# Self-Exciting Multifractional Processes

Fabian A. Harang Marc Lagunas-Merino Salvador Ortiz-Latorre

(Date: August 27, 2024)

###### Abstract.

We propose a new multifractional stochastic process which allows for self-
exciting behavior, similar to what can be seen for example in earthquakes and
other self-organizing phenomena. The process can be seen as an extension of a
multifractional Brownian motion, where the Hurst function is dependent on the
past of the process. We define this through a stochastic Volterra equation,
and we prove existence and uniqueness of this equation, as well as give bounds
on the $p-$order moments, for all $p\geq 1$. We show convergence of an Euler-
Maruyama scheme for the process, and also give the rate of convergence, which
is depending on the self-exciting dynamics of the process. Moreover, we
discuss different applications of this process, and give examples of different
functions to model self-exciting behavior.

##  1\. Introduction and Notation

In recent years, higher computer power and better tools from statistics show
that there are many natural phenomena which do not follow the standard normal
distribution, but rather exhibit different types of memory, and sometimes
changing these properties over time. Therefore several different types of
extensions of standard stochastic processes have been proposed to try to give
a more realistic picture of nature corresponding to what we observe. There are
several stochastic processes which are popular today for the modeling of
varying memory in a process, one of them is known as the Hawkes process, see
for example [9]. This is a point process which allows for self-exciting
behavior by letting the conditional intensity to be dependent on the past
events of the process. In this note, we will consider a continuous type of
process which is inspired by the multifractional Brownian motion. This process
is interesting for being a non-stationary Gaussian process which has
regularity properties changing in time. A simple version of this process is
known as the Riemann-Liouville multifractional Brownian motion and can be
represented by the integral

(1.1)
$B_{t}^{h}=\int_{0}^{t}\left(t-s\right)^{h\left(t\right)-\frac{1}{2}}dB_{s},$

where $\left\\{B_{t}\right\\}_{t\in[0,T]}$ is a Brownian motion and $h$ is a
deterministic function. Interestingly, if we restrict the process to a small
interval, say $\left[t-\epsilon,t+\epsilon\right]$, the local $\alpha$-Hölder
regularity of this process on that interval is of order $\alpha\sim h(t)$ if
$\epsilon$ is sufficiently small. Thus the regularity of the process is
depending on time. Applications of such processes have been found in fields
ranging from Internet traffic and image synthesis to finance, see for example
[2, 3, 4, 5, 6, 8, 11, 12]. In 2010 D. Sornette and V. Filimonov proposed a
self-excited multifractal process to be considered in the modeling of
earthquakes and financial market crashes, see [13]. By self-excited process,
the authors mean a process where the future state depends directly on all the
past states of the process. The model they proposed was defined in a discrete
manner. They also suggested a possible continuous time version of their model,
but they did not study its existence rigorously. This article is therefore
meant as an attempt to propose a continuous time version of a similar model to
that proposed by Sornette and Filimonov, and we will study its mathematical
properties.

We will first consider an extension of a multifractional Brownian process,
which is found as the solution to the stochastic differential equation

(1.2)
$X_{t}^{h}=\int_{0}^{t}\left(t-s\right)^{h(t,X_{s}^{h})-\frac{1}{2}}dB_{s},$

where $\left\\{B_{t}\right\\}_{t\in[0,T]}$ is a general $d$-dimensional
Brownian motion, and $h$ is bounded and takes values in $(0,1)$. Already at
this point we could think that the local regularity of the process $X$ would
be depending on the history of $X$ through $h$, in a similar manner as for the
multifractional Brownian motion in equation (1.1). As we can see, the
formulation of the process is through a stochastic Volterra equation with a
possibly singular kernel. We will therefore show the existence and uniqueness
of this equation, and then say that its solution is a Self-Exciting
Multifractional Process (SEM) $X^{h}$. We will study the probabilistic
properties, and discuss examples of functions $h$ which give different
dynamics for the process $X^{h}$. The process is neither stationary nor
Gaussian in general, and is therefore mathematically challenging to apply in
any standard model for example in finance but do, at this point, have some
interesting properties on its own. The study of such processes could also shed
some light on natural phenomena behaving outside of the scope of standard
stochastic processes, such as the self-excited dynamics of earthquakes as they
argue in [13].

We will first show the existence and uniqueness of the Equation (1.2) and then
study probabilistic and path properties such as variance and regularity of the
process. We will introduce an Euler-Maruyama scheme to approximate the
process, and show its strong convergence as well as estimate its rate of
convergence. Finally, we will discuss an extension of the process to a Gamma
type process, which might be interesting for various applications.

###  1.1. Notation and preliminaries

Let $T>0$ be a fixed constant. We will use the standard notation
$L^{\infty}\left(\left[0,T\right]\right)$ for essentially bounded functions on
the interval $\left[0,T\right]$. Furthermore, let
$\triangle^{(m)}\left[a,b\right]$ denote the $m$-simplex. That is, define
$\triangle^{(m)}[a,b]$ to be given by

$\triangle^{(m)}\left([a,b]\right):=\left\\{(s_{m},\ldots,s_{1}):a\leq
s_{1}<\ldots<s_{m}\leq b\right\\}.$

We will consider functions
$k:\triangle^{\left(2\right)}\left(\left[0,T\right]\right)\rightarrow\mathbb{R}_{+}$
which will be used as a kernel in an integral operator, in the sense that we
consider integrals of the form

$\int_{0}^{t}k\left(t,s\right)f\left(s\right)ds,$

whenever the integral is well defined. We call these functions Volterra
kernels.

######  Definition 1.

Let $k:\triangle^{(2)}\left(\left[0,T\right]\right)\rightarrow\mathbb{R}_{+}$
be a Volterra kernel. If $k$ satisfies

$t\mapsto\int_{0}^{t}k\left(t,s\right)ds\in
L^{\infty}\left(\left[0,T\right]\right)$

and

$\limsup_{\epsilon\downarrow
0}\parallel\int_{\cdot}^{\cdot+\epsilon}k\left(\cdot+\epsilon,s\right)ds\parallel_{L^{\infty}\left(\left[0,T\right]\right)}=0,$

then we say that $k\in\mathcal{K}_{0}.$

We will frequently use the constant $C$ to denote a general constant, which
might vary throughout the text. When it is important, we will mention what
this constant depends upon in subscript, i.e. $C=C_{T}$ to denote dependence
in $T$ .

##  2\. Zhang’s Existence and Uniqueness of Stochastic Volterra Equations

In this section we will assume that
$\left\\{B_{t}\right\\}_{t\in\left[0,T\right]}$ is a $d$-dimensional Brownian
motion defined on a filtered probability space
$(\Omega,\mathcal{F},\left\\{\mathcal{F}_{t}\right\\}_{t\in\left[0,T\right]},P)$.
Consider the following Volterra equation

(2.1)
$X_{t}=g\left(t\right)+\int_{0}^{t}\sigma\left(t,s,X_{s}\right)dB_{s},\qquad
0\leq t\leq T,$

where $g$ is a measurable, $\left\\{\mathcal{F}_{t}\right\\}$-adapted
stochastic process and
$\sigma:\triangle^{(2)}\left(\left[0,T\right]\right)\times\mathbb{R}^{n}\rightarrow\mathcal{L}\left(\mathbb{R}^{d},\mathbb{R}^{n}\right)$
is a measurable function, where
$\mathcal{L}\left(\mathbb{R}^{d},\mathbb{R}^{n}\right)$ is the linear space of
$d\times n$-matrices.

Next we write a simplified version of the hypotheses for $\sigma$ and $g$,
introduced previously by Zhang in [16], which will be used to prove that there
exists a unique solution to the equation $\left(\ref{eq:Stoch Volterra
equation}\right)$.

(H1):

There exists $k_{1}\in\mathcal{K}_{0}$ such that the function $\sigma$
satisfies the following linear growth inequality for all
$(s,t)\in\triangle^{(2)}\left([0,T]\right),$ and $x\in\mathbb{R}^{n}$,

$\left|\sigma\left(t,s,x\right)\right|^{2}\leq
k_{1}\left(t,s\right)\left(1+\left|x\right|^{2}\right).$

(H2):

There exists $k_{2}\in\mathcal{K}_{0}$ such that the function $\sigma$
satisfies the following Lipschitz inequality for all
$(s,t)\in\triangle^{(2)}\left([0,T]\right),$ $x,y\in\mathbb{R}^{n}$,

$\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}\leq
k_{2}\left(t,s\right)\left|x-y\right|^{2}.$

(H3):

For some $p\geq 2$, we have

$\sup_{t\in\left[0,T\right]}\int_{0}^{t}\left[k_{1}\left(t,s\right)+k_{2}\left(t,s\right)\right]\cdot\mathbb{E}\left[\left|g\left(s\right)\right|^{p}\right]ds<\infty,$

where $k_{1}$ and $k_{2}$ satisfy $\mathbf{H1}$ and $\mathbf{H2}.$

Based on the above hypotheses, we can use the following tailor made version of
the theorem on existence and uniqueness found in [16] to show that there
exists a unique solution to equation $\left(\ref{eq:Stoch Volterra
equation}\right)$.

######  Theorem 2.

$\left(\textnormal{Xicheng Zhang}\right)$ Assume that
$\sigma:\triangle^{(2)}\left([0,T]\right)\times\mathbb{R}^{n}\rightarrow\mathcal{L}\left(\mathbb{R}^{d},\mathbb{R}^{n}\right)$
is measurable, and $g$ is an $\mathbb{R}^{n}$-valued,
$\left\\{\mathcal{F}_{t}\right\\}$-adapted process satisfying
$\mathbf{H1}-\mathbf{H3}$. Then there exists a unique measurable,
$\mathbb{R}^{n}$-valued, $\left\\{\mathcal{F}_{t}\right\\}$-adapted process
$X_{t}$ satisfying for all $t\in[0,T]$ the equation

$X_{t}=g\left(t\right)+\int_{0}^{t}\sigma\left(t,s,X_{s}\right)dB_{s}.$

Furthermore, for some $C_{T,p,k_{1}}>0$ we have that

$\mathbb{E}\left[\left|X_{t}\right|^{p}\right]\leq
C_{T,p,k_{1}}\left(1+\mathbb{E}\left[\left|g\left(t\right)\right|^{p}\right]+\sup_{t\in[0,T]}\int_{0}^{t}k_{1}\left(t,s\right)\mathbb{E}\left[\left|g\left(s\right)\right|^{p}\right]ds\right),$

where $p$ is from $\mathbf{H3}.$

It will also be useful, in future sections, to consider the following
additional hypothesis.

