choosing A and c accordingly.

In some applications, instead of working with superprocesses, one has to work with enriched
versions of superprocesses that keep track of the underlying genealogy. These processes are
known as historical processes and go back to Perkins and his co-authors. In the second part
of this section, we introduce a specific example of a historical process, namely the historical
Brownian motion, which is the stochastic process studied in Section 3.2.

1.1.1 Superprocesses
While there are multiple equivalent ways to define a B(A, c)-superprocess, we define it via its
martingale problem as this turns out to be advantageous in the later sections. However, we
also briefly mention the derivation of superprocesses as scaling limits of branching diffusion
processes to provide an intuitive interpretation of superprocesses and state its Laplace trans-
form to highlight the connection of B(A, c)-superprocesses to critical Feller continuous state
branching processes.

We do not attempt to provide a full introduction to the wide field of superprocesses and thus
omit almost all proofs of the results presented in this brief introduction. Most of the proofs
as well as detailed introductions to the topic can be found in the lecture notes by Dawson
([Dawson, 1993]) and Perkins ([Perkins, 2002]).

Now, let (E, d) be a separable metric space, which we assume to be either compact or lo-
cally compact, and E its Borel-σ-algebra. Further, denote by M1(E) the space of probability
measures on (E, E) and by MF (E) the space of finite measure on the same measure space.
We equip these spaces with the topology of weak convergence. If we write 〈µ, f〉 =

∫
E fdµ

for a function f : E → R, a series µn ⊂ M1(E) (µn ⊂ MF (E)) converges to µ ∈ M1(E)
(µ ∈MF (E)) in the weak topology if and only if 〈µn, f〉 → 〈µ, f〉 for n→∞ for all bounded
and continuous f .

Denote by C0(E,R) = C0(E) the space of continuous functions from E to R which satisfy
f(x) → 0 if d(x, 0) → ∞. If E is compact, C0(E,R) = C(E,R), the space of continuous
functions from E to R, also denoted by C(E). By equipping it with the sup-norm ‖ · ‖,
‖f‖ = supx∈E |f(x)|, the space C0(E) becomes a Banach space.

Definition 1.1 (Feller semigroup). A Feller semigroup is a conservative, positive, contraction
semigroup (St)t∈[0,T ] on C0(E), i.e. a linear map which satisfies

(i) St+s = StSs for all s, t ∈ [0, T ] such that s+ t ∈ [0, T ] (semigroup property),

(ii) St1 = 1 for all t ∈ [0, T ] (conservativeness),

(iii) Stf ≥ 0 for all f ≥ 0 and t ∈ [0, T ] (positivity),



1.1. SUPERPROCESSES AND THE HISTORICAL BROWNIAN MOTION 3

(iv) ‖Stf‖ ≤ ‖f‖ for all t ∈ [0, T ] (contraction),

which also satisfies

(v) St : C0(E)→ C0(E) for all t ∈ [0, T ],

(vi) Stf(x)→ f(x) as t→ 0, for all f ∈ C0(E), x ∈ E (weakly continuous).

Remark 1.2. There are various definitions of Feller processes in the literature that slightly
deviate from each other. The one presented in Definition 1.1 is based on the one found in
[Kallenberg, 2002]. The conservativeness of the Feller semigroup is not part of the original
definition in [Kallenberg, 2002] but an additional assumption repeatedly used by the author.
For simplicity and as it is part of the definition of other authors (see e.g. [Ethier and Kurtz,
1986]), we assume all Feller semigroups to be conservative.

Proposition 1.3. Regardless of whether the conservativeness is included in its definition, a
Feller semigroup is always strongly continuous, i.e. it holds

Stf → f as t→ 0 for all f ∈ C0(E).

Proof. See Theorem 19.6 in [Kallenberg, 2002].

Definition 1.4 (Feller process). A Feller process is a Markov process whose transition semi-
group is a Feller semigroup.

In order to formulate the martingale problem defining B(A, c)-superprocesses, we have to
introduce the notion of generators of Feller processes. These generators uniquely determine
a Feller process. Thus, we can characterize Feller processes solely by their generator.

Definition 1.5 (Generator). Let (St)t∈[0,T ] be a Feller semigroup. The (infinitesimal) gener-
ator A of (St)t∈[0,T ] is defined by

Af = lim
t→0

Stf − f
t

if the limit exists. Its domain D(A) is the space of functions f in C0(E) for which the limit
exists.

Proposition 1.6. A Feller semigroup is uniquely characterized by its generator.

Proof. See Lemma 19.5 in [Kallenberg, 2002].

Proposition 1.7. Let (St)t∈[0,T ] be a Feller semigroup with generator A. Then, for all
f ∈ D(A),

Stf − f =
∫ t

0
SsAfds =

∫ t

0
ASsfds (1.1)

holds.

Proof. See Proposition 1.5 (Chapter 1) in [Ethier and Kurtz, 1986].

Proposition 1.8. Let A be the generator of a strongly continuous semigroup on C0(E) with
domain D(A). Then, the domain D(A) is dense in C0(E).



4 CHAPTER 1. PRELIMINARIES

Proof. See Corollary 1.6 (Chapter 1) in [Ethier and Kurtz, 1986].

We can now define a B(A, c)-superprocess via its martingale problem. To do so, consider
the process (X(t))t∈[0,T ] given by X(ω)(t) = ω(t) on the filtered probability space Ω̃ =
(Ω,F , (Ft)t∈[0,T ],Pm) with Ω = C([0, T ],MF (E)), the space of continuous functions from
[0, T ] to MF (E) equipped with the sup-norm, F the corresponding Borel-σ-algebra, Ft =⋂
s>tFos with Fos = σ(X(r) : r ≤ s) and Pm being the law of X. Further, let A be the

generator of a Feller process on E with domain D(A) and c > 0.

Definition 1.9 (B(A, c)-superprocess). The process X on Ω̃ is called a B(A, c)-superprocess
if its law Pm, for a fixed m ∈MF (E), is the unique solution of the martingale problem

Pm(X(0) = m) = 1 and for all φ ∈ D(A) the process

M(t)(φ) = 〈X(t), φ〉 − 〈X(0), φ〉 −
∫ t

0
〈X(s), Aφ〉ds, t ∈ [0, T ]

is a (Ft)t-local martingale with respect to Pm

and has quadratic variation [M(φ)]t =
∫ t

0
〈X(s), cφ2〉ds.

(MP)

The resulting martingale M(t)(φ) is a true martingale if 〈m, 1〉 < ∞. As we require that
m ∈MF (E), this is always the case in this monograph. In addition, it satisfies

[M(φ),M(ψ)]t = c

∫ t

0
〈X(s), φψ〉ds for all φ, ψ ∈ D(A) (1.2)

and induces a martingale measure (see Section 1.3). Note that, in the literature, it is not
uncommon to write that X solves the martingale problem instead of being more specific and
writing that its distribution Pm is a solution of the martingale problem.

An alternative way to define a B(A, c)-superprocess is given in the following theorem.

Theorem 1.10. The B(A, c)-superprocess X can also be characterized via its Laplace trans-
form

E[exp(−〈X(t), φ〉)|X(0) = m] = exp (−〈m,Vtφ〉) , (1.3)

where φ ∈ bpE, the space of non-negative, bounded, E-measurable functions, and Vt satisfies
the log-Laplace equation

Vtφ = Stφ−
1
2c
∫ t

0
St−s(Vsφ)2ds.

This characterization is equivalent to the characterization in Definition 1.9.

Proof. See Chapter 4 in [Dawson, 1993].

By setting Vtφ(x) = u(t, x) in (1.3), we get that

E[exp(−〈X(t), φ〉)|X(0) = m] = exp (−〈m,u(t, ·)〉) , (1.4)

holds with u being the unique solution to

∂u

∂t
= Au− 1

2cu
2, u(0) = φ.



1.1. SUPERPROCESSES AND THE HISTORICAL BROWNIAN MOTION 5

This allows us to prove the following result, which is an immediate consequence of the above
and needed in the subsequent chapters. The proof of this result is the motivation for in-
troducing the Laplace approach at this point. For more on the this characterization of a
B(A, c)-superprocess as well as Laplace transforms and log-Laplace equations for measure-
valued processes, we refer to Chapter 4 in [Dawson, 1993].

Proposition 1.11. If X is a B(A, c)-superprocess, its total mass process 〈X(t), 1〉 satisfies

d〈X(t), 1〉 =
√
c〈X(t), 1〉dW (t), t ∈ (0, T ], (1.5)

with 〈X(0), 1〉 = 〈m, 1〉 and W being a standard Brownian motion independent of X, i.e.
〈X(t), 1〉 is a critical Feller continuous state branching process, and it holds for all t ∈ [0, T ]

E[〈X(t), 1〉] = 〈m, 1〉 <∞

as well as
E[〈X(t), 1〉2] = ct〈m, 1〉+ 〈m, 1〉2 <∞.

Proof. A Critical Feller continuous state branching process X̃ is given by the Laplace trans-
form (see e.g. Section 4.3 in [Dawson, 2017])

E[exp(−θX̃(t))|X̃(0) = x] = exp(−vθ(t)x)

with vθ given by
vθ(t) = θ

1 + c
2θt

, c > 0.

For simplicity set Y (t) = 〈X(t), 1〉, which implies Y (0) = 〈m, 1〉 by (MP). Then, by (1.4), it
holds for α ∈ R and t ∈ [0, T ]

E[exp(−αY (t))|Y (0) = 〈m, 1〉] = E[exp(−〈X(t), α〉)|X(0) = m]
= exp(−〈m,u(t, ·)〉)

with u satisfying
∂u

∂t
= Au− 1

2cu
2, u(0) = α. (1.6)

As Au = 0 if u is constant in the x-argument, u(t, x) = vα(t) satisfies (1.6) and it holds

exp(−〈m,u(t, ·)〉) = exp(−u(t)〈m, 1〉) = exp(−u(t)Y (0))

Consequently, the Laplace transform of Y (t) coincides with the Laplace transform of a critical
Feller continuous state branching process, which proves the first part.

To prove the second part, note that we get from (1.5) that Y (t) is a martingale. Thus, we
obtain E[Y (t)] from

E[Y (t)] = E[Y (t)|F0] = Y (0) = 〈m, 1〉,

which is finite as m ∈MF (E).



6 CHAPTER 1. PRELIMINARIES

To obtain the second moment of Y (t), note that from the above we get

E[exp(θY (t))|Y (0) = 〈m, 1〉] = exp
(

θ

1− c
2θt
〈m, 1〉

)
,

which is the moment-generating function of Y (t). Consequently, for all t ∈ [0, T ], the second
moment of Y (t) is given by

E[Y (t)2] = d2

dθ2

(
θ

1− c
2θt
〈m, 1〉

)∣∣∣∣∣
θ=0

= ct〈m, 1〉+ 〈m, 1〉2,

which is also finite as m ∈MF (E).

The interpretation of superprocesses as scaling limits of critical branching processes has al-
ready been briefly outlined in the introduction. In the following, more details are provided.

Let ε > 0 and consider the following branching diffusion process. At time zero, a random
number of particles is placed in E according to a Poisson random measure with intensity
m
ε . As time goes on, the particles move around independently in E with the motion given
by a Feller motion process with generator A. The lifetime of each particle follows an inde-
pendent exponential distribution with rate c

ε . At the time of death, a particle leaves behind
either zero or two descendants, each with probability one half. The descendants start their in-
dependent motion at the place of death of the parent particle and act like their parent particle.

Denote by N(t) the number of particles alive at time t and denote their locations by Zi(t),
i = 1, . . . , N(t). Further, denote the Dirac measure at x ∈ E by δx. The process

Xε(t) = ε

N(t)∑
i=1

δZi(t) ∈MF (E)

is called a measure-valued branching process and assigns mass ε to each particle alive at time
t. Now, if ε goes to zero, the process Xε converges weakly to the B(A, c)-superprocess (see
e.g. [Dawson, 1993]).

A special case of branching diffusions are binary branching Brownian motions, obtained by
replacing the general Feller motion process on E in the above branching diffusion process
by a Brownian motion on Rd. The scaling limit of branching Brownian motions are the so-
called super-Brownian motion, which are B(1

2∆, 1)-superprocesses, as 1
2∆ is the generator of

a Brownian motion.

In the remainder of this work, both, the more general class of B(A, c)-superprocesses and
super-Brownian motions are of interest. While the results in Chapter 2 are proved for any
B(A, c)-superprocess satisfying some additional requirements, in Chapter 3 we restrict our
results to super-Brownian motions to make use of a particular property of the domain of 1

2∆.



