#  Functional Expansions

Bruno Dupire  Valentin Tissot-Daguette

###### Abstract

Path dependence is omnipresent in many disciplines such as engineering, system
theory and finance. It reflects the influence of the past on the future, often
expressed through functionals. However, non-Markovian problems are often
infinite-dimensional, thus challenging from a conceptual and computational
perspective. In this work, we shed light on expansions of functionals. First,
we treat static expansions made around paths of fixed length and propose a
generalization of the Wiener series$-$the intrinsic value expansion (IVE). In
the dynamic case, we revisit the functional Taylor expansion (FTE). The latter
connects the functional Itô calculus with the signature to quantify the effect
in a functional when a “perturbation” path is concatenated with the source
path. In particular, the FTE elegantly separates the functional from future
trajectories. The notions of real analyticity and radius of convergence are
also extended to the path space. We discuss other dynamic expansions arising
from Hilbert projections and the Wiener chaos, and finally show financial
applications of the FTE to the pricing and hedging of exotic contingent
claims.

Keywords — Functional Itô Calculus, Taylor Expansion, Path Signature, Wiener
Chaos, Contingent Claims

MSC (2020) Classification — 41A58, 91G20, 26E15, 60L10

###### Contents

  1. 1 Introduction
  2. 2 Static Expansions
    1. 2.1 $T-$functionals and Embeddings
    2. 2.2 Volterra Series
    3. 2.3 Wiener Series
    4. 2.4 Intrinsic Value Expansion
    5. 2.5 Wiener Chaos
    6. 2.6 Summary
  3. 3 Dynamic Expansions
    1. 3.1 Functional Taylor Expansion
    2. 3.2 Other Dynamic Expansions
  4. 4 Applications
    1. 4.1 Pricing of Path-dependent Claims
    2. 4.2 Static Hedging
  5. 5 Conclusion
  6. A Proofs
    1. A.1 Theorem 3.2
    2. A.2 Proposition 3.5
    3. A.3 Proposition 3.8
    4. A.4 Theorem 3.9
    5. A.5 Theorem 3.10
    6. A.6 Corollary 3.11
    7. A.7 Proposition 3.14
    8. A.8 Proposition 3.20
    9. A.9 Lemma 3.22

##  1 Introduction

Traditional problems in many disciplines rely on the strong hypothesis that
the future solely depends on the present. This is the Markov property, which
tremendously simplifies the study of causal relationships in physics, biology,
or social science. In finance, it allows to express the price and hedge of
vanilla options in terms of the spot value of the underlying. This is often no
longer the case when the payoff becomes path-dependent (also called exotic) or
the stock dynamics is more complex. The lack of Markovian representation can
have serious consequences from a computational perspective as the problem is
usually infinite-dimensional.

In some cases, a Markovian framework can be recovered by enlarging the state
process accordingly. As an illustration, consider an at-the-money lookback
option, i.e. $g(X_{T})=(\max_{0\leq u\leq T}x_{u}-x_{0})^{+}$ with underlying
$X$ and maturity $T$. If $X$ is Markov and
$Y\mathrel{\mathop{\ordinarycolon}}=(\max_{0\leq u\leq t}x_{u})_{t\in[0,T]}$,
then $(X,Y)$ is also Markov under mild assumptions. In turn, the price of the
lookback option at $t\in[0,T]$ will be function of $(t,x_{t},y_{t})$ only. The
same remedy may be applied when the dynamics of the stock depends on past
information, e.g. path-dependent volatility [24]. Taking the Hobson-Rogers
model [26], the underlying becomes Markov when the offset
processes$-$capturing historical trends of the stock$-$are added to the state.

Nevertheless, there are situations where path dependence cannot be absorbed by
finitely many features. We characterize such frameworks as fully non-
Markovian. Among others, we can mention American options whose reward depends
on a moving average of the underlying [6]. Also, the pricing problem in rough
volatility models belong to this category, even for the simplest instrument.
Either way, the option price becomes a functional of the path so far, calling
for the development of new mathematical tools.

One avenue consists of employing series expansions to project exotic payoffs
onto a finite-dimensional subspace arising, e.g., from orthogonal polynomials
[6], the Wiener chaos [30, 37], or the Karhuhnen-Loève expansion [47]. Series
expansions can also be applied to approximate stock processes with fully non-
Markovian dynamics; see, e.g., [11]. Similar in spirit, the path signature
showed promising results in option pricing and financial modeling problems
entailing path dependence [3, 14, 35]. These works motivate the present paper
as the use of the signature is justified by a deep yet overlooked result: the
functional Taylor expansion (FTE). First proposed by Fliess [18, 19, 20], it
is a generalization of the Taylor expansion where classical derivatives and
monomials are respectively replaced by functional derivatives and the
signature. Further historical notes and details will be given at the beginning
of Section 3.

Outline. We devote this study to the expansions of functionals by retracing
historical works and sharing new findings. In Section 2, we collect static
expansions in the sense that the path length remains unchanged. Starting in
the early 1900s, Volterra [48] expressed a functional defined on a fixed
horizon $[0,T]-$called $T-$functional and denoted by $g$ in this work$-$as
combination of homogeneous polynomials in integral form: If $X,Y$ are paths of
length $T$, then one can write

$g(X+Y)=\sum_{k<K}\int_{0}^{T}\int_{0}^{t_{k}}\cdots\int_{0}^{t_{2}}\psi_{k}(t_{1},...,t_{k})y_{t_{1}}\cdots
y_{t_{k}}dt_{1}\cdots dt_{k}+R_{K}(X,Y).$ (1.1)

The kernels $(\psi_{k})$ consists of higher-order Fréchet derivatives of $g$
evaluated at $X$; see [48, 41, 7] and Section 2.2. As can be seen, Volterra’s
approach relies on pointwise values of the added path $Y$. In contrast, the
Wiener series and chaos expansion [49, 50] encode a path in terms of its
infinitesimal increments. This gives, formally, the expansion

$g(X+Y)=\sum_{k<K}\int_{0}^{T}\int_{0}^{t_{k}}\cdots\int_{0}^{t_{2}}\phi_{k}(t_{1},...,t_{k})dy_{t_{1}}\cdots
dy_{t_{k}}+R_{K}(X,Y).$ (1.2)

In the chaos expansion, $X,Y$ are typical Brownian paths and as shown in
Stroock [45], the kernels $(\phi_{k})$ are expectations of iterated Malliavin
derivatives of $g$. It turns out that a pathwise expansion can be obtained by
computing the Malliavin derivatives of the stopped path and employing
Stratonovich integrals instead of Itô ones. This leads to our first
contribution: the intrinsic value expansion, set forth in Theorem 2.4. To the
best of our knowledge, neither the intrinsic value expansion nor a variation
of it has been discussed in the literature.

Alternatively, one can study the local behavior of a ”running”
functional$-$denoted by $f-$when extending the source path. This is the core
of Section 3, concerned with dynamic expansions. Let us give some insight. In
classical calculus, say on the real line, the Taylor expansion reads,

$f(x+y)=\sum_{k<K}f^{(k)}(x)\frac{y^{k}}{k!}+R_{K}(x,y),\quad
x,y\in\operatorname{\mathbb{R}},\quad
f\in\operatorname{\mathcal{C}}^{K}(\operatorname{\mathbb{R}}).$ (1.3)

The shock $(y)$ is fragmented into its scaled powers $(\frac{y^{k}}{k!})$,
weighted by the sensitivities of the function at the initial point $(x)$. In
the path space, a similar decomposition can be formulated: If $X,Y$ are paths
of arbitrary lengths, then

$f(X\oplus
Y)=\sum_{|\alpha|<K}\Delta_{\alpha}f(X)\operatorname{\mathcal{S}}_{\alpha}(Y)+R_{K}(X,Y),$
(1.4)

where $\oplus$ concatenates $X$ and $Y$. This is the functional Taylor
expansion (FTE)$-$our main object of interest in this work$-$stated in Theorem
3.10. Let us describe the right side of $\eqref{eq:FTE0}$ while keeping the
level of technicality to a minimum in this introduction. For that reason, we
also postpone the literature review to the beginning of Section 3.1. The first
summands $(\Delta_{\alpha}f(X))$ are higher order functional derivatives of
$f$, while $(\operatorname{\mathcal{S}}_{\alpha}(Y))$ are iterated integrals
of $Y$, also called signature functionals. The indexes
$\\{\alpha=\alpha_{1}\cdots\alpha_{k}\ \mathrel{\mathop{\ordinarycolon}}\
k<K\\}$ concurrently specify the order of differentiation for
$\Delta_{\alpha}f$ (with respect to the time or space variable) and
integration for $\operatorname{\mathcal{S}}_{\alpha}$. Unlike the Volterra and
Wiener expansion where the functional and perturbation path $Y$ are quite
entangled (see (1.1), (1.2)), the FTE provides a perfect separation between
them.

Fig. 1(c) compares the types of expansions discussed above. As can be seen,
static (respectively classical) expansions are made around a path (resp. a
point), whereas dynamic expansions describe the evolution of a functional
after the initial path.

Figure 1: Classification of expansions.

(a) Classical (around a point)

(b) Static (around a path)

(c) Dynamic (after a path)

The paper is organized as follows. In Section 2, we examine static expansions
and derive the novel intrinsic value expansion (IVE). Section 3.1 puts a
spotlight on the functional Itô calculus and path signature, culminating with
the functional Taylor expansion (FTE). We also derive remainder estimates of
the FTE and characterize the corresponding radius of convergence. Connections
are finally established between the FTE and static expansions. In Section 3.2,
we discuss other dynamic expansions arising from Hilbert projections and the
Wiener chaos. We present the pricing and dynamic hedging of exotic options in
Section 4 as possible applications of the FTE. Section 5 concludes the study
and Appendix A contains postponed proofs of the main results.

Notations and Definitions. We employ the framework from the functional Itô
calculus [16]. Fix throughout a horizon $T>0$, which can be interpreted as the
maturity of a financial derivative. Let
$\Lambda_{t}=\operatorname{\mathcal{D}}([0,t],\operatorname{\mathbb{R}})$ be
the Skorokhod space of càdlàg paths of length $t\in[0,T]$. Given
$X_{t}\in\Lambda_{t}$, $X_{s}$ denotes the whole trajectory up to time $s\leq
t$, while $x_{s}=X_{t}(s)$ is the value at time $s$. Moreover, write
$\Lambda\mathrel{\mathop{\ordinarycolon}}=\bigcup_{t\in[0,T]}\Lambda_{t}$ for
the collection of all càdlàg paths.

######  Definition 1.1.

A functional is a map
$f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$, while
a $T-$functional refers to any map
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$.

The distance between elements of $\Lambda$ is measured according to

$d_{\Lambda}(X_{t},Y_{s})=t-s+\sup_{0\leq u\leq t}|x_{u}-y_{u\wedge s}|,\quad
0\leq s\leq t\leq T.$ (1.5)

We say that a functional is $\Lambda-$continuous if it is continuous with
respect to $d_{\Lambda}$. The $\Lambda_{T}-$continuity of $T-$functionals is
defined similarly, where we remark that $d_{\Lambda}$ coincides with the
uniform topology. We now collect important notions:

  1. (i)

We recall the spatial and temporal functional derivatives introduced in [16],
namely

$\displaystyle\Delta_{x}f(X_{t})$ $\displaystyle=\lim_{h\to
0}\,\frac{f(X^{h}_{t})-f(X_{t})}{h},\ $ $\displaystyle
X^{h}_{t}(s)\,\,=x_{s}+h\,\mathds{1}_{\\{s\,=\,t\\}},\quad s\leq t,$
$\displaystyle\Delta_{t}f(X_{t})$ $\displaystyle=\lim_{\delta t\downarrow
0}\,\frac{f(X_{t,\delta t})-f(X_{t})}{\delta t},\ $ $\displaystyle X_{t,\delta
t}(s)=x_{s\wedge t},\hskip 29.02193pts\leq t+\delta t,$

whenever these quantities exist. We note that $\Delta_{x}$ and $\Delta_{t}$
need not commute; see Example 5.

  2. (ii)

Let $\oplus\mathrel{\mathop{\ordinarycolon}}\Lambda^{2}\to\Lambda$ be the
noncommutative binary operation that concatenates paths in a continuous
fashion. In other words, if $X\in\Lambda_{s}$, $Z\in\Lambda_{u}$ then
$Y=X\oplus Z\in\Lambda_{s+u}$ is given by

$y_{r}=x_{r\wedge s}+z_{(r-s)^{+}}-z_{0},\quad r\in[0,s+u].$

If $u>T-s$, we define instead $X\oplus Z=X\oplus Z_{T-s}\in\Lambda_{T}$.

  3. (iii)

If $\operatorname{\mathbb{Q}}$ is a measure on $\Lambda$ and
$f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$, we
write $\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[f(Y_{t})\ |\
X_{s}]$ for the conditioned expectation of $f(Y_{t})$ given that $Y_{t}$
coincides with $X_{s}$ up to $s<t$. In other words,

$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[f(Y_{t})\,|\,X_{s}]=\int_{\Lambda_{t-s}}f(\underbrace{X_{s}\oplus
Z}_{=\ Y_{t}})\operatorname{\mathbb{Q}}(dZ),\quad s\leq t.$ (1.6)

To avoid ambiguity, we favor the letters $Y,\ Z$ over $X$ to represent random
or future paths.

##  2 Static Expansions

Functional expansions are traditionally made for paths of fixed length.
Without loss of generality, we restrict ourselves to paths in $\Lambda_{T}$.
Contrary to $\Lambda$, the restriction $\Lambda_{T}$ has the advantage of
being a vector space: we can add paths, have a clear sense of directions, etc.
For simplicity, we often express static expansions around the null path of
length $T$.

###  2.1 $T-$functionals and Embeddings

It proves helpful to make the distinction between functionals defined on
$\Lambda$ and $T-$functionals (those restricted to $\Lambda_{T}$). If
$X\in\Lambda_{T}$ is a stock price trajectory, then
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$
can be viewed as the payoff of a claim with maturity $T$. Depending on the
term sheet of the claim, it is a priori unclear how to quantify the value of
the payoff having observed the price path only up to some intermediate time
$t<T$ only. We here address this question by embedding $T-$functionals into
the space of ”running” functionals.

######  Definition 2.1.

Let $\operatorname{\mathfrak{F}}$ be a set of functionals and
$\operatorname{\mathfrak{F}}_{T}$ is the restriction of
$\operatorname{\mathfrak{F}}$ to $\Lambda_{T}$. Then an embedding is an
operator
$\iota\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathfrak{F}}_{T}\hookrightarrow\operatorname{\mathfrak{F}}$
such that $f=\iota g\in\operatorname{\mathfrak{F}}$ satisfies
$f|_{\Lambda_{T}}=g$ for all $g\in\operatorname{\mathfrak{F}}_{T}$.

Notice that many embeddings may exist as the only constraint is that the
embedded functional coincides with a precribed $T-$functional on
$\Lambda_{T}$. We here focus our attention on the following class of
embeddings which resonates with many meanings in finance. Let
$\operatorname{\mathbb{Q}}$ be a probability measure on $\Lambda$ and define

$\iota_{\operatorname{\mathbb{Q}}}g(X_{t})\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})\,|\,X_{t}],$
(2.1)

with the conditioned expectation
$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\ \cdot\ |\ \cdot\ ]$
given in (1.6). In Definition 2.1, we may choose
$\operatorname{\mathfrak{F}}_{T}$, $\operatorname{\mathfrak{F}}$ to be the set
of uniformly bounded $(T-)$functionals so that clearly
$\iota_{\operatorname{\mathbb{Q}}}g\in\operatorname{\mathfrak{F}}$ and is
well-defined. Clearly, $\iota_{\operatorname{\mathbb{Q}}}g(X_{T})=g(X_{T})$ so
that $\iota_{\operatorname{\mathbb{Q}}}$ is indeed an embedding for every
$\operatorname{\mathbb{Q}}$. We stress that $X_{t}$ need not be in the support
of $\operatorname{\mathbb{Q}}$ as the latter is only introduced to extrapolate
the future. In financial terms and provided that $\operatorname{\mathbb{Q}}$
is a risk-neutral measure, $\iota_{\operatorname{\mathbb{Q}}}g$ is the price
functional of the claim $g$ under the ”model” $\operatorname{\mathbb{Q}}$.
Taking for instance the measure $\operatorname{\mathbb{Q}}_{\sigma}$ such that
the scaled canonical process $Y/\sigma$ is Brownian motion under
$\operatorname{\mathbb{Q}}_{\sigma}$, $\sigma>0$, then
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g$ gives the Bachelier price of $g$
with zero interest rates and volatility $\sigma$. An important case arises
when $\sigma\downarrow 0$, leading to the intrinsic functional,

$\iota_{0}g(X_{t})\mathrel{\mathop{\ordinarycolon}}=\iota_{\operatorname{\mathbb{Q}}_{0}}g(X_{t})=g(X_{t,T-t}).$
(2.2)

The intrinsic functional is central in Section 2.4 where we introduce the
intrinsic value expansion.

###  2.2 Volterra Series

Vito Volterra (1860-1940) was arguably the first mathematician to introduce
and study functional expansions. His findings have had tremendous influence in
nonlinear systems, where the Volterra expansion allows to express the solution
functional linking the input path to the output (see, e.g., [48, 41, 7]). His
idea was to approximate a Fréchet differentiable $T-$functional
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$ by
a multivariate function taking pointwise values of the input path.