(H4):

The process $g$ is continuous and satisfies for some $\delta>0$ and for any
$p\geq 2$,

$\mathbb{E}\left[\sup_{t\in\left[0,T\right]}\left|g\left(t\right)\right|^{p}\right]<\infty,$

and

$\mathbb{E}\left[\left|g\left(t\right)-g\left(s\right)\right|^{p}\right]\leq
C_{T,p}\left|t-s\right|^{\delta p}.$

##  3\. Self-Exciting Multifractional Stochastic Processes

Consider the stochastic process given formally by the Volterra equation

$X_{t}^{h}=g\left(t\right)+\int_{0}^{t}\left(t-s\right)^{h\left(t,X_{s}^{h}\right)-\frac{1}{2}}dB_{s},$

where $g$ is an $\left\\{\mathcal{F}_{t}\right\\}$-adapted, one-dimensional
process, $h:\left[0,T\right]\times\mathbb{R}\rightarrow\mathbb{R}_{+}$ and $B$
is a one-dimensional Brownian motion. In this section, we will show the
existence and uniqueness of the solution for the above equation by means of
Theorem 2. Moreover, we will discuss the continuity properties of the
solution.

######  Definition 3.

We say that a function
$h:\left[0,T\right]\times\mathbb{R}\rightarrow\mathbb{R}_{+}$ is a Hurst
function with parameters $(h_{*},h^{*})$, where $h_{*}\leq h^{*}$, if
$h\left(t,x\right)$ takes values in
$\left[h_{*},h^{*}\right]\subset\left(0,1\right)$ for all $x\in\mathbb{R}^{d}$
and $t\in\left[0,T\right]$ and $h$ satisfies the following Lipschitz
conditions for all $x,y\in\mathbb{R}$ and $t,t^{\prime}\in\left[0,T\right]$

$\left|h\left(t,x\right)-h\left(t,y\right)\right|\leq C\left|x-y\right|,$
$\left|h\left(t,x\right)-h\left(t^{\prime},x\right)\right|\leq
C\left|t-t^{\prime}\right|,$

for some $C>0$.

######  Lemma 4.

Let $\sigma\left(t,s,x\right)=\left(t-s\right)^{h(t,x)-\frac{1}{2}}$ and let
$h$ be a Hurst function with parameters $(h_{*},h^{*})$. Then

(3.1) $\left|\sigma\left(t,s,x\right)\right|^{2}\leq
k\left(t,s\right)\left(1+\left|x\right|^{2}\right),$

where

$k\left(t,s\right)=C_{T}\left(t-s\right)^{2h_{*}-1},$

and

(3.2) $\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}\leq
C_{T}k\left(t,s\right)\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}.$

Moreover, $\sigma$ satisfies $\mathbf{H1}$-$\mathbf{H2}$.

###### Proof.

We prove the three claims in the order they are stated in Lemma 4, and start
to prove equation $\left(\ref{eq:sigma linear growth}\right)$. Remember that

$h\left(t,x\right)\in\left[h_{*},h^{*}\right]\subset\left(0,1\right),$

for all $t\in[0,T]$ and $x\in\mathbb{R}$, therefore we can trivially find

(3.3)
$\left|\sigma\left(t,s,x\right)\right|^{2}=\left(t-s\right)^{2h(t,x)-1}=\left(t-s\right)^{2\left(h(t,x)-h_{*}\right)+2h_{*}-1}\leq
T^{2\left(h^{*}-h_{*}\right)}\left(t-s\right)^{2h_{*}-1},$

which yields equation $\left(\ref{eq:sigma linear growth}\right)$ with
$k\left(t,s\right)=C_{T}\left(t-s\right)^{2h_{*}-1}$. Next we consider
equation $\left(\ref{eq:Lipschitz sigma}\right)$, and using that
$x=\exp\left(\log\left(x\right)\right),$ we write

$\sigma\left(t,s,x\right)=\exp\left(\left(\log\left(t-s\right)\right)\left(h(t,x)-\frac{1}{2}\right)\right),$

where $(t,s)\in\triangle^{(2)}\left(\left[0,T\right]\right)$ and consider the
following inequality derived from the fundamental theorem of calculus

(3.4) $\left|e^{x}-e^{y}\right|\leq e^{\max(x,y)}\left|x-y\right|,\qquad
x,y\in\mathbb{R}.$

Using the Lipschitz assumption on $h$ together with the above inequality, we
have that

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}$
$\displaystyle\leq\exp\left(2\max\left(\log\left(t-s\right)\left(h\left(t,x\right)-\frac{1}{2}\right),\log\left(t-s\right)\left(h\left(t,y\right)-\frac{1}{2}\right)\right)\right)$
$\displaystyle\quad\times\left|\log\left(t-s\right)\right|^{2}\left|h\left(t,x\right)-h\left(t,y\right)\right|^{2}$
$\displaystyle\leq
C^{2}\exp\left(\max\left(\log\left(t-s\right)\left(2h\left(t,x\right)-1\right),\log\left(t-s\right)\left(2h\left(t,y\right)-1\right)\right)\right)$
$\displaystyle\quad\times\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}.$

If $\left|t-s\right|\geq 1$ then

$\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}\leq
C_{T}\left|x-y\right|^{2},$

since $h$ is bounded. If $\left|t-s\right|<1$ then $\log\left(t-s\right)<0$
and, using that if $\theta<0$ then $\max\left(\theta x,\theta
y\right)=\theta\min\left(x,y\right)$, we have

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}$
$\displaystyle\leq
C^{2}\exp\left(\log\left(t-s\right)\min\left(\left(2h\left(t,x\right)-1\right),\left(2h\left(t,y\right)-1\right)\right)\right)\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}$
$\displaystyle\leq
C^{2}\left|t-s\right|^{2h_{*}-1}\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}.$

These estimates yield equation $\left(\ref{eq:Lipschitz sigma}\right)$.

Let $k$ be defined as above and $\nu\geq 0$. Then, for any $0\leq a<T$, $0\leq
t<T-a$ and $\delta\in\left(0,1\right)$ fixed we have

$\displaystyle\phi_{\nu}\left(t,a\right)$
$\displaystyle:=\int_{a}^{a+t}k\left(a+t,s\right)\left|\log\left(a+t-s\right)\right|^{2\nu}ds$
$\displaystyle\leq
C_{T}\int_{a}^{a+t}\left(a+t-s\right)^{\left(2h_{*}-1\right)}\left|\log\left(a+t-s\right)\right|^{2\nu}ds$
$\displaystyle=C_{T}\int_{0}^{t}u^{2h_{*}-1}\left|\log\left(u\right)\right|^{2\nu}du$
$\displaystyle\leq
C_{T}C_{T,\delta}^{2\nu}\int_{0}^{t}u^{2h_{*}-1-2\nu\delta}du=C_{T,\delta,\nu,h_{*}}t^{2\left(h_{*}-\nu\delta\right)},$

where we have used that $\left|\log\left(u\right)\right|\leq
C_{T,\delta}u^{-\delta}$ for some constant $C_{T,\delta}>0$. Note that
$2\left(h_{*}-\nu\delta\right)>0$ if and only if $h_{*}>\nu\delta$. Therefore,
choosing $a=0$, we have that

$t\mapsto\int_{0}^{t}k\left(t,s\right)\left|\log\left(t-s\right)\right|^{2\nu}ds\in
L^{\infty}\left(\left[0,T\right]\right),$

if $h_{*}>0$, for $\nu=0$, and if $h_{*}>\delta$, for $\nu=1$. Furthermore, by
setting $a=t^{\prime}$ and $t=\epsilon$ in the estimate for
$\phi_{\nu}\left(t,a\right)$, we have

$\displaystyle\limsup_{\epsilon\rightarrow 0}$
$\displaystyle\parallel\int_{\cdot}^{\cdot+\epsilon}k\left(\cdot+\varepsilon,s\right)\left|\log\left(\cdot+\varepsilon-s\right)\right|^{2\nu}ds\parallel_{L^{\infty}\left(\left[0,T\right]\right)}\leq\limsup_{\epsilon\rightarrow
0}C_{T,\delta,\nu,h_{*}}\varepsilon^{2\left(h_{*}-\nu\delta\right)}=0,$

if $h^{*}>0$, for $\nu=0$, and if $h_{*}>\delta$, for $\nu=1$. Since $\delta$
can be chosen arbitrarily close to zero then $h_{*}$ can be arbitrarily close
to zero. These estimates yield that $\sigma$ satisfies
$\mathbf{H1\text{-}H3}$. ∎

Now, we can give the following theorem showing that the self-exciting
multifractional process from equation $\left(\ref{eq:SEM}\right)$ indeed
exists and is unique.

######  Theorem 5.

Let $\sigma\left(t,s,x\right)=\left(t-s\right)^{h(t,x)-\frac{1}{2}}$ and $h$
be a Hurst function with parameters $\left(h_{*},h^{*}\right)$. Moreover, let
$g$ be an $\left\\{\mathcal{F}_{t}\right\\}$-adapted, $\mathbb{R}$-valued
stochastic process satisfying $\mathbf{H3}.$ Then, there exists a unique
process $X_{t}^{h}$ satisfying the equation

(3.5)
$X_{t}^{h}=g\left(t\right)+\int_{0}^{t}\left(t-s\right)^{h\left(t,X_{t}^{h}\right)-\frac{1}{2}}dB_{s},$

where $\left\\{B_{t}\right\\}_{t\in\left[0,T\right]}$ is a one-dimensional
Brownian motion. Furthermore, we have the following inequality for some $p\geq
2$

$\mathbb{E}\left[\left|X_{t}^{h}\right|{}^{p}\right]\leq
C_{T,p,k_{1}}\left(1+\mathbb{E}\left[\left|g\left(t\right)\right|{}^{p}\right]+\sup_{t\in\left[0,T\right]}\int_{0}^{t}\left(t-s\right)^{h_{*}-\frac{1}{2}}\mathbb{E}\left[\left|g\left(s\right)\right|{}^{p}\right]ds\right)$

We call this process a Self-Exciting Multifractional process (SEM) .

###### Proof.

We have seen in Lemma 4 that
$\sigma\left(t,s,x\right)=\left(t-s\right)^{h\left(t,x\right)-\frac{1}{2}}$
satisfies $\mathbf{H1-H2}.$ Applying Zhang’s theorem gives the existence and
uniqueness and bounds on $p$-moments for the solution of (3.5). ∎

Next we will show the Hölder regularity for the solution of (3.5). We will
need some preliminary lemmas.

