# Aura: Privacy-preserving augmentation to improve test set diversity in noise
suppression applications

###### Abstract

Noise suppression models running in production environments are commonly
trained on publicly available datasets. However, this approach leads to
regressions in production environments due to the lack of training/testing on
representative customer data. Moreover, due to privacy reasons, developers
cannot listen to customer content. This ‘ears-off’ situation motivates
augmenting existing datasets in a privacy-preserving manner. In this paper, we
present Aura, a solution to make existing noise suppression test sets more
challenging and diverse while limiting the sampling budget. Aura is ‘ears-off’
because it relies on a feature extractor and a metric of speech quality,
DNSMOS P.835, both pre-trained on data obtained from public sources. As an
application of Aura, we augment a current benchmark test set in noise
suppression by sampling audio files from a new batch of data of 20K clean
speech clips from Librivox mixed with noise clips obtained from AudioSet. Aura
makes the existing benchmark test set harder by $100\%$ in DNSMOS P.835, a
$26\%$ improvement in Spearman’s rank correlation coefficient (SRCC) compared
to random sampling and, identifies $73\%$ out-of-distribution samples to
augment the test set.

Index Terms—  Machine Learning, Noise Suppression, Audio Datasets, Data
Privacy

##  1 INTRODUCTION

Speech enhancement includes removing background noise and reverberation [1,
2]. Deep learning based approaches provide state-of-the-art improvement but
typically involve supervised training on a large and diverse dataset of noisy
speech and clean speech pairs [3]. Since real-world applications contain
speech mixed with noise, most deep learning methods rely on synthetic data
that mix clean and noisy speech [4]. However, synthetic data cannot fully
capture real-world conditions leading to poor performance when deployed to
large, diverse customer bases [5]. One solution is to identify new conditions
to augment current test sets with additional representative data. However,
labeling a large batch of customer data is expensive, and often prohibitive
due to privacy and compliance reasons.

This paper presents Aura, an ‘ears-off’ methodology to identify scenarios in a
sample-efficient manner that is: (i) representative of the customer workload,
and (ii) challenging to noise suppression models. Real-world scenarios are
instrumental to generate a low-variance estimate of how noise suppressors
improve speech quality when used by customers. It is noteworthy that keeping
the test set small (i.e., sample efficiency) ensures agility of development
and deployment in production pipelines. Production workload datasets can be in
the order of millions of clips so sample efficiency is an important practical
challenge.

The main contribution of Aura is to operate in an ‘ears-off’ mode by relying
on a pre-trained audio feature extractor [6] and a pre-trained accurate
objective speech quality metric called DNSMOS P.835 [7]. Both the feature
extractor and speech quality metric are convolution-based models trained on
large-scale open-sourced processed data and do not risk memorizing customer
private information. Aura maps customer data into an embedding space,
partitions the data via clustering, and identifies the most challenging
scenarios for noise suppressors in each cluster.

Since we cannot share results on customer data, we evaluate Aura’s sampling on
simulated data. We apply Aura to rank $28$ noise suppression models from the
DNS Challenge [4] in terms of overall mean opinion scores (MOS) [7]. We show
that the Spearman’s rank correlation coefficient (SRCC) between a $1\%$ sample
produced by Aura and the entire mixture of noisy speech is $0.91$. This is an
improvement of $26\%$ compared to a $1\%$ random sampling of noisy speech. We
also apply _Aura_ to augment a current benchmark test set for noise
suppressors [4]. The resulting test set is more challenging to noise
suppressors with a predicted absolute decline of $0.36$ (a relative decline of
$100\%$) in DNSMOS P.835 [7]. Moreover, the _Aura_ -based test set captures
more diverse audio scenarios with an increase of $43\%$ in diversity (as
measured by $\chi^{2}-$ distance, and an addition of $73\%$ out-of-
distribution (OOD) samples compared to the current benchmark test set [4].

Related work. As part of the Deep Noise Suppression (DNS) challenge [4], Reddy
et al. created an extensive open-sourced noisy speech test set with about 20
noise types, 15 languages, 200+ speakers, emotions, and singing. However, the
resulting DNS benchmark test set does not cover all the scenarios that
customers who use real-time communication platforms experience. These gaps
motivate additional effort to identify challenging scenarios where DNS models
underperform.

Active learning is the common paradigm to select which data points to label
from a pool of unlabeled data [8] when the labeling budget is limited. Many
active learning heuristics are uncertainty-based and try to sample in priority
hard examples [9]. Kossen et al. show that heuristics in active learning lead
to biased estimates of model performances because they do not preclude
selecting redundant/outlier scenarios [10]. On the other hand, Sener et al.
propose a diversity-based solution identifying a set of data points that are
the most representative of the entire data set [11]. This paper integrates
diversity and uncertainty-based techniques to a pipeline that creates a test
set for noise suppression models while operating in an ‘ears-off’ environment.

Our ‘ears-off’ pipeline protects customers from privacy leakages, including
membership attacks [12] and the extraction of identifiable information [13]. A
growing body of work has developed differentially private learning algorithms
that do not expose private customer data [14]. Federated learning is a partial
solution to the ‘ears-off’ problem since it allows training models on
decentralized devices without the need for a central server to access private
data. Latif et al. and Guliani et al. present applications of the federated
learning paradigm to speech-related tasks [15, 16]. However, these methods
rely on an availability of labels at client endpoints.

We obtain an ‘ears-off’ pipeline by transferring knowledge from models trained
on open-source data. Self-supervision uses surrogate tasks to learn
representations or embeddings that are useful to real-world downstream tasks.
The method has been successful in image-based tasks (e.g., [17]), and
audio/video (e.g., [18]). In this paper, our pretext task is to label the type
of background noise in noisy speech. Moreover, we use DNSMOS P.835 [7] to
accurately measure the quality of speech without the need for clean reference.
DNSMOS P.835 has a $0.98$ correlation with human ratings for noise suppression
models and a 0.74 correlation per clip. This gives the flexibility to estimate
speech quality before and after noise suppression without listening to the
file.

##  2 Solution

Fig. 1: Overall structure of _Aura_. _Aura_ reduces the variance of the
performance metric $\rho$ (red) while maximizing the diversity component
(gray) to ensure good coverage of audio scenarios in the embedding space.

Our objective is to estimate a performance metric $\rho$ of noise suppression
on a target data $\mathcal{D}_{t}$ that represents real-world conditions. For
each speech clip, the P.835 protocol generates a MOS for signal quality SIG,
background noise BAK, and overall quality OVRL. We derive two performance
metrics $\rho$ from MOS: (i) differential MOS (DMOS) between after and before
denoising; (ii) stack ranking of competing noise suppressors according to
their average DMOS calculated on the target data $\mathcal{D}_{t}$.

The constraints are that only a small subset $S$ of files can be sampled out
of all the noisy speech in $\mathcal{D}_{t}$; and, that audio files in
$\mathcal{D}_{t}$ cannot be used to fit a model that could potentially encode
identifiable information in its parameters. The restriction
$|S|<<|\mathcal{D}_{t}|$ limits the size of the test set and allows rapid
testing of models during development. We want the sampling estimate of the
performance metric $\rho$ to have a small error $\varepsilon$ compared to the
expected value of $\rho$ on the target data $\mathcal{D}_{t}$. To minimize
$\varepsilon$, we trade-off bias and variance [19]. A random sample of
$\mathcal{D}_{t}$ audio files would result in zero bias, but high variance. On
the other hand, probability-proportional-to-size sampling (PPS) would minimize
the variance of the estimator by sampling audio files with a probability
proportional to $\rho$ [20]. One shortcoming of PPS sampling performed solely
on $\rho$ is that it does not consider the diversity of scenarios.

To trade-off variance and bias, we propose Aura, the pipeline shown in Figure
1. Aura learns a partition of the target data into $k$ clusters in an
embedding space obtained from a pre-trained feature extractor. Then, it
applies PPS to each cluster. Aura samples audio files in two steps. First, it
samples files in all the clusters of the embedding space to capture the
diversity represented in the target data. Second, within each cluster, it
favors audio files with the largest $\rho$ to reduce the variance of the
performance metric $\rho$.

Test set for noise suppression models. The first application of _Aura_ is to
sample $|S|$ audio files in the target data $\mathcal{D}_{t}$ to form a test
set for noise suppression models. In each of the $k$ clusters of the embedding
space, we sample $|S|/k$ files with a probability inversely proportional to
their predicted DMOS. Note that a smaller DMOS indicates a more challenging
(or noisier) speech scenario.

Model ranking. A model evaluation task in noise suppression involves comparing
the speech quality produced by different models on the same set of audio
files. Given $N$ noise suppressors, we adjust our sampling procedure in the
following manner: Within each cluster, an audio file is selected with a
probability proportional to the variance across models of the predicted DMOS
for that file. In this application, the sampling performance $\epsilon$ is
defined as the SRCC between the ranking of $N$ models obtained on the sample
$S$ and the one obtained with the entire target data $\mathcal{D}_{t}$.

##  3 Experiments

###  3.1 Pipeline Components

Choice of feature extractor. Our feature extractor is constructed from VGGish,
a model pre-trained on a 100M YouTube videos dataset [6]. It generates a 128
dimension embedding for each audio clip. VGGish embeddings are trained to
identifying the type of foreground sound in an audio clip. In the context of
noise suppression, we aim to detect the noise in the background of a speech
clip. Therefore, we fine-tune the feature extractor by using its embedding as
inputs to two fully connected layers and by classifying the type of background
noise.