For simplicity, suppose that $x_{0}=0$ and consider the regular partitions
$\Pi^{N}=\\{s_{n}=\frac{nT}{N},\ n=0,\ldots,N\\}$, $N\geq 1$. Then $g(X_{T})$
is estimated by $g^{N}(x_{s_{1}},...,x_{s_{N}})=g(X_{T}^{N})$ with the
piecewise constant path
$X_{T}^{N}=\sum_{n=0}^{N-1}x_{s_{n}}\mathds{1}_{[s_{n},s_{n+1})}+x_{T}\mathds{1}_{\\{T\\}}.$
If $g^{N}$ is real analytic and
$\bm{0}=(0,...,0)\in\operatorname{\mathbb{R}}^{N}$, a Taylor expansion of
$h\mapsto g^{N}(h\ x_{s_{1}},...,h\ x_{s_{N}})$ around $h=0$ gives

$\displaystyle g^{N}(x_{s_{1}},\ldots,x_{s_{N}})$
$\displaystyle=g^{N}(\bm{0})+\sum_{n=1}^{N}\partial_{n}g^{N}(\bm{0})x_{s_{n}}+\sum_{1\leq
n\leq m\leq N}\partial_{nm}g^{N}(\bm{0})\,x_{s_{n}}x_{s_{m}}+\ldots$ (2.3)

Put differently, $\eqref{eq:Volterra1}$ is an expansion of $g(X_{T}^{N})$
around the flat path $X_{0,T}$. Now consider the Fréchet derivatives of $g$ in
the direction of the Dirac mass $\delta(\cdot-t)$, $t\in[0,T]$, i.e. the
family $(F_{t})_{t\in[0,T]}$ such that

$D_{Y_{T}}g(X_{T})\mathrel{\mathop{\ordinarycolon}}=\lim_{h\to
0}\frac{g(X_{T}+hY_{T})-g(X_{T})}{h}=\int_{0}^{T}F_{t}g(X_{T})y_{t}dt,\quad
Y_{T}\in\Lambda_{T},$

when the above is well-defined. From the definition of $g^{N}$, choose
$Y_{T}=\mathds{1}_{[s_{n},s_{n+1})}$ to obtain

$\partial_{n}g^{N}(\bm{0})=D_{\mathds{1}_{[s_{n},s_{n+1})}}g(X_{0,T})=\int_{s_{n}}^{s_{n+1}}F_{t}g(X_{T})dt.$
(2.4)

Similarly, defining the iterated Fréchet derivatives $F_{t_{1}\cdots
t_{k}}=F_{t_{1}}\cdots F_{t_{k}}$ yields

$\partial_{n_{1}\cdots
n_{k}}g^{N}(\bm{0})=\int_{s_{n_{k}}}^{s_{n_{k}+1}}\cdots\int_{s_{n_{1}}}^{s_{{n_{1}}+1}}F_{t_{1}\cdots
t_{k}}g(X_{0,T})dt_{1}\cdots dt_{k}.$

Owing to $\eqref{eq:psi1}$ and the fact that $X^{N}$ equals $x_{s_{n}}$ on
$[s_{n},s_{n+1})$ for all $n\leq N$, we can rewrite $\eqref{eq:Volterra1}$ as

$\displaystyle g(X^{N}_{T})$
$\displaystyle=g(X_{0,T})+\int_{0}^{T}F_{t}g(X_{0,T})x_{t}^{N}dt+\int_{0}^{T}\int_{0}^{t_{2}}F_{t_{1}t_{2}}g(X_{0,T})x_{t_{1}}^{N}x^{N}_{t_{2}}dt_{1}dt_{2}+\ldots$

Letting $N\to\infty$ gives, at least formally, the Volterra series,

$g(X_{T})=g(X_{0,T})+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}F_{t_{1}\cdots
t_{k}}g(X_{0,T})\,x_{t_{1}}\cdots x_{t_{k}}\,dt^{\otimes k},$ (2.5)

with the notation $dt^{\otimes k}=dt_{1}\cdots dt_{k}$ and the simplexes,

$\triangle_{k,t}=\\{(t_{1},\ldots,t_{k})\in[0,t]^{k}\
\mathrel{\mathop{\ordinarycolon}}\ t_{1}\leq\cdots\leq t_{k}\\},\quad k\geq
1,\quad t\in[0,T].$ (2.6)

If $\psi_{0}=g(X_{0,T})$ and $\psi_{k}(t_{1},\ldots,t_{k})=F_{t_{1}\cdots
t_{k}}g(X_{0,T})$ denotes the Volterra kernels, then $\eqref{eq:Volterra}$ can
be written more compactly as
$g(X_{T})=\psi_{0}+\sum_{k=1}^{\infty}V_{k}\psi_{k}(X_{T}),$ with the
operators

$V_{k}\psi_{k}(X_{t})=\int_{\triangle_{k,t}}\psi_{k}\,x^{\otimes k}dt^{\otimes
k},\quad X_{t}\in\Lambda_{t}.$

Note that $V_{k}\psi_{k}$ is a homogeneous functional of degree $k$ in the
sense that $V_{k}\psi_{k}(\gamma X_{t})=\gamma^{k}V_{k}\psi_{k}(X_{t})$ with
$\gamma X_{t}\mathrel{\mathop{\ordinarycolon}}=(\gamma x_{s})_{s\in[0,t]}$,
$\gamma\neq 0$. For instance, $V_{1}\psi_{1}$ is a linear functional,
$V_{2}\psi_{2}$ is homogeneous quadratic, and so on. By construction, we see
that the kernels can be retrieved from Dirac impulses of $g(X_{T})$ at
specific dates. Fig. 2(a) shows the approximation of $\psi_{1}$ as in
$\eqref{eq:psi1}$ obtained by bumping a single value of the discretized path.

######  Example 1.

Let
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$
with Volterra expansion
$g(X_{T})=\psi_{0}+\sum_{k=1}^{\infty}V_{k}\psi_{k}(X_{T})$. Consider the
embedding (see Section 2.1) given by
$f(X_{t})=\psi_{0}+\sum_{k=1}^{\infty}V_{k}\psi_{k}(X_{t})$. In other words,
we truncate the simplexes of each homogeneous functional $(V_{k}\psi_{k})$. It
turns out that $f$ coincides with the
$f^{0}(X_{t})\mathrel{\mathop{\ordinarycolon}}=g(X^{0}_{t,T-t})$ where
$X^{0}_{t,T-t}\in\Lambda_{T}$ consists of the path $X_{t}$ glued with the null
path of length $T-t$; see Fig. 4(a). Indeed, observe that
$V_{k}\psi_{k}(X^{0}_{t,T-t})$ and $V_{k}\psi_{k}(X_{t})$ are equal as the
integrand of the former is zero as soon as $t_{k}>t$. We can thus regard
$f^{0}$ as the natural embedding of $g$ associated to the Volterra series.

Figure 2: Path deformation in Volterra and Wiener series.

(a) Volterra (Fréchet derivative)

(b) Wiener (Malliavin derivative)

###  2.3 Wiener Series

As seen above, the Volterra series expresses $T-$functionals$-$seen as the
output of a system$-$ in terms of pointwise values of the source path. In
contrast, the Wiener series and homogeneous chaos expansion [50,
49]$-$corresponding to the deterministic and stochastic case,
respectively$-$encode the path with its infinitesimal increments. We now
present a seemingly unconventional derivation of the Wiener series by
borrowing, deliberately, anachronistic elements from the Malliavin calculus.

Let $X\in\Lambda_{T}$ be a continuous path of finite variation. Assume again
that $x_{0}=0$, the knowledge of $(x_{s_{1}},...,x_{s_{N}})$ is equivalent to
the knowledge of the increments $\delta x_{s_{n}}=x_{s_{n}}-x_{s_{n-1}}$,
$n=1,\ldots,N$. We can therefore write $g^{N}(\delta x_{s_{1}},...,\delta
x_{s_{N}})\mathrel{\mathop{\ordinarycolon}}=g(X_{T}^{N})$ with $X_{T}^{N}$ as
in Section 2.2. Similarly, write the Taylor expansion of $h\mapsto g^{N}(h\
\delta x_{s_{1}},...,h\ \delta x_{s_{N}})$ at $h=0$. That is,

$\displaystyle g^{N}(x_{s_{1}},\ldots,x_{s_{N}})$
$\displaystyle=\phi^{N}_{0}+\sum_{n=1}^{N}\phi^{N}_{1}(s_{n})\delta
x_{s_{n}}+\sum_{1\leq n\leq m\leq N}\phi^{N}_{2}(s_{n},s_{m})\,\delta
x_{s_{n}}\delta x_{s_{m}}+\ldots$

with $\phi^{N}_{0}=g^{N}(\bm{0})\ (=g(X_{0,T})\ \forall N)$ and
$\phi^{N}_{k}(t_{1},\ldots,t_{k})=\partial_{n_{1}...n_{k}}g^{N}(\bm{0})$
whenever $t_{l}\in(s_{n_{l-1}},s_{n_{l}}]$, $l=1,\ldots,k$. Defining
$\phi_{0}=g(X_{0,T})$ and provided that
$\phi_{k}\mathrel{\mathop{\ordinarycolon}}=\lim_{N\to\infty}\phi_{k}^{N}$
exist for all $k\geq 1$, a passage to the limit yields, heuristically, the
Wiener series,

$\displaystyle
g(X_{T})=\phi_{0}+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}\phi_{k}\,dx^{\otimes
k}=\phi_{0}+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}\phi_{k}(t_{1},\ldots,t_{k})\,dx_{t_{1}}\cdots
dx_{t_{k}}.$ (2.7)

Let us express the kernels explicitly. First, note that
$\partial_{n}g^{N}(\delta x_{s_{1}},...,\delta
x_{s_{N}})=D_{s_{n}}g(X_{T}^{N})$, where $D_{\cdot}$ is the Malliavin
derivative

$D_{t}g(X_{T})\mathrel{\mathop{\ordinarycolon}}=D_{\mathds{1}_{[t,T]}}g(X_{T})=\lim_{h\downarrow
0}\frac{g(X_{T}+h\mathds{1}_{[t,T]})-g(X_{T})}{h}.$ (2.8)

Indeed, bumping an increment of $X^{N}_{T}$ induces a parallel shift in the
path following the shock; see Fig. 2(b). Letting $N\to\infty$ and evaluating
the result at the flat path $X_{0,T}$ leads to
$\phi_{1}(t_{1})=D_{t_{1}}g(X_{0,T})$. Similarly, writing $D_{t_{1}\cdots
t_{k}}=D_{t_{1}}\cdots D_{t_{k}}$ gives $\partial_{n_{1}...n_{k}}g^{N}(\delta
x_{s_{1}},...,\delta x_{s_{N}})=D_{s_{n_{1}}\cdots s_{n_{k}}}g(X_{T}^{N})$. In
conclusion, the kernels in the Wiener series are iterated Malliavin
derivatives, namely $\phi_{k}(t_{1},...,t_{k})=D_{t_{1}\cdots
t_{k}}g(X_{0,T})$.

######  Remark 2.2.

The Volterra and Wiener series share many similarities. For a comprehensive
comparison between the Volterra and Wiener chaos expansion, see Palm and
Poggio [41]. In particular, the authors show how $\psi_{k},\ \phi_{k}$ relate
to one another; when $\phi_{k}$ is smooth, then
$\psi_{k}=(-1)^{k}\partial_{t_{1}...t_{k}}\phi_{k}$ using integration by
parts. This can be seen from Fig. 2(b) in the case $k=1$. Alternatively,
observe that the Malliavin derivative is given by a Dirac impulse of the
”white noise” process $\dot{X}=\frac{d}{dt}X$. Consequently, the Volterra
kernels of $\dot{X}$ are precisely the Wiener kernels of $X$.

So far, the paths are of finite variation so that the iterated integrals in
$\eqref{eq:chaosStieltjes}$ are in the Riemann-Stieltjes sense. Before
proceeding with the stochastic case (Section 2.5) where the integrals are in
the Itô sense, we propose a generalization of the Wiener series to paths of
finite quadratic variation along a given sequence of partitions.

###  2.4 Intrinsic Value Expansion

Consider a smooth $T-$functional $g$ with associated intrinsic functional
$\iota_{0}g(X_{t})=g(X_{t,T-t})$ (see Section 2.1). Let us compute the
functional derivatives of $\iota_{0}g$. First, the temporal derivative
vanishes because the intrinsic value function is already defined through a
flat extension. Second, it is easily checked that the spatial derivative
coincides with the Malliavin derivative of $g$ evaluated at the stopped path
$X_{t,T-t}$. To sum up,

$\displaystyle\Delta_{t}\iota_{0}g$ $\displaystyle\equiv 0.$ (2.9)
$\displaystyle\Delta_{x}\iota_{0}g(X_{t})$ $\displaystyle=D_{t}g(X_{t,T-t}).$
(2.10)

These observations are illustrated in Fig. 3(b) and incidentally allow to
represent $g$ in a compact way. To keep the derivation short, we borrow
results from Section 3.1.

######  Proposition 2.3.

Let $g$ be a twice continuously Malliavin differentiable $T-$functional. If
$X_{T}$ is a continuous with finite quadratic variation along a given sequence
of partitions (see Definition 3.1), then

$g(X_{T})=g(X_{0,T})+\int_{0}^{T}D_{t}g(X_{t,T-t})\circ dx_{t},$ (2.11)

where the integral is in the sense of Stratonovich.

###### Proof.

From the regularity of $g$ in the statement, we can apply the pathwise
functional Stratonovich formula (see Theorem 3.2) to obtain

$\displaystyle g(X_{T})$
$\displaystyle=\iota_{0}g(X_{0})+\int_{0}^{T}\Delta_{t}\iota_{0}g(X_{t})dt+\int_{0}^{T}\Delta_{x}\iota_{0}g(X_{t})\circ
dx_{t}.$

The result follows from (2.10), $\iota_{0}g(X_{0})=g(X_{0,T})$, and
$\Delta_{t}\iota_{0}g\equiv 0$. ∎

Figure 3: Functional derivatives of the intrinsic value functional.

(a) $\Delta_{t}\iota_{0}g(X_{t})\equiv 0$

(b) $\Delta_{x}\iota_{0}g(X_{t})=D_{t}g(X_{t,T-t})$

We can now iteratively apply Proposition 2.3 to derive the intrinsic value
expansion (IVE) of $g$.

######  Theorem 2.4.

(Intrinsic Value Expansion (IVE)) Let $K\geq 1$ and
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$ be
$(K+1)-$times Malliavin differentiable and $X_{T}$ as in Proposition 2.3. Then

$\displaystyle g(X_{T})$
$\displaystyle=g(X_{0,T})+\sum_{k=1}^{K-1}\int_{\triangle_{k,T}}D_{t_{1}\cdots
t_{k}}g(X_{0,T})\,\circ dx^{\otimes k}+R_{K}(X_{T}),$ (2.12) $\displaystyle
R_{K}(X_{T})$ $\displaystyle=\int_{\triangle_{K,T}}D_{t_{1}\cdots
t_{K}}g(X_{t_{1},T-t_{1}})\,\circ dx^{\otimes K}.$ (2.13)

When expanding around a fixed auxiliary path $Y_{T}\in\Lambda_{T}$, the IVE
becomes (assuming $x_{0}=0$)

$\displaystyle g(X_{T}+Y_{T})$
$\displaystyle=g(Y_{T})+\sum_{k=1}^{K-1}\int_{\triangle_{k,T}}D_{t_{1}\cdots
t_{k}}g(Y_{T})\,\circ dx^{\otimes k}+R_{K}(X_{T},Y_{T}).$ (2.14)
$\displaystyle R_{K}(X_{T},Y_{T})$
$\displaystyle=\int_{\triangle_{K,T}}D_{t_{1}\cdots
t_{K}}g(Y_{T}+X_{t_{1},T-t_{1}})\,\circ dx^{\otimes K}.$ (2.15)

###### Proof.

The case $K=1$ corresponds to Proposition 2.3. If $K\geq 2$ and fixed
$t\in[0,T]$, set $g^{1}(X_{t})=D_{t}g(X_{t,T-t})$ on the new horizon $[0,t]$
and define $\iota_{0}g^{1}(X_{s})=g^{1}(X_{s,t-s})$. Again, the temporal
derivative is zero and
$\Delta_{x}\iota_{0}g^{1}(X_{s,t-s})=D_{st}g(X_{s,T-s})$. Proposition 2.3 thus
gives

$g^{1}(X_{t})=D_{t}g(X_{0,T})+\int_{0}^{t}D_{st}g(X_{s,T-s})\circ dx_{t},\quad
D_{st}g(X_{s,T-s})=\Delta_{x}g^{1}(X_{s,t-s}).$

Iterating and writing $\circ\ dx^{\otimes k}=\circ\ dx_{t_{1}}\cdots\circ
dx_{t_{k}}$ yields the claim. To show (2.14), fix $Y_{T}\in\Lambda_{T}$ and
apply (2.12) to the smooth $T-$functional $X_{T}\mapsto g(X_{T}+Y_{T})$. ∎

Given an infinitely Malliavin differentiable $T-$functional $g$, then its
intrinsic value series is obtained by letting $K\to\infty$ in Theorem 2.4,
namely

$\displaystyle
g(X_{T})=g(X_{0,T})+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}D_{t_{1}\cdots
t_{k}}g(X_{0,T})\,\circ dx^{\otimes k},$ (2.16)

whenever the series converge, i.e. $R_{K}(X_{T})\to 0$. Similarly if
$g(\cdot+Y_{T})$ is infinitely Malliavin differentiable, then the intrisic
value series of $g$ around $Y_{T}$ writes

$\displaystyle
g(X_{T}+Y_{T})=g(Y_{T})+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}D_{t_{1}\cdots
t_{k}}g(Y_{T})\,\circ dx^{\otimes k}.$ (2.17)

To the best of our knowledge, the IVE and Intrinsic value series has not
received attention in the current literature. It will be put into perspective
with other expansions in Section 2.6 and Section 3.1.6.

######  Example 2.

(Intrinsic functional) Let
$g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$
with intrinsic value series as in (2.16). As in Example 1, a natural embedding
of $g$ into the space of functionals consists of restricting the iterated
integrals appearing in the right side of (2.16) to $[0,t]\subseteq[0,T]$. In
other words, define

$f(X_{t})=g(X_{0,T})+\sum_{k=1}^{\infty}\int_{\triangle_{k,t}}D_{t_{1}\cdots
t_{k}}g(X_{0,T})\,\circ dx^{\otimes k},\quad t\in[0,T].$ (2.18)

Interestingly, $f$ is precisely the intrinsic functional of $g$. Indeed, the
IVE of $\iota_{0}g(X_{t})=g(X_{t,T-t})$ on $[0,t]$ has constant term
$\iota_{0}g(X_{0,t})$ and kernels given by $D_{t_{1}\cdots
t_{k}}\iota_{0}g(X_{0,t})$, $(t_{1},\ldots,t_{k})\in\triangle_{k,t}$.
Observing that $\iota_{0}g(X_{0,t})=g((X_{0,t})_{t,T-t})=g(X_{0,T})$, we
conclude from $\eqref{eq:IVEt}$ that $f=\iota_{0}g$. It comes as no surprise
that the intrinsic functional, illustrated in Fig. 4(b), is the natural
embedding of $g$ induced by the intrinsic value expansion (and in turn, the
Wiener series).

###  2.5 Wiener Chaos

In this section, we restate the Wiener chaos expansion as well as useful tools
from Malliavin calculus. We refer the interested reader to the excellent books
[15, 38, 36] for further details. Let us consider a stochastic basis
$(\Lambda,\operatorname{\mathcal{F}},\operatorname{\mathbb{F}},\operatorname{\mathbb{Q}})$
with $\sigma-$algebra $\operatorname{\mathcal{F}}$, filtration
$\operatorname{\mathbb{F}}$ and probability measure
$\operatorname{\mathbb{Q}}$. With a slight abuse of notation, We write $X$ for
the coordinate process while we rather use the letter $Y$ when taking
expectation under $\operatorname{\mathbb{Q}}$. For $p\in[1,\infty)$,
$t\in[0,T]$, let $L^{p}(\Lambda_{t})$ be the set of functionals on
$\Lambda_{t}$ such that

$\infty>\lVert
f\rVert_{L^{p}(\Lambda_{t})}\mathrel{\mathop{\ordinarycolon}}=\left(\int_{\Lambda_{t}}|f(Y)|^{p}\operatorname{\mathbb{Q}}(dY)\right)^{1/p}.$
(2.19)

Notice that $L^{p}(\Lambda_{t})$ implicitly depends on
$\operatorname{\mathbb{Q}}$. Furthermore, we define

$L^{p}(\Lambda)=\left\\{f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}\
\mathrel{\mathop{\ordinarycolon}}\ \infty>\lVert
f\rVert_{L^{p}(\Lambda)}\mathrel{\mathop{\ordinarycolon}}=\left(\int_{\Lambda_{T}}\int_{0}^{T}|f(Y_{t})|^{p}dt\,\operatorname{\mathbb{Q}}(dY)\right)^{1/p}\right\\}.$
(2.20)

Observe that $f\in L^{p}(\Lambda)$ if and only if the process
$f(X)=(f(X_{t}))_{t\in[0,T]}$ belongs to
$L^{p}(\operatorname{\mathbb{Q}}\otimes dt)$. For the special case $p=2$, we
see from $\eqref{eq:L(Lambda)}$ that
$L^{2}(\Lambda)=\int_{[0,T]}^{\oplus}L^{2}(\Lambda_{t})dt,$ i.e.
$L^{2}(\Lambda)$ is the direct integral of the Hilbert spaces
$(L^{2}(\Lambda_{t}))_{t\in[0,T]}$ with inner product
$(f,f^{\prime})_{L^{2}(\Lambda)}=\int_{0}^{T}(f,f^{\prime})_{L^{2}(\Lambda_{t})}dt$,
$\
(f,f^{\prime})_{L^{2}(\Lambda_{t})}=\int_{\Lambda_{t}}f(Y)f^{\prime}(Y)\operatorname{\mathbb{Q}}(dY)$.