######  Lemma 6.

Let $T>u>v>0.$ Then, for any $\alpha\leq 0$ and $\beta\in\left[0,1\right]$ we
have

$\left|u^{\alpha}-v^{\alpha}\right|\leq
2^{1-\beta}\left|\alpha\right|^{\beta}\left|u-v\right|^{\beta}\left|v\right|^{\alpha-\beta},$

and for $\alpha\in\left(0,1\right)$

$\left|u^{\alpha}-v^{\alpha}\right|\leq\left|\alpha\right|\left|u-v\right|^{\alpha+\beta\left(1-\alpha\right)}\left|v\right|^{-\beta\left(1-\alpha\right)}.$

###### Proof.

For $\alpha=0$ is clear. For $\alpha<1$ and $\alpha\neq 0,$ using the
remainder of Taylor’s formula in integral form we get

$\displaystyle\left|u^{\alpha}-v^{\alpha}\right|$
$\displaystyle=\left|\left(u-v\right)\int_{0}^{1}\alpha\left(v+\theta\left(u-v\right)\right)^{\alpha-1}\left(1-\theta\right)d\theta\right|$
(3.6)
$\displaystyle\leq\left|\alpha\right|\left|u-v\right|\int_{0}^{1}\left|v+\theta\left(u-v\right)\right|^{\alpha-1}d\theta\leq\left|\alpha\right|\left|u-v\right|\left|v\right|^{\alpha-1},$

where we have used that
$\left|v+\theta\left(u-v\right)\right|^{\alpha-1}\leq\left|v\right|^{\alpha-1}$.
Using that
$\left|v+\theta\left(u-v\right)\right|^{\alpha-1}\leq\theta^{\alpha-1}\left|u-v\right|^{\alpha-1}$
and assuming that $\alpha\in\left(0,1\right)$ we obtain

(3.7)
$\left|u^{\alpha}-v^{\alpha}\right|\leq\alpha\left|u-v\right|^{\alpha}\int_{0}^{1}\theta^{\alpha-1}d\theta=\left|u-v\right|^{\alpha}.$

In what follows we will use the interpolation inequality $a\wedge b\leq
a^{\beta}b^{1-\beta}$ for any $a,b>0$ and $\beta\in\left[0,1\right]$.

Consider the case $\alpha<0.$ Using the interpolation inequality with the
simple bound $\left|u^{\alpha}-v^{\alpha}\right|\leq 2\left|v\right|^{\alpha}$
and the bound (3.6) we get

$\displaystyle\left|u^{\alpha}-v^{\alpha}\right|$ $\displaystyle\leq
2^{1-\beta}\left|\alpha\right|^{\beta}\left|u-v\right|^{\beta}\left|v\right|^{\beta\left(\alpha-1\right)}\left|v\right|^{(1-\beta)\alpha}=2^{1-\beta}\left|\alpha\right|^{\beta}\left|u-v\right|^{\beta}\left|v\right|^{\alpha-\beta}.$

Consider the case $\alpha\in\left(0,1\right)$. Using the interpolation
inequality with the bounds (3.6) and (3.7) we can write

$\left|u^{\alpha}-v^{\alpha}\right|\leq\left|\alpha\right|\left|u-v\right|^{\beta}\left|v\right|^{\beta\left(\alpha-1\right)}\left|u-v\right|^{\left(1-\beta\right)\alpha}=\left|\alpha\right|\left|u-v\right|^{\alpha+\beta\left(1-\alpha\right)}\left|v\right|^{-\beta\left(1-\alpha\right)}.$

∎

######  Lemma 7.

Let $\sigma\left(t,s,x\right)=\left(t-s\right)^{h(t,x)-\frac{1}{2}}$ and $h$
be a Hurst function with parameters $\left(h_{*},h^{*}\right)$. Then, for any
$0<\gamma<2h_{*}$ there exists
$\lambda_{\gamma}:\triangle^{(3)}\left(\left[0,T\right]\right)\rightarrow\mathbb{R}$
such that

(3.8)
$\left|\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)\right|^{2}\leq\lambda_{\gamma}\left(t,t^{\prime},s\right),$

and

(3.9) $\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},$

for some constant $C_{T,\gamma}>0$.

###### Proof.

We have that

$\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)=\left(t-s\right)^{h\left(t,x\right)-\frac{1}{2}}-\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}.$

Furthermore, notice that for all $t>t^{\prime}>s>0$, we can add and subtract
the term $\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}$ to get

$\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)=J^{1}\left(t,t^{\prime},s,x\right)+J^{2}\left(t,t^{\prime},s,x\right),$

where

$\displaystyle J^{1}\left(t,t^{\prime},s,x\right)$
$\displaystyle:=\left(t-s\right)^{h(t,x)-\frac{1}{2}}-\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}},$
$\displaystyle J^{2}\left(t,t^{\prime},s,x\right)$
$\displaystyle:=\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}-\left(t^{\prime}-s\right)^{h(t^{\prime},x)-\frac{1}{2}}.$

First we bound $J^{1}$. Using the inequality $\left(\ref{eq:
expinequality}\right)$ and that $h$ is Lipschitz in the time argument, by
similar arguments as in Lemma 4, we obtain for any $\delta\in\left(0,1\right)$

$\displaystyle\left|J^{1}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq
C_{T}\left|t-t^{\prime}\right|\left|t-s\right|^{h_{*}-\frac{1}{2}}\left|\log\left(t-s\right)\right|$
$\displaystyle\leq
C_{T,\delta}\left|t-t^{\prime}\right|\left|t-s\right|^{h_{*}-\frac{1}{2}-\delta}$
$\displaystyle\leq
C_{T,\delta}\left|t-t^{\prime}\right|\left|t^{\prime}-s\right|^{h_{*}-\frac{1}{2}-\delta},$

since $s<t^{\prime}<t$. Next, in order to bound the term $J^{2}$, we apply
Lemma 6 with $u=t-s$,$v=t^{\prime}-s$ and $\alpha=h(t^{\prime},x)-\frac{1}{2}$
. Note that, since
$h(t^{\prime},x)\in\left[h_{*},h^{*}\right]\subset\left(0,1\right)$,
$\alpha\in\left[h_{*}-\frac{1}{2},h^{*}-\frac{1}{2}\right]\subset\left(-\frac{1}{2},\frac{1}{2}\right)$.
Hence, if $\alpha=$$h(t^{\prime},x)-\frac{1}{2}\leq 0$ (this implies
$h_{*}<1/2$ and $\alpha\left(-\frac{1}{2},0\right)$), we get

$\displaystyle\left|J^{2}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq
2\left|t-t^{\prime}\right|^{\beta_{1}}\left|t^{\prime}-s\right|^{h(t^{\prime},x)-\frac{1}{2}-\beta_{1}}$
$\displaystyle\leq
C_{T}\left|t-t^{\prime}\right|^{\beta_{1}}\left|t^{\prime}-s\right|^{h_{*}-\frac{1}{2}-\beta_{1}},$

for any $\beta_{1}\in\left(0,1\right)$. If
$\alpha=h(t^{\prime},x)-\frac{1}{2}>0$ (this implies $h^{*}>1/2$ and
$\alpha\in\left(0,\frac{1}{2}\right)$), we get

$\displaystyle\left|J^{2}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq\frac{1}{2}\left|t-t^{\prime}\right|^{\alpha+\beta_{2}\left(1-\alpha\right)}\left|t^{\prime}-s\right|^{-\beta_{2}\left(1-\alpha\right)}$
$\displaystyle\leq\frac{1}{2}\left|t-t^{\prime}\right|^{\alpha+\frac{1}{2}-\varepsilon\left(1-\alpha\right)}\left|t^{\prime}-s\right|^{-\frac{1}{2}+\varepsilon\left(1-\alpha\right)}$
$\displaystyle\leq\frac{1}{2}\left|t-t^{\prime}\right|^{h_{*}-\varepsilon}\left|t^{\prime}-s\right|^{-\frac{1}{2}+\frac{\varepsilon}{2}},$

where in the second inequality we have chosen
$\beta_{2}=\frac{1}{2\left(1-\alpha\right)}-\varepsilon,\varepsilon>0,$ and in
the third inequality we have used that
$(1-\alpha)\in\left(\frac{1}{2},1\right)$. Therefore, we can write the
following bound

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)\right|^{2}$
$\displaystyle\leq
2\left(\left|J^{1}\left(t,t^{\prime},s,x\right)\right|^{2}+\left|J^{2}\left(t,t^{\prime},s,x\right)\right|^{2}\right)$
$\displaystyle\leq
2\left(C_{T,\delta}\left|t-t^{\prime}\right|^{2}\left|t^{\prime}-s\right|^{2h_{*}-1-2\delta}\right.$
$\displaystyle\left.\quad+C_{T}\left|t-t^{\prime}\right|^{2\beta_{1}}\left|t^{\prime}-s\right|^{2h_{*}-1-2\beta_{1}}+\frac{1}{2}\left|t-t^{\prime}\right|^{2h_{*}-2\varepsilon}\left|t^{\prime}-s\right|^{-1+\varepsilon}\right)$
$\displaystyle\leq
C_{T,\beta_{1}}\left|t-t^{\prime}\right|^{2\beta_{1}}\left|t^{\prime}-s\right|^{-1+h_{*}-\beta_{1}},$

where to get the last inequality we have chosen $\delta=\beta_{1}$ and
$\varepsilon=h_{*}-\beta_{1}$. Therefore, defining

$\lambda_{\gamma}\left(t,t^{\prime},s\right):=C_{T,\gamma}\left(t-t^{\prime}\right)^{\gamma}\left(t^{\prime}-s\right)^{-1+h_{*}-\frac{\gamma}{2}},$

and choosing $\gamma$ such that $0<\gamma<2h_{*},$ we can compute

$\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left(t^{\prime}\right)^{h_{*}-\frac{\gamma}{2}}\left(t-t^{\prime}\right)^{\gamma},$

which concludes the proof. ∎

######  Proposition 8.

Let $\left\\{X_{t}^{h}\right\\}_{t\in\left[0,T\right]}$ be a SEM process
defined in Theorem 5, and assume that $g$ satisfies $\mathbf{H4}$ for some
$\delta>0$. Then there exists a set of paths $\mathcal{N}\subset\Omega$ with
$\mathbb{P}\left(\mathcal{N}\right)=0$, such that for all
$\omega\in\mathcal{N}^{c}$ the path $X_{t}^{h}\left(\omega\right)$ has
$\alpha$-Hölder continuous trajectories for any $\alpha<h_{*}\wedge\delta$. In
particular, we have