We follow Reddy et al. [4] to synthesize 1.5 million instances of 10 second
noisy speech containing $527$ categories of noise sounds from AudioSet [21].
We add each noise sound to the background of speech clips randomly drawn from
the bank of clean speech clips created by [4]. Our background noise type
classifier trained on this synthetic data achieved a mean average precision
(MAP) of $0.33$ and area-under-the-curve (AUC) of $0.96$ on a hold-out sample.
It provides us with a feature extractor that maps noisy speech to a 128
dimension embedding space that is semantically aligned with the type of
background noise. Using kmeans++ on the embedding space, we partition $1.5$
million noisy speech into $256$ clusters [22] 111The number of clusters is
selected as the number that achieve the lowest Davies-Bouldin index, where the
index is computed using Euclidean distance in the embedding space.. We
validated the quality of the clusters using subjective listening tests. The
raters were presented clips from 24 random clusters and asked to report
whether audio files in the cluster share a common audio property. We found
that 80$\%$ of the clusters share a common background noise.

Choice of MOS predictor. We use a pre-trained model, DNSMOS P.835 to predict
subjective speech quality [7].

###  3.2 Dataset

We will apply Aura to two target test sets.

Augmented DNS challenge test set. We create a target dataset of potential
noisy speech candidates to add to the INTERSPEECH 2021 DNS Challenge test set
[4]. We mix sounds from the balanced AudioSet data [21] with clean speech
clips [23]. We use segments from AudioSet as background noise and follow [23]
to generate noisy speech with signal-to-noise ratio (SNR) between -5 dB and 5
dB. The resulting target data combines 1.7K files from the DNS challenge test
set with 22K of 10 seconds clips from the newly synthesized noisy speech that
covers 527 noise types with at least 59 clips per class. The pool of noisy
speech candidates is at least partially out-of-distribution compared to [4],
which covers 120 noise types. Moreover, it does not overlap the dataset of
audio clips used to fine-tune our feature extractor and train DNSMOS P.835 and
thus, reproduces the conditions of an ‘ears-off’ environment.

Augmented DNS challenge test set + clean speech. To simulate real-world
conditions, for each noisy speech in the previous target data, we randomly
draw 10 clean speech clips from [4]. Clean speech presents a challenge to
stack rank models in development because it does not allow to discriminate the
performance of noise suppressors.

###  3.3 Augmenting DNS challenge test set

We evaluate the following: From the pool of noisy speech candidates, can
_Aura_ generate a new test set of the same size (1.7K) as the INTERSPEECH 2021
benchmark test set [4], but with audio scenarios that are more challenging and
diverse?

Protocol. We measure the diversity of a test set as the $\chi^{2}$ distance
between the distribution of audio segments across the embedding clusters and a
uniform distribution between clusters. We normalize the value by calculating
$\chi^{2}$ of the contingency table over the percentage of data points in each
cluster rather than raw frequency. The lower the $\chi^{2}$ distance, the more
audio properties encoded in the embedding space the resulting test set covers.

Test set | DMOS | $\chi^{2}$ | OOD
---|---|---|---
| SIG | BAK | OVRL |  | %
DNS | -0.24 | 0.99 | 0.18 | 498 | 0%
Challenge | $\pm 0.01$ | $\pm 0.01$ | $\pm 0.01$ | ($<0.001$) |
Proposed | -0.57 | 0.91 | -0.32 | 3843 | 69%
(no cluster) |  $\pm$ 0.01 | $\pm 0.01$ |  $\pm$ 0.001 | ($<0.001$) |
Proposed | -0.47 | 0.89 | -0.18 | 287 | 73%
\+ clustering | $\pm 0.01$ | $\pm 0.01$ | $\pm 0.01$ | ($\geq 0.2$) |
Table 1: Aura-based DNS test set. Sample provided by Aura on the new noisy
data set increases difficulty level (lower DMOS) and diversity (lower
$\chi^{2}$). The evaluation set is compared to the INTERSPEECH test set [4].
$\pm$ represents 95% conf. interval and P-values are in parenthesis. Low
p-values indicate sample does not cover the clusters formed by the target
data; large p-values mean almost uniform coverage.

Results. Table 1 compares DMOS and $\chi^{2}$ for the benchmark test set (top
row) and the _Aura_ -generated test set (bottom row). The proposed test set
replaces $73\%$ of noisy speech in the benchmark test set with a new noisy
speech from our synthetic data. _Aura_ forms a test set with clips for which
noise suppressors are more likely to degrade the signal and overall quality of
the speech than for clips in the benchmark dataset (lower DMOS). Moreover, its
$\chi^{2}$ distance is not significantly different from zero (p-value $\geq
0.05$), which indicates good coverage of all clusters in the embedding space
and thus, a diverse set of audio conditions. Table 1 also shows that sampling
the most challenging audio files in each cluster improves significantly
diversity ($\chi^{2}$ distance decreases from 3843 to 287) compared to
sampling the most challenging files without diversity constraint (middle row
in Table 1).

Figure 2 shows the top-10 noise types (in green) from the augmenting dataset
of noisy speech that _Aura_ adds to the test set. _Aura_ prioritizes new out-
of-distribution audio scenarios compared to the top-10 noise types present in
the benchmark test set.

###  3.4 Model ranking

Sampling | SRCC | $\chi^{2}$
---|---|---
method | SIG | BAK | OVRL | ($\times 10^{3})$
_Random_ |  $0.58$$\pm 0.02$ |  $0.72$$\pm 0.01$ |  $0.72$ $\pm 0.02$ |  $9$$\pm 0.01$
_Diversity_ |  $0.72$$\pm 0.02$ |  $0.89$ $\pm 0.01$ |  $0.82$$\pm 0.01$ |  $4$$\pm 0.02$
_Variance_ |  $0.80$$\pm 0.01$ |  $0.93$$\pm 0.01$ |  $0.88$$\pm 0.01$ |  $7$ $\pm 0.02$
_Aura_ | 0.84$\pm$ 0.01 |  0.93 $\pm$ 0.01 | 0.91$\pm$ 0.01 | 1$\pm 0.01$

Table 2: SRCC between the ranking of 28 NS models obtained from a $1\%$ Aura
sample with the model ranking obtained from using the entire data. Diversity
is measured using $\chi^{2}$ distance (lower is better). 95% conf interval
depicted using $\pm$.

We evaluate _Aura_ ’s sampling efficiency to accurately stack rank noise
suppression models with few samples $S$ from the target data
$\mathcal{D}_{t}$. The objective is to estimate a model ranking that has a
high SRCC with the ranking that would be obtained on the entire data.

Protocol. For each speech, we run $28$ noise suppression models from [4] and
on the DMOS predicted by our DNSMOS P.835 model as opposed to human ratings.
We bootstrap the sampling 200 times and report the mean and standard deviation
of the resulting rank correlation coefficients. We compare _Aura_ ’s sampling
performance to three alternative strategies: (i) _Random_ , which draws
randomly $1\%$ of data; (ii) _Diversity_ , which samples uniformly across
embedding clusters; (iii) _Variance_ , which samples proportionally to the
variance of predicted DMOS across the 28 models.

Results. Table 2 shows the SRCC for ranking based on signal, background and
overall DMOS. _Aura_ ’s sampling leads to a $26\%$ improvement over random
sampling for signal-based ranking. We also observe that the $95\%$ confidence
interval of the ranking obtained from _Aura_ -based samples is narrower than
the one obtained by random sampling. Compared to alternative approaches,
_Aura_ generates the sample with the lowest $\chi^{2}$ distance, which
indicates a better coverage of audio scenarios. On the other hand, _Random_
has the highest $\chi^{2}$ distance because it mostly samples clean speech.
Figure 3 shows that even for larger samples, _Aura_ still outperforms _Random_
in terms of rank correlation.

Fig. 2: Top-10 of out-of-distribution (green) and in-distribution (blue) noise
categories in Aura-based samples. In-distribution relates to categories in
INTERSPEECH 2021 test set. Fig. 3: SRRC of Aura-based rank estimates as a
function of sample size.

##  4 Conclusion

Aura presents an end-to-end system for improving the test set used to evaluate
deep noise suppression models. Aura is designed to work for ‘ears-off’
customer workloads to preserve privacy. Aura detects out-of-distribution
samples by relying on an objective quality metric and a pre-trained feature
extractor. Although targeted for noise suppression, the method is generic and
can be extended to other media (i.e., audio/video/image) given the successful
development of a powerful pre-trained feature extractor. This work is an
important step toward balancing customer privacy and measuring model’s
performance in real-world scenarios.

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Portfolios generated by contingent claim functions,

with applications to option pricing

Ricardo T. Fernholz111Claremont McKenna College, 500 E. Ninth St., Claremont,
CA 91711<EMAIL_ADDRESS>Robert Fernholz222Allocation Strategies, LLC,
Princeton, NJ<EMAIL_ADDRESS>

###### Abstract

This paper is a synthesis of the theories of portfolio generating functions
and rational option pricing. For a family of $n\geq 2$ assets with prices
represented by strictly positive continuous semimartingales, a contingent
claim function is a scalable positive $C^{2,1}$ function of the asset prices
and time. We extend the theory of portfolio generation to measure the value of
portfolios generated by contingent claim functions directly, with no numeraire
portfolio. We show that if a contingent claim function satisfies a particular
parabolic differential equation, then the value of the portfolio generated by
that contingent claim function will replicate the value of the function. This
differential equation is a general form of the Black-Scholes equation.

AMS 2000 Subject Classifications: 60H30, 91G10, 91G20. JEL Codes: G11, G13,
C58.

Keywords: Stochastic portfolio theory, functionally generated portfolios,
contingent claims, option pricing, Black-Scholes equation

##  1 Introduction

Functionally generated portfolios were introduced in Fernholz (1999b, 2002) as
one of the central concepts in stochastic portfolio theory. The theory of
rational option pricing began with the seminal paper of Black and Scholes
(1973), who used stochastic calculus to price European options on stocks. This
paper is a synthesis of portfolio generation and option pricing, and shows
that portfolio generation has a natural application to option pricing, while
option pricing might be considered an early example of portfolio generation.

