# SMAGEXP: a galaxy tool suite for transcriptomics data meta-analysis

Samuel Blanck 1,111to whom correspondence should be addressed , Guillemette
Marot 1,2

###### Abstract

Background: With the proliferation of available microarray and high throughput
sequencing experiments in the public domain, the use of meta-analysis methods
increases. In these experiments, where the sample size is often limited, meta-
analysis offers the possibility to considerably enhance the statistical power
and give more accurate results. For those purposes, it combines either effect
sizes or results of single studies in a appropriate manner. R packages metaMA
and metaRNASeq perform meta-analysis on microarray and NGS data, respectively.
They are not interchangeable as they rely on statistical modeling specific to
each technology.

Results: SMAGEXP (Statistical Meta-Analysis for Gene EXPression) integrates
metaMA and metaRNAseq packages into Galaxy. We aim to propose a unified way to
carry out meta-analysis of gene expression data, while taking care of their
specificities. We have developed this tool suite to analyse microarray data
from Gene Expression Omnibus (GEO) database or custom data from
affymetrix${}^{\mbox{\scriptsize{\copyright}}}$ microarrays. These data are
then combined to carry out meta-analysis using metaMA package. SMAGEXP also
offers to combine raw read counts from Next Generation Sequencing (NGS)
experiments using DESeq2 and metaRNASeq package. In both cases, key values,
independent from the technology type, are reported to judge the quality of the
meta-analysis. These tools are available on the Galaxy main tool shed. Source
code, help and installation instructions are available on github.

Conclusion: The use of Galaxy offers an easy-to-use gene expression meta-
analysis tool suite based on the metaMA and metaRNASeq packages.

1Univ. Lille Droit et Santé, EA 2694, F-59000 Lille, France

2 Inria Lille-Nord Europe, MODAL, F-59000 Lille, France

## Background

Meta-analyses are widely used in medicine and health policy to increase
statistical power in studies suffering from small sample sizes. Gene
expression experiments are a typical example of such designs. The R packages
metaMA and metaRNASeq are dedicated to gene expression microarray and NGS
meta-analysis, respectively. While metaMA and metaRNASeq are open source and
available on CRAN, they require coding skills in R to perform meta-analysis.
Thus, to facilitate the use and the dissemination of these packages, we
developed Galaxy wrappers. Galaxy [1, 2, 3] is an open, web-based platform for
data intensive biomedical research. It keeps tracks of history and all
analyses can be rerun. Galaxy community is very active and a lot of
bioinformatics tools are included in Galaxy thanks to a modular system based
on XML wrappers. These integrated tools can be shared via the Galaxy toolshed
which serves as an appstore.

## Methods

### Overview of R packages integrated into Galaxy

#### metaMA

Gene expression microarray data meta-analysis can be performed thanks to the
metaMA [4] R package. It proposes methods to combine either p-values or
moderated effect sizes from different studies to find differentially expressed
genes. In our pipeline we only keep the inverse normal method [5] to combine
the p-values calculated by limma [6] for each single study.

#### metaRNAseq

RNA-seq data meta-analysis can be performed thanks to the metaRNASeq [7] R
package. It implements two p-value combination techniques : the inverse normal
and Fisher methods [8]. Single study p-values are comptuted with DESeq2 [9].

#### Differences between metaMA and metaRNASeq

Main differences come from the statistical distributions used to model data
and from the manner to treat the genes exhibiting conflicting expression
patterns (i.e., under-expression when comparing one condition to another in
one study, and over-expression for the same comparison in another study).
Usually, microarray data are modelled by Gaussian distributions while NGS data
are modelled by Negative Binomial distributions. As explained in [4] and [7],
the trick which consists to use one-tailed p-values for each single study
before combination in metaMA avoids directional conflicts. In metaRNASeq, this
trick can not be used, which necessits a post-hoc identification of conflicts,
step which is also proposed in metaRNASeq.

### Description of Galaxy tools

SMAGEXP tool suite offers two distinct gene expression meta-analysis
functionalities : one dedicated to microarray data meta-analysis and one
dedicated to RNAseq data meta-analysis (see Table 1 and figure 1).

Figure 1: Overview of the tools from microarray data meta-analysis pipeline
integrated within Galaxy. Table 1: Summary of tools inputs and outputs.

Tool | Input | Output
---|---|---
GEOQuery | GEO database ID | Rdata object and .cond file
QCNormalization | Raw .CEL affymetrix files and .conf file | Rdata object and plots
Import custom data | Expression data in tabular .txt format | Rdata object and plots
Limma analysis | Rdata object from GEOQuery or QCNormalization | Rdata Object and HTML report
Microarray data meta-analysis | Rdata objects from Limma analyse | HTML report
RNA-seq data meta-analysis | Results text files from galaxy deseq2 tool | HTML report

#### Microarray data meta-analysis

##### GEOQuery tool

GEOQuery tool fetches microarray data directly from GEO database, based on the
GEOQuery [10] R package. Given a GSE accession ID, it returns an Rdata object
containing the data and a text file (.cond file) summarizing the conditions of
the experiment. The .cond file is a text file containing one line per sample
in the experiment. Each line is made of 3 columns:

  * •

Sample ID

  * •

Condition of the biological sample

  * •

Description of the biological sample

Figure 2: Example of .cond file.