$\left|\left(X_{t}^{h}-X_{s}^{h}\right)\left(\omega\right)\right|\leq
C\left(\omega\right)\left|t-s\right|^{\alpha},\qquad\omega\in\mathcal{N}^{c}.$

###### Proof.

By theorem 5, there exists a unique solution $X_{t}^{h}$ to Equation (3.5)
with bounded $p$-order moments. We will show that $X_{t}^{h}$ also have Hölder
continuous paths. To this end, we will show that for any $p\in\mathbb{N}$
there exists a constant $C>0$ and a function $\alpha$, both depending on $p$,
such that

$\mathbb{E}\left[\left|X_{s,t}^{h}\right|^{2p}\right]\leq
C_{p}\left|t-s\right|^{\alpha_{p}},$

where $X_{s,t}^{h}=X_{t}^{h}-X_{s}^{h}$. From this we apply Kolmogorov’s
continuity theorem (e.g. Theorem 2.8 in [10], page 53) in order to obtain the
claim. Note that the increment of $X_{s,t}$ minus the increment of $g$
satisfies

$X_{s,t}^{h}-\left(g\left(t\right)-g\left(s\right)\right)=\int_{s}^{t}\left(t-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}dB_{r}+\int_{0}^{s}\left(t-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}-\left(s-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}dB_{r},$

and thus using that

(3.10) $\left|a+b\right|^{q}\leq
2^{q-1}\left(\left|a\right|^{q}+\left|b\right|^{q}\right),$

for any $q\in\mathbb{N},$ we obtain

$\displaystyle\mathbb{E}\left[\left|X_{s,t}^{h}-\left(g\left(t\right)-g\left(s\right)\right)\right|^{2p}\right]$
$\displaystyle\leq
C_{p}\mathbb{E}\left[\left|\int_{s}^{t}\left(t-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}dB_{r}\right|^{2p}\right]$
$\displaystyle+C_{p}\mathbb{E}\left[\left|\int_{0}^{s}\left(t-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}-\left(s-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}dB_{r}\right|^{2}\right]$
$\displaystyle=:C_{p}\left(J_{s,t}^{1}+J_{s,t}^{2}\right).$

Clearly, as
$h\left(t,x\right)\in\left[h_{*},h^{*}\right]\subset\left(0,1\right)$, we have
by the Burkholder-Davis-Gundy (BDG) inequality that

(3.11) $\displaystyle J_{s,t}^{1}$ $\displaystyle\leq
C_{p}\mathbb{E}\left[\left|\int_{s}^{t}\left(t-r\right)^{2h\left(t,X_{r}\right)-1}dr\right|^{p}\right]$
$\displaystyle\leq
C_{p,T}\left|\int_{s}^{t}\left(t-r\right)^{2h_{*}-1}dr\right|^{p}=C_{p,T,h_{*}}\left|t-s\right|^{2ph_{*}}.$

Consider now the term $J_{s,t}^{2}$. Applying again BDG inequality together
with the bounds $\left(\ref{eq:time reg sigma}\right)$ and
$\left(\ref{eq:TimeLipschitzIntegralLambda}\right)$ from Lemma 7, we have that
for any $\gamma<2h_{*}$

$\displaystyle J_{s,t}^{2}$ $\displaystyle\leq
C_{p}\mathbb{E}\left[\left|\int_{0}^{s}\left[\left(\left(t-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}-\left(s-r\right)^{h\left(t,X_{r}\right)-\frac{1}{2}}\right)^{2}\right]dr\right|^{p}\right]$
(3.12) $\displaystyle\leq
C_{p}\mathbb{E}\left[\left|\int_{0}^{s}\lambda_{\gamma}\left(t,s,r\right)dr\right|^{p}\right]\leq
C_{p,T,\gamma}\left|t-s\right|^{p\gamma},$

Combining (3.11) and (3.12) we can see that

$\mathbb{E}\left[\left|X_{s,t}^{h}-\left(g\left(t\right)-g\left(s\right)\right)\right|^{2p}\right]\leq
C_{p,T,\gamma}\left|t-s\right|^{p\gamma}.$

Furthermore, again using (3.10) we see that

$\mathbb{E}\left[\left|X_{s,t}\right|^{2p}\right]\leq
2^{2p-1}\left(\mathbb{E}\left[\left|X_{s,t}^{h}-\left(g\left(t\right)-g\left(s\right)\right)\right|^{2p}\right]+\mathbb{E}\left[\left|\left(g\left(t\right)-g\left(s\right)\right)\right|^{2p}\right]\right).$

Thus invoking the bounds from $\left({\bf H4}\right)$ on $g$, we obtain that

$\mathbb{E}\left[\left|X_{s,t}\right|^{2p}\right]\leq
C_{p,T,\gamma}\left|t-s\right|^{2p\left(\frac{\gamma}{2}\wedge\delta\right)},$

and it follows from Kolmogorov’s continuity theorem that $X^{h}$ has
$\mathbb{P}$-a.s. $\alpha$-Hölder continuous trajectories with
$\alpha\in\left(0,h_{*}\wedge\delta\right)$. ∎

##  4\. Simulation of Self-Exciting Multifractional Stochastic Processes

The aim of this section is to study a discretization scheme for self-excited
multifractional (SEM) processes proposed in the previous sections. In
particular we will consider an Euler type discretization and prove that
converges strongly to the original process at a rate depending on $h_{*}$. We
end the section providing two examples of numerical simulations using the
Euler discretization.

###  4.1. Euler-Maruyama Approximation Scheme

Consider a time discretization of the interval $\left[0,T\right],$ using a
step-size $\Delta t=\frac{T}{N}>0$. The discrete time Euler-Maruyama scheme
(EM) is given by

(4.1) $\displaystyle\bar{X}_{0}^{h}$ $\displaystyle=X_{0}^{h}=0$ (4.2)
$\displaystyle\bar{X}_{k}^{h}$
$\displaystyle=\sum_{i=0}^{k-1}\left(t_{k}-t_{i}\right)^{h\left(t_{k},\bar{X}_{i}^{h}\right)-\frac{1}{2}}\Delta
B_{i},\qquad k\in\left\\{1,\ldots,N\right\\},$

where $\Delta B_{i}=B\left(t_{i+1}\right)-B\left(t_{i}\right)$, and yields an
approximation of $X_{t_{k}}^{h}$ for $t_{k}=k\Delta t$ with
$k\in\left\\{0,\ldots,N\right\\}.$

In order to study the approximation error, it is convenient to consider the
continuous time interpolation of
$\left\\{\bar{X}_{k}^{h}\right\\}_{k\in\left\\{0,\ldots,N\right\\}}$ given by

(4.3)
$\bar{X}_{t}^{h}=\int_{0}^{t}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}dB_{s},\qquad
t\in\left[0,T\right],$

where
$\eta\left(s\right):=t_{i}\cdot\boldsymbol{1}_{\left[t_{i},t_{i+1}\right)}\left(s\right)$.

The following theorem is the main result in this section and its proof uses
Lemmas 11 and 12, see below.

######  Theorem 9.

Let $h$ be a Hurst function with parameters $\left(h_{*},h^{*}\right)$ and let
$X_{t}^{h}$ be the solution of equation $\left(\ref{eq:SEM}\right)$. Then the
Euler-Maruyama scheme $\left(\ref{eq:DefContEuler}\right)$, satisfies

(4.4) $\sup_{0\leq t\leq
T}\mathbb{E}\left[\left|X_{t}^{h}-\bar{X}_{t}^{h}\right|^{2}\right]\leq
C_{T,\gamma,h_{*}}E_{h_{*}}\left(C_{T,\gamma,h_{*}}\Gamma\left(h_{*}\right)T^{h_{*}}\right)\left|\Delta
t\right|^{\gamma},$

where $\gamma\in\left(0,2h_{*}\right)$, and $C_{T,\gamma,h_{*}}$ is a positive
constant, which does not depend on $N$.

###### Proof.

Define

$\delta_{t}:=X_{t}^{h}-\bar{X}_{t}^{h},\qquad\varphi\left(t\right):=\sup_{0\leq
s\leq t}\mathbb{E}\left[\left|\delta_{s}\right|^{2}\right],\quad
t\in\left[0,T\right].$

For any $t\in\left[0,T\right],$ we can write

$\displaystyle\delta_{t}$
$\displaystyle=\int_{0}^{t}\left(\left(t-s\right)^{h\left(t,X_{s}^{h}\right)-\frac{1}{2}}-\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle=\int_{0}^{t}\left(\left(t-s\right)^{h\left(t,X_{s}^{h}\right)-\frac{1}{2}}-\left(t-s\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle\quad+\int_{0}^{t}\left(\left(t-s\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}-\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle=:I_{1}\left(t\right)+I_{2}\left(t\right).$

First we bound the second moment of $I_{1}\left(t\right)$ in terms of a
Volterra integral of $\varphi$. Using the Itô isometry, equation
$\left(\ref{eq:Lipschitz sigma}\right)$ and the Lipschitz property of $h$ we
get

$\displaystyle\mathbb{E}\left[\left|I_{1}\left(t\right)\right|^{2}\right]$
$\displaystyle\leq\int_{0}^{t}k\left(t,s\right)\left(\log\left(t-s\right)\right)^{2}\mathbb{E}\left[\left(h\left(t,X_{s}^{h}\right)-h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)\right)^{2}\right]ds$
$\displaystyle\leq
C_{T,\delta}\int_{0}^{t}\left(t-s\right)^{2\left(h_{*}-\delta\right)-1}\mathbb{E}\left[\left|X_{s}^{h}-\bar{X}_{\eta\left(s\right)}^{h}\right|^{2}\right]ds,$

for $\delta>0$, arbitrarily small. By adding and subtracting
$\bar{X}_{s}^{h}$, we easily get that

$\mathbb{E}\left[\left|X_{s}^{h}-\bar{X}_{\eta\left(s\right)}^{h}\right|^{2}\right]\leq
2\varphi\left(s\right)+2\mathbb{E}\left[\left|\bar{X}_{s}^{h}-\bar{X}_{\eta\left(s\right)}^{h}\right|^{2}\right],$

Moreover, combining equation $\left(\ref{eq:SecondMomentTLCE}\right)$ in Lemma
11, yields

$\int_{0}^{t}\left(t-s\right)^{2\left(h_{*}-\delta\right)-1}\mathbb{E}\left[\left|\bar{X}_{s}^{h}-\bar{X}_{\eta\left(s\right)}^{h}\right|^{2}\right]ds\leq
C_{T}\frac{T^{2\left(h_{*}-\delta\right)}}{2\left(h_{*}-\delta\right)}\left|\Delta
t\right|^{\gamma}.$