1.1. SUPERPROCESSES AND THE HISTORICAL BROWNIAN MOTION 7

1.1.2 Historical Brownian Motion
As mentioned previously, historical processes are enriched versions of superprocesses. In this
section, we introduce a specific historical process, namely the historical Brownian motion,
which is an enriched version of the super-Brownian motion. We once again introduce the
process via its martingale problem as this turns out to be advantageous in Section 3.2. How-
ever, it should be mentioned that historical processes can also be obtained as weak limits of
enriched branching processes, for which the particle motion is given by a motion on the path
space, as well as via their Laplace transform. We present the Laplace transform at a later
point in Section 3.2 but refer to Section II.3 in [Perkins, 2002] or Section 12 in [Dawson, 1993]
for details on the branching process approach.

Before we can state the martingale problem, some preparatory work is necessary. For this,
we mostly follow the notation introduced by Perkins in [Perkins, 1995]. In numerous pub-
lications, Perkins and his co-authors developed the theory of historical processes. For a
thorough introduction to historical processes in general and the historical Brownian motion
in particular, we refer to [Dawson and Perkins, 1991], [Perkins, 1995] as well as [Perkins, 2002].

Now, let C = C([0, T ],Rd) be the space of continuous functions mapping [0, T ] to Rd. Fol-
lowing the approach in [Perkins, 1995], we equip the space with the compact-open topology.
However, since [0, T ] is compact and Rd is a metric space, this topology coincides with the
topology of uniform convergence (see e.g. Chapter 7 in [Kelley, 1975]). Denote by C the
Borel-σ-algebra of C and let (Ct)t∈[0,T ] be the canonical filtration, which is given by

Ct = σ(y(s) : s ≤ t, y ∈ C).

Next, for y, w ∈ C and s ∈ [0, T ], define ys(t) = y(s ∧ t) and

(y/s/w)(t) =
{
y(t), if t < s,
w(t− s), if t ≥ s.

An element y ∈ C can also be viewed as a continuous path in Rd. Thus, the object ys is
the stopped path of y, a notion thoroughly studied in the next section. From Section V.2 in
[Perkins, 2002] we know that a function Φ : [0, T ]× C → R is (Ct)t-predictable if and only if
it is Borel-measurable and it holds Φ(t, y) = Φ(t, yt) for all t ∈ [τ, T ] and y ∈ C.

As before, denote by MF (C) the space of finite measures on C equipped with the topology
of weak convergence. For a t ∈ [0, T ] define

MF (C)t = {m ∈MF (C) : y = yt for m-almost all y}.

Further, define a measure Pτ,m ∈MF (C) by

Pτ,m(A) =
∫
C
Py(τ)({w : y/τ/w ∈ A})dm(y)

for all A ∈ C, where Px is the Wiener measure on (C, C) starting at x ∈ Rd, τ ∈ [0, T ] and
m ∈MF (C)τ .



8 CHAPTER 1. PRELIMINARIES

Let
ΩH = {H ∈ C([τ, T ],MF (C)) : H(t) ∈MF (C)t for all t ∈ [τ, T ]}

and let S̃ be the space of all starting points of the historical Brownian motion,

S̃ = {(τ,m) : τ ∈ [0, T ], m ∈MF (C)τ}.

Finally, let

Fτ,m = {Φ : [τ, T ]× C → R : Φ is (Ct)t-predictable, Pτ,m-a.s. right-continuous and
sup
s≥τ
|Φ(s, y)| ≤ K holds Pτ,m-a.s. for some K}

and

D(Aτ,m) = {Φ ∈ Fτ,m : there exists a Aτ,mΦ ∈ Fτ,m such that

Φ(t, y)− Φ(τ, y)−
∫ t

τ
Aτ,mΦ(s, y)ds

is a (Ct)t∈[τ,T ]-martingale under Pτ,m}.

Definition 1.12 (Historical Brownian motion). A predictable process H(t), t ∈ [τ, T ], on a
filtered probability space Ω̄ = (Ω,F , (Ft)t∈[τ,T ],P) and with sample paths almost surely in ΩH

is a historical Brownian motion with branching rate γ > 0 and starting at (τ,m) ∈ S̃ if and
only if its law Pτ,m solves the martingale problem

Pτ,m(X(τ) = m) = 1 and for all Φ ∈ D(Aτ,m)

Z(t)(Φ) = 〈H(t),Φ(t, ·)〉 − 〈m,Φ(τ, ·)〉 −
∫ t

τ
〈H(s), Aτ,mΦ(s, ·)〉ds, t ∈ [τ, T ],

is a continuous (Ft)t-martingale with respect to Pτ,m

and has quadratic variation [Z(Φ)]t =
∫ t

τ
〈H(s), γΦ(s, ·)2〉ds.

(MPHBM )

From [Perkins, 1995] we get that the historical Brownian motion can also be defined via a
more explicit martingale problem. To introduce this result, denote by C∞0 (Rd) the space of
infinitely continuously differentiable functions with compact support mapping Rd to R and
define

Dfd = {Φ : C → R : Φ(y) = Ψ(y(t1), . . . , y(tn)),
0 ≤ t1 ≤ . . . ≤ tn ≤ T, Ψ ∈ C∞0 (Rnd), n ∈ N}.

Thus, the space Dfd consists of functions mapping C to Rd that only take the values of y ∈ C
at a finite number of times into account. Next, set

D̃fd = {Φ : Φ(t, y) = Φ̃(yt) for some Φ̃ ∈ Dfd}

and for Ψ ∈ C∞0 (Rnd) let Ψi,j be the second order partial derivative of Ψ. For 1 ≤ i, j ≤ d
and 0 ≤ t1 ≤ . . . ≤ tn ≤ T define the (Ct)t-predictable process Ψ̄ by

Ψ̄i,j(t, y) =
n∑
k=1

n∑
`=1

1t≤tk∧t`Ψ(k−1)d+i,(`−1)+j(y(t1 ∧ t), . . . , y(tn ∧ t)).



1.2. FUNCTIONAL ITŌ-CALCULUS 9

Using this process, we define

∆̄Ψ(t, y) =
d∑
i=1

Ψ̄i,i(t, y),

which now allows us to formulate the following result.

Theorem 1.13 ([Perkins, 1995]). A (Ct)t-predictable process H(t), t ∈ [τ, T ] on Ω̄ is a
historical Brownian motion starting at (τ,m) ∈ S̃ and with branching rate γ > 0 if and only
if H(t) ∈ MF (C)t for all t ∈ [τ, T ] and the law Pτ,m of H is a solution to the following
martingale problem

Pτ,m(X(τ) = m) = 1 and for all Ψ ∈ Dfd

Z(t)(Ψ) = 〈H(t),Ψ〉 − 〈m,Ψ〉 −
∫ t

τ
〈H(s), 1

2∆̄Ψ(s, ·)〉ds, t ∈ [τ, T ],

is a continuous (Ft)t-martingale with respect to Pτ,m

and has quadratic variation [Z(Φ)]t =
∫ t

τ
〈H(s), γΨ2〉ds.

(MPHBM−fd)

Finally, let Pτ,m denote the law of the historical Brownian H motion starting at (τ,m) ∈ S̃
and set

H̃[s, t] = σ(H(u) : s ≤ u ≤ t).

By denoting the Pτ,m-completion of H̃[τ, T ] by H[τ, T ] and setting

Ht =
(⋂
s>t

H̃[τ, s]
)
∧ {Pτ,m-null sets},

we obtain a filtered probability space (ΩH ,H[τ, T ], (Ht)t∈[τ,T ],Pτ,m) on which the historical
Brownian motion is given by H(t)(ω) = ω(t) (see [Perkins, 1995]).

As the historical Brownian motion only comes into play in the final sections of this monograph,
we keep this introductory section on this process brief and thus conclude it with the above
result on the representation of the historical Brownian motion as a canonical process. Never-
theless, some further results, like the form of the Laplace transform of a historical Brownian
motion, are presented in Section 3.2, when we encounter this process for the first time.

1.2 Functional Itō-Calculus
In his landmark paper [Dupire, 2009], Dupire derives a functional version of Itō’s lemma to
model events that do not only depend on the current state X(t) of a stochastic process X
but on its whole past {X(s) : s ≤ t}. His approach has since been formalized in a series of
publications by Cont and Fournié ([Cont and Fournié, 2010], [Cont and Fournié, 2013], [Cont,
2016]) as well as Levental et al. ([Levental et al., 2013]).

Dupire defines the path process Xt of a process X by Xt(s) = X(s) for all s ∈ [0, t]. It is
assumed that the process X is such that the process Xt is an element in the space of bounded
right continuous functions from [0, t] to R with left limits, denoted by Λt. The functionals for
which the functional Itō-formula is derived [Dupire, 2009] map paths in Λ = ⋃

t∈[0,T ] Λt to R.



10 CHAPTER 1. PRELIMINARIES

0 t T 0 t T

Figure 1.1: Examples of path processes on R. Left: A path in the vector bundle considered by Dupire,
which is only defined on [0, t]. Right: A stopped path as it is considered by Cont as well as Levental
and their respective co-authors, which is defined on the whole interval [0, T ].

The space Λ is often referred to as a vector bundle and is not a vector space.

The main difference between the two versions of the functional Itō-formula introduced below
and the original work by Dupire is the underlying space of paths. Instead of considering the
vector bundle, Cont and Levental and their respective co-authors modify the notion of paths
of a process such that they are elements in D([0, T ],Rd), the space of right continuous func-
tions from [0, T ] to Rd with left limits, which is equipped with the sup-norm. More precisely,
the authors consider stopped paths. In contrast to the paths defined in [Dupire, 2009], the
path stopped at t is always a function from the whole time interval [0, T ] to Rd.

For simplicity, in the following we always assume that the process X has continuous paths.
However, to derive the functional Itō-formula, functionals defined on D([0, T ],Rd) have to be
considered. The reason for this is pointed out when we present the two versions of functional
derivatives below. Both versions of the functional Itō-formula introduced below also hold for
right continuous paths with left limits and we refer to the original works for the more general
versions and proofs.

1.2.1 The Approach by Levental et al.
Let X be a continuous process on a probability space (Ω,F ,P) and denote its value at time
t ∈ [0, T ] by X(t) ∈ Rd. Levental, Schroder and Sinha define the path of X stopped at time
t ∈ [0, T ] by

Xt(·) = X(t ∧ ·).

Consequently, it holds for all s ∈ [0, T ]

Xt(s) =
{
X(s), if s < t,
X(t), if s ≥ t.

The directional functional derivatives of functionals F : D([0, T ],Rd)→ R introduced by the
authors are defined for all paths in D([0, T ],Rd) and not just stopped paths.

Definition 1.14 ([Levental et al., 2013]). Let ei be the d-dimensional vector with a one
in the ith coordinate and zeros everywhere else. Let 1 ≤ i, j ≤ d, ω ∈ C([0, T ],Rd) and



1.2. FUNCTIONAL ITŌ-CALCULUS 11

0 t T 0 t T

Figure 1.2: The two path processes playing a role in the definition of the functional derivative by
Levental and co-authors. Left: The original path ω. Right: The shifted path ω + ε1[t,T ].

F : D([0, T ],Rd)→ R. Then the directional derivative of F in direction ei1[t,T ] is given by

DiF (ω; [t, T ]) = lim
ε→0

F (ω + εei1[t,T ])− F (ω)
ε

if the limit exists. The second order directional derivative in directions ei1[t,T ] and ej1[t,T ] is
given by

DijF (ω; [t, T ]) = lim
ε→0

DiF (ω + εej1[t,T ]; [t, T ])−DiF (ω; [t, T ])
ε

if the limit exists.

At this point it becomes clear why F has to be defined on D([0, T ],Rd). While the path ω
is continuous, the shifted path ω + εei1[t,T ] is no longer continuous but only right continuous
with left limits.

Next, the authors define a metric d̃ on [0, T ]×D([0, T ],Rd) by

d̃((t, ω), (t′, ω′)) = |t− t′|+ sup
s∈[0,T ]

‖ω(s)− ω′(s)‖.

The definition of the directional functional derivatives as well as the metric d̃ result in the
following version of the functional Itō-formula.

Theorem 1.15 ([Levental et al., 2013]). Assume the functional F : D([0, T ],Rd)→ R as well
as its first and second order directional derivatives are continuous in t and ω with respect to
the metric d̃. Further, let X be a continuous semimartingale. Then

F (Xt) = F (X0) +
d∑
i=1

∫ t

0
DiF (Xs; [s, T ])dXi(s)

+ 1
2

d∑
i, j=1

∫ t

0
DijF (Xs; [t, T ])d[Xi, Xj ](s).

(1.7)

1.2.2 The Approach by Cont and Fournié
In their first work on the functional Itō-formula ([Cont and Fournié, 2010]), Cont and Fournié
are still working with the vector bundle approach used in [Dupire, 2009]. In later publications,



12 CHAPTER 1. PRELIMINARIES

0 t T 0 t T

Figure 1.3: The two path processes playing a role in the definition of the functional derivative by
Cont and Fournié. Left: The original stopped path ωt. Right: The shifted stopped path ωt + ε1[t,T ].

summarized in [Cont, 2016], the authors no longer use the vector bundle approach but work
on a quotient space defined as follows.