####  2.5.1 Wiener Chaos Expansion of $T-$functionals

Let $\operatorname{\mathbb{Q}}$ be the Wiener measure so that the coordinate
process $X$ is Brownian motion.111Although the trajectories of $X_{T}$ are
$\operatorname{\mathbb{Q}}-$a.s. continuous, $T-$functionals are still defined
on $\Lambda_{T}$ to allow the presence of jumps when computing Malliavin or
functional derivatives. Moreover, let $\operatorname{\mathbb{F}}$ be the
filtration generated by $X$. One way to derive the chaos expansion is to
iterate Itô’s representation theorem. For every $g\in L^{2}(\Lambda_{T})$, we
recall that the latter guarantees the existence of a functional
$\varphi_{1}\in L^{2}(\Lambda)$ such that

$g(X_{T})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]+\int_{0}^{T}\varphi_{1}(X_{t})\
dx_{t},\quad\operatorname{\mathbb{Q}}-\text{a.s.}$ (2.21)

Next, set
$\phi_{0}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]$ and
$\phi_{1}(t)=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi_{1}(Y_{t})]$.
Since $\lVert\varphi_{1}\rVert_{L^{2}(\Lambda_{t})}<\infty$ for almost every
$t\in[0,T]$, we can apply (2.21) to $\varphi_{1}(X_{t})$ and obtain for some
functional $\varphi_{2}(t,\cdot)\in\int_{[0,t]}^{\oplus}L^{2}(\Lambda_{s})ds$,

$g(X_{T})=\phi_{0}+\int_{0}^{T}\phi_{1}(t)dx_{t}+\int_{\triangle_{2,T}}\varphi_{2}(t_{2},X_{t_{1}})\
dx_{t_{1}}dx_{t_{2}},\quad\operatorname{\mathbb{Q}}-\text{a.s.}$

Pursuing the same logic further leads to the Wiener chaos expansion of $g$,
stated in Theorem 2.5. Let us introduce the integral operators
$J_{k}\mathrel{\mathop{\ordinarycolon}}L^{2}(\triangle_{k,T})\to
L^{2}(\Lambda)$, given by222From the definition of
$\lVert\cdot\rVert_{L^{2}(\Lambda)}$ and repeated use of Itô isometry, note
that $\lVert J_{k}\phi_{k}\rVert_{L^{2}(\Lambda)}^{2}\leq T\
\lVert\phi_{k}\rVert_{L^{2}(\triangle_{k,T})}^{2}.$ Thus $J_{k}\phi_{k}\in
L^{2}(\Lambda)$ as implicitly claimed.

$\displaystyle J_{0}\phi_{0}(X_{t})$
$\displaystyle\equiv\phi_{0}\in\operatorname{\mathbb{R}},\
(=\mathrel{\mathop{\ordinarycolon}}L^{2}(\triangle_{0,T}))$ $\displaystyle
J_{k}\phi_{k}(X_{t})$
$\displaystyle=\int_{\triangle_{k,t}}\phi_{k}\,dx^{\otimes
k}=\int_{\triangle_{k,t}}\phi_{k}(t_{1},...,t_{k})\,dx_{t_{1}}\cdots
dx_{t_{k}},\quad k\geq 1.$ (2.22)

The image
$\operatorname{\mathfrak{I}}_{k}\mathrel{\mathop{\ordinarycolon}}=\\{J_{k}\phi_{k}\big{|}_{\Lambda_{T}}\mathrel{\mathop{\ordinarycolon}}\phi_{k}\in
L^{2}(\triangle_{k,T})\\}$ is called the Wiener chaos of order $k$. When
$\phi_{k}\equiv 1$, $J_{k}$ generates the $k-$fold Itô iterated integral,
which we simply denote by $J_{k}(X_{t})$. We restate the Wiener chaos
expansion in terms of $T-$functionals; see, e.g. [15, Theorem 1.10] for a more
classical formulation.

######  Theorem 2.5.

(Wiener Chaos Expansion for $T-$functionals) The space of square integrable
$T-$functionals can be expressed as a direct sum of chaos, namely
$L^{2}(\Lambda_{T})=\bigoplus_{k=0}^{\infty}\operatorname{\mathfrak{I}}_{k}.$
That is, for all $g\in L^{2}(\Lambda_{T})$, there exists a unique sequence
$\\{\phi_{k}\in L^{2}(\triangle_{k,T})\\}_{k\in\operatorname{\mathbb{N}}}$
such that

$\sum_{k=0}^{K}J_{k}\phi_{k}(X_{T})\
\underset{L^{2}(\Lambda_{T})}{\overset{K\uparrow\infty}{\xrightarrow{\hskip
34.14322pt}}}\ g(X_{T}).$ (2.23)

Furthermore, $\lVert
g\rVert^{2}_{L^{2}(\Lambda_{T})}=\sum_{k=0}^{\infty}\lVert\phi_{k}\rVert^{2}_{L^{2}(\triangle_{k,T})}$.

As shown by Stroock [45] (see also Theorem 1.32 in [36]), the kernels can be
made explicit when $g$ is smooth in the sense of Mallavin, namely
$g\in\operatorname{\mathbb{D}}^{K,2}(\Lambda_{T})$ for all
$K\in\operatorname{\mathbb{N}}$. We recall that
$g\in\operatorname{\mathbb{D}}^{K,2}(\Lambda_{T})$ if for all $0\leq k\leq K$,
the iterated Malliavin derivative $D_{t_{1}\cdots t_{k}}g=(D_{t_{1}}\cdots
D_{t_{k}})g$ exists and satisfies

$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{\triangle_{k,T}}|D_{t_{1}\cdots
t_{k}}g(Y_{T})|^{2}\ dt^{\otimes k}\right]<\infty.$ (2.24)

If $k=0$, (2.24) simply requires that $g\in L^{2}(\Lambda_{T})$. The
expression for $(\phi_{k})_{k=1}^{\infty}$ can be derived recursively from the
celebrated Clark-Ocone formula [12, 40] refining Itô’s representation theorem
when $g$ belongs to $\operatorname{\mathbb{D}}^{1,2}(\Lambda_{T})$.

######  Theorem 2.6.

(Clark-Ocone Formula) If $g\in\operatorname{\mathbb{D}}^{1,2}(\Lambda_{T})$,
then the integrand in (2.21) is $\operatorname{\mathbb{Q}}-$a.s. equal to
$\varphi_{1}(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t}g(Y_{T})|X_{t}]$,
that is

$g(X_{T})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]+\int_{0}^{T}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t}g(Y_{T})|X_{t}]\
dx_{t},\quad\operatorname{\mathbb{Q}}-\text{a.s.}$ (2.25)

We therefore conclude from Theorem 2.6 that
$\phi_{1}(t)=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t}g(Y_{T})]$.
When $g$ is smooth, iterating the argument together with the tower property
leads to Stroock’s representation of the Wiener kernels [45]:

$\phi_{k}(t_{1},...,t_{k})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t_{1}\cdots
t_{k}}g(Y_{T})],\quad k\geq 0.$ (2.26)

######  Example 3.

(Price functional) Similarly to Examples 1 and 2, we truncate the simplexes in
the chaos expansion to obtain a functional. Let $g\in L^{2}(\Lambda_{T})$ with
chaos expansion $g(X_{T})=\sum_{k=0}^{\infty}J_{k}\phi_{k}(X_{T})$ and define
$f(X_{t})=\sum_{k=0}^{\infty}J_{k}\phi_{k}(X_{t}).$ Then $f\in L^{2}(\Lambda)$
and we claim that $f$ coincides with the price functional
$\iota_{\operatorname{\mathbb{Q}}}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})\
|\ X_{t}]$. Indeed, the martingality of Itô iterated integrals gives

$\iota_{\operatorname{\mathbb{Q}}}g(X_{t})=\sum_{k=0}^{\infty}\iota_{\operatorname{\mathbb{Q}}}J_{k}\phi_{k}(X_{t})=\sum_{k=0}^{\infty}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[J_{k}\phi_{k}(Y_{T})\
|\ X_{t}]=\sum_{k=0}^{\infty}J_{k}\phi_{k}(X_{t}).$

The price functional is therefore the natural embedding associated to the
chaos expansion.

####  2.5.2 Expressing Wiener kernels with functional Derivatives

The Clark-Ocone formula (Theorem 2.6) allows to express a $T-$functional, up
to a constant, as a stochastic integral. The integrand is given by the the
optional projection of the (anticipative) Malliavin derivative of $g$. In
contrast, the functional Itô calculus permits to rewrite this integrand
directly in terms of a non-anticipative functional. This is summarized in the
next result, taken from [16].

######  Theorem 2.7.

For every $g\in L^{1}(\Lambda_{T})$ such that
$f(X_{t})=\iota_{\operatorname{\mathbb{Q}}}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})\,|\,X_{t}]$
is spatially differentiable, then

$g(X_{T})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]+\int_{0}^{T}\Delta_{x}f(X_{t})dx_{t},\quad\operatorname{\mathbb{Q}}-\text{a.s.}$
(2.27)

We shall see that the Wiener chaos expansion of $g$ can be recovered by
successive applications of Theorem 2.7. To this end, consider the family of
”martingale representation operators”

$\operatorname{\mathcal{M}}=(\operatorname{\mathcal{M}}_{t})_{t\in[0,T]},\quad\operatorname{\mathcal{M}}_{t}\\!f(X_{s})=\Delta_{x}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[f(Y_{t})|X_{s}],\quad
0\leq s\leq t,\quad f\in L^{1}(\Lambda_{t}),$ (2.28)

when the above expression is well-defined. It is worth noting that the domain
of $\operatorname{\mathcal{M}}_{T}$ contains
$\operatorname{\mathbb{D}}^{1,2}(\Lambda_{T})$, namely the set of
$T-$functionals with Clark-Ocone representation (2.25). This follows from the
general fact that integrating before differentiating gives more regularity
than the other way around. Besides, the uniqueness of the integrand in Itô’s
representation theorem implies that
$\operatorname{\mathcal{M}}_{T}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t}g(Y_{T})\
|\ X_{t}]$ $\operatorname{\mathbb{Q}}-$a.s. whenever
$g\in\operatorname{\mathbb{D}}^{1,2}(\Lambda_{T})$. In terms of the price
embedding $\iota_{\operatorname{\mathbb{Q}}}$, this reads

$\Delta_{x}\iota_{\operatorname{\mathbb{Q}}}=\iota_{\operatorname{\mathbb{Q}}}D_{t}\;\;\text{
on }\operatorname{\mathbb{D}}^{1,2}(\Lambda_{T}).$

Next, set
$\operatorname{\mathcal{M}}_{t_{1}...t_{k}}=\operatorname{\mathcal{M}}_{t_{1}}\cdots\operatorname{\mathcal{M}}_{t_{k}}$,
$(t_{1},...,t_{k})\in\triangle_{k,T}$, creating an entanglement of spatial
derivatives and conditioned expectations. If $k=2$, this reads
$\operatorname{\mathcal{M}}_{t_{2}t_{3}}\\!f(X_{t_{1}})=\Delta_{x}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\Delta_{x}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[f(Z_{t_{3}})\,|\,Y_{t_{2}}]\,\big{|}\,X_{t_{1}}\right]\\!.$
To safely apply the operator over and over, we introduce the domain of
$\operatorname{\mathcal{M}}$,

$\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}}=\bigcap_{k=1}^{\infty}\
\\{f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}\
\mathrel{\mathop{\ordinarycolon}}\
\,\operatorname{\mathcal{M}}_{t_{1}...t_{k}}\\!f\text{ is well-defined
}\forall\,(t_{1},...,t_{k})\in\triangle_{k,T}\\}.$ (2.29)

Hence
$\operatorname{\mathcal{M}}_{t}(\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}})\subseteq\operatorname{\mathcal{D}}_{\operatorname{\mathcal{M}}}$.
For a $T-$functional $g$, the operator $\operatorname{\mathcal{M}}$ is only
valid for $t=T$. The recursive use of $\operatorname{\mathcal{M}}$ is allowed
as long as $\operatorname{\mathcal{M}}_{T}\\!g$, henceforth a running
functional, belongs to
$\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}}$. The next
proposition highlights the role of the martingale representation operator to
recover the Wiener-Itô chaos expansion.

######  Proposition 2.8.

Let $g$ be a $T-$functional such that
$\operatorname{\mathcal{M}}_{T}g\in\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}}$.
Then the kernels in the Wiener chaos expansion of $g$ can be expressed as
$\phi_{0}=\operatorname{\mathcal{M}}_{T}\\!g(X_{0})$ and

$\phi_{k}(t_{1},...,t_{k})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\operatorname{\mathcal{M}}_{t_{2}...t_{k}T}\\!g(Y_{t_{1}})].$
(2.30)

###### Proof.

The martingale representation of $g(X_{T})$ can be rewritten as
$g(X_{T})=\phi_{0}+\int_{0}^{T}\operatorname{\mathcal{M}}_{T}\\!g(X_{t})dx_{t}.$
Next, for fixed $t\leq T$, the functional
$\operatorname{\mathcal{M}}_{T}\\!g(X_{t})$ admits a martingale representation
as well, namely

$\displaystyle\operatorname{\mathcal{M}}_{T}\\!g(X_{t})$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\operatorname{\mathcal{M}}_{T}\\!g(Y_{t})\,|\,X_{0}]+\int_{0}^{t}\Delta_{x}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\operatorname{\mathcal{M}}_{T}\\!g(Y_{t})\,|\,X_{s}]dx_{s}=\phi_{1}(t)+\int_{0}^{t}\operatorname{\mathcal{M}}_{tT}\\!g(X_{s})dx_{s}.$

Hence, a simple integration gives
$g(X_{T})=\phi_{0}+J_{1}\phi_{1}(X_{T})+\int_{0}^{T}\int_{0}^{t}(\operatorname{\mathcal{M}}_{tT}g)(X_{s})dx_{s}dx_{t}.$
Iterating the argument with the adequate time variable renaming yields the
claim. ∎

The expression in $\eqref{eq:integrandFunc}$ is in appearance similar to
Stroock’s representation $\eqref{eq:stroock}$ involving the Malliavin
derivative $D$. However, the operators $D$, $\operatorname{\mathcal{M}}$ are
drastically different in nature: $(D_{t}g(Y_{T}))_{t\in[0,T]}$ is a family of
$\operatorname{\mathcal{F}}_{T}-$measurable random variables while
$(\operatorname{\mathcal{M}}_{T}g(Y_{t}))_{t\in[0,T]}$ is an adapted process.

###  2.6 Summary

We recapitulate the static expansions seen so far in Table 1. The second
column recalls the natural embeddings discussed in Examples 1, 2 and 3 and are
also displayed in Fig. 4(c). Note that the Wiener series and intrinsic value
are very similar since the latter generalizes the former. Also, there is an
important difference between the constant terms: in the deterministic case, it
is the image of the flat path $X_{0,T}$ through $g$. On the other hand, in the
chaos expansion, it is replaced by the expected value of $g$ over all Brownian
paths.

Table 1: Classification of static expansions, i.e. for $T-$functionals $g\mathrel{\mathop{\ordinarycolon}}\Lambda_{T}\to\operatorname{\mathbb{R}}$ Expansion | Constant term | Kernel | Integral | Embedding
---|---|---|---|---
Volterra series | $g(X_{0,T})$ | $F_{t_{1}\cdots t_{k}}g(X_{0,T})$ | Riemann | $g(X_{t}\oplus\bm{0}_{T-t})$
Wiener series | $g(X_{0,T})$ | $D_{t_{1}\cdots t_{k}}g(X_{0,T})$ | Stieltjes | $g(X_{t,T-t})$
IVE | $"$ | $"$ | Stratonovich | $"$
Wiener chaos | $\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]$ | $\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t_{1}\cdots t_{k}}g(Y_{T})]$ | Itô | $\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})\ |\ X_{t}]$

Finally, let us compare the intrinsic value expansion and Wiener chaos
expansion in further depth. For convenience, the expansions are recalled:

$\displaystyle g(X_{T})\hskip 2.84544pt$ $\displaystyle=\hskip
2.84544ptg(X_{0,T})$ $\displaystyle+\hskip
17.07182pt\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}D_{t_{1}\cdots
t_{k}}g(X_{0,T})\,\circ\,dx^{\otimes k},\hskip 28.45274pt\quad$ (IVE)
$\displaystyle g(X_{T})\hskip 2.84544pt$ $\displaystyle=\hskip
2.84544pt\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]$
$\displaystyle+\hskip
17.07182pt\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t_{1}\cdots
t_{k}}g(Y_{T})]\;dx^{\otimes k},\quad\operatorname{\mathbb{Q}}-a.s.\quad$
(Wiener chaos) (2.31)

First, the nature of iterated integrals differ (Stratonovich versus Itô
sense). Second, although both expansions involve iterated Malliavin
derivatives of $g$, the kernels are computed according to the corresponding
natural embedding (intrinsic and price functional, respectively). But the main
difference comes from the paths themselves: In the intrinsic value expansion,
the paths can be quite arbitrary, as long as their roughness is controlled
along some given sequence of partitions. In the Wiener chaos expansion, they
must be ”typical” Brownian paths.

To bring the expansions closer, we compute the IVE of $X_{T}\to
g(X_{T}+Y_{T})$ as in Theorem 2.4 and thereafter take expectation with respect
to $Y_{T}$ under $\operatorname{\mathbb{Q}}$. This yields

$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(X_{T}+Y_{T})]=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]+\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t_{1}\cdots
t_{k}}g(Y_{T})]\,\circ dx^{\otimes k}.$ (2.32)

The right side of (2.32) closely resembles the Wiener chaos expansion of
$g(X_{T})$ (2.31), except that the integrals are in the sense of Stratonovich
instead of Itô. If $X_{T}$ is a typical Brownian path and admissible for the
IVE, subtracting (2.31) to (2.32) yields the bias representation

$\displaystyle\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(X_{T}+Y_{T})]-g(X_{T})=\sum_{k=1}^{\infty}\int_{\triangle_{k,T}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t_{1}\cdots
t_{k}}g(Y_{T})]\,[\circ\ dx^{\otimes k}-dx^{\otimes
k}],\quad\operatorname{\mathbb{Q}}-a.s.$ (2.33)

We remark that the first term on the right side of (2.33) vanishes when
$t\mapsto\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{t}g(Y_{T})]$
(equal to $\phi_{1}$) is of finite variation. Also, the second term simply
reads
$\frac{1}{2}\int_{0}^{T}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[D_{tT}g(Y_{T})]dt$
using Itô’s isometry.

Figure 4: Natural embeddings of static expansions. We recall that (b) is the
intrinsic functional, while (c) is the price functional under the Wiener
measure.

(a) Volterra

(b) Wiener series/IVE

(c) Wiener chaos

##  3 Dynamic Expansions

###  3.1 Functional Taylor Expansion

The Taylor expansion lies at the core of calculus and finds applications in
countless fields. It provides an explicit approximation of smooth functions by
polynomials. A natural question is whether the Taylor expansion can be
extended to more general spaces such as $\Lambda$. In particular, one may
wonder whether there are functionals echoing the monomials. A natural
candidate is the family of signature functionals. The path signature, first
studied by Chen [10], has gained much attention in recent years [3, 4, 14,
35], due notably to its universal approximation property. Indeed, density
results à la Stone-Weierstrass exist [31, 14] allowing, financially speaking,
to both price and replicate exotic payoffs with signature elements; see [35,
4]. However, the Stone-Weierstrass theorem is existential instead of
constructive. As such, a procedure that explicitly generates the replication
of a claim would be preferred.

In response to this appeal, we propose a pathwise Taylor expansion for path-
dependent payoffs: the functional Taylor expansion (FTE). To the best of our
knowledge, this generalization has been first proposed by Fliess [18, 19, 20],
termed the causal Taylor expansion, for $T-$functionals and continuously
differentiable paths. Interestingly, the causal derivative in [19] shares
similarities with the spatial derivative from the functional Itô calculus
[16]. The main difference is that the discontinuous ”bump” in the latter is
replaced by the vertical segment joining the terminal value of the original
path and the bumped one. Moreover, the causal derivative were defined on the
whole horizon, i.e. for $T-$functionals only. More recently, using the
functional derivatives in [16], Litterer and Oberhauser [31] establish a
functional Taylor expansion$-$called Chen-Fliess approximation$-$where the
underyling path comes from the strong solution of a stochastic differential
equation with smooth vector fields.