Column names are optional and only the columns order matters. As the GEO
dataset should already have been normalized, the GEOQuery tool does not
perform any normalization method, apart from an optional log2 transformation.

##### QCNormalization tool

It is possible to analyze .CEL files from affymetrix gene expression
microarray. The QCnormalization tool offers to ensure the quality of the data
and to normalize them. Several normalization methods are available :

  * •

rma normalization

  * •

quantile normalization + log2

  * •

background correction + log2

  * •

log2 only

This tool generates several quality figures : microarray images, boxplots and
MA plots. It also outputs an Rdata object containing the normalized data for
further analysis with the limma analysis tool.

##### Import custom data tool

This tool imports data stored in a tabular text file. Column titles (chip IDs)
must match the IDs of the .cond file. A few normalization methods are
proposed, but it is possible to skip the normalization step, by choosing
"none" in the normalization methods options. Therefore this tool is of special
interest when the input dataset has been previously normalized.

This tool also generates boxplots and MA plots and outputs an Rdata object
containing the data for further analysis with the limma analysis tool.

##### Limma analysis tool

The Limma analysis tool performs single analysis either of data previously
retrieved from GEO database or normalized affymetrix .CEL files data. Given a
.cond file, it runs a standard limma differential expression analysis. The
user choose two conditions extracted from the .cond file (see Figure 3). It
generates boxplots for rough quality control of normalization, p-value
histograms to ensure that statistical hypotheses are not violated and a
volcano plot to quickly identify the most-meaningful changes. This tool also
outputs a table summarizing the differentially expressed genes and their
annotations. Genes are sorted by ascending Benjamini-Hochberg adjusted
p-value, and annotations are retrieved via GEO database. This list of genes
can be exported to excel or to csv format. This table is sortable and
requestable. Furthermore it is possible to expand each row to display extended
annotations informations, including hypertext links to the National Center for
Biotechnology Information (NCBI) gene database. Finally, this tool outputs an
Rdata object to perform further meta-analysis and a tabular file containing
the all results and annotations of the differential analysis.

Figure 3: limma analysis tool form.

##### Microarray data meta-analysis tool

The meta-analysis relies on the metaMA R package. Prior to the meta-analysis
itself, a pre-processing is made in order to ensure compatiblity between
several sources of data. In fact, data could come from different types of
microarrays. First, we list the Entrez gene ID corresponding to each probe of
each dataset. Next, we keep the probes corresponding to the genes which are
shared by all the experiments of the meta-analysis. Then, for each dataset, we
merge the microarray probes originating from the same Entrez gene ID by
computing their mean. Note that the merging of different technologies induces
a loss of information and might generate several conflicts as probes do not
necessary reflect the same biological reality. Finally, the p-value
combination method of metaMA is run on the merged dataset. It generates a Venn
Diagram summarizing the results of the meta-analysis, and a list of indicators
to evaluate the quality of the performance of the meta-analysis :

  * •

DE : Number of differentially expressed genes

  * •

IDD (Integration Driven discoveries) : number of genes that are declared
differentially expressed in the meta-analysis that were not identified in any
of the single studies alone

  * •

Loss : Number of genes that are identified differentially expressed in single
studies but not in meta-analysis

  * •

IDR (Integration-driven Discovery Rate) : corresponding proportion of IDD

  * •

IRR (Integration-driven Revision) : corresponding proportion of Loss

It also outputs a fully sortable and requestable table, with gene annotations
and hypertext links to NCBI gene database.

#### RNA-seq data meta-analysis

The RNA-seq data meta-analysis tool relies on the deseq2 galaxy tool analysis
results. Given several text file resulting from the deseq2 tool, the
metaRNAseq tool performs a meta-analysis, generates the list of differentially
expressed genes, and outputs the DE, IDD, Loss, IDR and IRR indicators.

## Application

### Microarray meta-analysis example

SMAGEXP was applied to two GEO datasets identified with the following IDs :
GSE3524 and GSE13601. These two datasets contain human oral squamous cell
carcinoma (SCC) data. See Figure 4 for an overview of the worfklow of this
analysis.

Figure 4: Exemple of a galaxy workflow for microarray meta-analysis.

First, we fetch data from the GSE3524 using the GEOQuery tool (with parameter
"log2 transformation" = auto). Then we launch the limma analysis, using the
output from the GEOquery tool. It generates an Rdata output, which will be
usefull for the meta-analysis. Results can be seen on Figure 5 and Figure 6

Figure 5: limma analysis tool output plots.
Figure 6: limma analysis tool : table of top 10 genes for GSE3524 dataset.