Therefore, choosing $\delta=\frac{h_{*}}{2}$

(4.5) $\mathbb{E}\left[\left|I_{1}\right|^{2}\right]\leq
C_{T,h_{*}}\left\\{\int_{0}^{t}\left(t-s\right)^{h_{*}-1}\varphi\left(s\right)ds+\left|\Delta
t\right|^{\gamma}\right\\}.$

Next, we find a bound for the second moment of $I_{2}\left(t\right)$. Using
again the Itô isometry, equations $\left(\ref{eq:time reg sigma}\right)$ and
$\left(\ref{eq:TimeLipschitzIntegralLambda}\right),$ and Lemma 11 we can write

(4.6)
$\displaystyle\mathbb{E}\left[\left|I_{2}\left(t\right)\right|^{2}\right]$
$\displaystyle\leq\int_{0}^{t}\lambda_{\gamma}\left(t+\left(s-\eta\left(s\right)\right),t,s\right)ds\leq
C_{T,\gamma}\left|\Delta t\right|^{\gamma},$

for any $\gamma<2h_{*}$. Combining the inequalities
$\left(\ref{eq:I1Bound}\right)$ and $\left(\ref{eq:I2Bound}\right)$ we obtain

$\varphi\left(t\right)\leq
C_{T,\gamma,h_{*}}\left\\{\int_{0}^{t}\left(t-s\right)^{h_{*}-1}\varphi\left(s\right)ds+\left|\Delta
t\right|^{\gamma}\right\\}.$

Using Theorem 12 with $a\left(t\right)=C_{T,\gamma,h_{*}}\left|\Delta
t\right|^{\gamma},$$g\left(t\right)=C_{T,\gamma,h_{*}}$ and $\beta=h_{*}$ we
can conclude that

$\varphi\left(T\right)\leq
C_{T,\gamma,h_{*}}E_{h_{*}}\left(C_{T,\gamma,h_{*}}\Gamma\left(h_{*}\right)T^{h_{*}}\right)\left|\Delta
t\right|^{\gamma}.$

∎

######  Remark 10.

In [15], Zhang introduced an Euler type scheme for stochastic differential
equations of Volterra type and showed that his scheme converges at a certain
positive rate, without being very precise. A direct application of his result
to our case provides a worse rate than the one we obtain in Theorem 9. The
reason being that, due to our particular kernel, we are able to use a
fractional Gronwall lemma.

######  Lemma 11.

Let $h$ be a Hurst function with parameters $\left(h_{*},h^{*}\right)$ and let
$\bar{X}^{h}=\left\\{\bar{X}_{t}^{h}\right\\}_{t\in\left[0,T\right]}$ be given
by $\left(\ref{eq:DefContEuler}\right)$. Then

(4.7) $\mathbb{E}\left[\left|\bar{X}_{t}^{h}\right|^{2}\right]\leq C_{T},\quad
0\leq t\leq T,$

and

(4.8)
$\mathbb{E}\left[\left|\bar{X}_{t}^{h}-\bar{X}_{t^{\prime}}^{h}\right|^{2}\right]\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},\quad 0\leq t^{\prime}\leq
t\leq T,$

for any $\gamma<2h_{*}$, where $C_{T}$ and $C_{T,\gamma}$ are positive
constants.

###### Proof.

Recall that $k\left(t,s\right)=C_{T}\left(t-s\right)^{2h_{*}-1}$ and, since
$\eta\left(s\right)\leq s$, we have the following inequality

(4.9) $k\left(t,\eta\left(s\right)\right)\leq k\left(t,s\right).$

Using the Itô isometry, equation $\left(\ref{eq:BoundSigma2_No_x}\right)$ and
equation (4.9), we obtain

$\displaystyle\mathbb{E}\left[\left|\bar{X}_{t}^{h}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{t}\left(t-\eta\left(s\right)\right)^{2h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-1}ds\right]$
$\displaystyle\leq\int_{0}^{t}k\left(t,\eta\left(s\right)\right)ds\leq\int_{0}^{t}k\left(t,s\right)ds\leq
C_{T}.$

To prove the bound $\left(\ref{eq:SecondMomentTLCE}\right),$ note that

$\displaystyle\bar{X}_{t}^{h}-\bar{X}_{t^{\prime}}^{h}$
$\displaystyle=\int_{t^{\prime}}^{t}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}dB_{s},$
$\displaystyle\quad+\int_{0}^{t^{\prime}}\left\\{\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}-\left(t^{\prime}-\eta\left(s\right)\right)^{h\left(t^{\prime},\bar{X}_{\eta\left(s\right)}^{h}\right)-\frac{1}{2}}\right\\}dB_{s}$
$\displaystyle=:J_{1}+J_{2}.$

Due to the Itô isometry, equation $\left(\ref{eq:BoundSigma2_No_x}\right)$ and
$\left(\ref{eq: Volterrabound}\right)$, we obtain the bounds

$\displaystyle\mathbb{E}\left[\left|J_{1}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{t^{\prime}}^{t}\left(t-\eta\left(s\right)\right)^{2h\left(t,\bar{X}_{\eta\left(s\right)}^{h}\right)-1}ds\right]$
$\displaystyle\leq\int_{t^{\prime}}^{t}k\left(t,\eta\left(s\right)\right)ds\leq\int_{t^{\prime}}^{t}k\left(t,s\right)ds=C_{T}\left|t-t^{\prime}\right|^{2h_{*}}.$

Using again the Itô isometry, equation $\left(\ref{eq:time reg sigma}\right)$
and equation $\left(\ref{eq:TimeLipschitzIntegralLambda}\right)$ we can write,
for any $\gamma<2h_{*},$ that

$\displaystyle\mathbb{E}\left[\left|J_{2}\right|^{2}\right]$
$\displaystyle\leq\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},\eta\left(s\right)\right)ds\leq\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},$

where in the second inequality we have used
$\lambda_{\gamma}\left(t,t^{\prime},\eta\left(s\right)\right)\leq\lambda_{\gamma}\left(t,t^{\prime},s\right)$,
because $\lambda_{\gamma}$ is essentially a negative fractional power of
$(t-s)$ and $\eta\left(s\right)\leq s$. Combining the bounds for
$\mathbb{E}\left[\left|J_{1}\right|^{2}\right]$ and
$\mathbb{E}\left[\left|J_{2}\right|^{2}\right]$ the result follows. ∎

The following result is a combination of Theorem 1 and Corollary 2 in [14].

######  Theorem 12.

Suppose $\beta>0,$$a\left(t\right)$ is a nonnegative function locally
integrable on $0\leq t<T<+\infty$ and $g\left(t\right)$ is a nonnegative,
nondecreasing continuous function defined on $0\leq t<T$, $g\left(t\right)\leq
M$ (constant), and suppose $u\left(t\right)$ is nonnegative and locally
integrable on $0\leq t<T$ with

$u\left(t\right)\leq
a\left(t\right)+g\left(t\right)\int_{0}^{t}\left(t-s\right)^{\beta-1}u\left(s\right)ds,$

on this interval. Then,

$u\left(t\right)\leq
a\left(t\right)+\int_{0}^{t}\left(\sum_{n=1}^{\infty}\frac{\left(g\left(t\right)\Gamma\left(\beta\right)\right)^{n}}{\Gamma\left(n\beta\right)}\left(t-s\right)^{n\beta-1}a\left(s\right)\right)ds,\qquad
0\leq t<T.$

If in addition, $a\left(t\right)$ is a nondecreasing function on
$\left[0,T\right)$. Then,

$u\left(t\right)\leq
a\left(t\right)E_{\beta}\left(g\left(t\right)\Gamma\left(\beta\right)t^{\beta}\right),$

where $E_{\beta}$ is the Mittag-Leffler function defined by
$E_{\beta}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(k\beta+1\right)}$.

###  4.2. Examples

Let us now discuss some functions $h:\mathbb{R}\rightarrow\left(0,1\right)$
which produce some interesting self-exciting processes.

######  Example 13.

Let
$h\left(x\right)=\frac{1}{2}+\frac{1/2}{1+x^{2}}\in\left(\frac{1}{2},1\right)\subset\left(0,1\right),$
and $\left\\{B_{t}\right\\}_{t\in[0,T]}$ be a one-dimensional Brownian motion.
Assume $X_{t}^{h}$ starts at zero and define the process as given in equation
$\left(\ref{eq:SEM}\right)$. Figure $\left(\ref{plot: SEM_ProcessEX1}\right)$
shows the plot of $h$ on the left hand side and a sample path of the process,
on the right hand side, resulting from the implementation 111All simulations
were run with a step-size $\Delta t=1/100$. of the EM-approximation given by
equation $\left(\ref{eq:EM_1}\right)$. Notice the fact that this process is
smoother than a Brownian motion at the origin and rapidly converges to the
classical Brownian motion as as the process departs from zero. This implies
that $h\rightarrow\frac{1}{2}$ having only $h=1$ any time the sample path
crossed the $x$-axis again.

Figure 4.1. Numerical simulation of a trajectory of a SEM Process given the
Hurst function is $h\left(x\right)=\frac{1}{2}+\frac{1/2}{1+x^{2}}$.

Let
$h\left(x\right)=\frac{1}{2}-\frac{1/2}{1+x^{2}}\in\left(0,\frac{1}{2}\right)\subset\left(0,1\right),$
and $\left\\{B_{t}\right\\}_{t\in[0,T]}$ be a one-dimensional Brownian motion.
Assume $X_{t}^{h}$ starts at zero and define the process as given in equation
$\left(\ref{eq:SEM}\right)$. Figure $\left(\ref{plot: SEM_ProcessEX2}\right)$
shows the plot of $h$ on the left hand side and a sample path of the process,
on the right hand side, resulting from the implementation of the EM-
approximation given by equation $\left(\ref{eq:EM_1}\right)$. Is interesting
noticing in this case, contrary to the previous example, that we have a
rougher process than a Brownian motion at the origin, temporarily resembles
the classical Brownian motion as the sample path departs from zero and gets
rougher again whenever the process crosses the $x$-axis. This makes the
process go away from zero even faster due to the increased roughness.

Figure 4.2. Numerical simulation of a trajectory of a SEM Process given the
Hurst function is $h\left(x\right)=\frac{1}{2}-\frac{1/2}{1+x^{2}}$.