Another approach is given in Buckdahn et al. [8] where the authors build a
Taylor expansion for stochastic processes suitable to characterize viscosity
solutions of path-dependent partial differential equations (PPDEs). In
particular, a pathwise estimate of the remainder is given in [8], although
still requiring a stochastic framework. It is worth noting that the spatial
functional derivative in Buckdahn et al. [8] is slightly different from the
ones in [16] and the present article (in fact, weaker) and arises from the
martingale part in the Doob-Meyer decomposition. In a subsequent work,
Buckdahn et al. [9] introduce the rough functional Taylor expansion where the
path derivative is in the sense of Gubinelli [23]. The signature is thus given
by iterated rough integrals either defined pathwise in the one-dimensional
case or using higher order ”lifts” of the path otherwise; see Lyons [34], Friz
and Hairer [21].

In this work, we adopt the pathwise framework pioneered by Föllmer [22]
combined with tools from the functional Itô calculus [16]. Consequently, our
Taylor expansion is made ”$\omega$ by $\omega$” and thus model-free. Choosing
a similar avenue as Bally et al. [5] where pathwise functional Itô formulae
are derived, we establish a pathwise functional Stratonovich formula (Theorem
3.2) for one-dimensional continuous paths of finite quadratic variation along
a given sequence of partition. This provides, in passing, a way to construct
Stratonovich integrals as well as the signature in a pathwise manner. We
stress that our construction of the signature relies on limits of simple
Riemann sums and thus differs from the more abstract definition in rough path
theory [34, 21] given by imposed algebraic relations. In Section 3.1.4, we
eventually state the functional Taylor expansion in various forms. In
particular, the flexibility of the proposed framework allows, in Theorem 3.10,
to expand a functionals after a path which may not be regular at all.

The FTE presents several applications in control theory, engineering and
finance. In Section 4, we focus on the latter and outline the main use of the
FTE when pricing or hedging derivatives. As also pointed out in [31], the FTE
is of great relevance in cubature methods (see Lyons and Victoir [33], Crisan
et al. [13] and the references therein). In short, cubature is a
generalization of Gaussian quadrature to the path space; the exact integration
of polynomials turns into a perfect fit of expectation of signature elements.
This goes, however, beyond the scope of this work.

####  3.1.1 Functional Stratonovich Formula

One of the main ingredients in the FTE is a variation of the Functional Itô
formula [16]. In Theorem 3.2, we simultaneously employ a pathwise setting
similar to Föllmer [22] and Stratonovich integrals as in Litterer and
Oberhauser [31]. See also Bally et al. [5, Chapter 5] and the references
therein for pathwise functional Itô formulae, also along the lines of
Föllmer’s approach. First, we define a suitable class of integrators.

######  Definition 3.1.

Let $\Pi=(\Pi^{N})$ be a sequence of partitions of $[0,T]$ with vanishing mesh
size, that is
$|\Pi^{N}|\mathrel{\mathop{\ordinarycolon}}=\max_{t_{n}\in\Pi^{N}}|t_{n}-t_{n-1}|\overset{N\uparrow\infty}{\longrightarrow}0$.
For fixed $t\in[0,T]$, we say that $X\in\Lambda_{t}$ is a $\Pi$-integrator if
$X$ is continuous and such that the quadratic variation function along $\Pi$,
i.e.,

$[0,t]\ni s\mapsto\langle
X\rangle^{\Pi}_{s}\mathrel{\mathop{\ordinarycolon}}=\lim_{|\Pi_{s}^{N}|\downarrow
0}\sum_{t_{n}\in\Pi^{N}_{s}}(x_{t_{n}}-x_{t_{n-1}})^{2},\qquad\Pi^{N}_{s}\mathrel{\mathop{\ordinarycolon}}=\\{t_{n}\wedge
s\ |\ t_{n}\in\Pi^{N}\\},$ (3.1)

exists and is continuous. We write $\Omega^{\Pi}_{t}$ for the set of
$\Pi$-integrators in $\Lambda_{t}$ and
$\Omega^{\Pi}\mathrel{\mathop{\ordinarycolon}}=\bigcup_{t\in[0,T]}\Omega^{\Pi}_{t}$.

Notice that if $X\in\Omega^{\Pi}_{t}$, then $X_{s}\in\Omega^{\Pi}_{s}$ for all
$s\leq t$ using the sequence of ”stopped” partitions
$\Pi_{s}\mathrel{\mathop{\ordinarycolon}}=(\Pi_{s}^{N})$ defined in
$\eqref{eq:QV}$. The choice of $\Pi$ may considerably impact the richness of
$\Omega^{\Pi}$. Good candidates are refining partitions, that is
$\Pi^{1}\subseteq\Pi^{2}\subseteq\ldots$ such as the dyadic partition
$\Pi^{N}=\\{Tn2^{-N}\ \mathrel{\mathop{\ordinarycolon}}\ n=0,\ldots,2^{N}\\}$.
In effect, having a nested structure in the sequence of partitions guarantees
the almost sure convergence of the quadratic variation for some stochastic
processes, e.g. Brownian motion [43, Proposition 2.12.].

To compute the spatial derivatives of $f$ at a continuous path
$X\in\Omega^{\Pi}_{t}$, we need to make sure that $f$ is well-defined when
adding vertical bumps to $X$. To this end, we consider the enlargement
$\bar{\Omega}^{\Pi}_{t}=\Omega^{\Pi}_{t}+\operatorname{\mathcal{J}}_{t}$,
where $\operatorname{\mathcal{J}}_{t}$ contains all piecewise constant paths
in $\Lambda_{t}$ of finite variation. Defining as usual
$\bar{\Omega}^{\Pi}=\bigcup_{t\in[0,T]}\bar{\Omega}^{\Pi}_{t}$ gives the
sequence of strict inclusions
$\Omega^{\Pi}\subset\bar{\Omega}^{\Pi}\subset\Lambda$.

As is customary, regularity conditions are required on the function to expand.
More specifically, consider the classes of functionals defined on some subset
$\Lambda^{\prime}\subseteq\Lambda$,

$\operatorname{\mathbb{C}}^{K,L}(\Lambda^{\prime})=\\{f\mathrel{\mathop{\ordinarycolon}}\Lambda^{\prime}\to\operatorname{\mathbb{R}}\,\mathrel{\mathop{\ordinarycolon}}\,\Delta_{\alpha}f\text{
exists and is $\Lambda-$continuous }\,\forall\,\alpha\text{ s.t.
}\,|\alpha|_{0}\leq K,\,|\alpha|_{1}\leq L\\},\quad K,L\geq 0.$

Taking $K=L=\infty$ gives the class
$\operatorname{\mathbb{C}}^{\infty,\infty}(\Lambda^{\prime})$ of smooth
functionals in $\Lambda^{\prime}$.

######  Theorem 3.2.

(Functional Stratonovich formula) Let $X\in\Omega^{\Pi}_{t}$ for some
$t\in[0,T]$ and $f\in\operatorname{\mathbb{C}}^{1,2}(\bar{\Omega}^{\Pi})$.
Then,

$\displaystyle f(X_{t})$
$\displaystyle=f(X_{0})+\int_{0}^{t}\Delta_{t}f(X_{s})ds+\int_{0}^{t}\Delta_{x}f(X_{s})\circ
dx_{s},$ (3.2)

where the last integral implicitly depends on the sequence of partitions $\Pi$
associated to $X$.

###### Proof.

See Section A.1. ∎

######  Remark 3.3.

Although it is necessary to work with càdlàg trajectories to define the
spatial derivative, Theorem 3.2 and in turn, the functional Taylor expansion,
is established for continuous paths. Indeed, doing so prevents the presence of
jump terms which would make the FTE more general but much more intricate.

######  Example 4.

Let $X\in\Omega^{\Pi}_{t}$ and $f(X_{s})=h(x_{s})$ for some real function
$h\in\operatorname{\mathcal{C}}^{2}(\operatorname{\mathbb{R}})$. Then clearly
$f\in\operatorname{\mathbb{C}}^{1,2}(\bar{\Omega}^{\Pi}_{t})$ since
$\Delta_{t}f=0$, $\Delta_{x}f=h^{\prime}$, and
$\Delta_{xx}f=h^{\prime\prime}$. We can thus apply the pathwise Stratonovich
formula to obtain

$h(x_{t})=h(x_{0})+\int_{0}^{t}h^{\prime}(x_{s})\circ dx_{s}.$

We here recover the well-known fact that Stratonovich integrals yield a first
order calculus.

######  Example 5.

Let us verify the pathwise Stratonovich formula the time integral
$f(X_{t})=\int_{0}^{t}\varphi(X_{s})ds$ where the pair $(\varphi,X)$ satisfies
$\int_{0}^{t}|\varphi(X_{s})|ds<\infty$. Since
$\\{s\in[0,t]\,\mathrel{\mathop{\ordinarycolon}}\,X_{t}(s)\neq
X^{h}_{t}(s)\\}=\\{t\\}$ has Lebesgue measure zero for all $h\neq 0$, we
conclude that $\Delta_{x}f\equiv 0$. For the time derivative, we obtain
$\Delta_{t}f(X_{t})=\lim_{\delta t\downarrow 0}\,\frac{1}{\delta
t}\int_{0}^{\delta t}\varphi(X_{t,u})du=\varphi(X_{t}).$ As expected, this
gives

$f(X_{t})=\underbrace{f(X_{0})}_{=\
0}+\int_{0}^{t}\underbrace{\Delta_{t}f(X_{s})}_{\varphi(X_{s})}ds+\int_{0}^{t}\underbrace{\Delta_{x}f(X_{s})}_{=\
0}\circ\,dx_{s}=\int_{0}^{t}\varphi(X_{s})ds.$

We also remark that $\Delta_{t}\Delta_{x}f=0$ and
$\Delta_{x}\Delta_{t}f=\Delta_{x}\varphi$ provided that $\Delta_{x}\varphi$
exists. Hence $\Delta_{t},\Delta_{x}$ do not commute when
$\Delta_{x}\varphi\neq 0$.

####  3.1.2 Spatial Anti-derivative and Pathwise Integration

In the same spirit as Föllmer’s pathwise Itô calculus [22], Theorem 3.2 allows
to incidentally define pathwise Stratonovich integrals $\int y\circ dx$ where
the integrator may be of infinite variation. However, there are constraints on
the integrand: $Y$ must be the spatial derivative of some functional
$f\in\operatorname{\mathbb{C}}^{1,2}$ evaluated at $X$. In other words, the
admissible integrands correspond to all delta hedging strategies where $f$ is
interpreted as the price functional of an option.

In the path-independent case as in Föllmer [22], the integrands are simply the
continuously differentiable functions (both in time and space) of the path,
i.e. $y_{t}=h(t,x_{t}),\
h\in\operatorname{\mathcal{C}}^{1,1}(\operatorname{\mathbb{R}})$. Indeed, it
suffices to apply the Stratonovich formula to
$H(t,x_{t})\mathrel{\mathop{\ordinarycolon}}=\int^{x_{t}}_{x_{0}}h(t,y)dy$.
Back to our framework, we construct a spatial anti-derivative as follows.
Given $\varphi\in\operatorname{\mathbb{C}}^{1,1}(\bar{\Omega}^{\Pi})$, write
$X_{t}^{(\varepsilon)}\mathrel{\mathop{\ordinarycolon}}=X_{t}^{\varepsilon-
x_{t}}$ ($\Longrightarrow X_{t}^{(\varepsilon)}(t)=\varepsilon$) and define

$\Phi(X_{t})=\int_{x_{0}}^{x_{t}}\varphi(X_{t}^{(\varepsilon)})\
d\varepsilon.$ (3.3)

Then, we easily see that $\frac{\Phi(X^{\delta x}_{t})-\Phi(X_{t})}{\delta
x}=\frac{1}{\delta x}\int_{0}^{\delta x}\varphi(X_{t}^{(\varepsilon)})\
d\varepsilon\,\overset{\delta x\to 0}{\xrightarrow{\hskip
17.07182pt}}\,\varphi(X_{t})$ and $\Phi$ is a spatial anti-derivative as
desired. Moreover,
$\Delta_{t}\Phi(X_{t})=\int_{x_{0}}^{x_{t}}\Delta_{t}\varphi(X_{t}^{(\varepsilon)})d\varepsilon$.
Since $\Phi(X_{0})=0$, we can invoke and rearrange the functional Stratonovich
formula to obtain the pathwise integral

$\int_{0}^{t}\varphi(X_{s})\circ
dx_{s}=\int_{x_{0}}^{x_{t}}\varphi(X_{t}^{(\varepsilon)})\
d\varepsilon-\int_{0}^{t}\int_{x_{0}}^{x_{s}}\Delta_{t}\varphi(X_{s}^{(\varepsilon)})d\varepsilon\
ds.$ (3.4)

In particular, when $\Delta_{t}\varphi=0$, we simply have
$\int_{0}^{t}\varphi(X_{s})\circ
dx_{s}=\int_{x_{0}}^{x_{t}}\varphi(X_{t}^{(\varepsilon)})\ d\varepsilon.$
Thus, the Stratonovich integral can be made pathwise for all functionals in
$\operatorname{\mathbb{C}}^{1,1}(\bar{\Omega}^{\Pi})$. Incidentally, this
allows us to properly define signature functionals in $\bar{\Omega}^{\Pi}$;
see Proposition 3.5. It is important to point out that this simple
construction works only for one-dimensional paths.

######  Remark 3.4.

From the pathwise functional Itô formula [1, 5], note that

$\displaystyle\int_{0}^{t}\varphi(X_{s-})dx_{s}$
$\displaystyle=\int_{x_{0}}^{x_{t}}\varphi(X_{t}^{(\varepsilon)})\
d\varepsilon-\int_{0}^{t}\left(\Delta_{t}+\frac{1}{2}\Delta_{xx}\right)\Phi(X_{s})\
ds.$ (3.5)

The right side of (3.5) thus provides a pathwise definition of Itô integrals.

######  Example 6.

Similar to Example 5, we compute the functional derivatives of the
Stratonovich integral $f(X_{t})=\int_{0}^{t}\varphi(X_{s})\circ dx_{s}$
defined on $\bar{\Omega}^{\Pi}$. From the above discussion, we obtain as
expected that

$\Delta_{t}f(X_{t})=0,\quad\Delta_{x}f(X_{t})=\Delta_{x}\int_{x_{0}}^{x_{t}}\varphi(X_{t}^{(\varepsilon)})\
d\varepsilon=\varphi(X_{t}).$

####  3.1.3 Signature Functionals and Properties

The FTE entails a collection of objects that characterizes the underlying path
in its entirety. This is given by the signature [10, 34] which can be seen as
the infinite skeleton of a path.

Notations. A word is a sequence $\alpha=\alpha_{1}\ldots\alpha_{k}$ of letters
from the alphabet $\\{0,1\\}$. The number of $0$’s and $1$’s in $\alpha$ is
denoted by $|\alpha|_{0}$, $|\alpha|_{1}$, respectively. The length of
$\alpha$ is therefore
$|\alpha|\mathrel{\mathop{\ordinarycolon}}=|\alpha|_{0}+|\alpha|_{1}$. We will
often use the special words
$\bm{0}_{k}\mathrel{\mathop{\ordinarycolon}}=0\ldots 0$ such that
$|\bm{0}_{k}|=|\bm{0}_{k}|_{0}=k$ and
$\mathds{1}_{k}\mathrel{\mathop{\ordinarycolon}}=1\ldots 1$ such that
$|\mathds{1}_{k}|=|\mathds{1}_{k}|_{1}=k$. Also, for convenience, we write
$\alpha-j\mathrel{\mathop{\ordinarycolon}}=\alpha_{1}\ldots\alpha_{k-j}$ where
$|\alpha|=k$ and $0\leq j\leq k-1$. Finally, $\operatorname{\mathbb{A}}$
denotes the set containing all the words.

Next, we enlarge a path $X\in\Lambda$ with the time itself and henceforth set
$x^{0}_{t}=t$, $x^{1}_{t}=x_{t}$. This is a common procedure used to enrich
the signature, see [4, 14, 35] and Remark 3.6 below. The letters $0,1$ are
therefore identified with the time $t$ and path $x$, respectively. The
definition of the signature is a byproduct of the following proposition.

######  Proposition 3.5.

Let $\Pi$ be a sequence of partitions as in Definition 3.1. Recall also the
simplexes $(\triangle_{k,t})$ defined in $\eqref{eq:simplex}$. Then for all
$X_{t}\in\bar{\Omega}^{\Pi}$ and $\alpha\in\operatorname{\mathbb{A}}$, the
iterated Stratonovich integral

$\displaystyle\operatorname{\mathcal{S}}_{\alpha}(X_{t})\mathrel{\mathop{\ordinarycolon}}=\int_{\triangle_{k,t}}\circ\,dx^{\alpha}=\int_{0}^{t}\int_{0}^{t_{k}}\cdots\int_{0}^{t_{2}}\circ\,dx^{\alpha_{1}}_{t_{1}}\cdots\circ
dx^{\alpha_{k}}_{t_{k}},\qquad(k=|\alpha|)$ (3.6)

is well-defined pathwise. Moreover,
$\operatorname{\mathcal{S}}_{\alpha}\in\operatorname{\mathbb{C}}^{\infty,\infty}(\bar{\Omega}^{\Pi})$
for all $\alpha\in\operatorname{\mathbb{A}}$, and satisfies the recursion,

$\Delta_{t}\operatorname{\mathcal{S}}_{\alpha}=\begin{cases}\operatorname{\mathcal{S}}_{\alpha-1},&\alpha_{k}=0\\\
0,&\alpha_{k}=1\\\
\end{cases},\qquad\Delta_{x}\operatorname{\mathcal{S}}_{\alpha}=\begin{cases}\operatorname{\mathcal{S}}_{\alpha-1},&\alpha_{k}=1\\\
0,&\alpha_{k}=0\\\ \end{cases}.\qquad(k=|\alpha|)$ (3.7)

###### Proof.

See Section A.2 ∎

Together with the constant functional
$\operatorname{\mathcal{S}}_{\varnothing}\equiv 1$, we call the collection
$\operatorname{\mathcal{S}}=\\{\operatorname{\mathcal{S}}_{\alpha}\in\operatorname{\mathbb{C}}^{\infty,\infty}(\bar{\Omega}^{\Pi})\
\mathrel{\mathop{\ordinarycolon}}\ \alpha\in\operatorname{\mathbb{A}}\\}$ the
signature in $\Omega^{\Pi}$. When we are given a specific path
$X\in\Omega^{\Pi}_{t}$, then the signature of $X$ is understood as the image
$\operatorname{\mathcal{S}}(X)=\\{\operatorname{\mathcal{S}}_{\alpha}(X_{u})\
\mathrel{\mathop{\ordinarycolon}}\ u\in[0,t],\
\alpha\in\operatorname{\mathbb{A}}\\}$. Notice that the integrals in the above
definition are in the Riemann-Stieltjes (respectively Itô) sense when the
integrator (respectively integrand) is of bounded variation. In such
situations, the symbol $\circ$ may be removed. Assuming $x_{0}=0$ and writing
$\operatorname{\mathcal{S}}_{\alpha}$ instead of
$\operatorname{\mathcal{S}}_{\alpha}(X_{t})$ for brevity, the first signature
functionals are given by

$\displaystyle\begin{pmatrix}&&\operatorname{\mathcal{S}}_{\varnothing}&&\\\
&\operatorname{\mathcal{S}}_{0}&&\operatorname{\mathcal{S}}_{1}&\\\
\operatorname{\mathcal{S}}_{00}&\operatorname{\mathcal{S}}_{01}&&\operatorname{\mathcal{S}}_{10}&\operatorname{\mathcal{S}}_{11}\\\
\vdots&\vdots&&\vdots&\vdots\end{pmatrix}\;=\;\begin{pmatrix}&&1&&\\\
&t&&x_{t}&\\\
\frac{t^{2}}{2}&\int_{0}^{t}s\,dx_{s}&&\int_{0}^{t}x_{s}ds&\frac{x_{t}^{2}}{2}\\\
\vdots&\vdots&&\vdots&\vdots\end{pmatrix}.$ (3.8)

As can be seen, each entry in the above infinite pyramid generates two
descendants by either integrating the former with respect to $t\sim 0$ or
$x\sim 1$. Inversely, we gather from (3.7) that the functional derivatives
permit to retrieve the antecedent by removing the rightmost letter of a given
word.