Secondly, the same kind of analysis is run from raw .CEL files. We choose to
keep six .CEL files from the GSE13601 dataset (IDs from GSM342582 to
GSM342587). Quality control and normalization is done thanks to the
QCnormalization tool. Then, as previously, the limma analysis tool is run to
generate a HTML report and an Rdata output.

#### Run a metaMA analysis

To run the microarray meta-analysis tool, we only need the Rdata output of
each single study, generated by the limma analysis tool. It generates a Venn
diagram to compare the results of each study with the meta-analysis. It also
outputs several indicators as described in the description of the tool (see
Figure 7). As for the limma tool, annotated expressed genes are displayed in a
table which can be ordered and requested.

Figure 7: Venn diagram and summary of microarray data meta-analysis tool
results

### RNA-seq data meta-analysis example

The RNA-seq data meta-analysis tool relies on deseq2 results (see Figure 8).

Figure 8: Header of a deseq2 results file

It outputs a Venn diagram and the same indicators as in the microarray data
analysis tool for both Fisher and inverse normal p-values combinations. It
also generates a text file containing summarization of the results of each
single analysis and meta-analysis. Potential conflicts between single analyses
are indicated by zero values in the "signFC" column.

Figure 9: Header of a metaRNAseq results file

## Conclusion

We developed SMAGEXP, a toolsuite dedicated to gene-expression data meta-
analysis. This toolsuite proposes quality controls, single analyses and meta-
analyses of microarray and RNA-seq data, suggesting appropriate pipelines for
each type of data. It delivers fully annnotated results of differentially
expressed genes, exportable in several usual formats. Integrated into Galaxy,
SMAGEXP is easy to use for biologists and life scientists. R packages metaMA
and metaRNAseq thus inherit reproductibility and accessibility support from
Galaxy. SMAGEXP is available on the Galaxy main toolshed [11].
Source code is available on github at : https://github.com/sblanck/smagexp.
Furthermore, thanks to Docker, we made these Galaxy tools and their
dependencies easy to deploy. A fully dockerized instance of Galaxy containing
SMAGEXP is available at :
https://hub.docker.com/r/sblanck/galaxy-SMAGEXP/

## References

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  * [4] G. Marot, J.-L. Foulley, C.-D. Mayer, and F. Jaffrezic, “Moderated effect size and p-value combinations for microarray meta-analyses,” Bioinformatics, vol. 25, no. 20, pp. 2692–2699, 2009.
  * [5] L. Hedges and I. Olkin, Statistical Methods for Meta-Analysis.  London: Academic Press, 1985.
  * [6] M. E. Ritchie, B. Phipson, D. Wu, Y. Hu, C. W. Law, W. Shi, and G. K. Smyth, “limma powers differential expression analyses for RNA-sequencing and microarray studies,” Nucleic Acids Research, vol. 43, no. 7, p. e47, 2015\.
  * [7] A. Rau, G. Marot, and F. Jaffrézic, “Differential meta-analysis of rna-seq data from multiple studies,” BMC Bioinformatics, vol. 15, no. 1, pp. 1–10, 2014.
  * [8] R. A. Fisher, Statistical Methods for Research Workers.  Edinburgh: Oliver and Boyd, 1932.
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# A CLUSTER EXPANSION PROOF THAT THE STOCHASTIC EXPONENTIAL OF A BROWNIAN
MOTION IS A MARTINGALE

Steven D Miller<EMAIL_ADDRESS>Rytalix Capital

###### Abstract.

Let ${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and
continuous real function and $\psi\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let
${B}(t)$ be a standard Brownian motion defined with respect to a probability
space $(\Omega,\mathscr{F},\bm{\mathsf{P}})$ and where $d{B}(t)={\xi}(t)dt$
and $t\in\mathbb{R}^{+}$. The process $\xi(t)$ is a Gaussian white noise with
expectation $\bm{\mathsf{E}}~{}\xi(t)=0$ and with covariance
$\bm{\mathsf{E}}~{}\xi(t)\xi(s)=\delta(t-s)$. The Dolean-Dades stochastic
exponential ${Z}(t)$ is the solution to the linear stochastic differential
equation describing a geometric Brownian motion such that
$d{Z}(t)=\psi(t){Z}(t)d{B}(t)=\psi(t){Z}(t)\xi(t)dt$. Using a cluster
expansion method, and the moment and cumulant generating functions for
$\xi(t)$, it is shown that ${Z}(t)$ is a martingale. The original Novikov
criteria for ${Z}(t)$ being a true martingale are reproduced and exactly
satisfied, namely that

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\bm{\mathsf{E}}\exp\left(\int_{o}^{t}{\psi}(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)=1$

provided that $\exp\big{(}\int_{0}^{t}|{\psi}(u)|^{2}du\big{)}<\infty$ for all
$t>0$. However,
$\bm{\mathsf{E}}\big{[}|{Z}(t)|^{p}\big{]}=\exp(\tfrac{1}{2}p(p-1)\phi(t))$,
if $\phi(t)=\int_{0}^{t}|\psi(u)|^{2}du$ is monotone increasing and is a
submartingale for all $p>1$.