Let $h\left(x\right)=\frac{1}{1+x^{2}}\in\left(0,1\right),$ and
$\left\\{B_{t}\right\\}_{t\in[0,T]}$ be a one-dimensional Brownian motion.
Assume $X_{t}^{h}$ starts at zero and define the process as given in equation
$\left(\ref{eq:SEM}\right)$. Figure $\left(\ref{plot: SEM_ProcesEX3}\right)$
shows the plot of $h$ on the left hand side and a sample path of the process,
on the right hand side, resulting from the implementation 222All simulations
were run with a step-size $\Delta t=1/100$. of the EM-approximation given by
equation $\left(\ref{eq:EM_1}\right)$. Notice the fact that the Hurst function
collapses to zero as the process departs from zero, making the process be the
roughest possible. Therefore we would only recover smoother values, in
particular $h=1$ only the time the sample path crossed the $x$-axis again.

Figure 4.3. Numerical simulation of a trajectory of a SEM Process given the
Hurst function is $h\left(x\right)=\frac{1}{1+x^{2}}$.

##  5\. Self-Exciting Multifractional Gamma Processes

Barndorff-Nielsen in [1], introduces a class of self-exciting gamma type of
process, in order to model turbulence, because it captures the intermittency
effect observed in turbulent data. We would also like to extend our process in
order to capture the previously mentioned intermittency effect. One could
believe that if we were to choose a trigonometric function as a Hurst
function, i.e. $h\left(t,x\right)=\alpha+\beta\cdot\sin\left(\gamma x\right),$
for some $\alpha,\beta,\gamma\in\mathbb{R}$ in the SEM process, then we might
observe a regime switch in the Hurst parameters. Since the values of the
process $X_{t}$ may get very large, the oscillations may take place more and
more frequently. By introducing a type of gamma process (SEM-Gamma) we dampen
the Volterra kernel in (1.2) by an exponential function and make the process
oscillate around a mean value obtaining a more stable intermittency effect in
the Hurst function.

######  Definition 14.

We say that a function
$f:\left[0,T\right]\times\mathbb{R}\rightarrow\mathbb{R}_{+}$ is a dampening
function if it is nonnegative, satisfies the following Lipschitz conditions
for all $x,y\in\mathbb{R}$ and $t,t^{\prime}\in\left[0,T\right]$

$\left|f\left(t,x\right)-f\left(t,y\right)\right|\leq C\left|x-y\right|,$
$\left|f\left(t,x\right)-f\left(t^{\prime},x\right)\right|\leq
C\left|t-t^{\prime}\right|,$

and satisfies the following linear growth condition for all $x\in\mathbb{R}$
and $t\in\left[0,T\right]$

$\left|f\left(t,x\right)\right|\leq C\left(1+\left|x\right|\right),$

for some constant $C>0$.

Let $f$ be a dampening function and let $h$ be a Hurst function with
parameters $\left(h_{*},h^{*}\right)$. The self-excited multifractional gamma
process is given formally by the Volterra equation

(5.1)
$X_{t}^{h,f}=\int_{0}^{t}\exp\left(-f\left(t,X_{s}^{h,f}\right)\left(t-s\right)\right)\left(t-s\right)^{h(t,X_{s}^{h,f})-\frac{1}{2}}dB_{s}.$

The following lemma shows the existence and uniqueness of the above equation
by means of Theorem 2.

######  Lemma 15.

Let
$\sigma(t,s,x)=\exp\left(-f\left(t,x\right)\left(t-s\right)\right)\left(t-s\right)^{h(t,x)-\frac{1}{2}}$,
such that $f$ is a dampening function and $h$ is a Hurst function with
parameters $\left(h_{*},h^{*}\right)$. Then, we have that

(5.2) $\left|\sigma\left(t,s,x\right)\right|^{2}\leq
k\left(t,s\right)\left(1+\left|x\right|^{2}\right),$

where

$k\left(t,s\right)=C_{T}\left(t-s\right)^{2h_{*}-1},$

and

(5.3) $\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}\leq
C_{T}k\left(t,s\right)\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}.$

Moreover, $\sigma$ satisfies $\mathbf{H1}$-$\mathbf{H2}$.

Proofs for all the results in this section are reported in the appendix, since
they are analogous to the ones provided previously for the SEM process.

Now since the new $\sigma$ proposed for the SEM-Gamma process also satisfies
$\mathbf{H1}$-$\mathbf{H2}$ we can apply again Zhang’s theorem, in the same
way we did in Theorem 5, yielding the existence and uniqueness and bounds on
$p$-moments for the solution of

(5.4)
$X_{t}^{h,f}=g\left(t\right)+\int_{0}^{t}e^{-f\left(t,X_{t}^{h,f}\right)\left(t-s\right)}\left(t-s\right)^{h\left(t,X_{t}^{h,f}\right)-\frac{1}{2}}dB_{s}.$

We call this process a Self-Exciting Multifractional Gamma process (SEM-
Gamma).

The following Lemma is key to study the Hölder regularity for the solution of
$\left(\ref{eq: SEM-Gammaequation}\right)$, which coincides and can be derived
in the same way as for the SEM process. It is also useful for the discussion
of the strong convergence of the approximating scheme given in Theorem
$\left(\ref{thm:EM_StrongConvergenceGamma}\right).$

######  Lemma 16.

Let
$\sigma(t,s,x)=\exp\left(-f\left(t,x\right)\left(t-s\right)\right)\left(t-s\right)^{h(t,x)-\frac{1}{2}}$,
such that $f$ is a dampening function and $h$ is a Hurst function with
parameters $\left(h_{*},h^{*}\right)$. Then, for any $0<\gamma<2h_{*}$ there
exists
$\lambda_{\gamma}:\triangle^{(3)}\left(\left[0,T\right]\right)\rightarrow\mathbb{R}$
such that

(5.5)
$\left|\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)\right|^{2}\leq\lambda_{\gamma}\left(t,t^{\prime},s\right)\left(1+\left|x\right|^{2}\right),$

and

(5.6) $\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},$

for some constant $C_{T,\gamma}>0.$

In order to simulate this process we will use, again, the Euler-Maruyama
approximation to discretize the continuous equation $\left(\ref{eq: SEMP_OU-
Process}\right)$. Consider a time discretization of the interval
$\left[0,T\right],$ using a step-size $\Delta t=\frac{T}{N}>0$. The EM method
yields a discrete time approximation $\bar{X}_{k}^{h,f}$ of the process
$X_{t_{k}}^{h,f}$ for $t_{k}=k\Delta t$ with
$k\in\left\\{0,\ldots,N\right\\}.$ Therefore we have the following discrete
time equation

(5.7) $\displaystyle\bar{X}_{0}^{h,f}$ $\displaystyle=X_{0}^{h,f}=0$ (5.8)
$\displaystyle\bar{X}_{k}^{h,f}$
$\displaystyle=\sum_{i=0}^{k-1}\exp\left(-f\left(t_{k},\bar{X}_{i}^{h,f}\right)\left(t_{k}-t_{i}\right)\right)\left(t_{k}-t_{i}\right)^{h\left(t_{k},\bar{X}_{i}^{h,f}\right)-\frac{1}{2}}\Delta
B_{i}\quad\forall k\in\left\\{1,\ldots,N\right\\},$

where $\Delta B_{i}=B\left(t_{i+1}\right)-B\left(t_{i}\right)$.

Before trying to implement this approximation, in order to study the process
numerically we will have to prove the following theorem to ensure the
approximation is strongly converging to the process itself. It will be
convenient, just as we did with the SEM process, to consider a continuous time
interpolation of
$\left\\{\bar{X}_{k}^{h,f}\right\\}_{k\in\left\\{0,\ldots,N\right\\}}$ given
by

(5.9)
$\bar{X}_{t}^{h,f}=\int_{0}^{t}\exp\left(-f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)\right)\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}dB_{s},\quad\forall
t\in\left[0,T\right],$

where, again,
$\eta\left(s\right):=t_{i}\cdot\boldsymbol{1}_{\left[t_{i},t_{i+1}\right)}\left(s\right)$.
We also have the following technical result.

######  Lemma 17.

Let $f$ be a dampening function, $h$ be a Hurst function with parameters
$\left(h_{*},h^{*}\right)$ and
$\bar{X}^{h,f}=\left\\{\bar{X}_{t}^{h,f}\right\\}_{t\in\left[0,T\right]}$ be
given by $\left(\ref{eq:DefContEulerGamma}\right)$. Then

(5.10) $\mathbb{E}\left[\left|\bar{X}_{t}^{h,f}\right|^{2}\right]\leq
C_{T},\quad 0\leq t\leq T,$

and

(5.11)
$\mathbb{E}\left[\left|\bar{X}_{t}^{h,f}-\bar{X}_{t^{\prime}}^{h,f}\right|^{2}\right]\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},\quad 0\leq t^{\prime}\leq
t\leq T,$

for any $\gamma<2h_{*}$, where $C_{T}$ and $C_{T,\gamma}$ are positive
constants.

Using Lemma 17 and Theorem 12 we can show the order of convergence for the
approximating scheme.

######  Theorem 18.

Let $f$ be a dampening function, $h$ be a Hurst function with parameters
$\left(h_{*},h^{*}\right)$ and
$\bar{X}^{h,f}=\left\\{\bar{X}_{t}^{h,f}\right\\}_{t\in\left[0,T\right]}$ be
given by $\left(\ref{eq:DefContEulerGamma}\right)$. Then the Euler-Maruyama
scheme $\left(\ref{eq:DefContEulerGamma}\right)$, satisfies

(5.12) $\sup_{0\leq t\leq
T}\mathbb{E}\left[\left|X_{t}^{h,f}-\bar{X}_{t}^{h,f}\right|^{2}\right]\leq
C_{T,\gamma,h_{*}}E_{h_{*}}\left(C_{T,\gamma,h_{*}}\Gamma\left(h_{*}\right)T^{h_{*}}\right)\left|\Delta
t\right|^{\gamma},$

where $\gamma\in\left(0,2h_{*}\right)$ and $C_{T,\gamma,h_{*}}$ is a positive
constant, which does not depend on $N$.