######  Remark 3.6.

It is essential to enlarge one-dimensional paths with the time itself.
Otherwise, the signature would only consists of the rightmost diagonal of the
signature tree $\eqref{eq:tree}$, namely
$\operatorname{\mathcal{S}}_{\mathds{1}_{k}}(X_{t})=\frac{\delta
x_{t}^{k}}{k!}$, $k\geq 0$. Hence, the only information of the path contained
in the signature would be its endpoints. Moreover, adding the increasing path
$t\mapsto t$ ensures that the signature map
$X\mapsto\operatorname{\mathcal{S}}(X)$ is invertible; see [25]. In fact,
knowing the words
$\alpha^{(k)}\mathrel{\mathop{\ordinarycolon}}=1\bm{0}_{k+1}\,$, $k\geq 0$, is
enough to show the injectivity of the signature. Indeed, observe that
$\operatorname{\mathcal{S}}_{\alpha^{(k)}}(X_{t})=\int_{0}^{t}x_{s}\frac{(t-s)^{k}}{k!}ds,$
which are scaled $L^{2}([0,t])$ product of $X_{t}$ with the time-reversed
monomials $s\mapsto(t-s)^{k}$. As the latter form a complete basis of
$L^{2}([0,t])$, the path can thus be uniquely retrieved from
$(\operatorname{\mathcal{S}}_{\alpha^{(k)}}(X_{t}))_{k\geq 0}$. See Section
2.3 in [47] for further details.

######  Remark 3.7.

Contrary to rough path theory [34, 21] where iterated integrals arise from an
imposed algebraic property$-$namely, Chen’s identity [10]$-$our framework
allows to express the signature as limits of trapezoidal Riemann sums. This
exhibits a clear practical advantage as the integrals are directly
interpretable. Admittedly, a fair comparison is difficult as we here assume
one space dimension and different conditions on $X$ (finite quadratic
variation along a sequence versus $p-$variation metric in rough path theory).

We now collect several properties of signature functionals.

######  Proposition 3.8.

Let $\operatorname{\mathcal{S}}$ be the signature in $\Omega^{\Pi}$ and fix
two words $\alpha,\gamma$. Then the following properties holds:

  * (i)

(Higher derivatives) $\Delta_{\gamma}\operatorname{\mathcal{S}}_{\alpha}\neq
0$ if and only if $\alpha=\beta\gamma$ for some
$\beta\in\operatorname{\mathbb{A}}$. If so, then
$\Delta_{\gamma}\operatorname{\mathcal{S}}_{\alpha}=\operatorname{\mathcal{S}}_{\beta}$.

  * (ii)

(Linear independence of signature functionals) The signature functionals are
linearly independent in the sense that $\forall\
\operatorname{\mathbb{A}}^{\prime}\subseteq\operatorname{\mathbb{A}}$,
$|\operatorname{\mathbb{A}}^{\prime}|<\infty$, and coefficients
$(c_{\alpha})_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}$ such that
$\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}\equiv
0$, then $c_{\alpha}=0$ $\forall\
\alpha\in\operatorname{\mathbb{A}}^{\prime}$.

  * (iii)

(Linear dependence of signature $T-$functionals) The signature functionals
restricted to $\Omega_{T}^{\Pi}$ are linearly dependent.

  * (iv)

(Linear independence of signature $T-$functionals) The family
$\\{\operatorname{\mathcal{S}}_{\beta 1}|_{\Omega_{T}^{\Pi}}\
\mathrel{\mathop{\ordinarycolon}}\ \beta\in\operatorname{\mathbb{A}}\\}$ forms
a linearly independent system.

###### Proof.

See Section A.3. ∎

An alternative formulation of assertion (ii) is given in [18]. Assertion (iv)
states that essentially ”half” of the words are retained in order to obtain an
independent system. In fact, more can be said when considering words up to
some fixed length, as stated in the next Theorem.

######  Theorem 3.9.

The family $\\{\operatorname{\mathcal{S}}_{\beta
1}|_{\Omega_{T}^{\Pi}}\mathrel{\mathop{\ordinarycolon}}|\beta|<K\\}$ is a
basis for the space spanned by
$\\{\operatorname{\mathcal{S}}_{\alpha}|_{\Omega_{T}^{\Pi}}\
\mathrel{\mathop{\ordinarycolon}}\ |\alpha|\leq K\\}$.

###### Proof.

See Section A.4. ∎

####  3.1.4 Main Results

We first state the functional Taylor expansion (FTE) in general form (Theorem
3.10) and later discuss some important consequences (Corollaries 3.11 and
3.12). Multiple derivatives are denoted by
$\Delta_{\alpha}=\Delta_{\alpha_{1}}\cdots\Delta_{\alpha_{k}}$. For instance,
$\Delta_{001}f=\Delta_{t}(\Delta_{t}(\Delta_{x}f))$, where we stress again
that $0,1$ is identified with $t,x$, respectively. Besides if
$\alpha=\varnothing$, then $\Delta_{\alpha}$ is the identity operator, i.e.
$\Delta_{\varnothing}f=f$.

######  Theorem 3.10.

(Functional Taylor expansion (FTE)) Let
$f\in\operatorname{\mathbb{C}}^{K,K+1}(\Lambda)$ and fix two paths
$X_{s}\in\Lambda_{s}$, $Y_{u}\in\Omega^{\Pi}_{u}$ with $s,u\in[0,T]$. Then, we
have,

$\displaystyle f(X_{s}\oplus Y_{u})$
$\displaystyle=\sum_{|\alpha|<K}\Delta_{\alpha}f(X_{s})\operatorname{\mathcal{S}}_{\alpha}(Y_{u})+R_{K}(X_{s},Y_{u}),$
(3.9)

with the remainder functional,

$\displaystyle R_{K}(X_{s},Y_{u})$
$\displaystyle=\sum_{|\alpha|=K}\int_{\triangle_{K,u}}\Delta_{\alpha}f(X_{s}\oplus
Y_{t_{1}})\circ\,dy^{\alpha}.$ (3.10)

###### Proof.

See Section A.5. ∎

We stress that $X_{s}$ may be completely different from the added path $Y_{u}$
$-$ see Fig. 5. In particular, the signature of $X_{s}$ may well be undefined.
When $X_{s}\in\Omega^{\Pi}$, we can expand $f$ around a portion of $X_{s}$
itself. This is the essence of the next corollary. If $X\in\Lambda$ is of
length at least $t>0$, write $X|_{[s,t]}\in\Lambda_{t-s}$, $s\leq t$, for the
restriction of $X$ to $[s,t]$, i.e. $X|_{[s,t]}(u)=x_{s+u}$, $u\in[0,t-s]$.
Also, when $X$ is continuous, we can define $X|_{[t,s]}$ as the time-reversed
version of $X|_{[s,t]}$, that is

$X|_{[t,s]}(u)=\overleftarrow{X|_{[s,t]}}(u)\mathrel{\mathop{\ordinarycolon}}=x_{t-u},\quad
u\in[0,t-s].$

Note that the continuity of $X$ is essential as the time-reversed path would
be otherwise strictly càglàd, hence no longer in $\Lambda$. We adopt the
convention that $X_{t}\oplus X|_{[t,s]}=X_{s}$ so concatenating a time-
reversed path reduces the length.

######  Corollary 3.11.

Let $X\in\Omega^{\Pi}_{t}$ and
$f\in\operatorname{\mathbb{C}}^{K,K+1}(\bar{\Omega}^{\Pi})$. Then for all
$s<t$,

$\displaystyle f(X_{t})$
$\displaystyle=\sum_{|\alpha|<K}\Delta_{\alpha}f(X_{s})\operatorname{\mathcal{S}}_{\alpha}(X|_{[s,t]})+\sum_{|\alpha|=K}\int_{\triangle_{K,t-s}}\Delta_{\alpha}f(X_{s+t_{1}})\circ\,dx_{s+\cdot}^{\alpha}\;.$
(3.11)

Also, swapping the roles of $s$ and $t$ gives the anticipative expansion,

$\displaystyle f(X_{s})$
$\displaystyle=\sum_{|\alpha|<K}\Delta_{\alpha}f(X_{t})\operatorname{\mathcal{S}}_{\alpha}(X|_{[t,s]})+\sum_{|\alpha|=K}\int_{\triangle_{K,t-s}}\Delta_{\alpha}f(X_{t-t_{1}})\circ\,dx_{t-\cdot}^{\alpha}\;.$
(3.12)

###### Proof.

See Section A.6. ∎

Figure 5: Illustration of the pertubation paths ($Y$) added to the source path
($X$) appearing in the functional Taylor expansion (Theorem 3.10).

The expansion $\eqref{eq:CorFTE1}$ is in appearance similar to Litterer and
Oberhauser [31, Theorem 14]. The main difference is that our expansion is made
pathwise. Also, the word enumeration differs: Litterer and Oberhauser [31] use
the weighted length $\lVert\alpha\rVert=2|\alpha|_{0}+|\alpha|_{1}$ to
truncate the expansion. This is to reflect the ”$\sqrt{t}$ scale” of Itô
diffusions. This partition of the words will be used in Section 3.2.2 when
comparing the chaos expansion with the FTE.

We finally expand a functional around the initial value of the path. Borrowing
the terminology from differential calculus, we call such a series a functional
Maclaurin expansion.

######  Corollary 3.12.

(Functional Maclaurin expansion) Let $f$ be a functional in
$\operatorname{\mathbb{C}}^{K,K+1}(\bar{\Omega}^{\Pi})$. Then for all
$t\in[0,T]$, the following expansion holds,

$\displaystyle f(X_{t})$
$\displaystyle=\sum_{|\alpha|<K}\Delta_{\alpha}f(X_{0})\operatorname{\mathcal{S}}_{\alpha}(X_{t})+R_{K}(X_{t}),\quad
X_{t}\in\Omega_{t}^{\Pi},$ (3.13) $\displaystyle R_{K}(X_{t})$
$\displaystyle=\sum_{|\alpha|=K}\int_{\triangle_{K,t}}\Delta_{\alpha}f(X_{t_{1}})\circ\,dx^{\alpha}.$
(3.14)

######  Remark 3.13.

The functional Maclaurin expansion (or more broadly, the FTE) presents obvious
benefits to approximate functionals. Indeed, an estimate of
$f\in\operatorname{\mathbb{C}}^{K,K+1}(\bar{\Omega}^{\Pi})$ is given by the
truncation,

$f^{K}(X_{t})\mathrel{\mathop{\ordinarycolon}}=\sum_{|\alpha|\,\leq\,K}\Delta_{\alpha}f(X_{0})\operatorname{\mathcal{S}}_{\alpha}(X_{t}).$

In other words, $f^{K}(X)$ approximates the transformed path $f(X)$ by a
linear combination of signature elements. Fig. 6 provides an illustration. The
latter is referred to as a polynomial functional [31] and in finance, as a
signature payoff [35] or simply sig-payoff [14]. In calculus and differential
geometry, $f^{K}$ may also be regarded as the $K-$jet of $f$. By applying the
Stone–Weierstrass theorem, it can be shown that signature payoffs are dense in
the space of continuous functionals restricted to a compact subset of
$\Lambda$. Among others, the result is proved in Litterer and Oberhauser [31]
for continuous paths of bounded variation and in Cuchiero et al. [14] for
paths emanating from a continuous semimartingale. However, the Stone-
Weierstrass theorem only guarantees the existence of a signature payoff
arbitrarily close to a functional. The functional Taylor series, on the other
hand, makes the approximation explicit.

We now discuss several examples.

######  Example 7.

Let $f(X_{t})=\sqrt{2}\int_{0}^{t}\sin(x_{s})ds$ and assume that $x_{0}=\pi/4$
for reasons that will become clear below. Following from Example 5, we obtain
$\Delta_{t}f=\sqrt{2}\sin(x_{t})$ and $\Delta_{x}f=0$. The only non-zero,
second order derivative is therefore $\Delta_{xt}f=\sqrt{2}\cos(x_{t})$. Then
Corollary 3.12 gives the approximations

$\displaystyle f^{1}(X_{t})$
$\displaystyle=\Delta_{t}f(X_{0})\operatorname{\mathcal{S}}_{0}(X_{t})=\sqrt{2}\sin(\pi/4)t=t,$
$\displaystyle f^{2}(X_{t})$
$\displaystyle=f^{1}(X_{t})+\Delta_{tx}f(X_{0})\operatorname{\mathcal{S}}_{10}(X_{t})=f^{1}(X_{t})+\sqrt{2}\cos(\pi/4)\operatorname{\mathcal{S}}_{10}(X_{t})=\int_{0}^{t}(1+x_{s})ds.$

Fig. 6 provides an illustration in the $(t,X,f(X))$ space for two trajectories
of different lengths.

Figure 6: Taylor approximations of
$f(X_{t})=\sqrt{2}\int_{0}^{t}\sin(x_{s})ds$ of order $1$ and $2$, giving
$f^{1}$, $f^{2}$, respectively. Two realisations are shown in the $(t,X,f(X))$
space.

######  Example 8.

(Taylor expansion) Let
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$
be an analytic European payoff. Going back to the fundamentals, Taylor’s
theorem gives

$h(x_{t})=\sum_{k=0}^{K-1}h^{(k)}(x_{0})\frac{(x_{t}-x_{0})^{k}}{k!}+R_{K}(x_{t}),\quad
R_{K}(x_{t})=o(|x_{t}-x_{0}|^{K-1}).$ (3.15)

It is often overlooked that the scaled monomials appearing in the expansion
are in fact iterated integrals. Indeed, observe that
$\frac{(x_{t}-x_{0})^{k}}{k!}=\int_{0}^{t}\cdots\int_{0}^{t_{2}}\circ\,dx_{t_{1}}\ldots\circ
dx_{t_{k}}=\operatorname{\mathcal{S}}_{\mathds{1}_{k}}(X_{t}).$ Moreover, the
integral form of the remainder allows us to write

$\displaystyle
R_{K}(x_{t})=\int_{x_{0}}^{x_{t}}h^{(K)}(z)\frac{(z-x_{0})^{K-1}}{(K-1)!}dz=\int_{\triangle_{K,t}}h^{(K)}(x_{t_{1}})\circ\,dx^{\mathds{1}_{k}}.$

On the other hand, we can embed $h$ into the space of functionals by defining
$f(X_{t})=h(x_{t})$. Of course, $\Delta_{x}f=\partial_{x}h$ and
$\Delta_{t}f=\partial_{t}h=0$. Hence $f\in\operatorname{\mathbb{C}}^{K,K+1}$
and Corollary 3.12 yields

$\displaystyle f(X_{t})$
$\displaystyle=\sum_{k=0}^{K-1}\Delta_{\mathds{1}_{k}}f(X_{0})\operatorname{\mathcal{S}}_{\mathds{1}_{k}}(X_{t})+\int_{\triangle_{K,t}}\Delta_{\mathds{1}_{k}}f(X_{t_{1}})\circ\,dx^{\mathds{1}_{k}},$

which is consistent with $\eqref{eq:stdTaylor}$. We conclude that the elements
$\\{\operatorname{\mathcal{S}}_{\varnothing},\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{11},\operatorname{\mathcal{S}}_{111},\ldots\\}$
suffices to replicate European payoffs. Equivalently, they allow to recover
the volatility surface.

######  Example 9.

(Stochastic Taylor expansion) We slightly digress from our deterministic
setting and verify that Theorem 3.10 is a generalization of the stochastic
Stratonovich-Taylor expansion [2, 29]. For simplicity, let us stick to the
one-dimensional case and consider the nonautonomous Stratonovich stochastic
differential equation,

$dz_{t}=a(t,z_{t})dt+b(t,z_{t})\circ dx_{t},\quad
z_{0}\in\operatorname{\mathbb{R}},$ (3.16)

for smooth (hence Lipschitz) measurable functions
$a,b\mathrel{\mathop{\ordinarycolon}}[0,T]\times\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$,
and a standard Brownian motion $X$. If we further assume that $a,b$ have at
most linear growth, there exists a unique strong solution of $\eqref{eq:SDE}$
[29, Theorem 4.5.3.]. Moreover, the adaptedness of $Z$ ensures the existence
of a functional
$\operatorname{\mathcal{I}}\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}-$the
Itô map$-$such that

$z_{t}=\operatorname{\mathcal{I}}(X_{t})=z_{0}+\int_{0}^{t}a(s,\operatorname{\mathcal{I}}(X_{s}))ds+\int_{0}^{t}b(s,\operatorname{\mathcal{I}}(X_{s}))\circ
dx_{s}.$ (3.17)

Given
$h\in\operatorname{\mathcal{C}}^{K,K+1}([0,T],\operatorname{\mathbb{R}})$ and
$s\leq t$, the Stratonovich-Taylor expansion reads

$\displaystyle h(t,z_{t})$
$\displaystyle=\sum_{|\alpha|<K}\operatorname{\mathcal{V}}_{\alpha}h(s,z_{s})\operatorname{\mathcal{S}}_{\alpha}(X_{t})+\sum_{|\alpha|=K}\int_{\triangle_{K,t-s}}\operatorname{\mathcal{V}}_{\alpha}h(s+t_{1},z_{s+t_{1}})\circ\,dx^{\alpha},$
(3.18)

with the vector fields
$\operatorname{\mathcal{V}}_{0}=\partial_{t}+a\,\partial_{z}$,
$\operatorname{\mathcal{V}}_{1}=b\,\partial_{z}$, and
$\operatorname{\mathcal{V}}_{\alpha}=\operatorname{\mathcal{V}}_{\alpha_{1}}\cdots\operatorname{\mathcal{V}}_{\alpha_{k}}$.
Next, define the functional $f(X_{t})=h(t,\operatorname{\mathcal{I}}(X_{t}))$.
Comparing $\eqref{eq:STE}$ with Corollary 3.11, we gather that
$\operatorname{\mathcal{V}}_{\alpha}h=\Delta_{\alpha}f$ must hold. Even though
the regularity of $f$ is a priori not guaranteed, we will see that the
smoothness of $h$, $a$, $b$ carries over to $f$. It suffices to show that
$\operatorname{\mathcal{V}}_{\alpha}$ acts on $h$ exactly as $\Delta_{\alpha}$
acts on $f$ for $\alpha=0,1$. Indeed, the same would be true for longer words
due to the similar recursive structure of the differential operators. If
$\alpha=0$, the chain rule, equation $\eqref{eq:SDEFunc}$ and Example 5 give
respectively

$\displaystyle\Delta_{t}f=\partial_{t}h+\partial_{z}h\,\Delta_{t}\\!\operatorname{\mathcal{I}}=(\partial_{t}+a\,\partial_{z})h=\operatorname{\mathcal{V}}_{0}h.$

As $b$ is smooth and $X$ continuous $\operatorname{\mathbb{Q}}-$a.s., we have
$\Delta_{x}\operatorname{\mathcal{I}}=b$. Hence
$\Delta_{x}f=\partial_{z}h\,\Delta_{x}\\!\operatorname{\mathcal{I}}=b\,\partial_{z}h=\operatorname{\mathcal{V}}_{1}h.$
In conclusion, the two expansions coincide as claimed.

####  3.1.5 Remainder Estimates and Real Analytic Functionals

First, we give in Proposition 3.14 an upper bound for the remainder of the
FTE. In the diffusion case, one typically study the convergence of
(functional) Taylor expansions in the $L^{2}(\operatorname{\mathbb{Q}})$
sense; see Kloeden and Platen [29, Section 5.10.] or Litterer and Oberhauser
[31, Section 5.10.]. Indeed, Itô’s isometry greatly facilitates manipulation
of expected iterated integrals. However, this powerful tool is no longer
available when one is interested in the pathwise convergence of the Taylor
expansion. Another avenue pursued in Buckdahn et al. [8] consists of applying
the Burkholder–Davis–Gundy inequality several times to obtain an ”almost sure”
upper bound of the remainder. This procedure, although powerful, still
requires a stochastic framework. We here propose a purely pathwise remainder
estimate. For simplicity, we establish the result for the functional Maclaurin
expansion as in Corollary 3.12. To this end, we associate to each path
$X_{t}\in\Lambda$ the seminorm333Note that $[x_{0},x_{s}]$ is understood as
$[x_{0}\wedge x_{s},x_{0}\vee x_{s}]$ throughout. We also recall that
$X_{s}^{(\varepsilon)}=X_{s}^{\varepsilon-x_{s}}$ (so that
$X_{s}^{(\varepsilon)}(s)=\varepsilon$) as seen in Section 3.1.2.