##  1\. Introduction

Let ${B}(t)$ be a standard Brownian motion with respect to a probability space
$(\Omega,\mathscr{F},\bm{\mathsf{P}})$, and let
${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be smooth continuous real
function [1]. A classical problem in stochastic analysis is to prove that the
Dolean-Dades stochastic exponential

$\displaystyle{Z}(t)=\exp\left(\int_{o}^{t}{\psi}(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)$
(1.1) $\displaystyle{Z}(0)=0$ (1.2)

is a true martingale [2]. The DDSE is the exact solution of the geometric
Brownian motion [1]

$\displaystyle{d{Z}(t)}={\psi}(t){Z}(t)~{}\xi(t)dt\equiv{\psi}(t){Z}(t)~{}d{B}(t)$
(1.3)

where $d{B}(t)=\xi(t)dt$ and $\xi(t)$ is a Gaussian white noise. The solution
has the Ito stochastic integral representation

$\displaystyle{Z}(t)=\int_{0}^{t}\psi(u){Z}(u)d{B}(u)$ (1.4)

The explicit solution (1.1) is obtained [1] via the Ito expansion of
$\log{Z}(t)$. Proving that ${Z}(t)$ is a true martingale is actually a
somewhat difficult and subtle problem and is relevant to other theorems and
results [2-7]. Ito integrals of the form (1.4) are not always martingales. The
well-known necessary and sufficient conditions for ${Z}(t)$ to be a true
martingale are due to Novikov [2] and also Kazamaki [5]. The Novikov criteria
are

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\bm{\mathsf{E}}\exp\left(\int_{o}^{t}{\psi}(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)=1$
(1.5)

and the bound

$\displaystyle\exp\left(\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)<\infty,~{}~{}\forall
t>0$ (1.6)

or equivalently
$\frac{1}{2}\|\psi\|_{L_{2}(\mathbb{R})}^{2}=\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du<\infty$.
The function ${\psi}$ can also be a random process ${\psi}(t,\omega)$ with
respect to $\omega\in\Omega$ with Novikov criteria [6]

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\bm{\mathsf{E}}\exp\left(\int_{o}^{t}{\psi}(u,\omega)d{B}(u,\omega)-\frac{1}{2}\int_{0}^{t}|{\psi}(u,\omega)|^{2}du\right)=1$
(1.7)

and

$\displaystyle\bm{\mathsf{E}}\left\\{\exp\left(\int_{0}^{t}|{\psi}(u,\omega)|^{2}du\right)\right\\}<\infty,~{}~{}\forall
t>0$ (1.8)

In this note we only consider (1.5), and a short new proof is given to
establish that ${Z}(t)$ is a martingale. The exact criteria (1.5) and (1.6)
are established, via a cluster-expansion method and utilising moment and
cumulant-generating functions.

Geometric Brownian motion arises naturally in problems of stochastic
exponential growth, which occurs when a quantity of interest grows
exponentially but with a random multiplier and/or random waiting time between
spurts of growth. Such processes are ubiquitous in both Nature and in human-
created systems: bacterial growth in a sustaining medium; in nuclear and
cellular fission; tissue growth in embryonic and cancer biology; in viral
epidemics; in financial markets and bubbles, and the Black-Scholes option
pricing model; internet growth and propagation of viral social media posts;
population and extinction dynamics; Moore’s law for computer processing power;
and inflationary expansion of the very early Universe with random fluctuations
[8-17]. Noise can either boost stochastic exponential growth or drive a
population to extinction [8,11,12].

Most such models on random growth have focussed on the SDE for geometric
Brownian motion since it can be solved exactly with a strong solution. Suppose
a system evolves exponentially via the simple linear ODE $dX(t)=\alpha X(t)dt$
then $X(t)=X(0)\exp(\alpha t)$. If $\alpha>0$ then the system undergoes
exponential growth and $X(t)\rightarrow\infty$ as $t\rightarrow\infty$ and if
$\alpha<0$ then the system exponentially decays or collapses so that
$X(t)\rightarrow 0$ as $t\rightarrow\infty$. The system is stable or static
for $\alpha=0$. If the system is randomly perturbed by white noise
(additively) then

$\displaystyle
d{X}(t)=\alpha{X}(t)dt+\psi(t){X}(t)\xi(t)dt\equiv\alpha{X}(t)dt+\psi(t){X}(t)d{B}(t)$
(1.9)

A strong solution exists for this SDE [1]. Given any $C^{2}$-functional
$f({X}(t))$, the Ito Lemma is

