# A short memory condition for infinitely divisible random fields

Vitalii Makogin, Evgeny Spodarev Dr. Vitaly Makogin unexpectedly passed away
on the 8th of May 2024.Institute of Stochastics, Ulm University, Germany;
<EMAIL_ADDRESS>
###### Abstract

This small note yields a sufficient condition for the short range dependence
of measurable stationary infinitely divisible moving average random fields
with $d$–dimensional index space. Here, the short/long range dependence
concept in borrowed from the paper [2]. In the special case of symmetric
stable moving averages, our new condition coincides with the one from paper
[3].

Keywords: Stationary random field, moving average, infinitely divisible,
integral spectral representation, $\alpha$–stable, time series, heavy tails.

AMS subject classification 2020: Primary 60G60; Secondary 60E07, 60E10

##  1 Memory of random fields and their level sets

A random field $X=\\{X(t),\;t\in\mathbb{R}^{d}\\}$ is called infinitely
divisible (ID, for short) if all its finite dimensional distributions are
infinitely divisible, see e.g. [11, 9] for an introduction into infinite
divisibility. $X$ is said to be stationary if all its finite dimensional
distributions are translation invariant (with respect to space variable $t$).
In this note, we give a sufficient condition for stationary ID $X$ to have a
short memory. Many definitions of the memory of random functions are available
in research literature, confer e.g. [1, 7]. However, the only short/long range
dependence concept which is invariant with respect to monotone transformations
on the marginals of the field $X$ has been recently introduced in [2]. It
depends on the level sets $\\{t:X(t)>u\\}$ of $X$ in the following way:

######  Definition 1.

A measurable stationary random field $X$ is called short range dependent (SRD)
if

$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}}\int_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\,\mu(du)\mu(dv)\,dt<\infty$

for all probability measures $\mu$ on $\mathbb{R}$. If the above integral is
infinite for some choice of $\mu$ the field $X$ is named long range dependent
(LRD).

In addition, the above definition can be applied to heavy–tailed fields $X$
since it does not require the integrability of $X(0)$. The target statistic
from Definition 1 is the volume of level sets of $X$. In multivariate extreme
value theory, it appears naturally e.g. in the study of the asymptotic
behaviour of estimates of the tail dependence coefficient, cf. the recent
paper [4]. A statistical procedure for the detection of this kind of SRD/LRD
is discussed in [5].

In paper [3], implications of Definition 1 for $\alpha$– and max–stable
processes are shown. There, necessary and/or sufficient conditions of SRD/LRD
are given in terms of the kernel function $f$ in the spectral decomposition of
$X$.

Here, we prove a similar sufficient condition for stationary ID random fields
$X$ which possess a moving average integral representation. The paper is
organized as follows: preliminaries on ID fields are given in the next
section. The main SRD condition is formulated in Theorem 1 of Section 3. An
example of symmetric stable moving averages is considered there as well.
Finally, the proof of Theorem 1 (subdivided into a number of auxiliary
results) is given in Section 4.

##  2 Infinitely divisible random fields

Let $\Lambda$ be a stationary ID independently scattered random measure with
Lebesgue control measure and Lévy characteristics $(a_{0},b_{0},\nu_{0}),$
where $a_{0}\in\mathbb{R}$, $b_{0}\geq 0$ and $\nu_{0}$ is a Lévy measure. In
the sequel, we follow the exposition of ID random functions in [6]:

######  Definition 2.

A function $f:\mathbb{R}^{d}\to\mathbb{R}$ is called $\Lambda$-integrable if

$\int_{\mathbb{R}^{d}}|f(x)|\left|a_{0}+\int_{\mathbb{R}}\left(\mathbbm{1}_{[-1,1]}(yf(x))-\mathbbm{1}_{[-1,1]}(y)\right)y\nu_{0}(dy)\right|dx<\infty,$
$b_{0}^{2}\int_{\mathbb{R}^{d}}f^{2}(x)dx<\infty,$
$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}}\min\left(1,y^{2}f^{2}(x)\right)\nu_{0}(dy)dx<\infty.$

For $b_{0}>0$, $\Lambda$ possesses a non–trivial Gaussian part (in addition to
the jump part controlled by the Lévy measure $\nu_{0}$).

Let $X=\\{X(t),t\in\mathbb{R}^{d}\\}$ be a real–valued measurable stationary
ID moving average random field with spectral representation

$X(t)=\int_{\mathbb{R}^{d}}f(t-x)\Lambda(dx),\quad t\in\mathbb{R}^{d},$ (1)

where $f$ is $\Lambda$-integrable.