Once again, for ω ∈ D([0, T ],Rd) set ωt(·) = ω(t ∧ ·). Then, the space of stopped paths is
defined as the quotient space

Λd = {(t, ωt) : (t, ω) ∈ [0, T ]×D([0, T ],Rd)} = [0, T ]×D([0, T ],Rd)/ ∼

with the equivalence relation given by

(t, ω) ∼ (t′, ω) ⇔ {t = t′ and ωt = ω′t′}.

This space is equipped with a metric d∞ defined by

d∞((t, ω), (t′, ω′)) = |t− t′|+ sup
s∈[0,T ]

‖ωt(s)− ω′t′(s)‖.

Using these definitions, two kinds of derivatives are defined for non-anticipative functionals
on [0, T ] × D([0, T ],Rd), i.e. measurable maps F : (Λd, d∞) → (R,B(R)). The first kind of
derivative is with respect to time t, called horizontal derivative in [Cont, 2016].

Definition 1.16 ([Cont, 2016]). A non-anticipative functional F : Λd → R is said to be
horizontally differentiable at (t, ω) ∈ Λd if the limit

DF (t, ω) = lim
ε↓0

F (t+ ε, wt)− F (t, wt)
ε

exists. If F is horizontally differentiable for all (t, ω) ∈ Λd, the functional DF is called the
horizontal derivative of F .

The second kind of derivative is the actual functional derivative, called vertical derivative
in [Cont, 2016]. It compares to the derivative introduced in [Levental et al., 2013] with the
major difference being that the definition in [Cont, 2016] is only considering stopped paths.

Definition 1.17 ([Cont, 2016]). Let ei be the d-dimensional vector with a one in the ith
coordinate and zeros everywhere else. A non-anticipative functional F is said to be vertically
differentiable at (t, ω) ∈ Λd if the limit

∂iF (t, ω) = lim
ε→0

F (t, ωt + εei1[t,T ])− F (t, ωt)
ε

, i = 1, . . . , d,



1.3. MARTINGALE MEASURES 13

exists. If F is vertically differentiable for all (t, ω) ∈ Λd, the vector ∇ωF (t, ω) = (∂iF (t, ω))i=1,...,d
is called the vertical derivative of F . The second order directional vertical derivative is given
by

∂i∂jF (t, ω) = lim
ε→0

∂jF (t, ωt + εei1[t,T ])− ∂jF (t, ωt)
ε

, i, j = 1, . . . , d,

if the limit exists. If all derivatives exist, set ∇2
ωF (t, ω) = (∂i∂jF (t, ω))i,j=1,...,d.

As in the approach in [Levental et al., 2013], the definition of vertical derivatives requires
the definition of F for paths in D([0, T ],Rd). By comparing Figure 1.3 to Figure 1.2, the
differences in the paths considered in the definition of the derivatives becomes clear.

The following version of the functional Itō-formula is a slight simplification of the actual
formulation found in [Cont, 2016]. To formulate it, let X be a continuous process on a
probability space (Ω,F ,P) and denote its value at time t ∈ [0, T ] by X(t) ∈ Rd.
Theorem 1.18 ([Cont, 2016]). Assume the non-anticipative functional F : Λd → R as well as
the processes DF , ∇ωF and ∇2

ωF are continuous with respect to the metric d∞ and bounded.
Further, let X be a continuous semimartingale. Then

F (t,Xt) = F (0, X0) +
∫ t

0
DF (s,Xs)ds+

d∑
i=1

∫ t

0
∂iF (s,Xs)dX(s)

+ 1
2

d∑
i, j=1

∫ t

0
∂i∂jF (s,Xs)d[Xi, Xj ](s).

(1.8)

Note that the only difference between the two versions (1.7) and (1.8) is the addition of the
time argument in the functional F and the resulting DF term in (1.8). The difference in the
definition of the derivatives vanishes as in (1.7) the derivatives are only computed for stopped
paths.

1.3 Martingale Measures
The concept of martingale measures is introduced in [Walsh, 1986] as a measure-valued coun-
terpart to the traditional stochastic white noise and is used to study stochastic partial dif-
ferential equations. In our context, martingale measures play a fundamental role in the
formulation of the Itō-formulae for B(A, c)-superprocesses in Chapter 2 as well as the mar-
tingale representation formulae in Chapter 3.

Both, a B(A, c)-superprocess as well as a historical Brownian motion give rise to a martingale
measure. While we prove this result for B(A, c)-superprocesses in the second part of this
section, we refer to Chapter 2 in [Perkins, 1995] for the derivation of the martingale measure
corresponding to a historical Brownian motion.

We start this section with a summary of the relevant parts of the introduction of martingale
measures and the integration with respect to such measures in [Walsh, 1986]. The definition
of the stochastic integral with respect to a martingale measure in [Walsh, 1986] is restricted
to predictable integrands. However, to prove the results in Chapter 2, we have to compute the
integral for optional integrands. Therefore, we conclude this introductory chapter by proving
that we can extend the class of valid integrands to include such optional functions.



14 CHAPTER 1. PRELIMINARIES

1.3.1 Martingale Measures and Integration with respect to Martingale
Measures

For the sake of a brief introduction to martingale measures, we only consider the scenario rele-
vant for the remainder of this monograph. Among others restrictions, this implies a restriction
to a locally compact separable metric space E and the definition of stochastic integrals with
respect to orthogonal martingale measures. For the more general setting in which E is a Lusin
space and the stochastic integral with respect to a more general worthy martingale measure
as well as the proofs of the results stated, we refer to [Walsh, 1986].

Let E be a locally compact separable metric space with Borel-σ-algebra E . Further, as-
sume (Ω,F , (Ft)t∈[0,T ],P) is a filtered probability space with a right continuous filtration, set
L2 = L2(Ω,F ,P) and define the L2-norm by ‖f‖2 = E[f2] 1

2 .

Next, consider a function U defined on A × Ω with A being a subalgebra of E that satisfies
‖U(B)‖2 <∞ for all B ∈ A as well as U(B1 ∪B2) = U(B1) +U(B2) for all B1, B2 ∈ A with
B1 ∩B2 = ∅. Additionally, define a set function µ by

µ(B) = ‖U(B)‖22.

The function U is called σ-finite if there exists an increasing sequence (En)n ⊂ E with⋃
nEn = E and such that En = E|En ⊂ A as well as supB∈En

‖U(B)‖2 < ∞ for all n ∈ N. It
is called countably additive on (En)n if , in addition, for any sequence (Bj)j

Bj ∈ En for all n and Bj ↓ ∅ implies lim
j→∞

µ(Bj) = 0.

Further, if U is countably additive on (En)n, it can be extended to E by setting

U(B) =
{

limn→∞ U(B ∩ En), if the limit exists,
undefined, otherwise

(1.9)

for any B ∈ E.

Definition 1.19 (σ-finite L2-valued measure). A countably additive function U is called a
σ-finite L2-valued measure if it has been extended as in (1.9).

Definition 1.20 (Martingale measure). A process Mt(B), t ∈ [0, T ], B ∈ A, is called a
martingale measure if

(i) M0(B) = 0 for all B ∈ A,

(ii) Mt is a σ-finite L2-valued measure for all t ∈ (0, T ],

(iii) the process (Mt(B))t∈[0,T ] is a (Ft)t-martingale for all B ∈ A.

A martingale measure is called continuous if for all B ∈ A the mapping t 7→ Mt(B) is
continuous.

In order to define the stochastic integral with respect to a martingale measure M , further
conditions on M have to be imposed. One such condition is the following.



1.3. MARTINGALE MEASURES 15

Definition 1.21 (Orthogonal martingale measure). A martingale measure M is called or-
thogonal if B1, B2 ∈ A, B1 ∩ B2 = ∅ implies that the martingales {Mt(B1)}t∈[0,T ] and
{Mt(B2)}t∈[0,T ] are orthogonal, i.e. {Mt(B1)Mt(B2)}t∈[0,T ] is a martingale.

The definition of the stochastic integral with respect to an orthogonal martingale measure
relies on the fact that every orthogonal martingale measure is a worthy martingale measure.
Worthy martingale measures are martingale measures for which a dominating measure exists.
To define dominating measures, we first have to introduce the (co)variation Q of an orthogonal
martingale measure M . For such a martingale measure, define the set function Q for (s, t] ⊂
[0, T ] and B ∈ E by

Q((s, t]×B) = [M(B)]t − [M(B)]s
and extend Q by additivitiy to finite unions of disjoint sets (si, ti]×Bi, i = 1, . . . , n, by

Q

(
n⋃
i=1

(si, ti]×Bi
)

=
n∑
i=1

([M(Bi)]ti − [M(Bi)]si).

Definition 1.22 (Dominating measure). A random σ-finite measure K defined on B([0, T ])×
E × Ω is called dominating measure of an orthogonal martingale measure M if

(i) K is positive definite,

(ii) for fixed B ∈ E, {K((0, t]×B)}t∈[0,T ] is predictable,

(iii) for all n it holds E[K([0, T ]× En)] <∞ ,

(iv) for any (s, t]×B ⊂ [0, T ]× E it holds |Q((s, t]×B)| ≤ K((s, t]×B) almost surely.

The above definition is not the original version of the definition of dominating measures as it
can be found in [Walsh, 1986]. Instead, it has already been adjusted to account for the fact
that we only consider orthogonal martingale measures. For such martingale measures, as the
following proposition states, we immediately get the dominating measure from the covariation
Q defined above.

Proposition 1.23. For an orthogonal martingale measure, it holds for any t ∈ [0, T ] and
B ∈ E

P(Q((0, t]×B) = K((0, t]×B)) = 1.

We now have everything on hand to define the stochastic integral with respect to an orthogonal
martingale measure. The construction follows the standard steps known from the construction
of the regular Itō-integral.

Definition 1.24 (Elementary and Simple functions). A function f : Ω × [0, T ] × E → R is
called elementary if it can be written as

f(ω, s, x) = X(ω)1(a,b](s)1B(x)

with 0 ≤ a < b ≤ T , X a bounded, Fa-measurable random variable and B ∈ E. Functions
which can be written as a linear combination of elementary functions are called simple and
the class of simple functions is denoted by S.



16 CHAPTER 1. PRELIMINARIES

Definition 1.25 (Predictable functions). Denote by P the σ-algebra on Ω× E × [0, T ] gen-
erated by S. The σ-algebra P is called the predictable σ-algebra and functions that are
measurable with respect to P are called predictable functions.

Definition 1.26 (‖ · ‖M , PM ). For an orthogonal martingale measure with dominating mea-
sure K, define a norm on the set of predictable functions by

‖f‖M = E
[∫ T

0

∫
E
|f(s, x)|2K(dx, ds)

] 1
2

and denote by PM the set of predictable functions with finite ‖ · ‖M -norm.

For an elementary function f(ω, s, x) = X(ω)1(a,b](s)1B̃(x), the stochastic integral with re-
spect to an orthogonal martingale measure M , denoted by f •M , is defined by

f •Mt(B) = X(ω)(Mt∧b(B̃ ∩B)−Mt∧a(B̃ ∩B))

and the definition can be extend to f ∈ S by linearity.

Proposition 1.27. It holds for all f ∈ S and all orthogonal martingale measures M that

(i) f •M is an orthogonal martingale measure,

(ii) E[(f •Mt(B))2] ≤ ‖f‖2M for all B ∈ E and t ∈ [0, T ].

In order to extend the definition of the stochastic integral with respect to an orthogonal
martingale measure to function in PM , we need the following result.

Proposition 1.28. The class S is dense in PM with respect to the norm ‖ · ‖M .

The above proposition allows us to find, for every f ∈ PM , a sequence (fn)n ⊂ S such that
‖fn − f‖M → 0 as n→∞. From Proposition 1.27 we further get

E[(fm •Mt(B)− fn •Mt(B))2] ≤ ‖fm − fn‖2M .

As the series (fn)n converges to f with respect to ‖ · ‖M , we get that

E[(fm •Mt(B)− fn •Mt(B))2]→ 0 as m, n→∞.

Consequently, the sequence (fn •Mt(B))n is Cauchy and as L2(Ω,F ,P) is complete, the L2-
limit f • Mt(B) exists. This completes the construction of the integral with respect to a
martingale measure for functions in PM .