$\displaystyle
df({X}(t))=\nabla_{X}f({X}(t))dX(t)+\frac{1}{2}\nabla^{2}_{X}f({X}(t))d[{X},{X}](t)$
$\displaystyle=\nabla_{X}f({X}(t))d{X}(t)+\frac{1}{2}\nabla^{2}_{X}f({X}(t))|\psi(t)|^{2}dt$
$\displaystyle=\nabla_{X}f({X}(t))\big{\\{}\alpha{X}(t)dt+\psi(t){X}(t)d{B}(t)\big{\\}}+\frac{1}{2}\nabla^{2}_{X}f({X}(t))|\psi(t)|^{2}dt$
(1.10)

where $\nabla_{X}=d/d{X}(t)$ and $\nabla^{2}_{X}=d^{2}/d{X}(t)^{2}$. Using
$f({X}(t))=\log{X}(t)$

$\displaystyle d{U}(t)\equiv
d\log{X}(t)=\frac{1}{{X}(t)}d{X}(t)+\frac{1}{2}\left(-\frac{1}{|{X}(t)|^{2}}\right)|\psi(t)|^{2}|{X}(t)|^{2}dt$
$\displaystyle=\frac{1}{{X}(t)}\big{\\{}\alpha{X}(t)dt+\psi(t){X}(t)d{B}(t)\big{\\}}-\frac{1}{2}\psi(t)|^{2}dt$
$\displaystyle=\left(\alpha-\frac{1}{2}|\psi(t)|^{2}\right)dt+\psi(t)d{B}(t)$
(1.11)

Then the solution is

$\displaystyle X(t)=X(0)\exp(\alpha
t)\exp\left(\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)=\exp(\alpha
t){Z}(t)$ (1.12)

The expectation is

$\displaystyle\bm{\mathsf{E}}{X}(t)=X(0)\exp(\alpha
t)\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)=\exp(\alpha
t)\bm{\mathsf{E}}{Z}(t)$ (1.13)

which is $\bm{\mathsf{E}}{X}(t)=\exp(\alpha t)$ if $\bm{\mathsf{E}}Z(t)=1$.
The martingale property of ${Z}(t)$ is therefore desirable and ensures there
is no blow up at any finite time $t>0$.

##  2\. Main theorem and proof utilising a cluster expansion method

###  2.1. Preliminary definitions and lemmas

Prior to the proof, we first establish the following preliminary definitions
and lemmas. The proof utilises a cluster expansion method [18-21].

######  Definition 2.1.

White noise is (informally) defined as a zero-centred Gaussian process $\xi$
with covariance $\bm{\mathsf{E}}~{}\xi(t)\xi(s)=\delta(t-s)$ and expectation
$\bm{\mathsf{E}}\xi(t)=0$. A scalar or inner product
$\langle\bullet,\bullet\rangle$ in $L^{2}(\mathbb{R})$ can be defined as

$\displaystyle\langle\psi,\phi\rangle=\bm{\mathsf{E}}\big{\langle}\psi,\xi\big{\rangle}\big{\langle}\phi,\xi\big{\rangle}=\int\\!\\!\\!\\!\int\psi(s)\phi(t)\bm{\mathsf{E}}~{}\xi(s)\xi(t)dsdt=\int\\!\\!\\!\\!\int\psi(s)\phi(t)\delta(t-s)dsdt$
(2.1)

Then

$\displaystyle\langle\psi,\psi\rangle=\bm{\mathsf{E}}\big{\langle}\psi,\xi\big{\rangle}\big{\langle}\psi,\xi\big{\rangle}=\int\\!\\!\\!\\!\int\psi(s)\psi(t)\bm{\mathsf{E}}~{}\xi(s)\xi(t)dsdt=\int\\!\\!\\!\\!\int\psi(s)\psi(t)\delta(t-s)dsdt$
(2.2)

which is

$\displaystyle\langle\psi,\psi\rangle=\int|\psi(s)|^{2}ds=\|\psi\|_{L^{2}(\mathbb{R})}^{2}$
(2.3)

The noise $\xi$ can be formulated as a Gaussian random process on any space of
distributions containing $\mathrm{L}^{2}(\mathbb{R})$. Stochastic integrals of
the form $\int\psi(s)\xi(s)ds$ are well defined if
$\psi\in\mathrm{L}^{2}(\mathbb{R})$. If $\psi$ is an indicator function on
$[0,t]$ then the process $B(t)=\int_{0}^{t}\xi(s)ds$ is a Brownian motion and
$dB(t)=\xi(t)dt$.

######  Definition 2.2.

Given a time-ordered set $(t_{1},...t_{m})\in(0,T)$, with
$t_{1}<t_{2}<t_{3}<...<t_{m-1}<t_{m}$, the mth-order moments and cumulants for
the white noise $\xi(t)$ are given by

$\displaystyle\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}~{}\xi(t_{1})\otimes...\otimes\xi(t_{m})=\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}\xi(t_{q})$
$\displaystyle\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}~{}\xi(t_{1})\otimes...\otimes\xi(t_{m})=\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})$
(2.4)

where $\mathbb{T}$ is a ’time-ordering operator’. For example if
$t_{1}<t_{2}<t_{3}$ and $f(t)$ is a function of t then
$\overrightarrow{\mathbb{T}}f(t_{2})f(t_{1})f(t_{3}=f(t_{1})f(t_{2})f(t_{3})$.
Since $\xi(t)$ is a Gaussian, it is defined entirely by its first two moments
and all cumulants or order $m\geq 3$ vanish so that