The characteristic function $\varphi_{X(0)}$ of $X(0)$ is given by

$\varphi_{X(0)}(u)=\exp\left(-\int_{\mathbb{R}^{d}}K(uf(s))ds\right),$

where

$K(s)=-isa_{0}+\frac{1}{2}s^{2}b_{0}-\int_{\mathbb{R}}\left(e^{isy}-1-isy\mathbb{I}_{[-1,1]}(y)\right)\nu_{0}(dy),\quad
s\in\mathbb{R}.$

Functions $K$ and $\mathrm{Re}\\{K\\}$ are known to be continuous negative
definite functions. Obviously,
$\mathrm{Re}\\{K(s)\\}=\mathrm{Re}\\{K(|s|)\\}$, $s\in\mathbb{R}$ and
$\mathrm{Re}\\{K(s)\\}=\psi(s^{2}),s\geq 0,$ where $\psi$ is a Bernstein
function, see [10] for its definition and properties.

Due to the linearity of ID stochastic integrals, random variable

$s_{1}X(t)+s_{2}X(0)=\int_{\mathbb{R}^{d}}\left(s_{1}f(t-x)+s_{2}f(-x)\right)\Lambda(dx)$

is ID with characteristic function

$\varphi_{t}(s_{1},s_{2})=\exp\left(-\int_{\mathbb{R}^{d}}K(s_{1}f(t-x)+s_{2}f(-x))\,dx\right),\quad(s_{1},s_{2})\in\mathbb{R}^{2}.$

##  3 Main result

Let $|\cdot|$ be the Lebesgue measure in $\mathbb{R}^{d}$. Denote by
$L^{p}(\mathbb{R}^{d})$ the space of all functions
$f:\mathbb{R}^{d}\to\mathbb{R}$ such that $|f|^{p}$ is Lebesgue–integrable,
$p>0$. Let $a\wedge b$ denote the minimum of $a$ and $b$.

Introduce

$\sigma_{f}^{2}(s):=\int_{\mathbb{R}^{d}}\mathrm{Re}\\{K(sf(-x))\\}\,dx,\quad
s\in\mathbb{R},$
$\rho_{t}(s_{1},s_{2}):=\frac{1}{\sigma_{f}(s_{1})\sigma_{f}(s_{2})}\int_{\mathbb{R}^{d}}\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}\mathrm{Re}\\{K(s_{2}f(-x))\\}}\,dx$

for $s_{1},s_{2}\in\mathbb{R}$. Note that $0\leq\rho_{t}(s_{1},s_{2})\leq 1$
for all $s_{1},s_{2}\in\mathbb{R}$, $t\in\mathbb{R}^{d}.$ In addition, set

$\tilde{\rho}_{t}:=\sup_{(s_{1},s_{2})\in\mathbb{R}^{2}}\rho_{t}(s_{1},s_{2})$

and
$A_{\tilde{\rho}}=\\{t\in\mathbb{R}^{d}:\tilde{\rho}_{t}\leq\tilde{\rho}\\}$
for a fixed $\tilde{\rho}\in(0,1)$. Let
$A_{\tilde{\rho}}^{c}=\\{t\in\mathbb{R}^{d}:\tilde{\rho}_{t}>\tilde{\rho}\\}$
be the compliment of $A_{\tilde{\rho}}$ in $\mathbb{R}^{d}$.

######  Theorem 1.

Let $X$ be a measurable stationary ID moving average random field (1). Assume
that there exists $\tilde{\rho}<1$ such that $|A^{c}_{\tilde{\rho}}|<\infty$
and

$\int_{\mathbb{R}_{+}}\frac{\sigma_{f}(s)}{s}e^{-(1-\tilde{\rho})\sigma_{f}^{2}(s)}ds<\infty.$
(2)

Then, $X$ is SRD if $\int_{\mathbb{R}^{d}}\tilde{\rho}_{t}dt<\infty.$

The proof of this result including several auxiliary lemmata will be given in
the next section. Notice that $\sigma_{f}^{2}(s)=-\log|\varphi_{X(0)}(s)|.$
Then the integral within (2) rewrites as

$\int_{\mathbb{R}_{+}}\frac{|\log|\varphi_{X(0)}(s)||^{1/2}}{s}|\varphi_{X(0)}(s)|^{1-\tilde{\rho}}ds.$