Proposition 1.29. It holds for all f ∈ PM and all orthogonal martingale measures M that

(i) f •M is an orthogonal martingale measure,

(ii) E[(f •Mt(B))2] ≤ ‖f‖2M for all B ∈ E and t ∈ [0, T ].

To conclude the introductory part on martingale measures and integrals with respect to an
orthogonal martingale measure, we introduce the following notion which is in line with the
familiar notation of integrals and thus simplifies the representation of the results in Chapter
2 and Chapter 3:

f •Mt(B) =
∫ t

0

∫
B
f(s, x)M(ds, dx). (1.10)



1.3. MARTINGALE MEASURES 17

1.3.2 The Martingale Measure of the B(A, c)-Superprocess
We previously mentioned that there are different ways to define B(A, c)-superprocesses. In
this section, we prove the existence of a martingale measure associated with a B(A, c)-
superprocess, which is based on the process M(t)(φ) in (MP). This connection between
the martingale problem of a B(A, c)-superprocess and the martingale measure induced by it
is part of the motivation for defining B(A, c)-superprocesses via their martingale problems.

Consider the setting in Section 1.1.1 with E being a locally compact separable metric space
with Borel-σ-algebra E . Before we derive the martingale measure, recall the following concept.

Definition 1.30. A sequence (fn)n of functions from E to R converges bounded pointwise
(bp) to f if the sequence (fn)n converges pointwise to f and there exists a constant C ∈ R
such that |fn(x)| < C for all x ∈ E and n ∈ N.

As the domain D(A) of the generator A is dense in C(E) (see Proposition 1.8) and as the
bounded pointwise closure of C(E) is the set of bounded E-measurable functions, denoted
by bE , D(A) is bp-dense in bE . Consequently, as 1B ∈ bE , B ∈ E , there exists a sequence
(fn)n ⊂ D(A) such that fn

bp−→ 1B. By choosing the sequence (fn)n such that |fn| ≤ 1 for all
n, this allows us to define the following L2-limit

Mt(B) := M(t)(1B) := lim
n→∞

M(t)(fn), (1.11)

where M(t)(·) is the martingale arising from the martingale problem (MP) and the limit
exists by the dominated convergence theorem.

Theorem 1.31. The L2-limit M defined by (1.11) is a continuous orthogonal martingale
measure with dominating measure1 given by

ν((s, t]×B) =
∫ t

s
〈X(s), 1B〉ds for all 0 ≤ s < t ≤ T and B ∈ E .

Proof. To prove that M is a martingale measure, we have to show that M satisfies the three
properties in Definition 1.20. The first property is trivial as by definition

M(0)(B) = lim
n→∞

(
〈X(0), fn〉 − 〈X(0), fn〉 −

∫ 0

0
〈X(s), Afn〉ds

)
= 0

holds for every suitable sequence (fn)n.

To prove the second property, recall that we get from Proposition 1.11 that, for all t ∈ [0, T ],
E[(M(t)(E))2] < ∞ holds. Thus, E[(M(t)(B))2] < ∞ for all B ∈ E and we can pick the
subalgebra A to be E .

Now, let (Bj)j ⊂ E be a sequence with Bj → ∅ for j →∞ and set fj(ω, s) =
∫
Bj
X(ω, s)(dx).

Then, (fj)j is almost surely monotonically decreasing to zero and non-negative for all s ∈
1In the context of superprocesses, it is common to denote the dominating measure of the martingale measure

(and thus the covariation process if the martingale measure is orthogonal) by ν instead of K (or Q).



18 CHAPTER 1. PRELIMINARIES

[0, T ]. As, in addition,
∫ t

0 f(ω, s)ds < ∞ holds almost surely for any choice of B1 ∈ E , since
X(s) ∈MF (E) for all s ∈ [0, T ], we can apply the monotone convergence theorem to obtain

lim
j→∞

∫ t

0
fj(ω, s)ds =

∫ t

0
lim
j→∞

fj(ω, s)ds =
∫ t

0
0ds = 0.

Next, set gtj(ω) =
∫ t
0 fj(ω, s)ds. The sequence (gtj)j is also almost surely monotonically

decreasing to zero for all t ∈ [0, T ]. In addition, as

E[gt1] = E
[∫ t

0

∫
B1
X(s)(dx)ds

]
= E[(M(t)(B1))2] <∞,

the monotone convergence theorem can be applied a second time to obtain
lim
j→∞

E[gtj ] = E[ lim
j→∞

gtj ] = E[0] = 0.

Combining the above, we have
lim
j→∞

µ(Bj) = lim
j→∞

E[(M(t)(Bj))2]

= lim
j→∞

E
[
c

∫ t

0
〈X(s), (1Bj )2〉ds

]
= lim

j→∞
E
[
c

∫ t

0
〈X(s), 1Bj 〉ds

]
= 0.

By setting En = E for all n ∈ N, we obtain that M is σ-finite. As it is also finitely additive
and

lim
n→∞

M(t)(B ∩ En) = lim
n→∞

M(t)(B ∩ E) = lim
n→∞

M(t)(B) = M(t)(B)

holds for all t ∈ [0, T ], M is a σ-finite L2-valued measure.

The third property in Definition 1.20 follows from the fact that M(t)(B) is defined as a L2-
limit of processes M(t)(fn) with (fn)n ⊂ D(A). These processes are martingales by (MP)
and thus the L2-limit is also a martingale. As the processes M(t)(fn) are continuous, this also
yields the continuity of the L2-limit M(t)(B). Hence, M is a continuous martingale measure.

The martingale measure is also orthogonal as for any B1, B2 ∈ E with B1 ∩B2 = ∅ it holds

[M(B1),M(B2)]t = c

∫ t

0
〈X(s), 1B11B2〉ds = 0

by (1.2). From the above, we also get that the dominating measure ν has to be of the form
presented in the statement of the theorem, which completes the proof.

Notation (1.10) allows us to highlight that M is the martingale measure associated with a
B(A, c)-superprocess X by writing MX instead of M . Therefore,∫ t

0

∫
E
f(s, x)MX(ds, dx)

is the stochastic integral of f with respect to the martingale measure associated with the
B(A, c)-superprocess X. In particular, we get that the process M(t)(φ) in (MP) can be
written as

M(t)(φ) =
∫ t

0

∫
E
φ(s, x)MX(ds, dx).



1.3. MARTINGALE MEASURES 19

1.3.3 Extending the Class of Integrands
The results in Section 1.3.1 and Section 1.3.2 allow us to define the stochastic integral of a func-
tion f ∈ PM with respect to the martingale measure associated with a B(A, c)-superprocess.
However, some functions considered in Chapter 2 are not predictable but right continuous
with left limits and thus not in the class of functions for which Walsh defines the stochastic
integral with respect to a martingale measure. In this section, we extend the definition of the
integral to a wider class of functions to include all the functions we consider in this monograph.

More precisely, we introduce a class of integrands IMX
for which we can define the stochastic

integral with respect to the martingale measure associated with a B(A, c)-superprocess. This
class includes the class PM introduced in Section 1.3.1 but also bounded optional functions.
As bounded right continuous functions with left limits are optional (see Proposition 1.35),
this completion is sufficient to prove the results in the following chapters.

Once again, consider the setting in Section 1.1.1, a B(A, c)-superprocess X and denote the
distribution of X by P.

Definition 1.32 (Optional functions). Denote by O the σ-algebra generated by linear com-
binations of functions of the form

f(ω, s, x) = Y (ω)1[a,b)(s)1B(x)

with 0 ≤ a < b ≤ T , Y a bounded, Fa-measurable random variable and B ∈ E. The σ-algebra
O is called optional σ-algebra and a function is called optional if it is O-measurable.

Futher, let MX be the martingale measure associated with X and define a measure µMX
on

F × B([0, T ])× E by

µMX
(B1 ×B2 ×B3) = E

[
c

∫ T

0

∫
E

1B1×B2×B3(ω, s, x)X(ω, s)(dx)ds
]
.

This is the extension of the so-called Doléans measure of MX to F ×B([0, T ])×E . Denote by
L2
P the space L2(Ω× [0, T ]×E,P, µMX

), i.e. the space of P-measurable functions satisfying∫
f2dµMX

< ∞, and note that it coincides with the space PMX
introduced in Section 1.3.1.

Finally, denote the class of µMX
-null sets in F × B([0, T ])× E by N and set P̃ = P ∧ N . In

the remainder of this section we show that L2
P = L2

P̃ holds by following standard arguments
that can for example be found in [Chung and Williams, 2014]. This allows us to extend the
stochastic integral with respect MX to functions in L2

P̃ . For convenience we later denote this
space of functions (for which we can define the stochastic integral with respect to MX) by
IMX

.

Proposition 1.33. (i) A subset B̃ of Ω× [0, T ]×E belongs to P̃ if and only if there exists
a B ∈ P such that

B̃∆B = (B̃ \B) ∪ (B \ B̃) ∈ N .

(ii) If f : Ω× [0, T ]×E → R is F ×B([0, T ])×E-measurable, it is P̃-measurable if and only
if there exists a predictable function g such that

{f 6= g} ∈ N .



20 CHAPTER 1. PRELIMINARIES

Proof. (i) Set
A = {B̃ : ∃B ∈ P s.t. B̃∆B ∈ N}.

The proof is complete if we can prove that A = P̃. Thus, consider C = B̃∆B and
observe that B̃ = C∆B holds. Now, if C ∈ N ⊂ P̃ and B̃ ∈ P ⊂ P̃, this immediately
yields B̃ ∈ P̃. Consequently A ⊂ P̃.

As A contains N and P, to prove that P̃ ⊂ A also holds, it suffices to show that A is
a σ-algebra. This is the case as

• ∅ ∈ P and therefore ∅∆∅ = ∅ ∈ N , which implies ∅ ∈ A,
• B̃c∆Bc = B̃∆B and therefore B̃c ∈ A for all B̃ ∈ A,
• (⋃n B̃n)∆(⋃nBn) ⊂ ⋃n(B̃n∆Bn) and therefore ⋃n B̃n ∈ A for all (B̃n)n ⊂ A.

Thus, P̃ ⊂ A, which yields P̃ = A.

(ii) Let g be a predictable function and assume {f 6= g} ∈ N . Then, we have

{f ∈ S}∆{g ∈ S} ⊂ {f 6= g} for all S ∈ B(R).

By the first part of this proposition, as {g ∈ S} ∈ P, we get {f ∈ S} ∈ P̃ and conse-
quently f is P̃-measurable.

Now assume f is P̃-measurable and f = ∑∞
j=1 cj1B̃j

for disjoint B̃j ∈ P̃ and cj ∈ R.
For each B̃j there exists a Bj ∈ P such that B̃j∆Bj ∈ N by the first part of this
proposition. As the B̃j ’s are disjoint, B̃j ∩ Bj ⊂ B̃j and Bj ∩ B̃c

j = Bj \ B̃j ⊂ B̃j∆Bj ,
we have

Bi ∩Bj =
(
(Bi ∩ B̃i) ∪ (Bi ∩ B̃c

i )
)
∩
(
(Bj ∩ B̃j) ∪ (Bj ∩ B̃c

j )
)
∈ N (1.12)

if i 6= j. To obtain disjoint sets, set

B′1 = B1 and B′j =
j−1⋂
i=1

Bc
i ∩Bj for j ≥ 2.

Then, as Bj∆B′j = (Bj ∪B′j) \ (Bj ∩B′j), we get from (1.12) that

Bj∆B′j =

Bj ∪
j−1⋂
i=1

Bc
i ∩Bj

 \
Bj ∩

j−1⋂
i=1

Bc
i ∩Bj

 = Bj \

j−1⋂
i=1

Bc
i

 ∈ N
holds. In addition, for i 6= j,

B̃j∆B′j =

B̃j \
Bj ∩

j−1⋂
i=1

Bc
i

 ∪
Bj ∩

j−1⋂
i=1

Bc
i

 \ B̃j


= (B̃j \Bj) ∪

B̃j \ j−1⋂
i=1

Bc
i

 ∪
(Bj \ B̃j) ∩

j−1⋂
i=1

Bc
i

 ∈ N



1.3. MARTINGALE MEASURES 21

holds, as the union of the first and third term is a subset of B̃j∆Bj and the second term
can be written as

B̃j \
j−1⋂
i=1

Bc
i =

j−1⋃
i=1

Bi ∩ B̃j

and, if i 6= j, Bi ∩ B̃j ∈ N holds as the B̃j ’s are disjoint and (B̃j∆Bj) ∈ N .

Next, set g = ∑∞
j=1 cj1B′j . Then g is P-measurable and

{f 6= g} ⊂
∞⋃
j=1

(B̃j∆B′j) ∈ N .