$\displaystyle\left\\{\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})\right\\}_{m\geq
3}=0$ (2.5)

For example, at second order for a Gaussian process the binary cumulant is
equivalent to the binary moment

$\displaystyle\bm{\mathsf{C}}\xi(t_{1})\xi(t_{2})=\bm{\mathsf{E}}~{}\xi(t_{1})\xi(t_{2})-\bm{\mathsf{E}}~{}\xi(t_{1})\bm{\mathsf{E}}~{}\xi(t_{2})=\bm{\mathsf{E}}\xi(t_{1})\xi(t_{2})=\delta(t_{2}-t_{1})$
(2.6)

######  Definition 2.3.

The moment generating function (MGF) $\mathscr{M}[\xi(t)]$ and the cumulant-
generating function [CGF] $\mathscr{C}[\xi(t)]$ of the white noise $\xi(t)$
are given by

$\displaystyle{\mathscr{M}}[\xi(t)]=\sum_{m=0}^{\infty}\frac{\beta^{m}}{m!}\int_{0}^{t}...\int_{0}^{t_{m-1}}dt_{1}...dt_{m}\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}~{}\xi(t_{q})$
(2.7)
$\displaystyle{\mathscr{C}}[\xi(t)]=\sum_{m=1}^{\infty}\frac{\beta^{m}}{m!}\int_{0}^{t}...\int_{0}^{t_{m-1}}dt_{1}...dt_{m}\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})$
(2.8)

where ${\beta}$ is an arbitrary real constant. It is important to note that
the summation in (2.8) begins from $m=1$ and not $m=0$ and that

$\displaystyle\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}\xi(t_{q})\neq\overrightarrow{\mathbb{T}}\prod_{q=1}^{m}\bm{\mathsf{E}}\xi(t_{q})$
(2.9)
$\displaystyle\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})\neq\overrightarrow{\mathbb{T}}\prod_{q=1}^{m}\bm{\mathsf{C}}\xi(t_{q})$
(2.10)

Equations (2.7) and (2.8) can also be written as

$\displaystyle{\mathscr{M}}[\xi(t)]=\sum_{m=0}^{\infty}\frac{\beta^{m}}{m!}\int{\mathbf{D}}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}\xi(t_{q})$
(2.11)
$\displaystyle{\mathscr{C}}[\xi(t)]=\sum_{m=1}^{\infty}\frac{\beta^{m}}{m!}\int{\mathbf{D}}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})$
(2.12)

where $\int\bm{\mathrm{D}}_{m}[t]=\int...\int dt_{1}...dt_{m}$ is a ’path
integral’. Choosing ${\beta}=+1$

$\displaystyle{\mathscr{M}}[\xi(t)]=\sum_{m=0}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}\xi(t_{q})$
(2.13)
$\displaystyle{\mathscr{C}}[\xi(t)]=\sum_{m=1}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})$
(2.14)

######  Lemma 2.4.

The relation between the MGF and the CGF is

$\displaystyle{\log}{\mathscr{M}}[\xi(t)]={\mathscr{C}}[\xi(t)]$ (2.15)

so that

$\displaystyle{\mathscr{M}}[\xi(t)]=\exp\big{(}{\mathscr{C}}[\xi(t)]\big{)}$
(2.16)

Hence

$\displaystyle\sum_{m=0}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{E}}\prod_{q=1}^{m}\xi(t_{q})$
(2.17)
$\displaystyle\exp\left(\sum_{m=1}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\overrightarrow{\mathbb{T}}\bm{\mathsf{C}}\prod_{q=1}^{m}\xi(t_{q})\right)$
(2.18)

######  Proposition 2.5.

Given $\psi\in\mathrm{L}^{2}(\mathbb{R})$ and the white noise $\xi(t)$, then
for all $t>0$ define the Gaussian noise or random function

$\displaystyle\Xi(t)=\psi(t)\xi(t)$ (2.19)

Then $\bm{\mathsf{E}}\Xi(t)=0$ and

$\displaystyle\bm{\mathsf{E}}~{}\Xi(t)\Xi(s)=\psi(t)\psi(s)\bm{\mathsf{E}}~{}\xi(t)\xi(s)=\psi(t)\psi(s)\delta(t-s)$
(2.20)

######  Lemma 2.6.

Given $\Xi(t)=\psi(t)\xi(t)$ then for any $t_{2}>t_{1}$ and
$(t_{1},t_{2})\in(0,t)$

$\displaystyle\langle\psi,\psi\rangle\equiv\int_{0}^{t}\int_{0}^{t_{1}}\bm{\mathsf{E}}~{}\Xi(t_{1})\Xi(t_{2})dt_{1}dt_{2}=\int_{0}^{t}|\psi(t_{1})|^{2}dt_{1}=\|\psi\|_{L_{2}(\mathbb{R})}$
(2.21)