Now let us explain how to check the above condition (2) in the special case of
an $\alpha$–stable symmetric integrator measure $\Lambda$, cf. the book [8]:

######  Example 1 (Stable laws).

Let $X=\\{X(t),t\in\mathbb{R}^{d}\\}$ be a measurable stationary symmetric
$\alpha-$stable moving average random field (1) with $\alpha\in(0,2)$ and a
continuous bounded kernel $f\in L^{\alpha}(\mathbb{R}^{d})$. Then

$\sigma_{f}^{2}(s)=|s|^{\alpha}\|f\|_{\alpha}^{\alpha},\quad s\in\mathbb{R},$

where
$\|f\|_{\alpha}:=\left(\int_{\mathbb{R}^{d}}|f(x)|^{\alpha}\,dx\right)^{1/\alpha}$,

$\displaystyle\rho_{t}(s_{1},s_{2})$
$\displaystyle=\frac{1}{|s_{1}|^{\alpha/2}|s_{2}|^{\alpha/2}\|f\|_{\alpha}^{\alpha}}\int_{\mathbb{R}^{d}}\sqrt{|s_{1}|^{\alpha}|s_{2}|^{\alpha}|f(t-x)f(-x)|^{\alpha}}dx$
$\displaystyle=\frac{1}{\|f\|_{\alpha}^{\alpha}}\int_{\mathbb{R}^{d}}|f(t-x)f(-x)|^{\alpha/2}\,dx=\tilde{\rho}_{t},\quad
t\in\mathbb{R}^{d}.$

Moreover, if $f$ is a continuous bounded function then $\tilde{\rho}_{t}$ is a
continuous function at the origin as well (by Lebesgue dominated convergence
theorem). Hence, there exists $\tilde{\rho}<1$ such that
$|A_{\tilde{\rho}}^{c}|<\infty$ (the set $A_{\tilde{\rho}}^{c}$ is a
neighbourhood of the origin). Furthermore,

$\displaystyle\int_{\mathbb{R}_{+}}\frac{\sigma_{f}(s)}{s}e^{-(1-\tilde{\rho})\sigma_{f}^{2}(s)}ds$
$\displaystyle=\|f\|_{\alpha}^{\alpha/2}\int_{\mathbb{R}_{+}}s^{\alpha/2-1}e^{-(1-\tilde{\rho})s^{\alpha}\Sigma_{f}^{2}}ds<\infty,\quad\alpha\in(0,2).$

Hence, the assumptions of Theorem 1 hold true. The SRD condition

$\int_{\mathbb{R}^{d}}\tilde{\rho}_{t}dt<\infty$

reads (by Fubini theorem) as $f\in L^{\alpha/2}(\mathbb{R}^{d})$ which
coincides with [3, Theorem 3.4].

##  4 Proof of Theorem 1

Notice that by [10, Theorem 4.12] the function

$1-e^{itx}-itx\mathbb{I}_{[-1,1]}(x),\;t\in\mathbb{R}$

is negative definite for any $x\in\mathbb{R}$. We formulate the following

######  Lemma 1 ([10]).

Let $\psi:\mathbb{R}^{d}\to\mathbb{C}$ be a continuous negative definite
function. Then for any $x,y\in\mathbb{R}^{d}$ it holds

  * •

$\mathrm{Re}\\{\psi(x)\\}\geq 0,$ $\psi(x)=\overline{\psi(-x)}$,

  * •

$\left|\psi(x)+\psi(y)-\psi(x\pm y)\right|\leq
2\sqrt{\mathrm{Re}\\{\psi(x)\\}}\sqrt{\mathrm{Re}\\{\psi(y)\\}},$

  * •

$\left|\mathrm{Re}\\{\psi(x)\\}+\mathrm{Re}\\{\psi(x)\\}-\mathrm{Re}\\{\psi(x\pm
y)\\}\right|\leq
2\sqrt{\mathrm{Re}\\{\psi(x)\\}}\sqrt{\mathrm{Re}\\{\psi(y)\\}}.$

The next inequality is a corollary of the above statements:

######  Lemma 2.

For any continuous negative definite function
$\psi:\mathbb{R}^{d}\to\mathbb{C}$ it holds

$\mathrm{Re}\\{\psi(x+y)\\}\wedge(\mathrm{Re}\\{\psi(x)+\psi(y)\\})\geq\left(\sqrt{\mathrm{Re}\\{\psi(x)\\}}-\sqrt{\mathrm{Re}\\{\psi(y)\\}}\right)^{2}.$