For general P̃-measurable f , there exists a sequence (fn)n of P̃-measurable functions of
the above form such that fn → f µMX

-almost surely. Pick P-measurable gn’s such that
{fn 6= gn} ∈ N for all n ∈ N and set g = lim infn→∞ gn. Then g is also P-measurable
and ∞⋃

n=1
{fn 6= gn} ∪ { lim

n→∞
fn 6= f} ∈ N .

Set
Σ = (Ω× [0, T ]× E) \

( ∞⋃
n=1
{fn 6= gn} ∪ { lim

n→∞
fn 6= f}

)
.

Obviously, the set Σ has full mass, i.e. (Ω × [0, T ] × E) \ Σ ∈ N , and on Σ we have
fn(ω, s, x) = gn(ω, s, x) for all n ∈ N. Therefore

lim inf
n→∞

fn(ω, s, x) = lim inf
n→∞

gn(ω, s, x)

on Σ and

∅ = {(ω, s, x) ∈ Σ : f(ω, s, x) 6= lim
n→∞

fn(ω, s, x)}

= {(ω, s, x) ∈ Σ : f(ω, s, x) 6= lim inf
n→∞

fn(ω, s, x)}

= {(ω, s, x) ∈ Σ : f(ω, s, x) 6= lim inf
n→∞

gn(ω, s, x)}.

Thus, {(ω, s, x) ∈ Σ : f(ω, s, x) 6= lim infn→∞ gn(ω, s, x)} ∈ N and we obtain

{(ω, s, x) ∈ Ω× [0, T ]× E : f(ω, s, x) 6= lim inf
n→∞

gn(ω, s, x)}

⊂ {(ω, s, x) ∈ Σ : f(ω, s, x) 6= lim inf
n→∞

gn(ω, s, x)} ∪ (Ω× [0, T ]× E) \ Σ.

As both sets on the right hand side are null sets, we get

{f 6= g} ∈ N ,

which completes the proof.

The previous proposition allows us to prove the following result which establishes the con-
nection between P̃ and optional functions.



22 CHAPTER 1. PRELIMINARIES

Proposition 1.34. Any bounded optional function is P̃-measurable.

Proof. Let f(ω, s, x) = Y (ω)1[a,b)(s)1B(x) and g(ω, s, x) = Y (ω)1(a,b](s)1B(x) with 0 ≤ a <
b ≤ T , Y bounded and Fa-measurable and B ∈ E . As g is predictable, we get from Proposition
1.33 that f is P̃-measurable if {f 6= g} ∈ N . This is the case as

µM ({f 6= g}) = E
[
c

∫ T

0

∫
E

1{f 6=g}X(s)(dx)ds
]

= E
[
c

∫ T

0

∫
E

1{Y 1[a,b)(s)1B(x)6=Y 1(a,b](s)1B(s)}X(s)(dx)ds
]

= E
[
c

∫ T

0

∫
E

1{Ω×[a]×B}X(s)(dx)ds
]

= E
[
c

∫ T

0
1[a]X(s)(B)ds

]
= 0.

The last equality holds asX(s)(B) is almost surely finite. Consequently f is P̃-measurable and
as functions of this form generate O, we get that every optional function is P̃-measurable.

Since, by definition, P ⊂ P̃ holds, we have L2
P ⊂ L2

P̃ . To see that the two are actually equal,
recall that, by Proposition 1.33, there exists a function g ∈ L2

P for any f ∈ L2
P̃ such that

g = f µMX
-almost surely. This allows us to extend the stochastic integral with respect to

the martingale measure MX associated with the B(A, c)-superprocess X to square-integrable,
P̃-measurable integrands. This class of integrands includes bounded optional functions and
we denote it by IMX

.

As mentioned above, the extension of the class of integrands is necessary because some of the
functions we integrate with respect to the martingale measure MX in the later parts of this
monograph are not predictable but right continuous with left limits. To complete this section
on the extension of the class of valid integrands, we still have to prove that right continuous
functions with left limits are optional. In fact, it holds O = σ(r.c.l.l.), where r.c.l.l. is the set
of adapted right continuous functions with left limit. However, as we are only interested in
the inclusion σ(r.c.l.l.) ⊂ O and as the second inclusion follows from the definition of O, we
only prove the following.

Proposition 1.35. It holds:

(i) If f is Fa × E measurable, then f1[a,b) is optional for all 0 ≤ a < b ≤ T .

(ii) σ(r.c.l.l.) ⊂ O.

Proof. (i) As f is Fa×E measurable, we can approximate it pointwise by sums of indicator
functions 1F×B, F ∈ Fa, B ∈ E . Therefore, it is enough to consider f = 1F×B. Since
we can write

(f1[a,b))(ω, s, x) = 1F (ω)1[a,b)(s)1B(x),

the function f1[a,b) is optional as Y (ω) = 1F (ω) is Fa-measurable.



1.3. MARTINGALE MEASURES 23

(ii) Let f be an adapted right continuous function with left limits. Consider the parti-
tion {tni : 0 ≤ i ≤ n, 1 ≤ n ≤ ∞} with tn0 = 0, tnn = T , tni < tni+1 for all n and
maxi |tni+1 − tni | → 0 as n→∞. Further, define an approximation of the function f by

Appn(f)(ω, s, x) =
n−1∑
i=0

f(ω, tni , x)1[tni ,t
n
i+1)(s).

This approximation converges pointwise to f and as every summand of the above sum
is optional by the first part of this proposition, so is the approximation and thus the
limit f is also optional.





Chapter 2

Functional Itō-Calculus for
Superprocesses

In this chapter, we derive the Itō-formula as well as the functional Itō-formula for functions,
respectively functionals, of B(A, c)-superprocesses. One of the main steps towards deriving
the two formulae is the definition of the necessary derivatives of functions from [0, T ]×MF (E)
to R as well as the functional derivatives of functionals from [0, T ]×D([0, T ],MF (E)) to R.
For the later one, we choose to adapt the concept of horizontal and vertical derivatives as
introduced by Cont and Fournié. However, we also note that a result equivalent to Theorem
2.14 can be obtained if one uses the approach by Levental and co-authors.

In Section 2.1 and Section 2.2 the Itō-formula and the functional Itō-formula, respectively, are
obtained under the assumption that the underlying space E is compact. In the final section
of this chapter, we expand on how the two results can be extended to a setting with locally
compact E.

2.1 The Itō-Formula for Superprocesses

To derive the Itō-formula for a wide class of functions of B(A, c)-superprocesses, we use the
martingale measure associated with the underlying B(A, c)-superprocess to reformulate a re-
sult in [Jacka and Tribe, 2003]. Before summarizing the relevant results in [Jacka and Tribe,
2003], we introduce the class of finitely based functions of measure-valued processes – a class
of basic functions for which one can easily compute the Itō-formula (see Theorem 2.4) using
the traditional Itō-formula for Rd-valued processes.

Consider the setting in Section 1.1.1 with E compact and let X be a B(A, c)-superprocess
with associated martingale measure MX . When applying the generator A to a function G
with multiple arguments, we write A(x)G(x, y, z) to highlight that the generator is applied in
the x-coordinate, i.e. A(x)G(x, y, z) = A(G(·, y, z))(x). Finally, denote by C1,2([0, T ]×Rd) =
C1,2([0, T ]×Rd,R) the space of functions from [0, T ]×Rd to R which are continuous with one

25



26 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

partial derivative with respect to the first argument as well as two partial derivatives with
respect to the second argument, which are also continuous.

Definition 2.1 (Finitely based functions). Given a generator A, a function F : [0, T ] ×
MF (E)→ R is called finitely based if a function f ∈ C1,2([0, T ]×Rd) as well as φ1, . . . , φd ∈
D(A) exist such that

F (t, µ) = f(t, 〈µ, φ1〉, . . . , 〈µ, φd〉) (2.1)

holds for all t ∈ [0, T ], µ ∈MF (C).

Before we can formulate the Itō-formula for finitely based functions of a B(A, c)-superprocess,
we have to introduce the two following types of derivatives of a functions on finite measures.

Definition 2.2. A continuous function F : [0, T ]×MF (E)→ R is differentiable with respect
to time if the limit

D∗F (s, µ) = lim
ε→0

F (s+ ε, µ)− F (s, µ)
ε

exists.

Definition 2.3 (Directional derivatives). A continuous function F : [0, T ]×MF (E)→ R is
differentiable in direction δx, x ∈ E, if the limit

DxF (s, µ) = lim
ε→0

F (s, µ+ εδx)− F (s, µ)
ε

exists. We call DxF the directional derivative of F . Higher order directional derivatives are
defined iteratively.

Notation. We set DxyF (s, µ) = DxDyF (s, µ) and, if the derivative is continuous in all
arguments, we have DxyF (s, µ) = DyxF (s, µ). Further, we write D∗xF (s, µ) instead of
D∗DxF (s, µ) and deal with higher order mixed derivatives alike.

We can now formulate the Itō-formula for finitely based functions of a B(A, c)-superprocess
X. In the proof of this formula, the definition of X via its martingale problem comes in handy
once again.

Theorem 2.4 (Itō-formula for finitely based functions). Let X be a B(A, c)-superprocess and
F : [0, T ]×MF (E)→ R finitely based. Then, for t ∈ [0, T ], the following holds:

F (t,X(t)) = F (0, X(0)) +
∫ t

0
D∗F (s,X(s))ds

+
∫ t

0

∫
E
A(x)DxF (s,X(s))X(s)(dx)ds

+1
2

∫ t

0

∫
E
cDxxF (s,X(s))X(s)(dx)ds

+
∫ t

0

∫
E
DxF (s,X(s))MX(ds, dx).

(2.2)

Proof. As F is finitely based, it is of form (2.1). As the functions φi are in D(A), we get from
the martingale problem (MP) that the 〈X(t), φi〉’s are semimartingales. Thus, the traditional



2.1. ITŌ-FORMULA 27

Itō-formula for Rd-valued semimartingales yields

f(t, 〈X(t), φ1〉, . . . , 〈X(t), φd〉)
= f(0, 〈X(0), φ1〉, . . . , 〈X(0), φd〉)

+
∫ t

0
∂sf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)ds

+
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)d〈X(s), φi〉

+1
2

∫ t

0

d∑
i, j=1

∂ijf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)d[〈X,φi〉, 〈X,φj〉]s,

(2.3)

where ∂sf , ∂if and ∂ijf are the partial derivative of f .

From (MP) we further get

〈X(s), φi〉 = M(s)(φi) + 〈X(0), φi〉+
∫ s

0
〈X(r), Aφi〉dr.

As 〈X(0), φi〉 is constant and M(s)(φi) =
∫ s

0
∫
E φi(r, x)MX(dr, dx) (see Section 1.3.2), the

above yields
d〈X(s), φi〉 =

∫
E
φiM(ds, dx) + 〈X(s), Aφi〉ds.

In addition, as

[M(φ1),M(φ2)]t = c

∫ t

0
〈X(s), φ1φ2〉ds,

we have
d[M(φ1),M(φ2)]t = c〈X(s), φ1φ2〉dt.

Plugging these terms into (2.3) yields

f(t, 〈X(t), φ1〉, . . . , 〈X(t), φd〉)
= f(0, 〈X(0), φ1〉, . . . , 〈X(0), φd〉)

+
∫ t

0
∂sf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)ds

+
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)
∫
E
φi(x)M(ds, dx)

+
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)〈X(s), Aφi〉ds

+1
2

∫ t

0

d∑
i, j=1

∂ijf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)c〈X(s), φiφj〉ds.

(2.4)

The result now follows by computing the directional derivatives of finitely based functions
and identification with the expressions above. As ∂sf = D∗F holds, we get the equality of



28 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

the first integral in (2.4) and the first integral in (2.2). Further, as Dx〈µ, φ〉 = φ(x), the chain
rule of ordinary differentiation yields

DxF (t, µ) =
d∑
i=1

∂if(y1, . . . , yd)|y1=〈µ,φ1〉,...,yd=〈µ,φd〉φi(x).

Thus, ∫ t

0

∫
E
DxF (s,X(s))M(dx, ds)

=
∫ t

0

∫
E

d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)φi(x)M(ds, dx)

=
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)
∫
E
φi(x)M(ds, dx),

which is well-defined as φ, D·F ∈ IMX
, and further∫ t

0

∫
E
A(x)DxF (s,X(s))X(s)(dx)ds

=
∫ t

0

∫
E
A

(
d∑
i=1

∂if(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)D·〈X(s), φi〉
)

(x)X(s)(dx)ds

=
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φi〉, . . . , 〈X(s), φn〉)
∫
E
Aφi(x)X(s)(dx)ds

=
∫ t

0

d∑
i=1

∂if(s, 〈X(s), φi〉, . . . , 〈X(s), φn〉)〈X(s), Aφi〉ds.