###### Proof.

Using the sifting property of the delta function

$\displaystyle\int_{0}^{t}\int_{0}^{t_{1}}\bm{\mathsf{E}}~{}\Xi(t_{1})\Xi(t_{2})dt_{1}dt_{2}$
$\displaystyle=\int_{0}^{t}\int_{0}^{t_{1}}\psi(t_{1})\psi(t_{2})\bm{\mathsf{E}}~{}\Xi(t_{1})\Xi(t_{2})dt_{1}dt_{2}$
$\displaystyle=\int_{0}^{t}\psi(t_{1})\bigg{|}\int_{0}^{t_{1}}\psi(t_{2})\delta(t_{2}-t_{1})dt_{2}\bigg{|}dt_{1}$
$\displaystyle=\int_{0}^{t}\psi(t_{1})\psi(t_{1})dt_{1}=\int_{0}^{t}|\psi(t_{1})|^{2}dt_{1}\equiv\|\psi\|_{L_{2}(\mathbb{R^{+}})}$
(2.22)

∎

######  Proposition 2.7.

Given $\Xi(t)=\beta(t)\xi(t)$ then the Dolean-Dades stochastic exponential
${Z}(t)$ can be expressed as

$\displaystyle{Z}(t)=\exp\left(\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\exp\left(\int_{0}^{t}\psi(u)\xi(u)du-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\exp\left(\int_{0}^{t}\Xi(u)du-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
(2.23)

###  2.2. Main theorem

The main theorem and proof are now as follows:

######  Theorem 2.8.

The stochastic exponential ${Z}(t)$ is a true martingale iff

$\displaystyle\bm{\mathsf{E}}{Z}(t)\big{\\}}=\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)=1$
(2.24)

and requiring

$\displaystyle\exp(\|\psi\|_{L^{2}(\mathbb{R})}^{2})=\exp\left(\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)<\infty,~{}~{}\forall
t>0$ (2.25)

###### Proof.

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\psi(u)\xi(u)du-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\Xi(u)du-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\bm{\mathsf{E}}\exp\left(\int_{0}^{t}\Xi(u)du\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\bm{\mathsf{E}}\sum_{m=0}^{\infty}\frac{1}{m!}\int_{0}^{t}dt_{1}...\int_{0}^{t_{m-1}}dt_{m}\prod_{q=1}^{m}\Xi(t_{q})$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\bm{\mathsf{E}}\sum_{m=0}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\prod_{q=1}^{m}\Xi(t_{q})$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\sum_{m=0}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\bm{\mathsf{E}}\prod_{q=1}^{m}\Xi(t_{q})$
(2.26)

However, from (2.11), the MGF is

$\displaystyle{\mathscr{M}}[\Xi(t)]=\sum_{m=0}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\bm{\mathsf{E}}\prod_{q=1}^{m}\Xi(t_{q})$
(2.27)

so (2.26) becomes

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right){\mathscr{M}}[\Xi(t)]$
(2.28)

Now using Lemma (2.4)

$\displaystyle{\mathscr{M}}[\Xi(t)]=\exp({\mathscr{C}}[\Xi(t)])$ (2.29)

giving

$\displaystyle\bm{\mathsf{E}}{Z}(t)=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right){\mathscr{M}}[\Xi(t)]$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left({\mathscr{C}}[\Xi(t)]\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\sum_{m=1}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\bm{\mathsf{C}}\overrightarrow{\mathbb{T}}\prod_{q=1}^{m}\Xi(t_{q})\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\sum_{m=1}^{\infty}\frac{1}{m!}\int\mathbf{D}_{m}[t_{1}...t_{m}]\bm{\mathsf{C}}\overrightarrow{\mathbb{T}}\prod_{q=1}^{m}\psi(t_{q})\xi(t_{q})\right)$
(2.30)

Now since $\xi(t)$ is a Gaussian process, all cumulants of order $m\geq 3$
vanish so that

$\displaystyle\bm{\mathsf{C}}\overrightarrow{\mathbb{T}}\left\\{\prod_{q=1}^{m}\Xi(t_{q})\right\\}_{m\geq
3}=\bm{\mathsf{C}}\overrightarrow{\mathbb{T}}\left\\{\prod_{q=1}^{m}\psi(t_{q})\xi(t_{q})\right\\}_{m\geq
3}=0$ (2.31)