$\lVert\varphi\rVert_{X_{t}}=\sup_{s\leq
t}\sup_{\varepsilon\in[x_{0},x_{s}]}|\varphi(X_{s}^{(\varepsilon)})|,\quad\varphi\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}.$
(3.19)

As $\\{X_{s}^{(\varepsilon)}\ \mathrel{\mathop{\ordinarycolon}}\
\varepsilon\in[x_{0},x_{s}],\ s\leq t\\}$ is a compact subset of $\Lambda$,
then $\lVert\varphi\rVert_{X_{t}}<\infty$ whenever $\varphi$ is
$\Lambda-$continuous. Lastly, let $\lVert X_{t}\rVert_{\infty}=\sup_{s\leq
t}|x_{s}|$ and assume without loss of generality that $x_{0}=0$.

######  Proposition 3.14.

(Remainder Estimate) Let $X_{t}\in\Omega_{t}^{\Pi}$ and
$f\in\operatorname{\mathbb{C}}^{\infty,\infty}(\bar{\Omega}^{\Pi})$. For
$K\geq 1$, consider the functional Maclaurin expansion

$f(X_{t})=\sum_{|\alpha|<K}\Delta_{\alpha}f(\bm{0})\operatorname{\mathcal{S}}_{\alpha}(X_{t})+R_{K}(X_{t}).$

Moreover, write
$r_{\alpha}(X_{t})=\int_{\triangle_{K,t}}\Delta_{\alpha}f(X_{t_{1}})\circ
dx^{\alpha},$ so that $R_{K}(X_{t})=\sum_{|\alpha|=K}r_{\alpha}(X_{t}).$ Let
$|\alpha|_{01}$ be the number of occurrences of the word $01$ in $\alpha$.
Then for each $|\alpha|=K$, we have

$|r_{\alpha}(X_{t})|\leq\frac{2^{|\alpha|_{01}}}{|\alpha|_{0}!}c_{\alpha}\
t^{|\alpha|_{0}}\lVert X_{t}\rVert_{\infty}^{|\alpha|_{1}},$ (3.20)

with $c_{\alpha}=\lVert\Delta_{\alpha}f\rVert_{X_{t}}+(t\vee 1)\
\lVert\Delta_{0\alpha}f\rVert_{X_{t}}$. Letting
$C_{K}=\max_{|\alpha|=K}c_{\alpha}$ and $\rho(X_{t})=2(t\vee\lVert
X_{t}\rVert_{\infty})$, then the remainder is bounded by

$|R_{K}(X_{t})|\leq C_{K}\rho(X_{t})^{K}.$ (3.21)

###### Proof.

See Section A.7 ∎

In light of $\eqref{eq:remainder}$ and the definition of $\rho(\cdot)$, we
conclude that the functional Taylor expansion converges to its functional
Taylor series whenever both $t$ and $\lVert X_{t}\rVert_{\infty}$ are strictly
smaller than $1/2$. As $X_{t}$ is assumed to be uniformly continuous in
Proposition 3.14, we can therefore find $t$ sufficiently small such that the
FTE converges.

Note that Proposition 3.14 gives an error bound of the FTE at a single path.
Shifting gears, we now investigate functionals such that its FTE converges for
every path inside a neighborhood of some fixed $X\in\Lambda$. In line with the
terminology from calculus, we shall qualify these functionals as real
analytic.

What remains is to choose an adequate topology on an appropriate subspace of
$\Lambda$. Let $\Omega^{\text{Lip}}_{t}\subseteq\Omega^{\Pi}_{t}$ be the
subspace of Lipschitz paths, that is $Y\in\Omega^{\text{Lip}}_{t}$ if and only
if $[Y]_{\text{Lip}}\mathrel{\mathop{\ordinarycolon}}=\sup_{u\neq
s}\frac{|y_{s}-y_{u}|}{|s-u|}<\infty$. As usual, we write
$\Omega^{\text{Lip}}=\bigcup_{t\in[0,T]}\Omega^{\text{Lip}}_{t}$. As Lipschitz
paths have finite variation, the signature is therefore defined without
difficulty as Riemann-Stieljes integrals. Moreover, one has the upper bound,

$|\operatorname{\mathcal{S}}_{\alpha}(Y_{t})|\leq[Y_{t}]_{\text{Lip}}^{|\alpha|_{1}}\frac{t^{K}}{K!},\quad|\alpha|=K.$
(3.22)

For $s\leq t$ and Lipschitz paths
$Y_{t},Y_{s}^{\prime}\in\Omega^{\text{Lip}}$, define the metric

$d_{1}(Y_{t},Y_{s}^{\prime})=t-s+|y_{0}-y_{0}^{\prime}|+[Y_{t}-Y_{s,t-s}^{\prime}]_{\text{Lip}},$

and let $B_{\delta}^{\text{Lip}}(Y)$ be the Lipschitz ball centered at
$Y\in\Omega^{\text{Lip}}$ of radius $\delta$, i.e. the set of paths
$Y^{\prime}\in\Omega^{\text{Lip}}$ such that $d_{1}(Y,Y^{\prime})<\delta$.
When $Y=\bm{0}\in\Lambda_{0}$, we write $B_{\delta}^{\text{Lip}}$ instead of
$B_{\delta}^{\text{Lip}}(\bm{0})$.

######  Definition 3.15.

Let $\Lambda^{\prime}\subseteq\Lambda$. A functional
$f\in\operatorname{\mathbb{C}}^{\infty,\infty}(\Lambda^{\prime})$ is real
analytic at $X\in\Lambda^{\prime}$, if its functional Taylor expansion
converges in a neighborhood of $X$, i.e. there exists $\delta>0$ such that

$f(X\oplus
Y)=\sum_{\alpha\in\operatorname{\mathbb{A}}}\Delta_{\alpha}f(X)\operatorname{\mathcal{S}}_{\alpha}(Y)\mathrel{\mathop{\ordinarycolon}}=\lim_{K\to\infty}\sum_{|\alpha|\leq
K}\Delta_{\alpha}f(X)\operatorname{\mathcal{S}}_{\alpha}(Y)\quad\forall\ Y\in
B_{\delta}^{\text{Lip}}.$ (3.23)

The largest $\delta\geq 0$ such that $\eqref{eq:analytic}$ holds is called the
radius of convergence of the functional series of $f$ centered at $X$ and is
denoted by $\varrho(f,X)$. Finally, we say that $f$ is real analytic
(notation: $f\in\operatorname{\mathbb{C}}^{\omega}(\Lambda^{\prime})$) if $f$
is real analytic at every $X\in\Lambda^{\prime}$.

We emphasize that the center path $X$ in the above definition need not be
Lipschitz; what really matters is the regularity of the concatenated path $Y$.
Nevertheless, the functional has to be defined at $X\oplus Y$ which may
restrict the subspace $\Lambda^{\prime}\subseteq\Lambda$ as we shall see in
the next example.

######  Example 10.

It is easily seen that every signature functional belongs to
$\operatorname{\mathbb{C}}^{\omega}(\Lambda^{\prime})$ with
$\Lambda^{\prime}=\bar{\Omega}^{\Pi}$. Indeed for any word
$\alpha\neq\varnothing$ and $X\in\Omega^{\Pi}$, Chen’s identity [10] reads

$\operatorname{\mathcal{S}}_{\alpha}(X\oplus
Y)=\sum_{\beta\gamma=\alpha}\operatorname{\mathcal{S}}_{\beta}(X)\operatorname{\mathcal{S}}_{\gamma}(Y)\quad\forall\
Y\ \in\Omega^{\text{Lip}}.$ (3.24)

Since
$\Delta_{\gamma}\operatorname{\mathcal{S}}_{\beta\gamma}(X)=\operatorname{\mathcal{S}}_{\beta}(X)$
(see Proposition 3.8 (i)) we conclude that the right side of
$\eqref{eq:ChenIdentity}$ is the Taylor series of
$\operatorname{\mathcal{S}}_{\alpha}(X\oplus Y)$ and in turn that
$\operatorname{\mathcal{S}}_{\alpha}\in\operatorname{\mathbb{C}}^{\omega}(\bar{\Omega}^{\Pi})$.
Moreover, the radius of convergence at $X$ is infinite, i.e.
$\varrho(\operatorname{\mathcal{S}}_{\alpha},X)=\infty$.

Finally, we provide an explicit upper bound for the remainder term of smooth
functionals with Lipschitz pertubation paths and a characterization for the
corresponding radius of convergence.

######  Proposition 3.16.

Let $\Lambda^{\prime}\subseteq\Lambda$, $X\in\Lambda^{\prime}$, and
$f\in\operatorname{\mathbb{C}}^{K,K+1}(\Lambda^{\prime})$. Suppose that there
exist constants $C_{1},C_{2}\geq 0$ such that $|\Delta_{\alpha}f(X\oplus
Y^{\prime})|\leq C_{1}^{K}(K!)^{C_{2}}$ for all $|\alpha|=K$ and
$Y^{\prime}\in\Omega^{\textnormal{Lip}}$. Then, the remainder
$\eqref{eq:FTE2}$ of the FTE of $f$ satisfies

$\displaystyle|R_{K}(X,Y_{u})|$
$\displaystyle\leq(K!)^{C_{2}-1}\rho(Y_{u})^{K},\quad
Y_{u}\in\Omega_{u}^{\textnormal{Lip}},\quad u\geq 0,$ (3.25)

with $\rho(Y_{u})=2\ C_{1}([Y_{u}]_{\textnormal{Lip}}\vee 1)u$. If $f$ is also
smooth, then it is real analytic if and only if $C_{2}\leq 1$, with radius of
convergence

$\varrho(f,X)\geq\frac{1}{2C_{1}}\wedge 1\;\ \text{ if
$C_{2}<1$},\quad\varrho(f,X)=\infty\;\ \text{ if $C_{2}=1$}.$

###### Proof.

For all $|\alpha|=K$, note that
$|\int_{\triangle_{K,u}}\Delta_{\alpha}f(X\oplus Y_{t_{1}})dy^{\alpha}|\leq
C_{1}^{K}(K!)^{C_{2}}[Y]_{1}^{|\alpha|_{\textnormal{Lip}}}\frac{u^{K}}{K!}$
using $\eqref{eq:sigLipschitz}$ and the assumption on $\Delta_{\alpha}f$.
Hence $\eqref{eq:LipRemainder}$ follows from
$[Y]_{1}^{|\alpha|_{1}}\leq([Y]_{\textnormal{Lip}}\vee 1)^{K}$ and the
triangular inequality. Now if $C_{2}=1$ and $\delta=\frac{1}{2C_{1}}\wedge 1$,
then for all $Y_{u}\in B_{\delta}^{\text{Lip}}$,
$[Y_{u}]_{\textnormal{Lip}}<\delta\leq 1$ and similarly $u<1$. This implies
that $\rho(Y_{u})=2C_{1}u<1$ as desired. The cases $C_{2}<1$ and $C_{2}>1$ are
obvious. ∎

######  Remark 3.17.

Alternatively, one can choose
$\Omega^{\gamma}_{t}=\operatorname{\mathcal{C}}^{0,\gamma}([0,t],\operatorname{\mathbb{R}})$,
the space of $\gamma-$Hölder paths. When $\gamma>1/2$, the signature elements
are well-defined using Young integration [51]. More generally, one can
consider paths of finite $p-$variation with $p\in(1,2)$ and use estimates of
the iterated iterated as in Lyons [34]. In this work, we nevertheless stick to
Lipschitz paths for ease of presentation.

####  3.1.6 Connections with Static Expansions

Intrinsic Value Expansion (Section 2.4). Let $g$ be a smooth $T-$functional
and consider its IVE,

$g(X_{T})=g(X_{0,T})+\sum_{k=1}^{\infty}I_{k}\varphi_{k}(X_{T}),\qquad\
X_{T}\in\Omega^{\Pi}_{T},$ (3.26)

where we define $\varphi_{k}(t_{1},\ldots,t_{k})=D_{t_{1}\cdots
t_{k}}g(X_{0,T})$ and
$I_{k}\varphi_{k}(X_{T})=\int_{\triangle_{k,T}}\varphi_{k}\,\circ dx^{\otimes
k}$. On the other hand, assuming that the associated intrinsic functional
$\iota_{0}g(X_{t})=g(X_{t,T-t})$ is real analytic, one can compute its
functional Maclaurin series. In particular, for $X_{T}\in\Omega^{\Pi}_{T}$,

$\displaystyle g(X_{T})=\iota_{0}g(X_{T})=\underbrace{\iota_{0}g(\bm{0})}_{=\
g(X_{0,T})}+\sum_{\alpha\in\operatorname{\mathbb{A}}}\Delta_{\alpha}\iota_{0}g(\bm{0})\operatorname{\mathcal{S}}_{\alpha}(X_{T}).$
(3.27)

The aim of this section is to show how $\eqref{eq:IVEComp}$ relates to
$\eqref{eq:FTEComp}$. One way is to derive the intrinsic value expansion of
each signature $T-$functional
$\operatorname{\mathcal{S}}_{\alpha}|_{\Lambda_{T}}$. In other words, we seek
$(\varphi_{k}^{\alpha})_{k\geq 1}$ such that

$\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\operatorname{\mathcal{S}}_{\alpha}(X_{0,T})+\sum_{k\geq
1}I_{k}\varphi_{k}^{\alpha}(X_{T}).$

First, we conclude from a homogeneity argument that $\varphi^{\alpha}_{k}=0$
unless $k=|\alpha|_{1}$. Indeed, defining the scaled path $\lambda
X_{T}=(\lambda x_{t})_{t\in[0,T]}$, $\lambda>0$, we must have that

$\lambda^{|\alpha|_{1}}\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\operatorname{\mathcal{S}}_{\alpha}(\lambda
X_{T})=\operatorname{\mathcal{S}}_{\alpha}(X_{0,T})+\sum_{k\geq
1}I_{k}\varphi_{k}^{\alpha}(X_{T})\lambda^{k},\qquad\forall\ \lambda>0.$

The proposition gives an explicit formula for $(\varphi^{\alpha}_{k})$; see
the first example in [18] for a similar expression in the case of absolutely
continuous paths.

######  Proposition 3.18.

Let $\alpha\in\operatorname{\mathbb{A}}$ and $k=|\alpha|_{1}$. Moreover,
express $\alpha$ as

$\alpha=\bm{0}_{\gamma_{0}}1\bm{0}_{\gamma_{1}}1\cdots 1\bm{0}_{\gamma_{k}},$

where we allow $\gamma_{l}=0$ ($\Longrightarrow\
\bm{0}_{\gamma_{l}}=\varnothing$) for $l=0,\ldots,k$. Then
$\operatorname{\mathcal{S}}_{\alpha}(X_{T})=I_{k}\varphi_{k}^{\alpha}(X_{T})$
with the kernel

$\varphi_{k}^{\alpha}(t_{1},\ldots,t_{k})=\prod_{l=0}^{k}\frac{(t_{l+1}-t_{l})^{\gamma_{l}}}{\gamma_{l}!},\quad(t_{0}=0,\
t_{k+1}=T).$ (3.28)

###### Proof.

The case $k=0$ is trivial. For $k\geq 1$, the result can be established by
computing the $k-$th order Malliavin derivatives of
$\operatorname{\mathcal{S}}_{\alpha}$. However, this would be tedious.
Alternatively, we can use Fubini’s theorem iteratively to rearrange the order
of integration. Writing $\alpha=\beta 1\bm{0}_{\gamma_{k}}$ with
$\beta=\bm{0}_{\gamma_{0}}1\bm{0}_{\gamma_{1}}1\cdots 1\bm{0}_{\gamma_{k-1}}$,
observe that

$\displaystyle\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\int_{\triangle_{\gamma_{k}+1,T}}\operatorname{\mathcal{S}}_{\beta}(X_{t_{0}})\circ
dx_{t_{0}}dt^{\otimes\gamma_{k}}=\int_{0}^{T}\operatorname{\mathcal{S}}_{\beta}(X_{t_{0}})\frac{(T-t_{0})^{\gamma_{k}}}{\gamma_{k}!}\circ
dx_{t_{0}}.$

The result thus follows from a simple induction on $k$. ∎

It is worth noting that the kernel of signature functionals does not depend on
the precise location of the time points. Indeed, we see from
$\eqref{eq:kernelSig}$ that $\varphi_{k}^{\alpha}$ only entails time
increments. Finally, we connect the kernels $\varphi_{k}$ of $g$ with the
signature kernels $\varphi_{k}^{\alpha}$.

######  Corollary 3.19.

Let $g$ be a smooth $T-$function with analytic intrinsic functional
$\iota_{0}g$. Then the kernel of the intrinsic value expansion can be
expressed as,

$D_{t_{1}\cdots
t_{k}}g(X_{0,T})=\sum_{|\alpha|_{1}=k}\Delta_{\alpha}\iota_{0}g(X_{0})\prod_{l=0}^{k}\frac{(t_{l+1}-t_{l})^{\gamma_{l}}}{\gamma_{l}!}.$

###### Proof.

First, we recall from $\eqref{eq:IVEComp}$ and $\eqref{eq:FTEComp}$ that

$\sum_{k\geq
1}I_{k}\varphi_{k}(X_{T})=g(X_{T})-g(X_{0,T})=\sum_{\alpha\in\operatorname{\mathbb{A}}\setminus\\{\varnothing\\}}\Delta_{\alpha}\iota_{0}g(\bm{0})\operatorname{\mathcal{S}}_{\alpha}(X_{T})\\\
=\sum_{k\geq
1}\sum_{|\alpha|_{1}=k}\Delta_{\alpha}\iota_{0}g(\bm{0})I_{k}\varphi_{k}^{\alpha}(X_{T}).$
(3.29)

Using the linearity of $I_{k}$ and the fact that $\eqref{eq:IVEFTE}$ must hold
for all $X_{T}\in\Lambda_{T}$, then
$\varphi_{k}=\sum_{|\alpha|_{1}=k}\Delta_{\alpha}\iota_{0}g(\bm{0})\varphi_{k}^{\alpha}$.
We finally invoke the expression of $\varphi_{k}^{\alpha}$ given in
Proposition 3.18 to conclude. ∎

Wiener Chaos (Section 2.5). We show how the chaos expansion can be transformed
into the FTE for path-independent $T-$functionals. Let
$\operatorname{\mathbb{Q}}$ be the Wiener measure as usual and
$g(X_{T})=h(x_{T})$ for some real analytic function
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$.
The following result provides a characterization of path independence for
$T-$functionals. In passing, it brings us one step closer to the functional
Taylor expansion. We refer the reader to Section 2.5.2 for the definition of
the operator $\operatorname{\mathcal{M}}$ and its domain
$\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}}$ (given in
(2.29)).

######  Proposition 3.20.

Let $g$ be a $T-$functional such that
$\operatorname{\mathcal{M}}_{T}g\in\operatorname{\mathcal{D}}_{\\!\operatorname{\mathcal{M}}}$
and consider its price functional $f=\iota_{\operatorname{\mathbb{Q}}}g$
introduced in Example 3. Then the following are equivalent:

  * (i)

$g$ is path-independent.

  * (ii)

$\Delta_{\mathds{1}_{k}}f(X)$ is an $\operatorname{\mathbb{Q}}-$martingale for
all $k\geq 0$.

  * (iii)

The Wiener chaos expansion writes
$g(X_{T})=\sum_{k=0}^{\infty}\Delta_{\mathds{1}_{k}}f(\bm{0})J_{k}(X_{T})$.