Finally, as

DxxF (t, µ) =
d∑

i, j=1
∂ijf(s, 〈µ, φi〉, . . . , 〈µ, φd〉)φi(x)φj(x),

we obtain ∫ t

0

∫
E
cDxxF (s,X(s))X(s)(dx)ds

=
∫ t

0

∫
E
c

d∑
i, j=1

∂ijf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)φi(x)φj(x)X(s)(dx)ds

=
∫ t

0

d∑
i, j=1

∂ijf(s, 〈X(s), φ1〉, . . . , 〈X(s), φd〉)〈X(s), cφiφj〉ds,

which completes the proof.

Note that (2.2) resembles the Itō-formula Dawson proved for finitely based function of measure-
valued processes in [Dawson, 1978]. The introduction of martingale measures by Walsh
eight years later allows us to write the Itō-formula for finitely based functions of B(A, c)-
superprocesses in the presented form.



2.1. ITŌ-FORMULA 29

As mentioned above, the Itō-formula for a more general class of functions of the B(A, c)-
superprocess (Theorem 2.9) is based on results in [Jacka and Tribe, 2003]. In the following,
the relevant parts in [Jacka and Tribe, 2003] are introduced and one of the main theorems
(see Theorem 2.7) and an outline of its proof are presented.

Definition 2.5 (Good generator). Let (St)t∈[0,T ] be the semigroup of the generator A. The
generator A is called a good generator if a dense linear subspace D0 of C(E) that is an algebra
exists and St : D0 → D0 holds for all t ∈ [0, T ].

Remark 2.6. If A is a good generator, D0 is a core of A.
The following set of conditions, introduced in [Jacka and Tribe, 2003], is crucial for the
remainder of this chapter as it characterizes the class of functions for which we can formulate
the Itō-formula in Theorem 2.9.

Condition 1. The function F : [0, T ]×MF (E)→ R satisfies

(i) F (s, µ), DxF (s, µ), DxyF (s, µ), DxyzF (s, µ), D∗F (s, µ), D∗xF (s, µ), D∗xyF (s, µ) and
D∗xyzF (s, µ) exist and are continuous in s ∈ [0, T ], x, y, z ∈ E and µ ∈MF (E),

(ii) the maps x 7→ DxF (s, µ), x 7→ DxyF (s, µ) and x 7→ DxyzF (s, µ) are in the domain of
A for fixed s ∈ [0, T ], y, z ∈ E and µ ∈MF (E),

(iii) A(x)DxF (s, µ), A(x)DxyF (s, µ) and A(x)DxyzF (s, µ) are continuous in s ∈ [0, T ], x, y,
z ∈ E and µ ∈MF (E).

With all the preparatory work concluded, we can now introduce the main result from [Jacka
and Tribe, 2003], which is essential for the proof of Theorem 2.9. The class of processes con-
sidered in [Jacka and Tribe, 2003] is a slightly more general class of measure-valued processes
but contains the class of B(A, c)-superprocesses. In the following, we state the result and
present an outline of the proof.

Theorem 2.7 ([Jacka and Tribe, 2003]). Suppose F : [0, T ]×MF (E)→ R satisfies Condition
1, A is a good generator and X is a MF (E)-valued process with its law P being the solution
of the martingale problem

for all φ ∈ D(A) the process

M(t)(φ) = 〈X(t), φ〉 − 〈X(0), φ〉 −
∫ t

0
〈X(s), Aφ〉ds, t ∈ [0, T ]

is a (Ft)t-local martingale with respect to P

and has quadratic variation [M(φ)]t =
∫ t

0
〈X(s), σ(s)φ2〉ds,

where σ : Ω× [0, T ]× E → R is predictable and locally bounded. Then

F (t,X(t))−
∫ t

0
D∗F (s,X(s))ds

−
∫ t

0

∫
E
A(x)DxF (s,X(s)) + 1

2σ(s, x)DxxF (s,X(s))X(s)(dx)ds
(2.5)

is a (Ft)t-local martingale.



30 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

Outline of the proof. The proof can be broken down into the six following steps.
Step 1. To prove that (2.5) is a local martingale, let K > 0 and define the stopping times

τ1
K such that σ(t, ·)1t<τ1

K
is bounded by K as well as

τ2
K =

{
0, if 〈X(0), 1〉 ≥ K,
inf{t : 〈X(t), 1〉 ≤ K}, if 〈X(0), 1〉 < K,

(2.6)

with inf ∅ =∞. Now, set τK = τ1
K ∧ τ2

K . As τK is increasing with τK →∞ as K →∞,
the proof is complete if

F (t ∧ τK , X(t ∧ τK))−
∫ t∧τK

0
D∗F (s,X(s))ds

−
∫ t∧τK

0

∫
E
A(x)DxF (s,X(s)) + 1

2cDxxF (s,X(s))X(s)(dx)ds

is a martingale. To prove this, it is enough to consider the scenario in which there exists
a K > 0 such that

〈X(t), 1〉 ≤ K for all t ∈ [0, T ].

Step 2. Let K > 0 and define the subspace MK(E) of MF (E) by
MK(E) = {µ ∈MF (E) : 〈µ, 1〉 ≤ K}.

In contrast to the space MF (E), the space MK(E) is compact. Next, for a function
φ : Ek → R, define its symmetrization φsym by

φsym(x1, . . . , xk) = 1
k!
∑
π∈Sk

φ(xπ1, . . . , xπk),

where Sk is the space of permutations on {1, . . . , k}. Further, consider functions

φ(x1, . . . , xk) =
k∏
i=1

φi(xi)

with φi ∈ D0 for i = 1, . . . , k and denote their span by Dprod
0 (Ek). Finally, consider the

space of functions
Dsym

0 (Ek) = {φsym : φ ∈ Dprod
0 (Ek)},

which allows us to define the following set of functions on [0, T ]× En ×MK(E):

A sym
n =

{
m∑
i=1

∫
Eki

ψi(t)φi(x, z)µki(dz) : ψi ∈ C1([0, T ]), φi ∈ Dsym
0 (Eki+n), ki,m ≥ 0

}
.

If F ∈ A sym
0 , it is a function from [0, T ]×MK(E) to R and finitely based as it consists

of elements of form

ψ(t)
∫
Ek

1
k!
∑
π∈Sk

k∏
i=1

φi(zπi)µk(dz) = ψ(t) 1
k!
∑
π∈Sk

k∏
i=1

(∫
E
φi(zπi)µ(dzπi)

)

= ψ(t) 1
k!
∑
π∈Sk

k∏
i=1
〈µ, φi〉

= ψ(t)
k∏
i=1
〈µ, φi〉



2.1. ITŌ-FORMULA 31

and as linear combinations of finitely based functions are finitely based.

Step 3. The key part of the proof of the theorem is the fact that the space A sym
n is a dense

subset of {Dx1···xnF (t, µ) : F ∈ Cn([0, T ] ×MK(E))}. In particular this yields that
A sym

0 is dense in C([0, T ]×MK(E)).

This is by far the most involved step of the proof and includes the proof of the strong
continuity of the transition semigroup (Ut)t∈[0,T ], given by

UtΦ(µ) = E[Φ(X(t))|X(0) = µ]
for suitable Φ, as well as the existence and continuity of the derivatives Dx1···xnUtΦ(µ).
The authors refer to this as the smoothing property of the Dawson-Watanabe semi-
group. The proof of these properties relies on the branching structure of the B(A, c)-
superprocess, including its Poisson cluster representation.

Step 4. From the previous step, we know that F as well as DxyF can be approximated
by functions in A sym

0 and A sym
2 , respectively. In order to find approximations for the

remaining terms in (2.5), define a semigroup (V n
t )t∈[0,T ] on [0, T ]× En ×MK(E) by

V n
s F (t, x1, . . . , xn, µ) =

S
(x1)
s · · ·S(xn)

s F (t+ s, x1, . . . , xn, S
∗
sµ, ) if s+ t ≤ T ,

S
(x1)
T−t · · ·S

(xn)
T−tF (T, x1, . . . , xn, S

∗
T−tµ), if s+ t > T ,

where (S∗t )t∈[0,T ] denotes the dual semigroup of (St)t∈[0,T ]. The authors show that the
corresponding generator is given by

(D∗ +Qn)F (s, x1, . . . , xn, µ) = D∗F (s, x1, . . . , xn, µ) +
n∑
i=1

A(xi)F (t, x1, . . . , xn, µ)

+
∫
E
A(z)DzF (t, x1, . . . , xn, µ)µ(dz).

Next, the authors prove that A sym
n is a core for the generator D∗ + Qn, which yields

the approximation for the remaining terms in (2.5).

Step 5. The two previous steps yield individual approximations of the different terms in
(2.5). However, the existence of a sequence (Fn)n ⊂ A sym

0 approximating F that also
satisfies DxF

n → DxF as well as DxyF
n → DxyF as n → ∞ is not given. As the

existence of such an approximation is required to complete the proof, the authors prove
the following result.

Denote by ‖ · ‖Θ the sup-norm on the space C(Θ). Then, for any n > 0, K > 0, there
exists a Fn ∈ A sym

0 such that

‖F − Fn‖[0,T ]×MK(E) ≤
1
n
,

‖DxyF −DxyF
n‖[0,T ]×E2×MK(E) ≤

1
n
,

‖(D∗ +Q0)F − (D∗ +Q0)Fn‖[0,T ]×MK(E) ≤
1
n
.

Step 6. Let 〈X(0), 1〉 < K1 and consider the function Fn ∈ A sym
0 from the previous step.

1Recall that X(0) = m is a finite, deterministic measure.



32 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

From the second step we know that Fn is finitely based and thus an Itō-formula like
the one in Theorem 2.4 can be derived. This yields, as X(t ∧ τK) ∈MK(E),

Fn(0, X(0)) +M

= Fn(t ∧ τK , X(t ∧ τK))−
∫ t∧τK

0
D∗Fn(s,X(s))ds

−
∫ t∧τK

0

∫
E
A(x)DxF

n(s,X(s)) + 1
2cDxxF

n(s,X(s))X(s)(dx)ds,

where M is a (Ft)t-martingale. Using the uniform approximations from the previous
step and letting n go to infinity then yields that

F (t ∧ τK , X(t ∧ τK))−
∫ t∧τK

0
D∗F (s,X(s))ds

−
∫ t∧τK

0

∫
E
A(x)DxF (s,X(s)) + 1

2cDxxF (s,X(s))X(s)(dx)ds

is a martingale, which, by the first step, completes the proof.

By considering a deterministic instead of random branching rate in Theorem 2.7, i.e. by
setting σ ≡ c, we obtain the corresponding result for B(A, c)-superprocesses. To formulate
the Itō-formula , we have to find an explicit representation of the martingale. To derive this
representation, which is done in Theorem 2.9, we need the following proposition.

Proposition 2.8. Let F : MF (E)→ R be continuous and with a continuous derivative DxF .
Then

F (µ) = F (0) +
∫ 1

0

∫
E
DxF (θµ)µ(dx)dθ.

Proof. See Lemma 4 in [Jacka and Tribe, 2003].

Theorem 2.9 (Itō-formula). Let X be a B(A, c)-superprocess with good generator A and
assume F : [0, T ]×MF (E)→ R satisfies Condition 1. Then, for all t ∈ [0, T ], it holds

F (t,X(t)) = F (0, X(0)) +
∫ t

0
D∗F (s,X(s))ds

+
∫ t

0

∫
E
A(x)DxF (s,X(s))X(s)(dx)ds

+1
2

∫ t

0

∫
E
cDxxF (s,X(s))X(s)(dx)ds

+
∫ t

0

∫
E
DxF (s,X(s))MX(ds, dx).

Proof. As in the outline of the proof of Theorem 2.7 and in the proof of the traditional Itō-
formula (see e.g. [Karatzas and Shreve, 1998]), we have to localize the underlying process X.
Thus, consider a stopping time τK which we define as in (2.6).



2.1. ITŌ-FORMULA 33

Fix a K > 0. From the proof of Theorem 2.7, we get the existence of a function Fn ∈ A sym
0

such that
sup

t∈[0,T ], µ∈MK(E)
|F (t, µ)− Fn(t, µ)| → 0,

sup
t∈[0,T ], x, y∈E,
µ∈MK(E)

|DxyF (t, µ)−DxyF
n(t, µ)| → 0,

sup
t∈[0,T ], µ∈MK(E)

|D∗F (t, µ) +
∫
E
A(x)DxF (t, µ)µ(dx)

−D∗Fn(t, µ)−
∫
E
A(x)DzF

n(t, µ)µ(dx)| → 0

(2.7)

as n→∞. As functions in A sym
0 are finitely based and X(t∧ τK) ∈MK(E) for all t ∈ [0, T ],

Theorem 2.4 yields

Fn(t ∧ τK , X(t ∧ τK)) = Fn(0, X(0)) +
∫ t∧τK

0
D∗Fn(s,X(s))ds

+
∫ t∧τK

0

∫
E
A(x)DxF

n(s,X(s))X(s)(dx)ds

+1
2

∫ t∧τK

0

∫
E
cDxxF

n(s,X(s))X(s)(dx)ds

+
∫ t∧τK

0

∫
E
DxF

n(s,X(s))MX(ds, dx).