Functionally generated portfolios originally measured the value of a portfolio
versus the market portfolio as numeraire (Fernholz, 1999b), and later
functional generation with more general numeraire portfolios was developed
(Strong, 2014). The original functionally generated portfolios were defined
multiplicatively, and later the theory was extended to additive portfolio
generation (Karatzas and Ruf, 2017). Portfolios generated by functions that
depend on the sample paths of the assets rather than only on their current
values were introduced by Karatzas and Kim (2020). Portfolio generating
functions have been used for many purposes, from explaining the size effect
(Fernholz (2002), Example 4.3.2) to proving the existence of arbitrage in
stock markets (Fernholz et al. (2005) and Fernholz et al. (2018)).

In this paper we use multiplicative portfolio generation with definitions
based on asset prices rather than market weights. We extend the theory of
functionally generated portfolios to measure the value of portfolios directly,
with no numeraire portfolio. We restrict our attention to contingent claim
functions, which are defined to be _scalable_ $C^{2,1}$ functions of the asset
prices and time, and decompose the value of portfolios generated by these
functions into generating function and drift components. This decomposition is
similar to that of functionally generated portfolios with the market or other
portfolios as numeraire (Fernholz, 2002; Strong, 2014).

The modern theory of option pricing started with Black and Scholes (1973), who
derived the well-known Black-Scholes differential equation and then solved
this equation for the price of European put and call stock options. This
theory was advanced by Merton (1973), who developed the concept of a
dynamically rebalanced portfolio to replicate the prices of options following
the Black-Scholes model (see, e.g., the discussion of “perfect
substitutability” and “perfect hedge” on page 169 of Merton (1973)). The
theory of option pricing has developed considerably, and a review of much of
the early work can be found in Smith (1976).

Merton (1973) recognized scalability to be a natural property of options, and
referred to this property as _homogeneous of degree one_. Scalability requires
that a change of numeraire currency affects both the asset prices and the
value of the contingent claim in the same way. For example, if we switch from
measuring asset prices in euros rather than dollars, then the same exchange
rate should be applied to the price of a contingent claim as to the values of
the underlying assets.

This paper extends the theory of option pricing, since our methods and results
can be applied to generate portfolios that replicate the values of contingent
claims. We provide necessary and sufficient conditions for a scalable
$C^{2,1}$ contingent claim function to be _replicable_ , and show that these
conditions require that the contingent claim function satisfy a particular
parabolic differential equation. This differential equation is a general,
multivariate form of the Black-Scholes equation, and we provide several
examples with exact analytic solutions. Because any contingent claim function
that satisfies this general Black-Scholes differential equation is replicable,
it follows that any such function is equal to the present value of an option
with expiration payout equal to the terminal value of that same contingent
claim function. Furthermore, our results establish that the replicating
portfolio for this option is the portfolio generated by that contingent claim
function. In this way, we show that central concepts of two branches of
mathematical finance — stochastic portfolio theory and option pricing theory —
are closely related.

Finally, we generalize our framework to include assets that pay continuous
dividends and we apply this to several well-known examples. For two assets,
where one is riskless and pays a constant dividend and the other is risky and
pays no dividends, our general differential equation is identical to the
original Black-Scholes equation. If the terminal value of the contingent claim
function in this case is the payout of a European call option, then the
solution of the differential equation will be the same as in Black and Scholes
(1973). Similarly, if we consider a setting with one riskless asset and two or
more uncorrelated risky assets, then the solution of our general differential
equation will yield the same option price as in Stulz (1982) or Johnson
(1987). This setting can be further extended to include multiple correlated
risky assets as in Carmona and Durrleman (2006) and Guillaume (2019). Our
framework demonstrates that all of these examples can be understood as
different solutions to various cases of the same general differential
equation.

##  2 Setup

We consider a family of $n\geq 2$ stocks or other assets represented by the
price processes $X_{1},\ldots,X_{n}$, where the $X_{i}$ are strictly positive
continuous semimartingales that satisfy

$d\log
X_{i}(t)=\gamma_{i}(t)\,dt+\sum^{d}_{\ell=1}\zeta_{i,\ell}(t)\,dW_{\ell}(t),$
(2.1)

for $i=1,\ldots,n$, where $W=(W_{1},\ldots,W_{d})$, with $d\geq n$, is a
Brownian motion and the processes $\gamma_{i}$ and $\zeta_{i,\ell}$ are square
integrable and progressively measurable with respect to the Brownian
filtration ${\mathcal{F}}^{W}$. Let $\langle\log X_{i},\log X_{j}\rangle_{t}$
represent the cross variation process for $\log X_{i}$ and $\log X_{j}$, with
processes $\sigma_{ij}$ such that

$\sigma_{ij}(t)\triangleq\sum_{\ell=1}^{d}\zeta_{i,\ell}(t)\zeta_{j,\ell}(t)=\frac{d\langle\log
X_{i},\log X_{j}\rangle_{t}}{dt},$

for $i,j=1,\ldots,n$ and $t\in[0,T]$. A riskless asset has zero quadratic
variation, and we assume neither the existence nor the nonexistence of a
riskless asset. If a riskless asset exists, we shall denote it by $X_{1}$,
with $\langle X_{1}\rangle_{t}\equiv 0$, for $t\in[0,T]$ . In Sections 2
through 5, we assume that there are no dividends.

A portfolio $\pi$ is identified by its weight processes
$\pi_{1},\ldots,\pi_{n}$, which are bounded processes progressively measurable
with respect to ${\mathcal{F}}^{W}$, and which add up to one. For a portfolio
$\pi$, the portfolio value process $Z_{\pi}$ will satisfy

$dZ_{\pi}(t)\triangleq
Z_{\pi}(t)\sum_{i=1}^{n}\pi_{i}(t)\frac{dX_{i}(t)}{X_{i}(t)}.$

In logarithmic form this becomes

$d\log Z_{\pi}(t)=\sum_{i=1}^{n}\pi_{i}(t)\,d\log
X_{i}(t)+\gamma^{*}_{\pi}(t)dt,\quad\text{{\rm a.s.}},$ (2.2)

with the excess growth rate process

$\gamma^{*}_{\pi}(t)\triangleq\frac{1}{2}\bigg{(}\sum_{i=1}^{n}\pi_{i}(t)\sigma_{ii}(t)-\sum_{i,j=1}^{n}\pi_{i}(t)\pi_{j}(t)\sigma_{ij}(t)\bigg{)}.$
(2.3)

Details regarding this logarithmic representation for portfolios can be found
in Fernholz (2002), Section 1.1.

In some contexts the assets $X_{1},\ldots,X_{n}$ comprise a market, e.g., if
the $X_{i}$ represent the capitalizations of companies. In this case, an
important portfolio is the market portfolio $\mu$ with market weight processes
$\mu_{i}$, for $i=1,\ldots,n$, defined by

$\mu_{i}(t)\triangleq X_{i}(t)/Z(t),$

for $t\in[0,T]$, with

$Z(t)\triangleq X_{1}(t)+\cdots+X_{n}(t),$

so that $\mu(t)\in\Delta^{n}\triangleq\\{x\in{\mathbb{R}}^{+n}\colon
x_{1}+\cdots+x_{n}=1\\}$, a.s., for $t\in[0,T]$ (with
${\mathbb{R}}^{+}=\\{x\in{\mathbb{R}}\colon x>0\\}$). A short calculation
shows that the value process $Z_{\mu}$ for the market portfolio satisfies

$d\log Z_{\mu}(t)=d\log Z(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$.

We use partial differentiation of $C^{2,1}$ functions of the form
$f\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$, and we use the
notation $D_{i}f$ for the partial derivative of $f$ with respect to the $i$th
variable, for $i=1,\ldots,n$, and $D_{t}f$ for the partial derivative of $f$
with respect to the last variable, time. In order to avoid confusion when we
consider derivatives of composed functions $f\circ g$, we sometimes use
classical notation for partial derivatives. For example, for $C^{1}$ functions
$f\colon{\mathbb{R}}^{+}{}^{n}\to{\mathbb{R}}^{+}$ and
$g=(g_{1},\ldots,g_{n})\colon{\mathbb{R}}^{+}{}^{m}\to{\mathbb{R}}^{+}{}^{n}$
the partial derivative $D_{i}(f\circ g)(x)$ with respect to $x_{i}$, for
$x=(x_{1},\ldots,x_{m})\in{\mathbb{R}}^{+}{}^{m}$, will be written with the
notation

$\frac{\partial}{\partial x_{i}}\big{(}f(g(x)\big{)}=D_{i}(f\circ
g)(x)=\sum_{j=1}^{n}D_{j}f(g(x))D_{i}g_{j}(x),$

for $i=1,\ldots,m$, by the chain rule.

##  3 Contingent claim functions

Contingent claim functions provide a present value for a combination of assets
that is contingent on some future event. Perhaps the prototype contingent
claim function was introduced in the option-pricing model developed by Fischer
Black and Myron Scholes (Black and Scholes, 1973). This model generated a
present value for a European call option, which at the expiration date pays
the amount by which a stock price exceeds a given strike price, or zero, if
the stock price is below the strike price. The pricing function in the Black-
Scholes model is a function ${\mathbf{V}}$ of two asset values, $X_{1}$ and
$X_{2}$, where $X_{1}\equiv K$ is the fixed strike price and $X_{2}$ is the
variable price of a stock. After time $T$ has elapsed, the option pays the
greater of zero and the amount by which the stock price $X_{2}(T)$ exceeds the
strike price $X_{1}(T)=K$.

A natural property of a contingent claim function is scalability, which states
that a change in numeraire currency affects both asset prices and the value of
a contingent claim in the same way. For a European call, if both the stock
price $X_{2}$ and the strike price $X_{1}\equiv K$ are measured in dollars,
then the option value ${\mathbf{V}}$ is also measured in dollars. Now suppose
that we measure all the prices in cents rather than dollars, so the stock
prices now change by a factor of 100. For consistency, the price of the call
in cents must now also change by a factor of 100, so

${\mathbf{V}}(cX_{1}(t),cX_{2}(t),t)=c{\mathbf{V}}(X_{1}(t),X_{2}(t),t),$
(3.1)

for $t\in[0,T]$, with $c=100$. The same transformation would be necessary if
we changed from dollars to euros, for example, with $c$ now equal to the
appropriate exchange rate. Merton (1973) referred to this property as
homogeneous of degree one, and showed that it is a property inherent in
rationally priced options (see Merton (1973), Theorem 6). We shall see in
Example 7.2 below that the Black-Scholes option-pricing formula satisfies
(3.1). We now formalize this property.