###### Proof.

See Section A.8. ∎

######  Remark 3.21.

Although $f=\iota_{\operatorname{\mathbb{Q}}}g$ in Proposition 3.20 is termed
price functional, the spatial derivatives
$(\Delta_{\mathds{1}_{k}}f(X))_{k\geq 0}$ may not correspond to Greeks in
derivatives pricing. Indeed, $X$ has to be interpreted as the source of risk,
which usually differs from the underlying. For instance, a call option in the
Black-Scholes with zero dividend and interest rate would read
$g(X_{T})=(\operatorname{\mathcal{I}}(X_{T})-K)^{+}$, where $X$ is Brownian
motion and $\operatorname{\mathcal{I}}$ the Itô map
$\operatorname{\mathcal{I}}(X_{T})=x_{0}e^{\sigma
x_{T}-\frac{1}{2}\sigma^{2}T}$. Exceptions include the Bachelier model, where
the (scaled) coordinate process $X$ is the asset itself. In this case,
$(\Delta_{\mathds{1}_{k}}f(X))_{k\geq 0}$ indeed correspond to the option
Delta, Gamma, and so on.

Next, we express $k-$fold Itô iterated integrals as linear combinations of
finitely many signature functionals. First, we recall that

$J_{k}(X_{t})=h_{k}(t,x_{t}),\quad
h_{k}(t,x)=\frac{t^{k/2}}{k!}H_{k}\left(\frac{x}{\sqrt{t}}\right),$ (3.30)

where $H_{k}$ is the $k-$th probabilist’s Hermite polynomial; see [15].
Although $X_{t}\mapsto J_{k}(X_{t})$ is indistinguishable from the time-space
Hermite polynomial $h_{k}(t,x_{t})$, their functional derivatives may differ
as nicely explained in Oberhauser [39]. However, using the functional Itô
formula and the uniqueness of the Doob-Meyer decomposition, the operator
$\operatorname{\mathcal{L}}\mathrel{\mathop{\ordinarycolon}}=\Delta_{t}+\frac{1}{2}\Delta_{xx},$
must act in the same way, regardless of the representation chosen; see again
[39]. For obvious reasons, $\operatorname{\mathcal{L}}$ is often called the
causal heat operator. The next simple lemma relates the Itô iterated integrals
to the signature.

######  Lemma 3.22.

The Itô iterated integrals are smooth functionals and admit the Maclaurin
expansions

$\displaystyle
J_{k}(X_{t})=\sum_{\alpha\in\operatorname{\mathbb{A}}_{k}}(-2)^{-|\alpha|_{0}}\operatorname{\mathcal{S}}_{\alpha}(X_{t}),\quad
k\geq 0,$

with the disjoint subsets
$\operatorname{\mathbb{A}}_{k}=\left\\{\alpha\in\operatorname{\mathbb{A}}\
\mathrel{\mathop{\ordinarycolon}}\ \lVert\alpha\rVert=k\right\\}$, $\
\lVert\alpha\rVert\mathrel{\mathop{\ordinarycolon}}=2|\alpha|_{0}+|\alpha|_{1}$.

###### Proof.

See Section A.9. ∎

We observe from Lemma 3.22 that Itô iterated integrals induce a partition of
the words in $\operatorname{\mathbb{A}}$ based on the ”weighted length”
$\lVert\cdot\rVert$. Note that the latter arises naturally when establishing
convergence results about iterated Itô or Stratonovich integrals [29, 2, 33].
Finally, we bridge the gap between the FTE and Wiener-Itô chaos expansion in
this simple case.

######  Proposition 3.23.

Let $g$ be an analytic path-independent $T-$functional with price functional
$f=\iota_{\operatorname{\mathbb{Q}}}g$. Under the assumptions of Proposition
3.20, then
$\Delta_{\mathds{1}_{k}}f(\bm{0})(-2)^{-|\alpha|_{0}}=\Delta_{\alpha}f(\bm{0})$.
Consequently,

$g(X_{T})=\sum_{k=0}^{\infty}J_{k}\phi_{k}(X_{T})=\sum_{k=0}^{\infty}\sum_{\alpha\in\operatorname{\mathbb{A}}_{k}}\Delta_{\mathds{1}_{k}}f(\bm{0})(-2)^{-|\alpha|_{0}}\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\sum_{\alpha}\Delta_{\alpha}f(\bm{0})\operatorname{\mathcal{S}}_{\alpha}(X_{T}).$

###### Proof.

As $g$ is path-independent, Proposition 3.20 gives
$g(X_{T})=\sum_{k=0}^{\infty}\Delta_{\mathds{1}_{k}}f(\bm{0})J_{k}(X_{T})$ and
$\Delta_{\mathds{1}_{k}}f(X)$ is an $\operatorname{\mathbb{Q}}-$ martingale
for all $k\geq 0$. As $X$ is Brownian motion, the functional Itô formula [16]
implies that each $\Delta_{\mathds{1}_{k}}f$ solves the path-dependent PDE
$\operatorname{\mathcal{L}}\varphi=\left(\Delta_{t}+\frac{1}{2}\Delta_{xx}\right)\varphi=0.$
Put differently, any second spatial derivative can be converted into a
temporal derivative by multiplying by the factor $-2$. Thus,
$\Delta_{\mathds{1}_{k}}f=(-2)^{|\alpha|_{0}}\ \Delta_{\alpha}f$ for every
word $\alpha\in\operatorname{\mathbb{A}}_{k}$. The result follows from Lemma
3.22. ∎

######  Remark 3.24.

In the general case where the kernels are time-dependent, it is delicate to
compare the Wiener chaos expansion with the FTE. Indeed, we would need to
expand the terms $J_{k}\phi_{k}$ which does not have the required regularity
if expressed pathwise; see the discussion in [39]. Inversely, one can compute
the Wiener chaos expansion of each signature functional, namely
$\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\sum_{k=0}^{\infty}J_{k}\phi_{k}(X_{T})$
for some kernels $(\phi_{k})$ to be determined. In light of Stroock’s formula
$\eqref{eq:stroock}$, one has to compute the expected higher-order Malliavin
derivatives of the signature to obtain $(\phi_{k})$. This was conducted
recently by Ferrucci and Cass [17] for a wide class of Gaussian processes
encompassing fractional Brownian motion with Hurst parameter $H\in(1/4,1)$.

###  3.2 Other Dynamic Expansions

####  3.2.1 Hilbert Projections

Another procedure to expand functionals consists of projecting its image seen
as a collection of transformed paths. More precisely, let
$f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$,
$X\in\Lambda_{T}$ and consider $Z=f(X)$ given by $z_{t}=f(X_{t})$,
$t\in[0,T]$. If $Z\in\operatorname{\mathcal{H}}$ where
$\operatorname{\mathcal{H}}$ is a separable Hilbert space, then we can write
$Z=\sum_{k=1}^{\infty}(Z,F_{k})_{\operatorname{\mathcal{H}}}F_{k}$ for an
orthonormal basis (ONB) $\operatorname{\mathfrak{F}}=(F_{k})_{k=1}^{\infty}$
of $\operatorname{\mathcal{H}}$. Put differently,

$f(X_{t})=\sum_{k=1}^{\infty}\ (f(X),F_{k})_{\operatorname{\mathcal{H}}}\
F_{k}(t).$ (3.31)

Truncating $\eqref{eq:Hilbert}$ at some level $K\geq 1$ gives the projection
$f^{K,\operatorname{\mathfrak{F}}}(X)=\sum_{k\leq
K}(f(X),F_{k})_{\operatorname{\mathcal{H}}}\ F_{k}.$ An illustration is given
Fig. 7. One may wonder which ONB yields the best approximation$-$in some
suitable sense$-$for a fixed truncation level. For instance, given a measure
$\operatorname{\mathbb{Q}}$ on $\Lambda$ and
$\operatorname{\mathcal{H}}=L^{2}([0,T],dt)$, one can minimize the projection
error, that is

$\min_{\operatorname{\mathfrak{F}},\,\text{ONB}}\;\lVert
f-f^{K,\operatorname{\mathfrak{F}}}\rVert_{L^{2}(\Lambda)}=\min_{\operatorname{\mathfrak{F}},\,\text{ONB}}\;\lVert
f(Y)-f^{K,\operatorname{\mathfrak{F}}}(Y)\rVert_{L^{2}(\operatorname{\mathbb{Q}}\otimes
dt)},\quad K\geq 1.$

The unique optimal ONB turns out to be, the same for all truncation level $K$
and given by the eigenfunctions of the covariance kernel
$\kappa(s,t)=\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Q}}}(f(X_{s}),f(X_{t}))$.
This is a classical result established by Karhunen [28] and Loève [32] and
employing the optimal basis in (3.31) leads to the Karhunen-Loève (KL)
expansion of $f$.

Figure 7: Hilbert projection of functionals in the $(t,X,f(X))$ space.

Among other financial applications, the KL expansion allows to directly
simulate the functional(s) appearing in the payoff of exotic options [47].
Also, a market generator can be devised when $f$ represents the map linking
the random source (e.g. Brownian motion) to the stock price process. Indeed,
the Fourier coefficient in (3.31) can be ”learned” from historical data
permitting the simulation of new scenarios.

######  Remark 3.25.

If we are only given a $T-$functional $g\in L^{2}(\Lambda_{T})$, one can
consider an embedding
$\iota\mathrel{\mathop{\ordinarycolon}}L^{2}(\Lambda_{T})\hookrightarrow
L^{2}(\Lambda)$ (such as $\iota_{\operatorname{\mathbb{Q}}}$ defined in
Section 2.1) and apply the above methodology to $\iota g$. We thereafter
choose $t=T$ in (3.31) to obtain the representation

$g^{K,\operatorname{\mathfrak{F}}}(X_{T})=\sum_{k\leq K}(\iota
g(X),F_{k})_{L^{2}(dt)}F_{k}(T).$

####  3.2.2 Dynamic Wiener Chaos Expansion

We refer the reader to the beginning of Section 2.5 for some notations
pertaining to the Wiener chaos expansion. In particular, we consider a
stochastic basis
$(\Lambda,\operatorname{\mathcal{F}},\operatorname{\mathbb{F}},\operatorname{\mathbb{Q}})$
and take $\operatorname{\mathbb{Q}}$ to be the Wiener measure throughout.

It is well-known that the Wiener chaos expansion can be extended from random
variables to stochastic processes [15, Chapter 2]. Indeed, if $Y$ is a
measurable square integrable process (possibly non-adapted to the Brownian
filtration), then one can apply Theorem 2.5 at each intermediate time. The
chaos expansion for processes is of great importance in Malliavin calculus as
it allows to define, among other things, the Skorokhod integral [44]. As our
focus is on non-anticipative functionals, we henceforth suppose that $Y$ is
adapted. An application of Doob’s functional representation [27, Lemma 1.13.]
shows that every right-continuous process $Y$ adapted to the Brownian
filtration is paired with a functional
$f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$ such
that $y_{t}=f(X_{t})\,$ $\forall\ t\in[0,T],$ $\operatorname{\mathbb{Q}}-$a.s.
We therefore expect that a ”dynamic” Wiener chaos expansion is available,
namely a chaos expansion for functionals in $L^{2}(\Lambda)$. This is
confirmed in the next theorem.

######  Theorem 3.26.

(Wiener Chaos Expansion for Functionals) Every functional $f\in
L^{2}(\Lambda)$ is associated with a unique sequence $\\{\phi_{k}\in
L^{2}(\triangle_{k,T})\\}_{k\geq 1}$ such that for all $t\in[0,T]$

$\sum_{k=0}^{K}J_{k}\phi_{k+1}(\cdot,t)(X_{t})=\sum_{k=0}^{K}\int_{\triangle_{k,t}}\phi_{k+1}(t_{1},\ldots,t_{k},t)\,dx^{\otimes
k}\
\underset{L^{2}(\Lambda_{t})}{\overset{K\uparrow\infty}{\xrightarrow{\hskip
34.14322pt}}}\ f(X_{t}).$ (3.32)

###### Proof.

See [15, Chapter 2]. ∎

######  Example 11.

Let $g$ be a square integrable $T-$functional with Wiener chaos expansion
$g=\sum_{k\geq 0}J_{k}\phi_{k}$ and consider the price functional
$f(X_{t})=\iota_{\operatorname{\mathbb{Q}}}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})\
|\ X_{t}]$. Applying Theorem 3.26 to $f$, we obtain that

$f(X_{t})=\sum_{k=0}^{\infty}\int_{\triangle_{k,t}}\tilde{\phi}_{k+1}(t_{1},\ldots,t_{k},t)\,dx^{\otimes
k},$ (3.33)

for some kernels $\\{\tilde{\phi}_{k}\in L^{2}(\triangle_{k,T})\\}_{k\geq 1}$.
On the other hand, we have seen in Example 3 that $f(X_{t})$ coincides with
truncated Wiener chaos of $g$ on $[0,t]$, namely $f(X_{t})=\sum_{k\geq
0}J_{k}\phi_{k}(X_{t})$. We conclude from the uniqueness (in
$L^{2}(\operatorname{\mathbb{Q}})$) of the Wiener kernels that
$\tilde{\phi}_{k+1}(\cdot,t)=\phi_{k}|_{\triangle_{k,t}}$.

##  4 Applications

###  4.1 Pricing of Path-dependent Claims

The functional Taylor expansion has obvious applications in the pricing of
exotic options. In what follows, fix a risk-neutral measure
$\operatorname{\mathbb{Q}}$ and assume zero interest rate. Moreover, the paths
represent the evolution of the underlying stock price. If $g\in
L^{1}(\Lambda_{T})$ represents the payoff of a path-dependent claim of
European type, then its value (or price) with respect to
$\operatorname{\mathbb{Q}}$ is given by
$v_{0}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]$. If
$g$ can be nearly replicated by a signature payoff, i.e.
$g\approx\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}$
where $\operatorname{\mathbb{A}}^{\prime}$ is a finite subset of
$\operatorname{\mathbb{A}}$, then

$v_{0}\approx\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\operatorname{\mathcal{S}}_{\alpha}(Y_{T})].$
(4.1)

Thus, a pricing method can be devised provided that both the coefficients
$(c_{\alpha})$ and risk-neutral prices of the ”primitive” securities
$(\operatorname{\mathcal{S}}_{\alpha})$ are computed efficiently.

This was carried out by Arribas et al. [4] and Lyons et al. [35], where
$(c_{\alpha})$ are calculated by regressing realizations of the payoff against
the signature functionals. The expected signature is either known explicitly
[4] or implied by the market [35].

If a functional Taylor (or Maclaurin) expansion was available for $g$, we
would set $c_{\alpha}=\Delta_{\alpha}g(X_{0})$ in view of Eq. 4.1. However,
$g$ is a priori defined only for paths of length $T$ of its temporal
derivative may not exist. A remedy is to consider an embedding
$\iota\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathfrak{F}}_{T}\hookrightarrow\operatorname{\mathfrak{F}}$
(see Definition 2.1) with
$\operatorname{\mathfrak{F}}_{T},\operatorname{\mathfrak{F}}$ to be specified.
In the context of pricing, we see that a natural choice is
$\operatorname{\mathfrak{F}}_{K}=L^{1}(\Lambda_{T})$ and
$\operatorname{\mathfrak{F}}=\bar{L}^{1}(\Lambda)$, the subspace of
functionals
$f\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$ such
that $f(Y)$ is bounded in $L^{1}(\operatorname{\mathbb{Q}})$. Although other
embeddings can be used, we here choose the family of ”Bachelier embeddings”
$(\iota_{\operatorname{\mathbb{Q}}_{\sigma}})_{\sigma\geq 0}$ seen in Section
2.1, namely

$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}_{\sigma}}[g(Y_{T})\,|\,X_{t}],$

where the scaled canonical process $Y/\sigma$ is
$\operatorname{\mathbb{Q}}_{\sigma}-$Brownian motion. For all $\sigma\geq 0$,
then $\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g\in\bar{L}^{1}(\Lambda)$ from
Jensen’s inequality and the tower property of expectations. Hence
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}$ embeds $L^{1}(\Lambda_{T})$ into
$\bar{L}^{1}(\Lambda)$. If
$\iota_{{\operatorname{\mathbb{Q}}_{\sigma}}}g\in\operatorname{\mathbb{C}}^{K,K+1}$
for some $K\geq 1$, we can choose
$\operatorname{\mathbb{A}}^{\prime}=\\{\alpha\in\operatorname{\mathbb{A}}\
\mathrel{\mathop{\ordinarycolon}}\ |\alpha|\leq K\\}$ and apply the FTE to
obtain

$\displaystyle
v_{0}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[g(Y_{T})]=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g(Y_{T})]\approx\sum_{|\alpha|\leq
K}\underbrace{\Delta_{\alpha}\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g(X_{0})}_{=\mathrel{\mathop{\ordinarycolon}}\
c_{\alpha,\sigma}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\operatorname{\mathcal{S}}_{\alpha}(Y_{T})].$
(4.2)

Provided that the signature claims are priced correctly (namely the ”model”
$\operatorname{\mathbb{Q}}$ is calibrated to the market), then (4.2) gives an
explicit approximation for the value of $g$. We emphasize that the Bachelier
model is solely used to compute the coefficients. Note that each value of
$\sigma$ will generate a distinct sequence of coefficients
$(c_{\alpha,\sigma})$ and in turn a distinct approximation of the price. In
view of Theorem 3.9, the coefficients will nevertheless be unique once
expressed in the basis $\\{\operatorname{\mathcal{S}}_{\beta
1}|_{\Omega_{T}^{\Pi}}\ \mathrel{\mathop{\ordinarycolon}}\ |\beta|<K\\}$.
Because expectations with respect to an atomless measure tend to smooth out
the integrand, we favor $\sigma>0$ to compute $(c_{\alpha,\sigma})$ over the
degenerate case $\sigma=0$ associated to the intrinsic functional.

######  Remark 4.1.

If $g$ is a path-independent payoff, i.e. $g(X_{T})=h(x_{T})$ for some
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$,
then
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g(X_{t})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}_{1}}[h(x_{t}+\sigma
Y_{T-t})]$, $t<T$, is a smooth function of $x_{t}$. However, the regularity of
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g$ is no longer guaranteed for
general path-dependent payoffs. As an illustration, consider a fixed strike,
at-the-money lookback call option, i.e. $g(X_{T})=\max_{0\leq t\leq
T}x_{t}-x_{0}$. Assuming no interest rates, a simple calculation shows that
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g(X_{t})=u(t,x_{t},\max_{0\leq
s\leq t}x_{s})$, where

$\displaystyle u(t,x,m)$
$\displaystyle=x-x_{0}+(m-x)\left[2\Phi\left(\frac{m-x}{\sigma\sqrt{T-t}}\right)-1\right]+2\sigma\sqrt{T-t}\
\phi\left(\frac{m-x}{\sigma\sqrt{T-t}}\right),$

where $\phi,\ \Phi$ is the PDF and CDF of the standard normal distribution,
respectively. Although $u$ is clearly a smooth function in $m$,
$\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g$ does not even belong to
$\operatorname{\mathbb{C}}^{0,1}(\Lambda)$ as the functional spatial
derivative of $\max_{0\leq s\leq t}x_{s}$ is not $\Lambda-$continuous. One way
to sort out this issue is to regularize the running maximum as in [16, Example
2] and compute thereafter the needed functional derivatives in (4.2).