This leaves

$\displaystyle\bm{\mathbf{E}}{Z}(t)=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle\times\exp\left(\int\mathbf{D}_{m}[t_{1}]\psi(t_{1})\bm{\mathsf{C}}\xi(t_{1})\big{\\}}+\frac{1}{2}\int\mathbf{D}[t_{1},t_{2}]\psi(t_{1})\psi(t_{2})\bm{\mathsf{C}}\big{\\{}\xi(t_{1})\xi(t_{2})\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle\times\exp\left(\int_{0}^{t}dt_{1}\psi(t_{1})\bm{\mathsf{C}}\xi(t_{1})+\frac{1}{2}\int_{0}^{t}\int_{0}^{t_{1}}dt_{1}dt_{2}\psi(t_{1})\psi(t_{2})\bm{\mathsf{C}}\xi(t_{1})\xi(t_{2})\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle\times\exp\left(\int_{0}^{t}dt_{1}\psi(t_{1})\underbrace{\bm{\mathsf{E}}~{}\\{\xi(t_{1})\big{\\}}}_{=0}+\frac{1}{2}\int_{0}^{t}\int_{0}^{t_{1}}dt_{1}dt_{2}\psi(t_{1})\psi(t_{2})\bm{\mathsf{E}}~{}\xi(t_{1})\xi(t_{2})\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}\int_{0}^{t_{1}}dt_{1}dt_{2}\psi(t_{1})\psi(t_{2})\bm{\mathsf{E}}~{}\xi(t_{1})\xi(t_{2})\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}\int_{0}^{t_{1}}dt_{1}dt_{2}{\psi}(t_{1}){\psi}(t_{2}){\delta}(t_{2}-t_{1})\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}dt_{1}{\psi}(t_{1})\bigg{|}\int_{0}^{t_{1}}dt_{2}{\psi}(t_{2}){\delta}(t_{2}-t_{1})\bigg{|}\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}dt_{1}\psi(t_{1})\psi(t_{1})\right)$
$\displaystyle=\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}|\psi(t_{1})|^{2}dt_{1}\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle\equiv\exp\left(-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du+\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)=1$
(2.32)

iff

$\displaystyle\exp\left(\dfrac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)<\infty$
(2.33)

which are the Novikov criteria. If for some function
$\psi(t),~{}\exists~{}T>0$ such that
$\frac{1}{2}\int_{0}^{T}|\psi(u)|^{2}du=\infty$ then
$\exp\left(\tfrac{1}{2}\int_{0}^{T}|\psi(u)|^{2}du\right)=\infty$ and
$\exp\left(-\tfrac{1}{2}\int_{0}^{T}|\psi(u)|^{2}du\right)=0$ so that
$\bm{\mathsf{E}}Z(t)=0$. Hence ${Z}(t)$ is a martingale for these criteria and
the proof is complete. ∎

As a corollary, it follows easily that
$\bm{\mathsf{M}}(t,p)=\bm{\mathsf{E}}[|{Z}(t)|^{p}]$ is a submartingale for
all $p>1$ if $\phi(t)=\int_{0}^{t}|\psi(u)|^{2}du$ is monotone increasing with
t.

######  Corollary 2.9.

Given $\bm{\mathsf{E}}{Z}(t)=1$ it follows that

$\displaystyle\bm{\mathsf{M}}(t,p)=\bm{\mathsf{E}}[|{Z}(t)|^{p}]=\exp\left(\tfrac{1}{2}[p(p-1)\phi(t)\right)$
(2.34)

is a submartingale for all $p>1$, if $\phi(t))=\int_{0}^{t}|\psi(u)|^{2}du$ is
bounded but monotone increasing with t.

###### Proof.

$\displaystyle|{Z}(t)|^{p}=\left(\exp\left|\int_{0}^{t}\psi(u)d{B}(u)-\frac{1}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\right|^{p}$
$\displaystyle=\exp\left(p\int_{0}^{t}\psi(u)d{B}(u)-\frac{p}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\exp\left(-\frac{p}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(p\int_{0}^{t}\psi(u)d{B}(u)\right)$
(2.35)

Then

$\displaystyle\bm{\mathsf{M}}(t,p)=$
$\displaystyle\bm{\mathsf{E}}[|{Z}(t)|^{p}]=\exp\left(-\frac{p}{2}\int_{0}^{t}|\psi(u)|^{2}du\right)\bm{\mathsf{E}}\exp\left(p\int_{0}^{t}\psi(u)d{B}(u)\right)$
$\displaystyle=\exp\left(-\frac{1}{2}p\int_{0}^{t}|\psi(u)|^{2}du\right)\exp\left(\frac{1}{2}p^{2}\int_{0}^{t}|\psi(u)|^{2}du\right)$
$\displaystyle=\exp\left(\frac{1}{2}p(p-1)\int_{0}^{t}|\psi(u)|^{2}du\right)=\exp\left(\tfrac{1}{2}p(p-1)\phi(t)\right)$
(2.36)

If $\phi(t)>\phi(t^{\prime})$ for all $t>t^{\prime}$ then
$\bm{\mathsf{M}}(t,p)>\bm{\mathsf{M}}(t^{\prime},p)$ so that
$\bm{\mathsf{M}}(t,p)$ is monotone increasing. Hence, $\bm{\mathsf{M}}(t,p)$
is a submartingale on $\mathbb{R}^{+}$. If $p=1$ then
$\bm{\mathsf{M}}(t,1)=\bm{\mathsf{E}}{Z}(t)=1$. ∎

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