######  Definition 3.1.

For $n\geq 1$, a function $V\colon{\mathbb{R}}^{+}{}^{n}\to{\mathbb{R}}^{+}$
is scalable if for $c\in{\mathbb{R}}^{+}$ and $x\in{\mathbb{R}}^{+}{}^{n}$,
$V(cx)=cV(x)$. A function
$V\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ is scalable if
for $c\in{\mathbb{R}}^{+}$, $x\in{\mathbb{R}}^{+}{}^{n}$, and $t\in[0,T]$,
$V(cx,t)=cV(x,t)$.

######  Definition 3.2.

A contingent claim function is a continuous scalable function
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ that
is $C^{2,1}$ on ${\mathbb{R}}^{+}{}^{n}\times[0,T)$. The terminal value
function for the contingent claim function ${\mathbf{V}}$ is the function
${\mathbf{V}}(\,\cdot\,,T)\colon{\mathbb{R}}^{+n}\to{\mathbb{R}}^{+}$.

The definition restricts contingent claim functions to be functions with
values in ${\mathbb{R}}^{+}$, which excludes the Black-Scholes valuation
function. However, we show in Example 7.2 below that in practice this apparent
restriction is easily accommodated.

The following lemma is the key to unlocking portfolio generation by contingent
claim functions.

######  Lemma 3.3.

Let $V\colon{\mathbb{R}}^{+}{}^{n}\to{\mathbb{R}}^{+}$ be a scalable $C^{1}$
function. Then, for $x\in{\mathbb{R}}^{+}{}^{n}$,

$\sum_{i=1}^{n}x_{i}D_{i}V(x)=V(x).$ (3.2)

###### Proof.

For $n=1$, and $x\in{\mathbb{R}}^{+}$, $V(x)=xV(1)$, so

$DV(x)=\frac{d}{dx}\big{(}xV(1)\big{)}=V(1),$

and $xDV(x)=xV(1)=V(x)$.

Now suppose $n\geq 2$. For $x\in{\mathbb{R}}^{+}{}^{n}$, let
$z=x_{1}+\cdots+x_{n}$ and let $\xi_{j}=x_{j}z^{-1}$, for $j=1,\ldots,n$, so
$\xi=(\xi_{1}\ldots,\xi_{n})\in{\mathbb{R}}^{+}{}^{n}$. Then $x=z\xi$, and
since $V$ is scalable,

$V(x)=V(z\xi)=zV(\xi).$

The calculations

$\frac{\partial z}{\partial
x_{i}}=1,\quad\text{and}\quad\frac{\partial\xi_{j}}{\partial
x_{i}}=\frac{\partial}{\partial
x_{i}}(x_{j}z^{-1})=z^{-1}({\mathbbm{1}}_{i=j}-\xi_{j}),\vspace{5pt}$

for $i,j=1,\ldots,n$, give us

$\frac{\partial}{\partial
x_{i}}V(\xi)=\sum_{j=1}^{n}D_{j}V(\xi)\frac{\partial\xi_{j}}{\partial
x_{i}}=z^{-1}\Big{(}D_{i}V(\xi)-\sum_{j=1}^{n}\xi_{j}D_{j}V(\xi)\Big{)},$

and

$\displaystyle D_{i}V(x)$ $\displaystyle=\frac{\partial}{\partial
x_{i}}\big{(}zV(\xi)\big{)}$ $\displaystyle=V(\xi)+z\frac{\partial}{\partial
x_{i}}V(\xi)$
$\displaystyle=V(\xi)+D_{i}V(\xi)-\sum_{j=1}^{n}\xi_{j}D_{j}V(\xi),$ (3.3)

for $i=1,\ldots,n$. It follows that

$\displaystyle\sum_{i=1}^{n}x_{i}D_{i}V(x)$
$\displaystyle=\sum_{i=1}^{n}x_{i}\Big{(}V(\xi)+D_{i}V(\xi)-\sum_{j=1}^{n}\xi_{j}D_{j}V(\xi)\Big{)}$
$\displaystyle=zV(\xi)+\sum_{i=1}^{n}x_{i}D_{i}V(\xi)-z\sum_{j=1}^{n}\xi_{j}D_{j}V(\xi)$
$\displaystyle=V(x).$

∎

A condition similar to (3.2) applied to semimartingales appears in Karatzas
and Ruf (2017), Definition 3.2. Here (3.2) is not a definition, but rather the
consequence of scalability, a natural property of the class of functions that
we consider.

##  4 Portfolios generated by contingent claim functions

Functional generation of portfolios was introduced in Fernholz (1999b, 2002),
where a continuous function ${\mathbf{S}}\colon\Delta^{n}\to{\mathbb{R}}^{+}$
was defined to generate a portfolio $\pi$ if

$d\log\big{(}Z_{\pi}(t)/Z_{\mu}(t)\big{)}=d\log{\mathbf{S}}(\mu(t))+d\Theta_{\mathbf{S}}(t),\quad\text{{\rm
a.s.}},$ (4.1)

for $t\in[0,T]$, where $\Theta_{\mathbf{S}}$ is a function of finite
variation. More recently, Strong (2014) generalized this definition to the
case where the market portfolio $\mu$ could be replaced by an arbitrary
portfolio $\nu$, with the value function $Z_{\nu}$ replacing $Z_{\mu}$ in
(4.1) . Time-dependent portfolio generating functions were defined in Fernholz
(2002), Section 3.2, to be positive continuous functions
${\mathbf{S}}\colon\Delta^{n}\times[0,T]\to{\mathbb{R}}^{+}$. It was shown
that if ${\mathbf{S}}$ is $C^{2,1}$ it generates the portfolio $\pi$ with
weights

$\pi_{i}(t)=\Big{(}D_{i}\log{\mathbf{S}}(\mu(t),t)+1-\sum_{j=1}^{n}\mu_{j}(t)D_{j}\log{\mathbf{S}}(\mu(t),t)\Big{)}\mu_{i}(t),\quad\text{{\rm
a.s.}},$ (4.2)

for $t\in[0,T]$. The portfolio $\pi$ satisfies

$d\log\big{(}Z_{\pi}(t)/Z_{\mu}(t)\big{)}=d\log{\mathbf{S}}(\mu(t),t)+d\Theta_{\mathbf{S}}(t),\quad\text{{\rm
a.s.}},$ (4.3)

for $t\in[0,T]$, where $\Theta_{\mathbf{S}}$ is a function of finite
variation. Here we define time-dependent generating functions directly in
terms of the asset values rather than the market weights.

######  Definition 4.1.

A continuous function
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$
generates the portfolio $\pi$ if

$d\log
Z_{\pi}(t)=d\log{\mathbf{V}}(X(t),t)+d\Phi_{\mathbf{V}}(t),\quad\text{{\rm
a.s.}},$ (4.4)

for $t\in[0,T)$, where $\Phi_{\mathbf{V}}$ is a process of finite variation.
In this case, ${\mathbf{V}}$ is called a portfolio generating function and
$\Phi_{\mathbf{V}}$ the drift process for ${\mathbf{V}}$.

######  Example 4.2.

Let ${\mathbf{V}}(x_{1},x_{2},t)=x_{1}\lor x_{2}$. Then $V$ generates the
portfolio $\pi$ with weights

$\pi_{1}(t)={\mathbbm{1}}_{X_{1}(t)\geq X_{2}(t)}(t)\quad\text{ and
}\quad\pi_{2}(t)=1-\pi_{1}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$, with value process

$Z_{\pi}(t)=X_{1}(t)\lor X_{2}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$, and drift process

$\Phi_{\mathbf{V}}(t)=-\Lambda_{\log(X_{1}/X_{2})}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$, where $\Lambda_{\log(X_{1}/X_{2})}$ is the semimartingale
local time at zero for $\log(X_{1}/X_{2})$ (see Fernholz (2002), Lemma 4.1.10
and Example 4.3.1 regarding this example, and Section 3.7 of Karatzas and
Shreve (1991) regarding semimartingale local time). Let us note that
${\mathbf{V}}$ is scalable here, as required. ∎

In Example 4.2, ${\mathbf{V}}$ is not differentiable, but now we restrict our
attention to contingent claim functions.

######  Theorem 4.3.

Let ${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$
be a contingent claim function, and suppose that, for $i=1,\ldots,n$, the
functions $x_{i}D_{i}{\mathbf{V}}(x,t)/{\mathbf{V}}(x,t)$ are bounded for
$(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$. Then ${\mathbf{V}}$ generates the
portfolio $\pi$ with weights

$\pi_{i}(t)=\frac{X_{i}(t)D_{i}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)},\quad\text{{\rm
a.s.}},$ (4.5)

for $i=1,\ldots,n$ and $t\in[0,T)$, and drift process

$\Phi_{\mathbf{V}}(t)=-\int_{0}^{t}\bigg{(}\frac{1}{2}\sum_{i,j=1}^{n}\frac{D_{ij}{\mathbf{V}}(X(s),s)}{{\mathbf{V}}(X(s),s)}X_{i}(s)X_{j}(s)\sigma_{ij}(s)+\frac{D_{t}{\mathbf{V}}(X(s),s)}{{\mathbf{V}}(X(s),s)}\bigg{)}ds,\quad\text{{\rm
a.s.}},$ (4.6)

for $t\in[0,T)$.