###### Proof.

By Lemma 1, we get

$\displaystyle\mathrm{Re}\\{\psi(x+y)\\}\wedge(\mathrm{Re}\\{\psi(x)+\psi(y)\\})$
$\displaystyle=\mathrm{Re}\\{\psi(x)+\psi(y)\\}-(\mathrm{Re}\\{\psi(x)\\}+\mathrm{Re}\\{\psi(y)\\}-\mathrm{Re}\\{\psi(x+y)\\})_{+}$
$\displaystyle\geq\mathrm{Re}\\{\psi(x)+\psi(y)\\}-2\sqrt{\mathrm{Re}\\{\psi(x)\\}}\sqrt{\mathrm{Re}\\{\psi(y)\\}}.$

∎

The following evident bound will be used in the sequel: for any
$z_{1},z_{2}\in\mathbb{C}$

$\left|e^{-z_{1}}-e^{-z_{2}}\right|\leq
e^{-(\mathrm{Re}\\{z_{1}\\}\wedge\mathrm{Re}\\{z_{2}\\})}|z_{1}-z_{2}|.$ (3)

######  Lemma 3.

For all $s_{1},s_{2}\in\mathbb{R}$ it holds

$\displaystyle\left|\mathbf{E}e^{i\left(s_{1}X(t)+s_{2}X(0)\right)}-\mathbf{E}e^{is_{1}X(t)}\mathbf{E}e^{is_{2}X(0)}\right|$
$\displaystyle\leq\exp\left[-\int_{\mathbb{R}^{d}}\left(\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}-\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\right)^{2}dx\right]$
$\displaystyle\times
2\int_{\mathbb{R}^{d}}\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}dx.$

###### Proof.

Employing inequality (3), we get

$\displaystyle\left|\mathbf{E}e^{i\left(s_{1}X(t)+s_{2}X(0)\right)}-\mathbf{E}e^{is_{1}X(t)}\mathbf{E}e^{is_{2}X(0)}\right|$
$\displaystyle=\left|\exp\left[-\int_{\mathbb{R}^{d}}K\left(s_{1}f(t-x)+s_{2}f(-x)\right)dx\right]\right.$
$\displaystyle\left.-\exp\left[-\int_{\mathbb{R}^{d}}\left(K(s_{1}f(t-x))+K(s_{2}f(-x))\right)dx\right]\right|$
$\displaystyle\leq\exp\left[-\int_{\mathbb{R}^{d}}\mathrm{Re}\\{K(s_{1}f(t-x)+s_{2}f(-x))\\}dx\right.$
$\displaystyle\left.\wedge\int_{\mathbb{R}^{d}}\left(\mathrm{Re}\\{K(s_{1}f(t-x))\\}+\mathrm{Re}\\{K(s_{2}f(-x))\\}\right)dx\right]$
$\displaystyle\times\int_{\mathbb{R}^{d}}\left|K(s_{1}f(t-x)+s_{2}f(-x))-K(s_{1}f(t-x))-K(s_{2}f(-x))\right|dx.$
(4)

Recall that $\mathrm{Re}\\{K\\}\geq 0.$ Apply Lemma 2 to

$\mathrm{Re}\\{K(s_{1}f(t-x)+s_{2}f(-x))\\}\wedge\mathrm{Re}\\{K(s_{1}f(t-x))+K(s_{2}f(-x))\\}$

to get

$\displaystyle\int_{\mathbb{R}^{d}}\left[\mathrm{Re}\\{K(s_{1}f(t-x)+s_{2}f(-x))\\}\wedge\mathrm{Re}\\{K(s_{1}f(t-x))+K(s_{2}f(-x))\\}\right]dx$
$\displaystyle\geq\int_{\mathbb{R}^{d}}\left(\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}-\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\right)^{2}dx,$

Then the right–hand side of (4) is bounded from above by

$\displaystyle\exp\left[-\int_{\mathbb{R}^{d}}\left(\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}-\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\right)^{2}dx\right]$
$\displaystyle\times\int_{\mathbb{R}^{d}}\left|K(s_{1}f(t-x)+s_{2}f(-x))-K(s_{1}f(t-x))-K(s_{2}f(-x))\right|dx$
$\displaystyle\leq\exp\left[-\int_{\mathbb{R}^{d}}\left(\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}-\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\right)^{2}dx\right]$
$\displaystyle\times
2\int_{\mathbb{R}^{d}}\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}dx,$

where the last inequality follows from Lemma 1.