######  Example 19.

We will continue the previous example (4.2). In [13], D. Sornette and V.
Filimonov suggested a class of self-excited processes that may exhibit all
stylized facts found in financial time series as heavy tails (asset return
distribution displays heavy tails with positive excess kurtosis), absence of
autocorrelations (autocorrelations in asset returns are negligible, except for
very short time scales $\simeq 20$ minutes), volatility clustering (absolute
returns display a positive, significant and slowly decaying autocorrelation
function) and the leverage effect (volatility measures of an asset are
negatively correlated with its returns) among others stated in [7]. As we will
see, the SEM-Gamma process resembles this properties for some choices of $h$.
The SEM-Gamma process could also be interesting for modeling commodity markets
given that it mean reversion property, clustering in its increments and also
stationary increments, given by the dampening through the exponential
function. The right plot in Figure $\left(\ref{plot: SEM-Gamma}\right)$,
corresponds to a simulation of a sample path of the process (5.8), given the
Hurst function $h$ is the same as in example (4.2) given by
$h\left(x\right)=\frac{1}{1+x^{2}}$. Notice also we have taken in this first
example of the SEM-Gamma process $f\left(x\right)=0,$ which provides the
regular SEM process of the previous section just to show the left plot looks
very similar to the left plot in Figure $\left(\ref{plot:
SEM_ProcesEX3}\right)$.

Figure 5.1. Numerical simulation of a trajectory of a SEM-Gamma Process given
the Hurst function is $f=0$ and $h(x)=\frac{1}{1+x^{2}}$.

Figure $\left(\ref{plot: SEM-Gamma_Comparative}\right)$ shows the change in
the behavior of the Hurst exponent (a transition from rougher values to
smoother values, i.e. $h\approx 0$ to $h\approx 1$) as we shift from lower
values for speed of mean reversion, i.e. $f$, to higher values. In particular
we compare $f\in\left\\{0,0.5,1,10\right\\}$

Figure 5.2. Numerical simulations of trajectories of SEM-Gamma processes given
$h\left(x\right)=\frac{1}{1+x^{2}}$ and $f\in\left\\{0,0.5,1,10\right\\}$.

######  Remark 20.

Notice one can control the clustering effect of the increments and the varying
regularity of the process by controlling the parameter $f$, regardless of the
Hurst function chosen as $h$. This is desirable in numerous fields, for
example in financial markets modeling, when trying to capture shocks in asset
prices. It is also important to remark the fact that using
$f\left(x\right)=5,$ we reduced the amount of spikes to none, shifting the
process nature from very rough and big drift, to a very smooth and driftless
process. The following Figure $\left(\ref{plot: scale comparison}\right)$
shows how by zooming in the case $f\left(x\right)=5$ close enough we observe
the rough nature hidden at a lower scale.

Figure 5.3. Scale comparative of the SEM-Gamma process with
$f\left(x\right)=5$ and $h\left(x\right)=\frac{1}{1+x^{2}}$.

It also makes sense to let $f\left(x\right)$ be a function of $x$, rather than
a constant and in particular, if we take
$f\left(x\right)=h\left(x\right)=\frac{1}{1+x^{2}}$, we can see in the
following Figure $\left(\ref{plot: SEM-Gamma h1equalh2}\right)$ how the regime
switch in the Hurst exponent is less abrupt favoring sustained difference of
roughness in time.

Figure 5.4. Numerical simulation of a trajectory of a SEM-Gamma Process given
the Hurst function is $f\left(x\right)=h(x)=\frac{1}{1+x^{2}}$

######  Remark 21.

The plots in Figure $\left(\ref{plot: ACF}\right)$ show the autocorrelation
function of the absolute value in the time series of the increments in the SEM
process (left graph) from example (4.2) and in the SEM-Gamma process (right
graph) with $f\left(x\right)=0.1$. Notice that autocorrelation in the second
case is clearly much higher.

Figure 5.5. SEM and SEM-Gamma Processes Autocorrelation Function.

##  6\. Appendix

In this appendix we have placed the proofs for the results related with SEM-
Gamma process since they are analogous to the proofs in previous sections.

###  6.1. Proof of Lemma 15

###### Proof.

We will again proof the three results stated in the lemma by reducing them to
the case proved in Lemma 4. To do so we start by proving equation
$\left(\ref{eq:sigma linear growth-gamma}\right)$. By definition, we have that

$\displaystyle\sigma\left(t,s,x\right)$
$\displaystyle=\exp\left(-f\left(t,x\right)\left(t-s\right)\right)\left(t-s\right)^{h\left(t,x\right)-\frac{1}{2}}.$

Note that

$\exp\left(-f\left(t,x\right)\left(t-s\right)\right)\leq 1,$

since $f\geq 0$ and $s<t$, for all
$\left(t,s\right)\in\triangle^{\left(2\right)}\left(\left[0,T\right]\right)$.
We also have that
$h\left(t,x\right)\in\left[h_{*},h^{*}\right]\subset\left(0,1\right)$ for all
$\left(t,x\right)\in\left[0,T\right]\times\mathbb{R}$. Therefore the result
trivially follows from Lemma 4.

Next we consider equation $\left(\ref{eq:Lipschitz sigma-gamma}\right)$, using
the fact that we can rewrite $\sigma\left(t,s,x\right)$ as

$\sigma\left(t,s,x\right)=\exp\left(-f\left(t,x\right)\left(t-s\right)+\log\left(t-s\right)\left(h\left(t,x\right)-\frac{1}{2}\right)\right),$

and make use again of inequality $\left(\ref{eq: expinequality}\right)$, for
all $x,y\in\mathbb{R}.$ Now we can write the following upper bound

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|$
$\displaystyle\leq\exp\left(\max\left(-f\left(t,x\right),-f\left(t,y\right)\right)\left(t-s\right)\right)\exp\left(\left(\max\left(h\left(t,x\right),h\left(t,y\right)\right)-\frac{1}{2}\right)\cdot\log\left(t-s\right)\right)$
$\displaystyle\quad\times\left(\left|f\left(t,x\right)-f\left(t,y\right)\right|\left|t-s\right|+\left|\log\left(t-s\right)\right|\left|h\left(t,x\right)-h\left(t,y\right)\right|\right).$

Recalling that $\left|e^{-f\left(t,x\right)\left(t-s\right)}\right|\leq 1$ and
that $f$ and $h$ are uniformly Lipschitz, we have that

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t,s,y\right)\right|^{2}$
$\displaystyle\leq
C^{2}\exp\left(\max\left(\log\left(t-s\right)\left(2h\left(t,x\right)-1\right),\log\left(t-s\right)\left(2h\left(t,y\right)-1\right)\right)\right)$
$\displaystyle\quad\times\left|\log\left(t-s\right)\right|^{2}\left|x-y\right|^{2}.$

This reduces the proof to the previous case of a SEM process, see Lemma 4. ∎

###  6.2. Proof of Lemma 16

###### Proof.

In order to prove equation $\left(\ref{eq:time reg sigma-gamma}\right)$, it is
clear that

$\displaystyle\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)$
$\displaystyle=e^{-f\left(t,x\right)\left(t-s\right)}\left(t-s\right)^{h\left(t,x\right)-\frac{1}{2}}-e^{-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)}\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}.$

Furthermore, notice that for all $t>t^{\prime}>s>0,$ we can add and subtract
the term

$e^{-f\left(t,x\right)\left(t-s\right)}\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}},$

to get

$\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)=\tilde{J}^{1}\left(t,t^{\prime},s,x\right)+\tilde{J}^{2}\left(t,t^{\prime},s,x\right),$

where

$\displaystyle\tilde{J}^{1}\left(t,t^{\prime},s,x\right)$
$\displaystyle:=e^{-f\left(t,x\right)\left(t-s\right)}\left(\left(t-s\right)^{h\left(t,x\right)-\frac{1}{2}}-\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}\right),$
$\displaystyle\tilde{J}^{2}\left(t,t^{\prime},s,x\right)$
$\displaystyle:=e^{-f\left(t,x\right)\left(t-s\right)}\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}-e^{-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)}\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}.$

First we bound $\tilde{J}^{1}$ by using that
$e^{-f\left(t,x\right)\left(t-s\right)}\leq 1,$

$\displaystyle\left|\tilde{J}^{1}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq\left|J^{1}\left(t,t^{\prime},s,x\right)\right|\leq
C_{T,\delta}\left|t-t^{\prime}\right|\left|t^{\prime}-s\right|^{h_{*}-\frac{1}{2}-\delta},$

since $s<t^{\prime}<t,$ and where $J^{1}$ is the terms appearing in the proof
of Lemma 7. Next, in order to bound the term $\tilde{J}^{2}$, we add and
subtract the quantity

$e^{-f\left(t,x\right)\left(t-s\right)}\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}},$

to obtain

$\displaystyle\left|\tilde{J}^{2}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq\left|e^{-f\left(t,x\right)\left(t-s\right)}\left(\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}-\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\right)\right.$
$\displaystyle\qquad\left.-\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\left(e^{-f\left(t,x\right)\left(t-s\right)}-e^{-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)}\right)\right|$
$\displaystyle\leq\left|e^{-f\left(t,x\right)\left(t-s\right)}\right|\left|\left(t-s\right)^{h(t^{\prime},x)-\frac{1}{2}}-\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\right|$
$\displaystyle\qquad+\left|\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\right|\left|e^{-f\left(t,x\right)\left(t-s\right)}-e^{-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)}\right|$
$\displaystyle\leq\left|J^{2}\left(t,t^{\prime},s,x\right)\right|+\left|\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\right|\left|e^{-f\left(t,x\right)\left(t-s\right)}-e^{-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)}\right|,$

where $J^{2}$ is the term appearing in the proof of Lemma 7. Using inequality
$\left(\ref{eq: expinequality}\right)$ we can rewrite the previous expression
as

$\displaystyle\left|\tilde{J}^{2}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq\left|J^{2}\left(t,t^{\prime},s,x\right)\right|+\left|\left(t^{\prime}-s\right)^{h\left(t^{\prime},x\right)-\frac{1}{2}}\right|$
$\displaystyle\qquad\times\left|e^{\max\left(-f\left(t,x\right)\left(t-s\right),-f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)\right)}\right|\left|f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)-f\left(t,x\right)\left(t-s\right)\right|$
$\displaystyle\leq\left|J^{2}\left(t,t^{\prime},s,x\right)\right|+C_{T}\left|t^{\prime}-s\right|^{h\left(t^{\prime},x\right)-\frac{1}{2}}\left|t-t^{\prime}\right|\left|f\left(t^{\prime},x\right)-f\left(t,x\right)\right|.$

Then, adding and subtracting $f\left(t,x\right)\left(t^{\prime}-s\right)$, and
using the linear growth and Lipschitz conditions on $f$, we obtain