###### Proof.

By Itô’s rule,

$\displaystyle d\log{\mathbf{V}}(X(t),t)$
$\displaystyle=\sum_{i=1}^{n}D_{i}\log{\mathbf{V}}(X(t),t)dX_{i}(t)+D_{t}\log{\mathbf{V}}(X(t),t)dt+\frac{1}{2}\sum_{i,j=1}^{n}D_{ij}\log{\mathbf{V}}(X(t),t)d\big{\langle}X_{i},X_{j}\big{\rangle}_{t}$
$\displaystyle=\sum_{i=1}^{n}X_{i}(t)D_{i}\log{\mathbf{V}}(X(t),t)d\log
X_{i}(t)+\frac{1}{2}\sum_{i=1}^{n}X_{i}(t)D_{i}\log{\mathbf{V}}(X(t),t)\sigma_{ii}(t)dt$
$\displaystyle\qquad+D_{t}\log{\mathbf{V}}(X(t),t)dt+\frac{1}{2}\sum_{i,j=1}^{n}X_{i}(t)X_{j}(t)D_{ij}\log{\mathbf{V}}(X(t),t)\sigma_{ij}(t)dt,\quad\text{{\rm
a.s.}},$ (4.7)

for $t\in[0,T)$, where we applied Itô’s rule again for the second equation.
Note that in $D_{i}\log{\mathbf{V}}=D_{i}(\log\circ{\mathbf{V}})$, etc., the
partial derivatives are applied to the composition of $\log$ and
${\mathbf{V}}$.

Lemma 3.3 shows that the $\pi_{i}$ defined by (4.5) add up to one, so they
define a portfolio $\pi$, and with this portfolio (4.7) becomes

$\displaystyle d\log{\mathbf{V}}(X(t),t)$
$\displaystyle=\sum_{i=1}^{n}\pi_{i}(t)d\log
X_{i}(t)+\frac{1}{2}\sum_{i=1}^{n}\pi_{i}(t)\sigma_{ii}(t)dt+\frac{D_{t}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}dt$
$\displaystyle\qquad+\frac{1}{2}\sum_{i,j=1}^{n}\frac{D_{ij}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}X_{i}(t)X_{j}(t)\sigma_{ij}(t)dt-\frac{1}{2}\sum_{i,j=1}^{n}\pi_{i}(t)\pi_{j}(t)\sigma_{ij}(t)dt$
$\displaystyle=\sum_{i=1}^{n}\pi_{i}(t)d\log
X_{i}(t)+\gamma^{*}_{\pi}(t)dt+\frac{D_{t}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}dt$
$\displaystyle\qquad+\frac{1}{2}\sum_{i,j=1}^{n}\frac{D_{ij}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}X_{i}(t)X_{j}(t)\sigma_{ij}(t)dt$
$\displaystyle=d\log Z_{\pi}(t)-d\Phi_{\mathbf{V}}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T)$, as in (2.2) and (2.3), and (4.4) follows. ∎

The portfolio weight processes $\pi_{i}$ in (4.5) are implicit functions of
the price process $X$, with

$\pi_{i}(t)=\pi_{i}(X(t),t),$

for $i=1,\ldots,n$, and $t\in[0,T]$.

######  Definition 4.4.

A portfolio $\pi$ is scale-invariant if for $c\in{\mathbb{R}}^{+}$ and price
process $X$

$\pi_{i}(cX(t),t)=\pi_{i}(X(t),t),$

for $i=1,\ldots,n$ and $t\in[0,T]$.

######  Corollary 4.5.

The portfolio generated by a contingent claim function is scale-invariant.

###### Proof.

Let ${\mathbf{V}}$ be a contingent claim function and let $\pi$ be defined as
in (4.5). Since ${\mathbf{V}}$ is scalable, for $c\in{\mathbb{R}}^{+}$,
$x\in{\mathbb{R}}^{+}{}^{n}$, and $t\in[0,T)$,

$D_{i}{\mathbf{V}}(cx,t)=\frac{\partial}{\partial
cx_{i}}{\mathbf{V}}(cx,t)=c^{-1}\frac{\partial}{\partial
x_{i}}\big{(}c{\mathbf{V}}(x,t)\big{)}=D_{i}{\mathbf{V}}(x,t).$

Hence, for price process $X$,

$\pi_{i}(cX(t),t)=\frac{cX_{i}(t)D_{i}{\mathbf{V}}(cX(t),t)}{{\mathbf{V}}(cX(t),t)}=\frac{X_{i}(t)D_{i}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}=\pi_{i}(X(t),t),\quad\text{{\rm
a.s.}},$

for $i=1,\ldots,n$, and $t\in[0,T)$. This equation can be extended by
continuity to $t\in[0,T]$. ∎

A $C^{2,1}$ function ${\mathbf{S}}=\Delta^{n}\times[0,T]\to{\mathbb{R}}^{+}$
can be extended to a contingent claim function
$\SS\colon{\mathbb{R}}^{+n}\times[0,T]\to{\mathbb{R}}^{+}$, where

$\SS(x,t)=z{\mathbf{S}}(z^{-1}x,t)=z{\mathbf{S}}(\mu,t),$ (4.8)

for $x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{+n}$, $t\in[0,T]$,
$z=x_{1}+\cdots+x_{n}$, and $\mu=z^{-1}x\in\Delta^{n}$. For
$c\in{\mathbb{R}}^{+}$,

$\SS(cx,t)=cz{\mathbf{S}}((cz)^{-1}cx,t)=cz{\mathbf{S}}(z^{-1}x,t)=c\SS(x,t),$

for $x\in{\mathbb{R}}^{+n}$ and $t\in[0,T]$, so $\SS$ is scalable. We now show
that ${\mathbf{S}}$ and $\SS$ generate the same portfolio.

######  Proposition 4.6.

Let ${\mathbf{S}}\colon\Delta^{n}\times[0,T]\to{\mathbb{R}}^{+}$ be a
$C^{2,1}$ function and let
$\SS\colon{\mathbb{R}}^{+n}\times[0,T]\to{\mathbb{R}}^{+}$ be its extension by
(4.8). Then the weight processes of the portfolios generated by ${\mathbf{S}}$
and $\SS$ are almost surely equal, and their drift processes satisfy

$d\Phi_{\SS}(t)=d\Theta_{\mathbf{S}}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$.

###### Proof.

First, let us show that the weight processes for the portfolios generated by
${\mathbf{S}}$ and $\SS$ are almost surely equal. It follows from (3.3) that

$D_{i}\SS(X(t),t)={\mathbf{S}}(\mu(t),t)+D_{i}{\mathbf{S}}(\mu(t),t)-\sum_{j=1}^{n}\mu_{j}(t)D_{j}{\mathbf{S}}(\mu(t),t),\quad\text{{\rm
a.s.}},$

for $i=1,\ldots,n$, and $t\in[0,T]$, and $\mu(t)=Z^{-1}(t)X(t)$. By (4.5), the
weight processes $\pi_{1},\ldots,\pi_{n}$ for $\SS$ are given by

$\displaystyle\pi_{i}(t)$
$\displaystyle=\frac{X_{i}(t)D_{i}\SS(X(t),t)}{\SS(X(t),t)}=\frac{X_{i}(t)D_{i}\SS(X(t),t)}{Z(t){\mathbf{S}}(\mu(t),t)}$
$\displaystyle=\mu_{i}(t)\Big{(}1+D_{i}\log{\mathbf{S}}(\mu(t),t)-\sum_{j=1}^{n}\mu_{j}(t)D_{j}\log{\mathbf{S}}(\mu(t),t)\Big{)},\quad\text{{\rm
a.s.}},$

for $i=1,\ldots,n$, and $t\in[0,T]$, which are almost surely equal to the
weight processes for ${\mathbf{S}}$ in (4.2). Hence, the term $d\log
Z_{\pi}(t)$ will be almost surely equal in (4.4) and (4.3), and

$\displaystyle d\Phi_{\SS}(t)$ $\displaystyle=d\log
Z_{\pi}(t)-d\log\SS(X(t),t)$ $\displaystyle=d\log
Z_{\pi}(t)-d\log\big{(}Z(t){\mathbf{S}}(\mu(t),t)\big{)}$
$\displaystyle=d\log\big{(}Z_{\pi}(t)/Z(t)\big{)}-d\log{\mathbf{S}}(\mu(t),t)$
$\displaystyle=d\Theta_{\mathbf{S}}(t),\quad\text{{\rm a.s.}},$

for $t\in[0,T]$. ∎

######  Example 4.7.

The extended entropy function. The entropy function

$S(x)=-\sum_{i=1}^{n}x_{i}\log x_{i},$

for $x\in\Delta^{n}$, served as a prototype for portfolio generating functions
in Fernholz (1999a), Definition 4.2, and it generates the portfolio $\pi$ with
weight processes

$\pi_{i}(t)=-\frac{\mu_{i}(t)\log\mu_{i}(t)}{S(\mu(t))},\quad\text{{\rm
a.s.}},$ (4.9)

for $i=1,\ldots,n$ and $t\in[0,T]$, where the $\mu_{i}$ are the market weight
processes (see also Fernholz (1999b), Example 4.2, and Fernholz (2002),
Section 2.3). In our context we can let ${\mathbf{S}}(x,t)=S(x)$, for
$x\in\Delta^{n}$ and $t\in[0,T]$, with a dummy time variable, but for some
$x\in{\mathbb{R}}^{+n}$ this function ${\mathbf{S}}$ will take on negative
values, so it cannot serve as a generating function. However, we can extend
${\mathbf{S}}$ to $\SS$ as in (4.8), with

$\SS(x,t)=z{\mathbf{S}}(z^{-1}x,t),$

for $x\in{\mathbb{R}}^{+n}$, $t\in[0,T]$, and $z=x_{1}+\cdots+x_{n}$. If we
let $\mu=z^{-1}x$, then $\mu\in\Delta^{n}$ and

$\SS(x,t)=z{\mathbf{S}}(\mu,t)=-z\sum_{i=1}^{n}\mu_{i}\log\mu_{i}=z\log
z-\sum_{i=1}^{n}x_{i}\log x_{i},$

for $t\in[0,T]$. In this case (4.5) shows that the extended entropy function
$\SS$ generates the portfolio $\pi$ with weights

$\pi_{i}(t)=\frac{X_{i}(t)D_{i}\SS(X(t),t)}{\SS(X(t),t)}=\frac{X_{i}(t)\big{(}\log
Z(t)-\log
X_{i}(t)\big{)}}{\SS(X(t),t)}=-\frac{\mu_{i}(t)\log\mu_{i}(t)}{{\mathbf{S}}(\mu(t),t)},\quad\text{{\rm
a.s.}},$

for $i=1,\ldots,n$, and $t\in[0,T]$, and this agrees with (4.9) above. ∎

##  5 Replicable contingent claim functions

The concept of replicability was introduced by Merton (1973), where he
considered combinations of a stock, an option on the stock, and a riskless
bond (see Merton (1973), the last paragraph of page 168). Here we formalize
this concept for contingent claim functions.