Combing the limits in (2.7) with the above equation, we obtain
F (t ∧ τK , X(t ∧ τK)) = lim

n→∞
Fn(t ∧ τK , X(t ∧ τK))

= lim
n→∞

(
Fn(0, X(0))

+
∫ t∧τK

0
D∗Fn(s,X(s))ds

+
∫ t∧τK

0

∫
E
A(x)DxF

n(s,X(s))X(s)(dx)ds

+ 1
2

∫ t∧τK

0

∫
E
cDxxF

n(s,X(s))X(s)(dx)ds

+
∫ t∧τK

0

∫
E
DxF

n(s,X(s))MX(ds, dx)
)

= F (0, X(0))

+
∫ t∧τK

0
D∗F (s,X(s))ds

+
∫ t∧τK

0

∫
E
A(x)DxF (s,X(s))X(s)(dx)ds

+ 1
2

∫ t∧τK

0

∫
E
cDxxF (s,X(s))X(s)(dx)ds

+ lim
n→∞

∫ t∧τK

0

∫
E
DxF

n(s,X(s))MX(ds, dx).



34 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

By Proposition 2.8, the convergence of the second order derivatives of Fn yields
sup

t∈[0,T ], x∈E
µ∈MK(E)

|DxF (t, µ)−DxF
n(t, µ)|

≤ sup
t∈[0,T ], x∈E
µ∈MK(E)

|
∫ 1

0

∫
E
DxyF (t, θµ)µ(dy)dθ −

∫ t

0

∫
E
DxyF

n(t, θµ)µ(dy)dθ|

+ sup
t∈[0,T ], x∈E
µ∈MK(E)

|DxF (t, 0)−DxF
n(t, 0)|

= sup
t∈[0,T ], x∈E
µ∈MK(E)

|
∫ 1

0

∫
E
DxyF (t, θµ)−DxyF

n(t, θµ)µ(dy)dθ|

+ sup
t∈[0,T ], x∈E
µ∈MK(E)

|DxF (t, 0)−DxF
n(t, 0)|

≤
∫ 1

0

∫
E

sup
t∈[0,T ], x, y∈E
µ∈MK(E)

|DxyF (t, θµ)−DxyF
n(t, θµ)|µ(dy)dθ

+ sup
t∈[0,T ], x∈E
µ∈MK(E)

|DxF (t, 0)−DxF
n(t, 0)|.

Because of (2.7), the upper bound on the right hand side of the equation goes to zero if n
goes to infinity. Thus, DxF

n(t, µ) converges to DxF (t, µ) in the sup-norm, which yields the
convergence with respect to ‖ ·‖M . Therefore, by the definition of the stochastic integral with
respect to a martingale measure,

lim
n→∞

∫ t∧τK

0

∫
E
DxF

n(s,X(s))MX(ds, dx) =
∫ t∧τK

0

∫
E
DxF (s,X(s))MX(ds, dx).

Letting K go to infinity completes the proof.

The following elementary example illustrates how the Itō-formula can be applied.
Example 2.10. Let X be a B(A, c)-superprocess with good generator A and consider the
function F (t, µ) = ψ(t) + 〈µ, φ〉 with ψ ∈ C1([0, T ]) and φ ∈ D(A) such that Aφ ∈ D(A).
Then

D∗F (s, µ) = ψ′(s), DxF (s, µ) = φ(x) and DxxF (s, µ) = 0.
Therefore, Condition 1 is satisfied and the Itō-formula in Theorem 2.9 yields

F (t,X(t)) = ψ(0) + 〈X(0), φ〉+
∫ t

0
ψ′(s)ds

+
∫ t

0

∫
E
Aφ(x)X(s)(dx)ds

+
∫ t

0

∫
E
φ(x)MX(ds, dx)

= ψ(t) + 〈X(0), φ〉+
∫ t

0
〈X(s), Aφ〉ds+M(t)(φ)

= ψ(t) + 〈X(t), φ〉,
with the last equation following from the martingale problem (MP).



2.2. FUNCTIONAL ITŌ-FORMULA 35

2.2 The Functional Itō-Formula for Superprocesses
Based on the Itō-formula for B(A, c)-superprocesses in Theorem 2.9, we can now derive the
functional Itō-formula for B(A, c)-superprocesses. The approach presented is based on the
work by Cont and Fournié (see Section 1.2.2) as the functionals considered by the authors
contain a time argument and thus the approach adopts more natural to our setting. However,
if one prefers to define derivatives as in [Levental et al., 2013] (see Section 1.2.1), the result
obtained is the same, as in the present setting, the two definitions of functional derivatives
coincide. For more on this, check the remarks after Example 2.15 and the alternative formu-
lation of the Itō-formula in Theorem 2.16.

As mentioned above, the setting in this section follows the ideas by Cont and Fournié. More
precisely, we adjust the setting in [Cont, 2016] to measure-valued processes. Therefore, denote
by D([0, T ],MF (E)) the space of right continuous functions with left limits from [0, T ] to
MF (E) and equip the space with a metric d̃ given by

d̃(ω, ω′) = sup
s∈[0,T ]

dP (ω(s), ω′(s))

for ω, ω′ ∈ D([0, T ],MF (E)), where dP is the Prokhorov metric on MF (E). As in Section 1.2,
the stopped path ωt for ω ∈ D([0, T ],MF (E)) is given by ωt(s) = ω(t ∧ s). Further, define
for s ∈ [0, T ]

ωt−(s) =
{
ω(s), if s ∈ [0, t),
ω(t−), if s ∈ [t, T ].

The notion of stopped paths allows us to define an equivalence relation on the space [0, T ]×
D([0, T ],MF (E)) by

(t, ω) ∼ (t′, ω′) ⇔ t = t′ and ωt = ω′t′ ,

which gives rise to the quotient space

ΛT := {(t, ωt) : (t, ω) ∈ [0, T ]×D([0, T ],MF (E))} = [0, T ]×D([0, T ],MF (E))/ ∼ .

Next, define a metric d∞ on ΛT by

d∞((t, ω), (t′, ω′)) = d̃(ωt, ω′t′) + |t− t′| = sup
s∈[0,T ]

dP ((ω(t ∧ s), ω′(t′ ∧ s)) + |t− t′|.

Definition 2.11 (Continuity with respect to d∞). A functional F : ΛT → R is continuous
with respect to d∞ if for all (t, ω) ∈ ΛT and every ε > 0 there exists an η > 0 such that for
all (t′, ω′) ∈ ΛT with d∞((t, ω), (t′, ω′)) < η we have

|F (t, ω)− F (t′, ω′)| < ε.

Definition 2.12 (Non-anticipative). A measurable functional F on [0, T ]×D([0, T ],MF (E))
is non-anticipative if

F (t, ω) = F (t, ωt) for all ω ∈ D([0, T ],MF (E)),

which is the case if F : ΛT → R.



36 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

As the setting presented in Section 1.2.2 transfers nicely to real-valued functionals F on
[0, T ]×D([0, T ],MF (E)), we can define the two following types of derivatives.

Definition 2.13 (Functional derivatives). A continuous non-anticipative functional F :
ΛT → R is

(i) horizontally differentiable at (t, ω) ∈ ΛT if the limit

D∗F (t, ω) = lim
ε→0

F (t+ ε, ωt)− F (t, ωt)
ε

exists. If this is the case for all (t, ω) ∈ ΛT , we call D∗F the horizontal derivative of F .

(ii) vertically differentiable at (t, ω) ∈ ΛT in direction δx1[t,T ], x ∈ E, if the limit

DxF (t, ω) = lim
ε→0

F (t, ωt + εδx1[t,T ])− F (t, ωt)
ε

exists. If this is the case for all (t, ω) ∈ ΛT , we call DxF the vertical derivative of F in
direction δx1[t,T ]. Higher order vertical derivatives are defined iteratively.

Notation. As in Section 2.1, we set DxyF (t, ω) = DxDyF (t, ω), write D∗xF (t, ω) instead of
D∗DxF (t, ω) and so on. Additionally, we denote by DF the functional

DF : [0, T ]×D([0, T ],MF (E))× E 3 (t, ω, x) 7→ DxF (t, ω) ∈ R.

The definition of horizontal and vertical derivatives allows us to define the following set of
conditions on a functional F .

Condition 2. The functional F : ΛT → R satisfies

(i) F is bounded and continuous,

(ii) the horizontal derivative D∗F (t, ω) is continuous and bounded in (t, ω) ∈ ΛT ,

(iii) the vertical derivatives Dx1F (t, ω), Dx1x2F (t, ω), Dx1x2x3F (t, ω) and the mixed deriva-
tives D∗x1F (t, ω), D∗x1x2F (t, ω), D∗x1x2x3F (t, ω) are bounded and continuous in (t, ω) ∈
ΛT and x1, x2, x3 ∈ E,

(iv) for fixed (t, ω) ∈ ΛT , x1, x2 ∈ E, the maps x 7→ DxF (t, ω), x 7→ Dxx1F (t, ω) and
x 7→ Dxx1x2F (t, ω) are in the domain of A,

(v) A(x)Dx1F (t, ω), A(x)Dx1x2F (t, ω) and A(x)Dx1x2x3F (t, ω) are continuous in (t, ω) ∈ ΛT
and x1, x2, x3 ∈ E.

For functionals F satisfying the above condition, we can now formulate the functional Itō-
formula for B(A, c)-superprocesses.



2.2. FUNCTIONAL ITŌ-FORMULA 37

Theorem 2.14 (Functional Itō-formula). Let X be a B(A, c)-superprocess with good generator
A and assume F : ΛT → R satisfies Condition 2. Then, for all t ∈ [0, T ], it holds

F (t,Xt) = F (0, X0) +
∫ t

0
D∗F (s,Xs)ds

+
∫ t

0

∫
E
A(x)DxF (s,Xs)X(s)(dx)ds

+1
2

∫ t

0

∫
E
cDxxF (s,Xs)X(s)(dx)ds

+
∫ t

0

∫
E
DxF (s,Xs)MX(ds, dx).

Proof. As in the proof of Theorem 2.9, we can use the stopping time τK defined by (2.6) to
localize X such that X(τK ∧ t) ∈MK(E) for all t ∈ [0, T ]. However, to keep the notation sim-
ple, we assume, without loss of generality, that there exists a K > 0 such that X(t) ∈MK(E)
for all t ∈ [0, T ].

We start by defining a mesh {τnk : k = 1, . . . , k(n)} on [0, t] by

τn0 = 0, τnk = inf{s > τnk−1 : 2ns ∈ N} ∧ t

for all n ∈ N and use this mesh to define a stepwise approximation of the mapping s 7→ Xt(s)
by

Appn(Xt)(s) =
k(n)∑
i=1

X(τni+1)1[τn
i ,τ

n
i+1)(s) +X(t)1[t,T ](s).

Note that, while X itself has continuous path, Appn(Xt) is a piecewise constant approximation
of the path Xt which is right continuous with left limits. It holds

F (τni+1, App
n(Xt)τn

i+1−)− F (τni , Appn(Xt)τn
i −)

= F (τni+1, App
n(Xt)τn

i+1−)− F (τni , Appn(Xt)τn
i

)
+ F (τni , Appn(Xt)τn

i
)− F (τni , Appn(Xt)τn

i −).
(2.8)

To complete the proof, we proceed in two steps. In the first step, we consider the two differ-
ences on the right hand side of (2.8) and use the fundamental theorem of calculus as well as
the Itō-formula in Theorem 2.9 to rewrite the two terms. In the second step, we let n go to
infinity and consider the limits of the individual terms.

By setting hni = τni+1 − τni and ψ(s) = F (τni + s,Appn(Xt)τn
i

), we get that the first part of
(2.8) equals ψ(hn)− ψ(0) as

ψ(hni )− ψ(0) = F (τni + hni , App
n(Xt)τn

i
)− F (τni , Appn(Xt)τn

i
)



38 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

and, for all s ∈ [0, T ],

Appn(Xt)τn
i+1−(u) =

{
Appn(Xt)(u), if u ∈ [0, τni+1),
Appn(Xt)(τni+1−), if u ∈ [τni+1, T ],

=
{
Appn(Xt)(u), if u ∈ [0, τni+1),
Appn(Xt)(τni ), if u ∈ [τni+1, T ],

=
{
Appn(Xt)(u), if u ∈ [0, τni ),
Appn(Xt)(τni ), if u ∈ [τni , T ],

= Appn(Xt)τn
i

(u).