$\displaystyle\left|f\left(t^{\prime},x\right)\left(t^{\prime}-s\right)-f\left(t,x\right)\left(t-s\right)\right|$
$\displaystyle\leq\left|f\left(t,x\right)\right|\left|t^{\prime}-t\right|+\left|t^{\prime}-s\right|\left|f\left(t^{\prime},x\right)-f\left(t,x\right)\right|$
$\displaystyle\leq
C\left|t^{\prime}-t\right|\left(1+\left|x\right|\right)+C\left|t^{\prime}-s\right|\left|t^{\prime}-t\right|,$

and we can conclude that

$\displaystyle\left|\tilde{J}^{2}\left(t,t^{\prime},s,x\right)\right|$
$\displaystyle\leq\left|J^{2}\left(t,t^{\prime},s,x\right)\right|+C\left|t^{\prime}-s\right|^{h\left(t^{\prime},x\right)-\frac{1}{2}}\left|t-t^{\prime}\right|\left(1+\left|x\right|\right)$
$\displaystyle\leq\left|J^{2}\left(t,t^{\prime},s,x\right)\right|+C_{T}\left|t^{\prime}-s\right|^{h_{*}-\frac{1}{2}}\left|t-t^{\prime}\right|\left(1+\left|x\right|\right).$

Therefore, if we define

$\lambda_{\gamma}\left(t,t^{\prime},s\right):=C_{T,\gamma}\left(t-t^{\prime}\right)^{\gamma}\left(t^{\prime}-s\right)^{-1+h_{*}-\frac{\gamma}{2}},$

for $0<\gamma<2h_{*},$ and use the final bounds for $J^{1}$ and $J^{2}$ in
Lemma 7, we get that

$\displaystyle\left|\sigma\left(t,s,x\right)-\sigma\left(t^{\prime},s,x\right)\right|^{2}$
$\displaystyle\leq
4\left(\left|J^{1}\left(t,t^{\prime},s,x\right)\right|^{2}+\left|J^{2}\left(t,t^{\prime},s,x\right)\right|^{2}+\left|C_{T}\left|t^{\prime}-s\right|^{h_{*}-\frac{1}{2}}\left|t-t^{\prime}\right|\left(1+\left|x\right|\right)\right|^{2}\right)$
$\displaystyle\leq\lambda_{\gamma}\left(t,t^{\prime},s\right)\left(1+\left|x\right|^{2}\right),$

and

$\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left(t^{\prime}\right)^{h_{*}-\frac{\gamma}{2}}\left(t-t^{\prime}\right)^{\gamma},$

which concludes the proof. ∎

###  6.3. Proof of Lemma 17

###### Proof.

Recall that $k\left(t,s\right)=C_{T}\left(t-s\right)^{2h_{*}-1}$ and, since
$\eta\left(s\right)\leq s$, we have the following inequality

(6.1) $k\left(t,\eta\left(s\right)\right)\leq k\left(t,s\right).$

Using the Itô isometry, that $e^{-2f\left(t,x\right)}\leq 1$, equation
$\left(\ref{eq:BoundSigma2_No_x}\right)$ and equation (6.1), we obtain

$\displaystyle\mathbb{E}\left[\left|\bar{X}_{t}^{h,f}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{t}e^{-2f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{2h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-1}ds\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{t}\left(t-\eta\left(s\right)\right)^{2h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-1}ds\right]$
$\displaystyle\leq\int_{0}^{t}k\left(t,\eta\left(s\right)\right)ds\leq\int_{0}^{t}k\left(t,s\right)ds\leq
C_{T}.$

To prove the bound $\left(\ref{eq:SecondMomentTLCE}\right),$ note that

$\displaystyle\bar{X}_{t}^{h,f}-\bar{X}_{t^{\prime}}^{h,f}$
$\displaystyle=\int_{t^{\prime}}^{t}e^{-f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}dB_{s},$
$\displaystyle\quad+\int_{0}^{t^{\prime}}\left\\{e^{-f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right.$
$\displaystyle\qquad\qquad\left.-e^{-f\left(t^{\prime},\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t^{\prime}-\eta\left(s\right)\right)^{h\left(t^{\prime},\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right\\}dB_{s}$
$\displaystyle=:J_{1}+J_{2}.$

Due to the Itô isometry, that $e^{-2f\left(t,x\right)}\leq 1$, equation
$\left(\ref{eq:BoundSigma2_No_x}\right)$ and $\left(\ref{eq:
Volterrabound-1}\right)$, we obtain the bounds

$\displaystyle\mathbb{E}\left[\left|J_{1}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{t^{\prime}}^{t}e^{-2f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{2h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-1}ds\right]$
$\displaystyle\leq\int_{t^{\prime}}^{t}k\left(t,\eta\left(s\right)\right)ds\leq\int_{t^{\prime}}^{t}k\left(t,s\right)ds=C_{T}\left|t-t^{\prime}\right|^{2h_{*}}.$

Using again the Itô isometry, equation $\left(\ref{eq:time reg sigma-
gamma}\right)$ and equation $\left(\ref{eq:TimeLispchitzIntegralLambda-
gamma}\right)$ we can write, for any $\gamma<2h_{*},$ that

$\displaystyle\mathbb{E}\left[\left|J_{2}\right|^{2}\right]$
$\displaystyle\leq\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},\eta\left(s\right)\right)\left(1+\mathbb{E}\left[\left|\bar{X}_{\eta\left(s\right)}^{h,f}\right|^{2}\right]\right)ds\leq
C_{T}\int_{0}^{t^{\prime}}\lambda_{\gamma}\left(t,t^{\prime},s\right)ds\leq
C_{T,\gamma}\left|t-t^{\prime}\right|^{\gamma},$

where in the second inequality we have used that
$\lambda_{\gamma}\left(t,t^{\prime},\eta\left(s\right)\right)\leq\lambda_{\gamma}\left(t,t^{\prime},s\right)$,
because $\lambda_{\gamma}$ is essentially a negative fractional power of
$(t-s)$ and $\eta\left(s\right)\leq s$ and also that
$\mathbb{E}\left[\left|\bar{X}_{t}^{h,f}\right|^{2}\right]\leq C_{T}$, $0\leq
t\leq T$, which we just have proved above. Combining the bounds for
$\mathbb{E}\left[\left|J_{1}\right|^{2}\right]$ and
$\mathbb{E}\left[\left|J_{2}\right|^{2}\right]$ the result follows. ∎

###  6.4. Proof of Theorem 18

###### Proof.

We will reduce the proof to the case in Theorem 9. To do so, in the same way
we did, we define

$\delta_{t}:=X_{t}^{h,f}-\bar{X}_{t}^{h,f},\qquad\varphi\left(t\right):=\sup_{0\leq
s\leq t}\mathbb{E}\left[\left|\delta_{s}\right|^{2}\right],\quad
t\in\left[0,T\right].$

For any $t\in\left[0,T\right]$, we can write

$\displaystyle\delta_{t}$
$\displaystyle=\int_{0}^{t}\left(e^{-f\left(t,X_{s}^{h,f}\right)\left(t-s\right)}\left(t-s\right)^{h(t,X_{s}^{h,f})-\frac{1}{2}}\right.$
$\displaystyle\qquad-\left.e^{-f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle=\int_{0}^{t}\left(e^{-f\left(t,X_{s}^{h,f}\right)\left(t-s\right)}\left(t-s\right)^{h(t,X_{s}^{h,f})-\frac{1}{2}}\right.$
$\displaystyle\qquad-\left.e^{-f\left(t,X_{s}^{h,f}\right)\left(t-s\right)}\left(t-s\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle+\int_{0}^{t}\left(e^{-f\left(t,X_{s}^{h,f}\right)\left(t-s\right)}\left(t-s\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right.$
$\displaystyle\qquad-\left.e^{-f\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\left(t-\eta\left(s\right)\right)}\left(t-\eta\left(s\right)\right)^{h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)-\frac{1}{2}}\right)dB_{s}$
$\displaystyle=:\tilde{I}_{1}\left(t\right)+\tilde{I}_{2}\left(t\right).$

First we bound the second moment of $\tilde{I}_{1}\left(t\right)$ in terms of
a certain integral of $\varphi$. Using the Itô isometry, equation
$\left(\ref{eq:Lipschitz sigma-gamma}\right)$ and the Lipschitz property of
$h$ we get

$\displaystyle\mathbb{E}\left[\left|\tilde{I}_{1}\left(t\right)\right|^{2}\right]$
$\displaystyle\leq\int_{0}^{t}k\left(t,s\right)\left(\log\left(t-s\right)\right)^{2}\mathbb{E}\left[\left(h\left(t,X_{s}^{h,f}\right)-h\left(t,\bar{X}_{\eta\left(s\right)}^{h,f}\right)\right)^{2}\right]ds$
$\displaystyle\leq
C_{T,\delta}\int_{0}^{t}\left(t-s\right)^{2\left(h_{*}-\delta\right)-1}\mathbb{E}\left[\left|X_{s}^{h,f}-\bar{X}_{\eta\left(s\right)}^{h,f}\right|^{2}\right]ds,$

for $\delta>0,$ arbitrarily small. By the same arguments as in the proof of
Theorem 9 we obtain the following bound

(6.2) $\mathbb{E}\left[\left|\tilde{I}_{1}\right|^{2}\right]\leq
C_{T,h_{*}}\left\\{\int_{0}^{t}\left(t-s\right)^{h_{*}-1}\varphi\left(s\right)ds+\left|\Delta
t\right|^{\gamma}\right\\}.$

Next, we find a bound for the second moment of $\tilde{I}_{2}\left(t\right)$.
Using again the Itô isometry, equations $\left(\ref{eq:time reg sigma-
gamma}\right)$ and $\left(\ref{eq:TimeLispchitzIntegralLambda-gamma}\right)$,
and Lemma 11 we can write

(6.3)
$\mathbb{E}\left[\left|\tilde{I}_{2}\right|^{2}\right]\leq\int_{0}^{t}\lambda_{\gamma}\left(t+\left(s-\eta\left(s\right)\right),t,s\right)\left(1+\mathbb{E}\left[\left|\bar{X}_{\eta\left(s\right)}^{h,f}\right|^{2}\right]\right)ds\leq
C_{T,\gamma}\left|\Delta t\right|^{\gamma},$

for any $\gamma<2h_{*}$, and where we have used that

$\mathbb{E}\left[\left|\bar{X}_{s}^{h,f}\right|^{2}\right]\leq C_{T},\qquad
0\leq s\leq T.$

Combining the inequalities $\left(\ref{eq: barI1Bound}\right)$ and
$\left(\ref{eq: barI2Bound}\right)$ we obtain

$\tilde{\varphi}\left(t\right)\leq
C_{T,\gamma,h_{*}}\left\\{\int_{0}^{t}\left(t-s\right)^{h_{*}-1}\varphi\left(s\right)ds+\left|\Delta
t\right|^{\gamma}\right\\}.$

Using again Lemma 12 we can conclude. ∎

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