######  Definition 5.1.

Let ${\mathbf{V}}$ be a contingent claim function and let $\pi$ be the
portfolio it generates by Theorem 4.3. Then ${\mathbf{V}}$ is replicable if

$Z_{\pi}(t)={\mathbf{V}}(X(t),t),\quad\text{{\rm a.s.}},$ (5.1)

for $t\in[0,T)$.

This definition is suggested in the Remark following Example 3.2.3 of Fernholz
(2002). A financial instrument priced at the value of a non-replicable
contingent claim function would create an instant arbitrage between this
instrument and the portfolio it generates. Accordingly, our definition of
replicable could be considered a local version of the term viable in Section
2.2.2 of Karatzas and Kardaras (2021).

The following proposition follows immediately from Definitions 4.1 and 5.1.

######  Proposition 5.2.

A contingent claim function ${\mathbf{V}}$ is replicable if and only if its
drift process

$\Phi_{\mathbf{V}}(t)=0,\quad\text{{\rm a.s.}},$ (5.2)

for $t\in[0,T)$.

######  Corollary 5.3.

A contingent claim function
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ is
replicable if and only if

$D_{t}{\mathbf{V}}(X(t),t)+\frac{1}{2}\sum_{i,j=1}^{n}\sigma_{ij}(t)X_{i}(t)X_{j}(t)D_{ij}{\mathbf{V}}(X(t),t)=0,\quad\text{{\rm
a.s.}},$ (5.3)

for $t\in[0,T)$.

###### Proof.

This follows immediately from Proposition 5.2 and (4.6) of Theorem 4.3. ∎

If a contingent claim function ${\mathbf{V}}$ is replicable, then for each
$(x,t)\in{\mathbb{R}}^{+}{}^{n}\times[0,T)$, ${\mathbf{V}}(x,t)$ is the
present value of an asset with value ${\mathbf{V}}(x,T)$ at time $T$.
Corollary 5.3 thus provides necessary and sufficient conditions for a
contingent claim function to correctly price an option with an expiration
payout of ${\mathbf{V}}(\,\cdot\,,T)$. Furthermore, the replicating portfolio
for this option is the portfolio generated by the contingent claim function
${\mathbf{V}}$ according to (4.5) from Theorem 4.3.

######  Example 5.4.

For real constants $p_{1},\ldots,p_{n}$, with $p_{1}+\cdots+p_{n}=1$, let
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ be
the $C^{2,1}$ function defined by

${\mathbf{V}}(x,t)=x_{1}^{p_{1}}\cdots x_{n}^{p_{n}},$

for $x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{+n}$ and $t\in[0,T]$. Then
${\mathbf{V}}$ is scalable, so it is a contingent claim function and generates
the portfolio $\pi$ with weights

$\pi_{i}(t)=\frac{X_{i}(t)D_{i}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}=\frac{X_{i}(t)\big{(}p_{i}X_{1}^{p_{1}}(t)\cdots
X_{i}^{p_{i}-1}(t)\cdots X_{n}^{p_{n}}(t)\big{)}}{X_{1}^{p_{1}}(t)\cdots
X_{n}^{p_{n}}(t)}=p_{i},$

for $i=1,\ldots,n$ and $t\in[0,T]$, where the $X_{i}$ are defined as in (2.1).
Here $\pi$ is a constant-weighted portfolio, and by Theorem 4.3 we have the
decomposition (4.4) with drift process

$\displaystyle\Phi_{\mathbf{V}}(t)$
$\displaystyle=\frac{1}{2}\int_{0}^{t}\bigg{(}\sum_{i=1}^{n}p_{i}\sigma_{ii}(s)-\sum_{i,j=1}^{n}p_{i}p_{j}\sigma_{ij}(s)\bigg{)}ds$
$\displaystyle=\int_{0}^{t}\gamma_{\pi}^{*}(s)ds,\quad\text{{\rm a.s.}},$
(5.4)

for $t\in[0,T]$. If, for example, $p_{i}>0$, for $i=1,\ldots,n$, and the
covariance matrix $(\sigma_{ij})$ is nonsingular, then $\gamma^{*}_{\pi}>0$,
a.s., (see Fernholz (2002), Proposition 1.3.7), so (5.2) is not satisfied, and
${\mathbf{V}}$ is not replicable.

It would be convenient to define
${\mathbf{\widetilde{V}}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$
by

${\mathbf{\widetilde{V}}}(x,t)={\mathbf{V}}(x,t)\exp\Big{(}\int_{0}^{t}\gamma_{\pi}^{*}(s)ds\Big{)},$
(5.5)

for $(x,t)\in{\mathbb{R}}^{+}{}^{n}\times[0,T]$, which would compensate for
the term (5.4), but in this case ${\mathbf{\widetilde{V}}}$ could be a random
process rather than a function. However, if the $\sigma_{ij}$ are constant,
then $\gamma^{*}_{\pi}(t)=\gamma^{*}_{\pi}(0)$, for $t\in[0,T]$, is also
constant, and

${\mathbf{\widetilde{V}}}(x,t)={\mathbf{V}}(x,t)e^{\gamma_{\pi}^{*}(0)t},$
(5.6)

for $(x,t)\in{\mathbb{R}}^{+}{}^{n}\times[0,T]$, so ${\mathbf{\widetilde{V}}}$
is a $C^{2,1}$ function. Now, for $t\in[0,T]$,

$\displaystyle d\log{\mathbf{\widetilde{V}}}(X(t),t)$
$\displaystyle=d\log{\mathbf{V}}(X(t),t)+\gamma_{\pi}^{*}(0)dt$
$\displaystyle=d\log Z_{\pi}(t)-d\Phi_{\mathbf{V}}(t)+\gamma_{\pi}^{*}(0)dt$
$\displaystyle=d\log Z_{\pi}(t),\quad\text{{\rm a.s.}},$

so ${\mathbf{\widetilde{V}}}$ is a replicable contingent claim function with
drift process

$\Phi_{\mathbf{\widetilde{V}}}(t)=0,\quad\text{{\rm a.s.}},$

for $t\in[0,T)$, as required by Proposition 5.2. ∎

######  Example 5.5.

For $p\in(0,1)$ let

${\mathbf{V}}(x,t)=\big{(}x_{1}^{p}+\cdots+x_{n}^{p}\big{)}^{1/p},$ (5.7)

for $x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{+}{}^{n}$ and $t\in[0,T]$. Then
${\mathbf{V}}$ is $C^{2,1}$ and scalable, so it is a contingent claim function
and generates the portfolio $\pi$ with weights

$\pi_{i}(t)=\frac{X_{i}(t)D_{i}{\mathbf{V}}(X(t),t)}{{\mathbf{V}}(X(t),t)}=\frac{X_{i}^{p}(t)}{X_{1}^{p}(t)+\cdots+X_{n}^{p}(t)},$

for $i=1,\ldots,n$ and $t\in[0,T]$. In this case, the decomposition (4.4) has
drift process

$\Phi_{\mathbf{V}}(t)=(1-p)\int_{0}^{t}\gamma_{\pi}^{*}(s)ds,\quad\text{{\rm
a.s.}},$

for $t\in[0,T]$ (see Fernholz (2002), Example 3.4.4), so as in Example 5.4,
${\mathbf{V}}$ is not replicable. By modifying ${\mathbf{V}}$ as in (5.5), we
have

${\mathbf{\widetilde{V}}}(x,t)={\mathbf{V}}(x,t)\exp\Big{(}(1-p)\int_{0}^{t}\gamma_{\pi}^{*}(s)ds\Big{)},$

for $(x,t)\in{\mathbb{R}}^{+}{}^{n}\times[0,T]$, but since the portfolio
weights $\pi_{i}$ are variable, $\gamma^{*}_{\pi}$ will not be constant even
if all the covariances are. Hence, ${\mathbf{\widetilde{V}}}$ remains a
process rather than a function, and this modification fails to create a
replicable contingent claim function.∎

Although in Example 5.5 we cannot use the simple adjustment of Example 5.4,
nevertheless with non-random covariances this generating function can be
modified to become replicable. The modification needed to make the contingent
claim function (5.7) replicable can be constructed by first using a
logarithmic transformation to convert (5.3) into a heat equation, and then
finding a convolutional solution to this heat equation (see Evans (2010),
Section 2.3, Theorem 1). This procedure is carried out for the simple case of
$p=1/2$ and $\sigma_{ij}=0$ for $i\neq j$ in Example A.1 of the Appendix.

##  6 Dividends

It is convenient to use continuous dividend payments to represent the
continuous interest payments from a riskless bank account. However, the
treatment we propose here applies to all continuous dividend payments, whether
they are from riskless assets or from stocks.

######  Definition 6.1.

A dividend rate process $\delta$ is a bounded process defined on $[0,T]$ that
is progressively measurable with respect to ${\mathcal{F}}^{W}$.

In this section and the next, price processes $X_{i}$ may have corresponding
dividend rate processes $\delta_{i}$, which measure the rate at which
dividends are paid by the assets. A dividend rate process contributes to the
“total return” of the asset (see Fernholz (2002), (1.1.31)). For a portfolio
$\pi$, the dividend rate process for the portfolio is defined by

$\delta_{\pi}(t)\triangleq\sum_{i=1}^{n}\pi_{i}(t)\delta_{i}(t),$

for $t\in[0,T]$, so $\delta_{\pi}$ measures the rate at which the portfolio
pays dividends.