Thus, we have

F (τni+1, App
n(Xt)τn

i
)− F (τni , Appn(Xt)τn

i
)

=
∫ τn

i+1−τ
n
i

0
D∗F (τni + s,Appn(Xt)τn

i
)ds

=
∫ τn

i+1

τn
i

D∗F (s,Appn(Xt)τn
i

)ds

as ψ(hni )− ψ(0) =
∫ hn

i
0 ψ′(s)ds and

ψ′(u) = lim
ε→0

ψ(u+ ε)− ψ(u)
ε

= lim
ε→0

F (τni + u+ ε,Appn(Xt)τn
i

)− F (τni + u,Appn(Xt)τn
i

)
ε

= D∗F (τni + u,Appn(Xt)τn
i

).

By setting φ(µ) = φ̃(µ−X(τni )) with φ̃(µ) = F (τni , Appn(Xt)τn
i − + µ1[τn

i ,T ]), we get that the
second term on the right hand side of (2.8) is equal to φ(X(τni+1))− φ(X(τni )) as

φ(X(τni+1))− φ(X(τni ))
= F (τni , Appn(Xt)τn

i − + (X(τni+1)−X(τni ))1[τn
i ,T ])− F (τni , Appn(Xt)τn

i −)

and

Appn(Xt)τn
i − + (X(τni+1)−X(τni ))1[τn

i ,T ](u)

=
{
Appn(Xt)(u), if u ∈ [0, τni ),
Appn(Xt)(τni −) +X(τni+1)−X(τni ), if u ∈ [τni , T ],

=
{
Appn(Xt)(u), if u ∈ [0, τni ),
X(τni+1), if u ∈ [τni , T ],

= Appn(Xt)τn
i
.

We now want to apply Theorem 2.9 to φ. To do so, we have to check if φ satisfies Condition



2.2. FUNCTIONAL ITŌ-FORMULA 39

1. From

Dxφ(µ) = lim
ε→0

φ(µ+ εδx)− φ(µ)
ε

= lim
ε→0

1
ε

(
F (τni , Appn(Xt)τn

i − + (µ−X(τni ))1[τn
i ,T ] + εδx1[τn

i ,T ])

− F (τni , Appn(Xt)τn
i − + (µ−X(τni ))1[τn

i ,T ])
)

= DxF (τni , Appn(Xt)τn
i − + (µ−X(τni ))1[τn

i ,T ])

we get the existence of DxF (s, µ) as F satisfies Condition 2. Analogously, we get

Dx1x2φ(µ) = Dx1x2F (τni , Appn(Xt)τn
i − + (µ−X(τni ))1[τn

i ,T ])
Dx1x2x3φ(µ) = Dx1x2x3F (τni , Appn(Xt)τn

i − + (µ−X(τni ))1[τn
i ,T ])

and thus, as F satisfies Condition 2, we get the existence of the higher order derivatives. To
show that φ is continuous, consider

Appn(Xt)τn
i −(s) + (µ−X(τni ))1[τn

i ,T ](s)

=
{
Appn(Xt)(s), if s ∈ [0, τni ),
Appn(Xt)(τni −) + µ−X(τni ), if s ∈ [τni , T ],

=
{
Appn(Xt)τn

i −(s), if s ∈ [0, τni ),
µ, if s ∈ [τni , T ].

Now, as

d∞((τni , Appn(Xt)τn
i − + (µ−X(τni ))1[τn

i ,T ]),
(τni , Appn(Xt)τn

i − + (µm −X(τni ))1[τn
i ,T ]))

= sup
u∈[0,T ]

dP (Appn(Xt)τn
i −(u) + (µ−X(τni ))1[τn

i ,T ](u),

Appn(Xt)τn
i −(u) + (µm −X(τni ))1[τn

i ,T ](u))
= dP (µm, µ),

we get the continuity of φ with respect to µ from

{µm
m→∞−−−−→ µ}

⇒ {dP (µm, µ) m→∞−−−−→ 0}
⇒ {d∞((τni , Appn(Xt)τn

i − + (µ−X(τni ))1[τn
i ,T ]),

(τni , Appn(Xt)τn
i − + (µm −X(τni ))1[τn

i ,T ]))
m→∞−−−−→ 0}

⇒ {|F (τni , Appn(Xt)τn
i − + (µ−X(τni ))1[τn

i ,T ])

− F (τni , Appn(Xt)τn
i − + (µm −X(τni ))1[τn

i ,T ])|
m→∞−−−−→ 0},

where the last part follows from the continuity of F . Analogously, we obtain the continuity of
the derivatives of φ with respect to xi and µ as well as the remaining conditions in Condition
1 from the conditions on F .



40 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

Consequently, we can apply Theorem 2.9 to φ and obtain, as φ has no time argument,

φ(X(τni+1))− φ(X(τni )) =
∫ τn

i+1

τn
i

∫
E
A(x)Dxφ(X(s))X(s)(dx)ds

+ 1
2

∫ τn
i+1

τn
i

∫
E
cDxxφ(X(s))X(s)(dx)ds

+
∫ τn

i+1

τn
i

∫
E
Dxφ(X(s))MX(ds, dx).

Plugging in the definition of φ, we end up with

F (τni , Appn(Xt)τn
i

)− F (τni , Appn(Xt)τn
i −)

=
∫ τn

i+1

τn
i

∫
E
A(x)DxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])X(s)(dx)ds

+ 1
2

∫ τn
i+1

τn
i

∫
E
cDxxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])X(s)(dx)ds

+
∫ τn

i+1

τn
i

∫
E
DxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])M(ds, dx).

Combining this with the result for the first part of the sum in (2.8) yields the following
expression for the left hand side in (2.8):

F (τni+1, App
n(Xt)τn

i+1−)− F (τni , Appn(Xt)τn
i −)

=
∫ τn

i+1

τn
i

D∗F (s,Appn(Xt)τn
i

)ds

+
∫ τn

i+1

τn
i

∫
E
A(x)DxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])X(s)(dx)ds

+ 1
2

∫ τn
i+1

τn
i

∫
E
cDxxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])X(s)(dx)ds

+
∫ τn

i+1

τn
i

∫
E
DxF (τni , Appn(Xt)τn

i − + (X(s)−X(τni ))1[τn
i ,T ])M(ds, dx).

Define the index in(s) such that s ∈ [τnin(s), τ
n
in(s)+1). Then, summation of the above terms

over i yields

F (t, Appn(Xt)t−)− F (0, X0)

=
∫ t

0
D∗F (s,Appn(Xt)τn

in(s)
)ds

+
∫ t

0

∫
E
A(x)DxF (τnin(s), App

n(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])X(s)(dx)ds

+ 1
2

∫ t

0

∫
E
cDxxF (τnin(s), App

n(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])X(s)(dx)ds

+
∫ t

0

∫
E
DxF (τnin(s), App

n(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])M(ds, dx).
(2.9)



2.2. FUNCTIONAL ITŌ-FORMULA 41

Note that while the function

(ω, s, x) 7→ DxF (τnin(s), App
n(Xt)τn

in(s)− + (X(s)−X(τnin(s)))1[τn
in(s),T ])(ω)

is not predictable, it is in IMX
due to the results in Section 1.3.3. Hence, the integral with

respect to the martingale measure MX in (2.9) is well-defined. This completes the first of the
two steps.

For the convergence of the terms on the left hand side of (2.9) consider

d∞((s,Xs), (τnin(s), App
n(Xt)τn

in(s)
))

= |s− τnin(s)|+ sup
u∈[0,T ]

dP (X(s ∧ u), Appn(Xt)τn
in(s)

(u))

≤ 1
2n + sup

0≤i≤k(n)
sup

u∈[τn
i ,τ

n
i+1)

dP (X(s ∧ u), X(τnin(s) ∧ τ
n
i+1)),

which goes to zero due to the continuity of the paths of X. In addition, the continuity of the
paths of X and the continuity of F yield

lim
n→∞

F (t, Appn(Xt)t−) = F (t,Xt−) = F (t,Xt).

From the continuity assumptions on D∗F and

d∞((s,Xs), (s,Appn(Xt)τn
in(s)

))→ 0,

we get
lim
n→∞

D∗F (s,Appn(Xt)τn
in(s)

) = D∗F (s,Xs).

In combination with the boundedness assumption on D∗F this allows us to apply the domi-
nated convergence theorem to get the convergence of the first term on the right hand side of
(2.9), namely

lim
n→∞

∫ t

0
D∗F (s,Appn(Xt)τn

in(s)
)ds =

∫ t

0
D∗F (s,Xs)ds.

To prove the convergence of the second term on the right hand side, consider

d∞((s,Xs)(τnin(s), App
n(Xt)τn

in(s)− + (X(s)−X(τnin(s)))1[τn
in(s),T ]))

= |s− τnin(s)|+ sup
u∈[0,T ]

dP (Xs(u), Appn(Xt)τn
in(s)−(u) + (X(s)−X(τnin(s)))1[τn

in(s),T ](u)).

(2.10)

By the triangle inequality

sup
u∈[0,T ]

dP (Xs(u), Appn(Xt)τn
in(s)−(u) + (X(s)−X(τnin(s)))1[τn

in(s),T ](u))

≤ sup
u∈[0,T ]

dP (Xs(u), Appn(Xt)τn
in(s)−(u))

+ sup
u∈[0,T ]

dP (Appn(Xt)τn
in(s)−(u), Appn(Xt)τn

in(s)−(u) + (X(s)−X(τnin(s)))1[τn
in(s),T ](u))



42 CHAPTER 2. FUNCTIONAL ITŌ-CALCULUS FOR SUPERPROCESSES

holds. Further, it holds

sup
u∈[0,T ]

dP (Appn(Xt)τn
in(s)−(u), Appn(Xt)τn

in(s)−(u) + (X(s)−X(τnin(s)))1[τn
in(s),T ](u))

= dP (X(τnin(s)), X(τnin(s)) +X(s)−X(τnin(s)))
= dP (X(τnin(s)), X(s)),

which goes to zero as n goes to infinity. In combination with

sup
u∈[0,T ]

dP (Xs(u), Appn(Xt)τn
in(s)−(u))→ 0 and |s− τnin(s)| → 0 as n→∞,

this implies that (2.10) goes to zero as n goes to inifnity. The continuity assumption on ADF
then yields

lim
n→∞

A(x)DxF (τnin(s), App
n(Xt)τn

in(s)− + (X(s)−X(τnin(s)))1[τn
in(s),T ]) = A(x)DxF (s,Xs).

To get the convergence of the integrals, set

αn(s) = τnin(s) and βn(s) = Appn(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])

and assume the bound of ADF is given by 0 < B <∞. Then∫
E
|A(x)DxF (αn(s), βn(s))|X(s)(dx) ≤

∫
E
BX(s)(dx) ≤ BK <∞

for all n and thus we can apply the dominated convergence theorem to obtain

lim
n→∞

∫
E
A(x)DxF (αn(s), βn(s))X(s)(dx) =

∫
E
A(x)DxF (s,Xs)X(s)(dx)

for all s ∈ [0, T ]. Combining this with the fact that

|
∫
E
A(x)DxF (αn(s), βn(s))X(s)(dx)| ≤

∫
E
|A(x)DxF (αn(s), βn(s))|X(s)(dx) ≤ BK <∞

holds for all n allows us to apply the dominated convergence theorem once again to end up
with

lim
n→∞

∫ t

0

∫
E
A(x)DxF (αn(s), βn(s))X(s)(dx)ds

= lim
n→∞

∫ t

0

∫
E
A(x)DxF (τnin(s), App

n(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])X(s)(dx)ds

=
∫ t

0

∫
E
A(x)DxF (s,Xs)X(s)(dx)ds

and thus get the convergence of the second term on the right hand side of (2.9).

By using the same arguments, we get

lim
n→∞

∫ t

0

∫
E
DxxF (τnin(s), App

n(Xt)τn
in(s)− + (X(s)−X(τnin(s)))1[τn

in(s),T ])X(s)(dx)ds

=
∫ t

0

∫
E
DxxF (s,Xs)X(s)(dx)ds,



2.2. FUNCTIONAL ITŌ-FORMULA 43

i.e. the convergence of the third term on the right hand side of (2.9).

For the convergence of the last term, the integral with respect to the martingale measure
MX , assume (ωn)n ⊂ D([0, T ],MF (E)) with ωn → ω as n goes to infinity. Th