###  4.2 Static Hedging

The FTE possesses also immediate applications to static hedging. Consider a
claim $g$ with embedding
$f=\iota_{\operatorname{\mathbb{Q}}_{\sigma}}g\in\operatorname{\mathbb{C}}^{K,K+1}$,
$\sigma>0$. Suppose we are given a replicating portfolio
$\varphi=\sum_{|\alpha|\leq K}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}$
of $g$ such that $\Delta_{\alpha}f(X_{0})=c_{\alpha}$ for all $|\alpha|\leq
K$. We shall see that the FTE allows to quantify the hedging error
$|g-\varphi|$. First, applying Corollary 3.12 to the difference $f-\varphi$
gives

$f(X_{t})-\varphi(X_{t})=R_{K}(X_{t})=\sum_{|\alpha|=K}\int_{\triangle_{K,t}}\Delta_{\alpha}f(X_{t_{1}})\circ\,dx^{\alpha},$
(4.3)

for any price path $X\in\Omega^{\Pi}$. If $g$ belongs to the space spanned by
the signature functionals up to order $K$, we would obtain a perfect hedge,
i.e. $R_{K}\equiv 0$. Otherwise, the hedging error can be bounded as we now
explain. In line with financial markets, we assume that $X$ is the piecewise
interpolation of tick data and is in particular Lipschitz continuous. If also
$\sup_{|\alpha|=K}|\Delta_{\alpha}f(X)|\leq C^{K}$ for all
$X\in\Omega^{\text{Lip}}$, $C<\infty$, then Proposition 3.16 gives
$R_{K}(X_{t})=R_{K}(X_{0},X_{t})=\operatorname{\mathcal{O}}(\frac{t^{K}}{K!})$.
Choosing $t=T$, the hedging error for the claim $g$ is therefore bounded by

$|g(X_{T})-\varphi(X_{T})|\leq|R_{K}(X_{T})|=\operatorname{\mathcal{O}}\left({\textstyle\frac{T^{K}}{K!}}\right),\quad\forall\
X_{T}\in\Omega^{\text{Lip}}_{T}.$

We emphasize that the above upper bound holds pathwise, matching the needs of
exotic option traders to be protected against any future scenario.

##  5 Conclusion

This paper gathers and intends to elucidate expansions in the path space. In
particular, we draw a distinction between static expansions (Volterra, Wiener,
IVE) and dynamic ones$-$in particular the functional Taylor expansion (FTE)
lying at the heart of this work. We can summarize the latter as a pathwise
tool combining the functional Itô calculus and the signature to unravel path
dependence. We then establish parallels between the FTE and static expansions
such as the intrinsic value expansion and the celebrated Wiener chaos. As seen
in the applications, the FTE proves useful for the pricing and hedging of
exotic claims as it effectively separates the payoff functional from the price
path. For instance, this separation can be incorporated into numerical methods
such as cubature schemes to guarantee pricing accuracy and speed up
computation.

We deem the study of path functionals a necessity in modern quantitative
finance, given the omnipresence of path dependence. As such, it seems that
many works can be undertaken in this direction. In particular, one can
investigate financial applications of the Volterra and Wiener series, used
extensively in nonlinear systems but barely in finance. Another avenue would
be to tackle non-Markovian problems with the FTE, such as the pricing of
American options involving path-dependent features in the payoff, the
dynamics, or both.

### Acknowledgments

We would like to thank Josef Teichmann, Harald Oberhauser, Harvey Stein as
well as the participants of the Research in Options conference (RiO 2021 and
RiO 2022) and the 2022 CFMAR Workshop at UCSB for precious comments. We are
also grateful to our colleagues at Bloomberg (Bryan Liang, Guixin Liu) for
fruitful discussions.

##  Appendix A Proofs

###  A.1 Theorem 3.2

###### Proof.

Recall that $X\in\Omega^{\Pi}_{t}$ implies $X_{s}\in\Omega^{\Pi}_{s}$ for all
$s\leq t$. We can therefore show the result for $s=t$ without loss of
generality. For fixed $N$, write $\delta x_{t_{n}}=x_{t_{n}}-x_{t_{n-1}}$,
$\delta t_{n}=t_{n}-t_{n-1}$ with $1\leq n\leq N$. Note that we omit the
dependence of $\delta x_{t_{n}},\ \delta t_{n}$ on $N$ for ease of
presentation. Let $X^{N}\in\bar{\Omega}^{\Pi}_{t}$ be the càdlàg piecewise
constant approximation of $X$ along $\Pi^{N}$, i.e.
$X^{N}=\sum_{t_{n}\in\Pi_{t}^{N}}x_{t_{n-1}}\mathds{1}_{[t_{n-1},t_{n})}+x_{t}\mathds{1}_{\\{t\\}}.$
Since $X^{N}_{0}=X_{0}$, we can write

$\displaystyle
f(X_{t})-f(X_{0})=f(X_{t})-f(X_{t}^{N})+\sum_{t_{n}\in\Pi_{t}^{N}}(f(X_{t_{n}}^{N})-f(X_{t_{n-1}}^{N})).$

Clearly, $X^{N}\overset{N\uparrow\infty}{\longrightarrow}X$ uniformly, thus
$f(X_{t}^{N})\overset{N\uparrow\infty}{\longrightarrow}f(X_{t})$ as well since
$f$ is $\Lambda-$continuous. Next, we decompose the summands in the above
telescopic sum as

$f(X_{t_{n}}^{N})-f(X_{t_{n-1}}^{N})=\underbrace{f(X_{t_{n}-}^{N})-f(X_{t_{n-1}}^{N})}_{\text{flat
extension}}+\underbrace{f(X_{t_{n}}^{N})-f(X_{t_{n}-}^{N})}_{\text{vertical
bump}}.$

We thus retrieve from $X^{N}$ an alternation of flat extensions and bumps. As
$f\in\operatorname{\mathbb{C}}^{1,2}(\bar{\Omega}^{\Pi}_{t})$, the mean value
theorem applied to $u\mapsto f(X^{N}_{t_{n-1},u})$, $h\mapsto
f((X^{N}_{t_{n}-})^{h})$ gives the existence of $(u_{n},h^{+}_{n})\in(0,\delta
t_{n})\times(0,\delta{x}_{t_{n}})$ such that

$\displaystyle f(X_{t_{n}-}^{N})-f(X_{t_{n-1}}^{N})$
$\displaystyle=\Delta_{t}f(X^{N}_{t_{n-1},u_{n}})\delta t_{n},$ $\displaystyle
f(X_{t_{n}}^{N})-f(X_{t_{n}-}^{N})$
$\displaystyle=\Delta_{x}f(X_{t_{n}-}^{N})\delta
x_{t_{n}}+\frac{1}{2}\Delta_{xx}f((X_{t_{n}-}^{N})^{h^{+}_{n}})\delta
x^{2}_{t_{n}}.$ (A.1)

Similarly, using the function $h\mapsto f((X^{N}_{t_{n}})^{-h})$, there exists
$h_{n}\in(0,\delta{x}_{t_{n}})$ such that

$f(X_{t_{n}-}^{N})-f(X_{t_{n}}^{N})=-\Delta_{x}f(X_{t_{n}}^{N})\delta
x_{t_{n}}+\frac{1}{2}\Delta_{xx}f((X_{t_{n}}^{N})^{-h_{n}})\delta
x^{2}_{t_{n}}.$ (A.2)

Hence, subtracting (A.2) to (A.1) yields

$2(f(X_{t_{n}}^{N})-f(X_{t_{n}-}^{N}))=(\Delta_{x}f(X_{t_{n}-}^{N})+\Delta_{x}f(X_{t_{n}}^{N}))\delta
x_{t_{n}}+\frac{\Delta_{xx}f((X_{t_{n}-}^{N})^{h_{n}^{+}})-\Delta_{xx}f((X_{t_{n}}^{N})^{-h_{n}})}{2}\delta
x^{2}_{t_{n}}.$

Noticing that $(X_{t_{n}}^{N})^{-h_{n}}=(X_{t_{n}-}^{N})^{h_{n}^{-}}$ with
$h_{n}^{-}\mathrel{\mathop{\ordinarycolon}}=\delta x_{t_{n}}-h_{n}$, we obtain

$\displaystyle\sum_{t_{n}\in\Pi_{t}^{N}}(f(X_{t_{n}}^{N})-f(X_{t_{n-1}}^{N}))$
$\displaystyle=\sum_{t_{n}\in\Pi_{t}^{N}}\Delta_{t}f(X^{N}_{t_{n-1},u_{n}})\
\delta t_{n}$ (A.3)
$\displaystyle+\sum_{t_{n}\in\Pi_{t}^{N}}\frac{\Delta_{x}f(X_{t_{n}-}^{N})+\Delta_{x}f(X_{t_{n}}^{N})}{2}\
\delta x_{t_{n}}$ (A.4)
$\displaystyle+\sum_{t_{n}\in\Pi_{t}^{N}}\frac{\Delta_{xx}f((X_{t_{n}-}^{N})^{h_{n}^{+}})-\Delta_{xx}f((X_{t_{n}-}^{N})^{h_{n}^{-}})}{4}\
\delta x^{2}_{t_{n}}$ (A.5)

As $t\to t$ is a smooth integrator and $\Delta_{t}f$ is $\Lambda-$continuous,
the right side of $\eqref{eq:riemann}$ converges to the Riemann integral

$\lim_{|\Pi^{N}_{t}|\downarrow
0}\sum_{t_{n}\in\Pi_{t}^{N}}\Delta_{t}f(X^{N}_{t_{n-1},u_{n}})\ \delta
t_{n}=\lim_{|\Pi^{N}_{t}|\downarrow
0}\sum_{t_{n}\in\Pi_{t}^{N}}\Delta_{t}f(X^{N}_{t_{n-1}})\ \delta
t_{n}=\int_{0}^{t}\Delta_{t}f(X_{s})ds.$

Now consider the compact set
$\operatorname{\mathcal{K}}_{N}=\\{(X_{s}^{N})^{h}\ |\
(s,h)\in[0,t]\times[-\delta_{N},\delta_{N}]\\}\subset\Lambda$ with
$\delta_{N}=\max_{n=1,...,N}|\delta x_{t_{n}}|$. Then $\Delta_{xx}f$ is
uniformly $\Lambda-$continuous in $\operatorname{\mathcal{K}}_{N}$. In turn,
the function $(s,h)\mapsto\Delta_{xx}f((X_{s}^{N})^{h})$,
$(s,h)\in[0,t]\times[-\delta_{N},\delta_{N}]$, admits a modulus of continuity
$\omega_{N}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}_{+}\to\operatorname{\mathbb{R}}_{+}.$
Thus,

$|\Delta_{xx}f((X_{t_{n}-}^{N})^{h_{n}^{+}})-\Delta_{xx}f((X_{t_{n}-}^{N})^{h_{n}^{-}})|\leq\omega_{N}(|h_{n}^{+}-h_{n}^{-}|)\leq\omega_{N}(\delta_{N})\quad\forall\
n=1,\ldots,N.$

As $\sum_{t_{n}\in\Pi_{t}^{N}}\delta x_{t_{n}}^{2}$ converges to $\langle
X\rangle^{\Pi}_{t}<\infty$, there exists $N_{0}\in\operatorname{\mathbb{N}}$
such that $\sum_{t_{n}\in\Pi_{t}^{N}}\delta x_{t_{n}}^{2}\leq 2\langle
X\rangle^{\Pi}_{t}$ $\ \forall\ N\geq N_{0}$. Then $\eqref{eq:remain}$ is
dominated by $\frac{1}{2}\omega_{N}(\delta_{N})\ \langle X\rangle^{\Pi}_{t}$
for all $N\geq N_{0}$, and thus vanishes as $N\uparrow\infty$ since $X$ is
uniformly continuous. Altogether, we have shown that

$\displaystyle f(X_{t})-f(X_{0})$
$\displaystyle=\int_{0}^{t}\Delta_{t}f(X_{s})ds+\lim_{|\Pi^{N}_{t}|\downarrow
0}\sum_{t_{n}\in\Pi_{t}^{N}}\frac{\Delta_{x}f(X^{N}_{t_{n}-})+\Delta_{x}f(X^{N}_{t_{n}})}{2}\
\delta x_{t_{n}}.$

In fine, the limit of $\eqref{eq:strat}$ exists and coincides with the
Stratonovich integral $\int_{0}^{t}\Delta_{x}f(X_{s})\circ dx_{s}$. ∎

###  A.2 Proposition 3.5

###### Proof.

We fix a path $X\in\Omega^{\Pi}_{t}$ and show the result by induction on
$k=|\alpha|\geq 1$. If $k=1$, then clearly
$\operatorname{\mathcal{S}}_{0}(X_{t})=t$,
$\operatorname{\mathcal{S}}_{1}(X_{t})=x_{t}-x_{0}$ are well-defined smooth
functionals. Now suppose that $\operatorname{\mathcal{S}}_{\beta}$ exists for
all $|\beta|<k$, $k\geq 2$. Pick any word $\alpha$ such that
$k=|\alpha|>|\alpha|_{1}$. If $\alpha_{k}=0$, then
$\operatorname{\mathcal{S}}_{\alpha}=\int_{0}^{t}\operatorname{\mathcal{S}}_{\alpha-1}(X_{s})ds$
is a well-defined Riemann integral. Moreover,
$\Delta_{t}\operatorname{\mathcal{S}}_{\alpha}=\operatorname{\mathcal{S}}_{\alpha-1}$,
$\Delta_{x}\operatorname{\mathcal{S}}_{\alpha}\equiv 0$. The smoothness
$\operatorname{\mathcal{S}}_{\alpha}$ follows by induction.

Finally, we treat the case when $\alpha_{k}=1$. We would like to make sense of
$\operatorname{\mathcal{S}}_{\alpha}(X_{t})\mathrel{\mathop{\ordinarycolon}}=\int_{0}^{t}\operatorname{\mathcal{S}}_{\alpha_{-1}}(X_{s})\circ
dx_{s}$ by showing that the integrand is the spatial derivative of some smooth
functional to apply Theorem 3.2. In light of the discussion in Section 3.1.2
and the smoothness of $\operatorname{\mathcal{S}}_{\alpha-1}$, let us
introduce
$\Phi(X_{t})=\int_{x_{0}}^{x_{t}}\operatorname{\mathcal{S}}_{\alpha-1}(X_{t}^{(\varepsilon)})\
d\varepsilon.$ First,
$\Delta_{x}\Phi=\operatorname{\mathcal{S}}_{\alpha-1}\in\operatorname{\mathbb{C}}^{\infty,\infty}(\bar{\Omega}^{\Pi})$
by construction. If $\alpha_{k-1}=1$, then $\Delta_{t}\Phi(X_{t})=0$ and in
turn
$\Phi(X_{t})=\int_{0}^{t}\operatorname{\mathcal{S}}_{\alpha-1}(X_{s})\circ
dx_{s}=\operatorname{\mathcal{S}}_{\alpha}(X_{t})$ using the functional
Stratonovich formula. When $\alpha_{k-1}=1$, then
$\Delta_{t}\Phi(X_{t})=\int_{x_{0}}^{x_{t}}\operatorname{\mathcal{S}}_{\alpha-2}(X_{t}^{(\varepsilon)})\
d\varepsilon$, so $\eqref{eq:sigDefStrat}$ reads
$\operatorname{\mathcal{S}}_{\alpha}(X_{t})=\Phi(X_{t})-\int_{0}^{t}\int_{x_{0}}^{x_{s}}\operatorname{\mathcal{S}}_{\alpha-2}(X_{s}^{(\varepsilon)})\
d\varepsilon ds.$ Hence $\operatorname{\mathcal{S}}_{\alpha}$ is well-defined
in both cases and taking the spatial (respectively temporal) derivative gives
$\Delta_{x}\operatorname{\mathcal{S}}_{\alpha}=\operatorname{\mathcal{S}}_{\alpha-1}$
(resp. $\Delta_{t}\operatorname{\mathcal{S}}_{\alpha}=0$), which is
$\eqref{eq:derivSig}$. ∎

###  A.3 Proposition 3.8

###### Proof.

(i) This follows from an iterative application of $\eqref{eq:derivSig}$. (ii)
Suppose that there exists a finite subset
$\operatorname{\mathbb{A}}^{\prime}\subseteq\operatorname{\mathbb{A}}$ and
$(c_{\alpha})_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}$ such that
$f\mathrel{\mathop{\ordinarycolon}}=\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}\equiv
0.$ As $f$ is a combination of finitely many signature functionals, then $f$
is smooth using Proposition 3.5. Moreover, assertion (i) implies that
$\Delta_{\alpha}\operatorname{\mathcal{S}}_{\beta}(X_{0})=\mathds{1}_{\\{\alpha=\beta\\}}$
for all $\alpha,\beta\in\operatorname{\mathbb{A}}$. We conclude that
$c_{\alpha}=\Delta_{\alpha}f(X_{0})=0$ for all
$\alpha\in\operatorname{\mathbb{A}}^{\prime}$ as desired. (iii) There are many
examples. A simple one follows from the product rule: define
$g(X_{T})=T(x_{T}-x_{0})$, which is equal to
$\operatorname{\mathcal{S}}_{01}(X_{T})+\operatorname{\mathcal{S}}_{10}(X_{T})$.
With $T$ being fixed, it can be treated as a coefficient. The linear
dependence follows from the fact that $X_{T}\mapsto x_{T}-x_{0}$ is a
signature functional, namely $\operatorname{\mathcal{S}}_{1}(X_{T})$. (iv) We
start with a Lemma.

######  Lemma A.1.

Let $f\in\operatorname{\mathbb{C}}(\Omega^{\Pi})$ such that
$\operatorname{\mathcal{I}}\\!f(X_{T})\mathrel{\mathop{\ordinarycolon}}=\int_{0}^{T}f(X_{t})\circ
dx_{t}=0$ on $\Omega^{\Pi}_{T}$. Then $f$ vanishes on $\Omega^{\Pi}$.

###### Proof.

Take $X_{t}\in\Omega_{t}^{\Pi}$. Consider the family $(Y_{T}^{(h,\delta
t)})\subseteq\Omega_{T}^{\Pi}$ defined as

$Y_{T}^{(h,\delta t)}(s)=x_{t\wedge s}+h\left(\frac{(t-s)^{+}}{\delta t}\wedge
1\right),\quad s\in[0,T],\quad(h,\delta
t)\in\operatorname{\mathbb{R}}_{+}\times(0,\infty).$

Note in particular that $Y_{T}^{(0,\delta t)}=X_{t,T-t}$. By assumption
$\operatorname{\mathcal{I}}\\!f(X_{t,T-t})=\operatorname{\mathcal{I}}\\!f(Y_{T}^{(h,\delta
t)})=0$. Hence,

$\displaystyle 0=\frac{1}{h}\left(\operatorname{\mathcal{I}}f(Y_{T}^{(h,\delta
t)})-\operatorname{\mathcal{I}}f(X_{t,T-t})\right)=\frac{1}{\delta
t}\int_{t}^{t+\delta t}f(Y_{s}^{(h,\delta t)})\ ds\;\,\overset{h\ \downarrow\
0}{\xrightarrow{\hskip 22.76228pt}}\;\,\frac{1}{\delta t}\int_{0}^{\delta
t}f(X_{t,u})\ du.$

Letting $\delta t$ go to zero yields the result. ∎

To prove (iv), suppose that there exists
$\operatorname{\mathbb{A}}^{\prime}\subseteq\\{\operatorname{\mathcal{S}}_{\beta
1}|_{\Omega_{T}^{\Pi}}\ \mathrel{\mathop{\ordinarycolon}}\
\beta\in\operatorname{\mathbb{A}}\\}$,
$|\operatorname{\mathbb{A}}^{\prime}|<\infty$, and
$(c_{\alpha})_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}$ such that
$\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}(X_{T})=0$
for all $X_{T}\in\Lambda_{T}$. As every word in
$\operatorname{\mathbb{A}}^{\prime}$ ends with a ”$1$”, we can write

$\sum_{\alpha\in\operatorname{\mathbb{A}}^{\prime}}c_{\alpha}\operatorname{\mathcal{S}}_{\alpha}(X_{T})=\sum_{\beta
1\in\operatorname{\mathbb{A}}^{\prime}}c_{\beta
1}\operatorname{\mathcal{S}}_{\beta 1}(X_{T})=\int_{0}^{T}\sum_{\beta
1\in\operatorname{\mathbb{A}}^{\prime}}c_{\beta
1}\operatorname{\mathcal{S}}_{\beta}(X_{t})\circ dx_{t}.$

Applying Lemma A.1 to $f=\sum_{\beta
1\in\operatorname{\mathbb{A}}^{\prime}}c_{\beta
1}\operatorname{\mathcal{S}}_{\beta}$ and conclude that $f\equiv 0$. Using
(ii), this gives $c_{\alpha}=0$ for all
$\alpha\in\operatorname{\mathbb{A}}^{\prime}$ as desired. ∎

###  A.4 Theorem 3.9

###### Proof.

The independence is already shown in Proposition 3.8 (iv). We prove the
spanning property by induction. The result is clearly true for $K=1$ as