######  Definition 6.2.

For price processes $X_{1},\ldots,X_{n}$ with corresponding dividend processes
$\delta_{1},\ldots,\delta_{n}$, let ${\mathbf{V}}$ be a contingent claim
function and let $\pi$ be the portfolio it generates. Then ${\mathbf{V}}$ is
replicable if

$\log{\mathbf{V}}(X(t),t)=\log
Z_{\pi}(t)+\int_{0}^{t}\delta_{\pi}(s)ds,\quad\text{{\rm a.s.}},$ (6.1)

for $t\in[0,T)$, where $\delta_{\pi}$ is the dividend rate process for $\pi$.

The next proposition follows immediately from Definitions 4.1 and 6.2.

######  Proposition 6.3.

Let ${\mathbf{V}}$ be a contingent claim function and $\pi$ be the portfolio
it generates. Then ${\mathbf{V}}$ is replicable if and only if its drift
process

$\Phi_{\mathbf{V}}(t)=\int_{0}^{t}\delta_{\pi}(s)ds,\quad\text{{\rm a.s.}},$

for $t\in[0,T)$, where $\delta_{\pi}$ is the dividend rate process for $\pi$.

######  Corollary 6.4.

For price processes $X_{1},\ldots,X_{n}$ with corresponding dividend rate
processes $\delta_{1},\ldots,\delta_{n}$, a contingent claim function
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ is
replicable if and only if

$D_{t}{\mathbf{V}}(X(t),t)+\frac{1}{2}\sum_{i,j=1}^{n}\sigma_{ij}(t)X_{i}(t)X_{j}(t)D_{ij}{\mathbf{V}}(X(t),t)=\sum_{i=1}^{n}\delta_{i}(t)X_{i}(t)D_{i}{\mathbf{V}}(X(t),t),\quad\text{{\rm
a.s.}},$ (6.2)

for $t\in[0,T)$.

###### Proof.

Follows from Proposition 6.3 and Theorem 4.3. ∎

We shall see in the next section that (6.2) is a general version of the Black-
Scholes equation (see Black and Scholes (1973), Equation (7)).

##  7 Black-Scholes and the option-pricing problem

The option-pricing problem for European options is: given a continuous and
scalable terminal value function
$f\colon{\mathbb{R}}^{+}{}^{n}\to{\mathbb{R}}^{+}$ and an expiration date
$T>0$, find a replicable contingent claim function
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ such
that ${\mathbf{V}}(x,T)=f(x)$, for $x\in{\mathbb{R}}^{+}{}^{n}$. The theory of
option pricing we consider here was pioneered by Black and Scholes (1973) and
Merton (1973).

######  Remark 7.1.

With certain restrictions, our setup in Section 2 will produce a _complete_
market (see Karatzas and Shreve (1998), Section 1.6, or Karatzas and Kardaras
(2021), Section 3.3). In a complete market, for a wide range of measurable
functions $f\colon{\mathbb{R}}^{+n}\to{\mathbb{R}}^{+}$, there exists for each
such function a corresponding portfolio $\pi$ with terminal value
$Z_{\pi}(T)=f(X(T))$, a.s. (Karatzas and Shreve (1998), Proposition 6.2). Here
our selection of terminal value functions is more restricted, since these
functions must be scalable. It is an open question whether or not in a
complete market every portfolio with a scalable terminal value function is
functionally generated.∎

In order to avoid the type of difficulty encountered in Example 5.4 for
general random processes $\sigma_{ij}$, we assume henceforth that the
parameters $\delta_{i}$ and $\sigma_{ij}$, for $i,j=1,\ldots,n$, are constant.
The growth rates $\gamma_{i}$ do not appear in (6.2), so there is no need to
assume that they are constant. Since the $\gamma_{i}$ may not be bounded, we
cannot assume the absence of arbitrage (see Fernholz et al. (2018), Section
5). Although the original motivation for the Black-Scholes model may have been
to avoid arbitrage, here we base option pricing on replicability. We should
note that it is not necessary here to assume that the $\delta_{i}$ and
$\sigma_{ij}$ are constant, but only that they are non-random functions of
time. However, in this more general setting the differential equation (6.2)
could become intractable, so we prefer to leave this generalization for future
work.

A (European) option is defined to have a specified payout at a given
expiration date $T>0$. The corresponding contingent claim function is defined
to have a specified continuous terminal value function
${\mathbf{V}}(\,\cdot\,,T)\colon{\mathbb{R}}^{+}{}^{n}\to[0,\infty)$. In this
case, we can write (6.2) as

$\frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sum_{i,j=1}^{n}\sigma_{ij}x_{i}x_{j}\frac{\partial^{2}{\mathbf{V}}}{\partial
x_{i}\partial
x_{j}}-\sum_{i=1}^{n}\delta_{i}x_{i}\frac{\partial{\mathbf{V}}}{\partial
x_{i}}=0,$ (7.1)

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$, which is a form of the Kolmogorov
backward equation (see, e.g., Karatzas and Shreve (1991), Chapter 5, Equation
(1.7)).

######  Example 7.2.

The Black-Scholes model is a solution to the option-pricing problem for a
(European) call option (Black and Scholes, 1973). For $T>0$, a call option has
terminal value ${\mathbf{V}}(x,K,T)=(x-K)^{+}$, for $x,K\in{\mathbb{R}}^{+}$,
where $K$ is a constant riskless asset called the strike price of the call.
Black and Scholes (1973) showed that ${\mathbf{V}}$ follows the differential
equation

$\frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}{\mathbf{V}}}{\partial
x^{2}}+rx\frac{\partial{\mathbf{V}}}{\partial x}-r{\mathbf{V}}=0,$ (7.2)

for $(x,t)\in{\mathbb{R}}^{+}\times[0,T)$, where the constant
$r\in{\mathbb{R}}$ is the riskless interest rate.

Let us consider the Black-Scholes model in our setting. For $n=2$, we define
the riskless asset $X_{1}$ with constant price $X_{1}\equiv K>0$ and constant
dividend rate $\delta_{1}\equiv r\in{\mathbb{R}}$. We define a second asset
$X_{2}$ that follows (2.1) with $\langle\log X_{2}\rangle_{t}=\sigma^{2}t$,
$\sigma^{2}>0$, for $t\in[0,T]$, and $\delta_{2}\equiv 0$. Since
$(x-K)^{+}\notin{\mathbb{R}}^{+}$ for $x\leq K$, we need to modify the
terminal value to, e.g., ${\mathbf{V}}(x_{1},x_{2},T)=x_{2}\lor x_{1}$ for
$x_{1},x_{2}\in{\mathbb{R}}^{+}$, and then subtract $x_{1}=K$ from the result.

For $n=2$ we have $\pi_{1}(t)=1-\pi_{2}(t)$, a.s., so for
$(x,t)\in{\mathbb{R}}^{+2}\times[0,T)$, equation (7.1) becomes

$\displaystyle 0$ $\displaystyle=\frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sigma_{22}x_{2}^{2}\frac{\partial^{2}{\mathbf{V}}}{\partial
x_{2}^{2}}-rx_{1}\frac{\partial{\mathbf{V}}}{\partial x_{1}}$
$\displaystyle=\frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sigma^{2}x_{2}^{2}\frac{\partial^{2}{\mathbf{V}}}{\partial
x_{2}^{2}}+rx_{2}\frac{\partial{\mathbf{V}}}{\partial x_{2}}-r{\mathbf{V}},$
(7.3)

which for $(x_{2},t)\in{\mathbb{R}}^{+}\times[0,T)$ gives us (7.2).

Black and Scholes (1973) showed that the solution of the differential equation
(7.3) with the terminal value ${\mathbf{V}}(K,x_{2},T)=(x_{2}-K)^{+}$, for
$x_{2}\in{\mathbb{R}}^{+}$, is

${\mathbf{V}}(K,x_{2},t)=N(z_{1})x_{2}-N(z_{2})Ke^{r(t-T)},$ (7.4)

for $t\in[0,T)$, with

$z_{1}=\frac{1}{\sigma\sqrt{T-t}}\Big{(}\log\big{(}x_{2}/Ke^{r(t-T)}\big{)}+\sigma^{2}\big{(}T-t\big{)}/2\Big{)}\quad\text{
and }\quad z_{2}=z_{1}-\sigma\sqrt{T-t},$ (7.5)

for $t\in[0,T)$, where $N$ is the Gaussian cumulative distribution function,

$N(z)\triangleq\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-s^{2}/2}ds,$ (7.6)

for $z\in{\mathbb{R}}$. We see that the function ${\mathbf{V}}$ defined by
(7.4) is scalable, since a change of scale by $c>0$ will change $x_{2}$ to
$cx_{2}$ and $x_{1}=K$ to $cx_{1}=cK$, so
${\mathbf{V}}(cx_{1},cx_{2},t)=c{\mathbf{V}}(x_{1},x_{2},t)$. Hence,
${\mathbf{V}}$ is a replicable contingent claim function.∎

The Black-Scholes value (7.4) can also be calculated by defining the riskless
asset $X_{1}$ with zero dividend rate $\delta_{1}\equiv 0$ and price that
grows at constant rate $r\in{\mathbb{R}}$, so that

$X_{1}(t)=Ke^{r(t-T)},$ (7.7)

for $t\in[0,T]$ and $K\in{\mathbb{R}}^{+}$. Here the value of $X_{1}$ grows
risklessly at the rate of $r$ and $X_{1}(T)=K$ at expiration. Equation (7.3)
becomes

$\ \frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sigma^{2}x_{2}^{2}\frac{\partial^{2}{\mathbf{V}}}{\partial
x_{2}^{2}}=0,$ (7.8)

for $(x,t)\in{\mathbb{R}}^{+2}\times[0,T)$, and the solution of this with
terminal value ${\mathbf{V}}(x_{1},x_{2},T)=(x_{2}-x_{1})^{+}$, for
$x_{1},x_{2}\in{\mathbb{R}}^{+}$, will be

${\mathbf{V}}(Ke^{r(t-T)},x_{2},t)=N(z_{1})x_{2}-N(z_{2})Ke^{r(t-T)},$ (7.9)

for $x_{2}\in{\mathbb{R}}^{+}$ and $t\in[0,T)$, with $z_{1}$ and $z_{2}$ as in
(7.5) and $N(z)$ as in (7.6). This solution is the same as (7.4), although the
riskless rate $r$ does not appear explicitly in the equation, but rather
enters implicitly with the asset $X_{1}$ as defined in (7.7).

The solution (7.9) follows from Margrabe (1978), Equation (7), and can also be
derived using the methods of Example A.1 in the Appendix. Margrabe finds the
price of an option on the maximum of the terminal prices of two risky assets,
and it remains valid if one of the assets is riskless. Note that the portfolio
generating function ${\mathbf{V}}$ of Example 4.2 has the same terminal value
as that of Margrabe (1978), but ${\mathbf{V}}$ is not a contingent claim
function since it is not $C^{2,1}$.

Stulz (1982) and Johnson (1987) extend the approach of Black and Scholes
(1973) to solve for the prices of options on the maximum or minimum of the
terminal prices of several uncorrelated assets. We can derive a multivariate
version of (7.2) for the price processes $X_{1},X_{2},\ldots,X_{n}$, for
$n\geq 2$, if we let the dividend rate for the constant process $X_{1}$ be the
riskless interest rate $r\in{\mathbb{R}}$, and let the rest of the dividend
rates be zero, with $\delta_{i}\equiv 0$, for $i=2,\ldots,n$. In this case
(7.1) becomes

$\frac{\partial{\mathbf{V}}}{\partial
t}+\frac{1}{2}\sum_{i,j=2}^{n}\sigma_{ij}x_{i}x_{j}\frac{\partial^{2}{\mathbf{V}}}{\partial
x_{i}\partial x_{j}}+r\sum_{i=2}^{n}x_{i}\frac{\partial{\mathbf{V}}}{\partial
x_{i}}-r{\mathbf{V}}=0,$ (7.10)

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$ (see Section 2.4 of Karatzas and
Shreve (1998) for a general exposition of these ideas). Both Carmona and
Durrleman (2006) and Guillaume (2019) present results about the solution to
(7.10) for different terminal value functions.

Acknowledgements. The authors thank Ioannis Karatzas for his invaluable
comments and suggestions, and the second author thanks the Institute for
Advanced Study for inspiring this research.

##  Appendix A Appendix

######  Example A.1.

Let us consider the contingent claim function (5.7) in the simplest case,
where $p=1/2$ and

$d\log X_{i}(t)=\gamma_{i}(t)dt+\sigma_{i}dW_{i}(t),$

for $i=1,\ldots,n$ and $t\in[0,T]$, with $W$ a Brownian motion and constants
$\sigma_{i}>0$ and ${\mathcal{F}}^{W}$-progressively measurable processes
$\gamma_{i}$, for $i=1,\ldots,n$. In this case (5.3) will be

$D_{t}{\mathbf{V}}(x,t)+\frac{1}{2}\sum_{i=1}^{n}\sigma_{i}^{2}x^{2}_{i}D_{ii}{\mathbf{V}}(x,t)=0,$
(A.1)

where $x\in{\mathbb{R}}^{+}{}^{n}$, $t\in[0,T]$, and
${\mathbf{V}}\colon{\mathbb{R}}^{+}{}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ is
$C^{2,1}$. We wish to solve this equation with the condition

${\mathbf{V}}(x,T)=\big{(}x_{1}^{1/2}+\cdots+x_{n}^{1/2}\big{)}^{2},$

for $x\in{\mathbb{R}}^{+n}$, which corresponds to the $L^{1/2}$-norm on
${\mathbb{R}}^{n}$. We can change variables, with

$y_{i}=\log x_{i}+\sigma_{i}^{2}t/2,\qquad\text{ or,}\qquad
x_{i}=e^{y_{i}}e^{-\sigma_{i}^{2}t/2},$

for $t\in[0,T)$ and $i=1,\ldots,n$, so $y\in{\mathbb{R}}^{n}$ and
$U\colon{\mathbb{R}}^{n}\times[0,T]\to{\mathbb{R}}^{+}$ defined by

$U(y,t)={\mathbf{V}}(x,t)$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$, will be $C^{2,1}$. Hence, we have

$\displaystyle D_{t}{\mathbf{V}}(x,t)$ $\displaystyle=\frac{\partial}{\partial
t}U(\log x+\sigma^{2}t/2,t)$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{n}\sigma_{i}^{2}D_{i}U(y,t)+D_{t}U(y,t),$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$, and since

$\displaystyle D_{i}{\mathbf{V}}(x,t)$ $\displaystyle=\frac{\partial}{\partial
x_{i}}U(\log x+\sigma^{2}t/2,t)$ $\displaystyle=x_{i}^{-1}D_{i}U(y,t),$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$ and $i=1,\ldots,n$, we also have

$\displaystyle D_{ii}{\mathbf{V}}(x,t)$
$\displaystyle=\frac{\partial}{\partial x_{i}}D_{i}{\mathbf{V}}(x,t)$
$\displaystyle=\frac{\partial}{\partial
x_{i}}\big{(}x_{i}^{-1}D_{i}U(y,t)\big{)}$
$\displaystyle=-x_{i}^{-2}D_{i}U(y,t)+x_{i}^{-1}\frac{\partial}{\partial
x_{i}}D_{i}U(\log x+\sigma^{2}t/2,t)$
$\displaystyle=-x_{i}^{-2}D_{i}U(y,t)+x_{i}^{-2}D_{ii}U(y,t).$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$ and $i=1,\ldots,n$. Hence, (A.1)
becomes

$D_{t}U(y,t)+\frac{1}{2}\sum_{i=1}^{n}\sigma_{i}^{2}D_{ii}U(y,t)=0,$ (A.2)

for $(y,t)\in{\mathbb{R}}^{n}\times[0,T)$, and if we let

$\tau=T-t\qquad\text{ and
}\qquad{\widetilde{U}}(y,\tau)=U(y,t)={\mathbf{V}}(x,t),$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T)$, we have

$\frac{\partial{\widetilde{U}}}{\partial\tau}=\frac{1}{2}\sum_{i=1}^{n}\sigma_{i}^{2}\frac{\partial^{2}{\widetilde{U}}}{\partial
y_{i}^{2}},$ (A.3)

for $\tau\in[0,T)$, which is a heat equation (see Evans (2010), Section 2.3).

We must solve (A.3) with the initial condition

${\widetilde{U}}(y,0)={\mathbf{V}}(x,T)=\big{(}x_{1}^{1/2}+\cdots+x_{n}^{1/2}\big{)}^{2}=\big{(}e^{y_{1}/2}e^{-\sigma_{1}^{2}T/4}+\cdots+e^{y_{n}/2}e^{-\sigma_{n}^{2}T/4}\big{)}^{2},$

for $y\in{\mathbb{R}}^{n}$. The solution to this initial value problem is

$\displaystyle{\widetilde{U}}(y,\tau)$
$\displaystyle=\sum_{i=1}^{n}\frac{e^{-\sigma_{i}^{2}T/2}}{(2\pi\sigma_{i}^{2}\tau)^{1/2}}\int_{{\mathbb{R}}}e^{-(y_{i}-z_{i})^{2}/2\sigma_{i}^{2}\tau}e^{z_{i}}dz_{i}$
$\displaystyle\quad+\sum^{n}_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}\bigg{(}\frac{e^{-\sigma_{i}^{2}T/4}}{(2\pi\sigma_{i}^{2}\tau)^{1/2}}\int_{{\mathbb{R}}}e^{-(y_{i}-z_{i})^{2}/2\sigma_{i}^{2}\tau}e^{z_{i}/2}dz_{i}\bigg{)}\bigg{(}\frac{e^{-\sigma_{j}^{2}T/4}}{(2\pi\sigma_{j}^{2}\tau)^{1/2}}\int_{{\mathbb{R}}}e^{-(y_{j}-z_{j})^{2}/2\sigma_{j}^{2}\tau}e^{z_{j}/2}dz_{j}\bigg{)}$
$\displaystyle=\sum_{i=1}^{n}e^{-\sigma_{i}^{2}(T-\tau)/2}e^{y_{i}}+\sum^{n}_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}e^{-\sigma_{i}^{2}\tau/8}e^{-\sigma_{j}^{2}\tau/8}\big{(}e^{-\sigma_{i}^{2}(T-\tau)/4}e^{y_{i}/2}\big{)}\big{(}e^{-\sigma_{j}^{2}(T-\tau)/4}e^{y_{j}/2}\big{)},$

for $(y,\tau)\in{\mathbb{R}}^{n}\times[0,T]$ (see Evans (2010), Section 2.3,
Theorem 1). With this we have

${\mathbf{V}}(x,t)=\sum_{i=1}^{n}x_{i}+\sum^{n}_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}e^{\sigma_{i}^{2}(t-T)/8}e^{\sigma_{j}^{2}(t-T)/8}x_{i}^{1/2}x_{j}^{1/2},$

for $(x,t)\in{\mathbb{R}}^{+n}\times[0,T]$, which generates the portfolio
$\pi$ with weights

$\pi_{i}(t)=\frac{X_{i}(t)+e^{\sigma_{i}^{2}(t-T)/8}X^{1/2}_{i}(t)\sum^{n}_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}e^{\sigma_{j}^{2}(t-T)/8}X_{j}^{1/2}(t)}{{\mathbf{V}}(X(t),t)}.$

for $t\in[0,T]$ and $i=1,\ldots,n$. Since (A.1) is satisfied, the portfolio
$\pi$ satisfies (5.1), and ${\mathbf{V}}$ is replicable. ∎

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