∎

Finally, Theorem 1 follows immediately from the lemma below:

######  Lemma 4.

Let $\tilde{\rho}<1$ such that $|A_{\tilde{\rho}}|>0.$ Then for any $t\in
A_{\tilde{\rho}}$ we have

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)$
$\displaystyle\leq\frac{2}{\pi^{2}}\left(\int_{\mathbb{R}_{+}}\frac{\sigma_{f}(s)}{s}e^{-(1-\tilde{\rho})\sigma_{f}^{2}(s)}ds\right)^{2}\tilde{\rho_{t}}.$
(5)

###### Proof.

By [3, Theorem 2.1] we have

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\mu(du)\mu(dv)$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{\varphi_{(X(0),X(t))}(s_{1},s_{2})-\varphi_{X(0)}(s_{1})\varphi_{X(0)}(s_{2})}{-s_{1}s_{2}}\hat{\mu}(s_{1})\hat{\mu}(s_{2})ds_{1}ds_{2},$

where $\hat{\mu}(s)=\int_{\mathbb{R}}e^{isx}\mu(dx)$ is the characteristic
function of a probability measure $\mu$ satisfying the inequality
$|\hat{\mu}(s)|\leq 1$ for all real $s$. Applying Lemma 3 to the right–hand
side of

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)$
$\displaystyle\leq\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{\left|\varphi_{(X(0),X(t))}(s_{1},s_{2})-\varphi_{X(0)}(s_{1})\varphi_{X(0)}(s_{2})\right|}{|s_{1}s_{2}|}ds_{1}ds_{2},$

we get that

$4\pi^{2}\int_{\mathbb{R}}\int_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)$

is bounded from above by

$\displaystyle\int_{\mathbb{R}^{2}}\frac{\exp\left\\{-\int_{\mathbb{R}^{d}}\left(\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}-\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\right)^{2}dx\right\\}}{|s_{1}||s_{2}|}$
$\displaystyle\times
2\int_{\mathbb{R}^{d}}\sqrt{\mathrm{Re}\\{K(s_{1}f(t-x))\\}}\sqrt{\mathrm{Re}\\{K(s_{2}f(-x))\\}}\,dxds_{1}ds_{2}$
$\displaystyle=2\int_{\mathbb{R}^{2}}\exp\left\\{-\sigma_{f}^{2}(s_{1})-\sigma_{f}^{2}(s_{2})+2\sigma_{f}(s_{1})\sigma_{f}(s_{2})\rho_{t}(s_{1},s_{2})\right\\}\frac{\sigma_{f}(s_{1})\sigma_{f}(s_{2})\rho_{t}(s_{1},s_{2})}{|s_{1}||s_{2}|}ds_{1}ds_{2}$
$\displaystyle\leq
2\tilde{\rho_{t}}\int_{\mathbb{R}^{2}}\exp\left\\{-\sigma_{f}^{2}(s_{1})-\sigma_{f}^{2}(s_{2})+2\sigma_{f}(s_{1})\sigma_{f}(s_{2})\tilde{\rho}\right\\}\frac{\sigma_{f}(s_{1})\sigma_{f}(s_{2})}{|s_{1}||s_{2}|}ds_{1}ds_{2}$
$\displaystyle=2\tilde{\rho_{t}}\int_{\mathbb{R}^{2}}\exp\left\\{-(1-\tilde{\rho})\sigma_{f}^{2}(s_{1})-(1-\tilde{\rho})\sigma_{f}^{2}(s_{2})-(\sigma_{f}(s_{1})-\sigma_{f}(s_{2}))^{2}\tilde{\rho}\right\\}\frac{\sigma_{f}(s_{1})\sigma_{f}(s_{2})}{|s_{1}||s_{2}|}ds_{1}ds_{2}$
$\displaystyle\leq
2\tilde{\rho_{t}}\int_{\mathbb{R}^{2}}\exp\left\\{-(1-\tilde{\rho})\sigma_{f}^{2}(s_{1})-(1-\tilde{\rho})\sigma_{f}^{2}(s_{2})\right\\}\frac{\sigma_{f}(s_{1})\sigma_{f}(s_{2})}{|s_{1}||s_{2}|}ds_{1}ds_{2}$
$\displaystyle=8\left(\int_{\mathbb{R}_{+}}\frac{\sigma_{f}(s)}{s}e^{-(1-\tilde{\rho})\sigma_{f}^{2}(s)}ds\right)^{2}\tilde{\rho_{t}}.$

∎

The statement of Theorem 1 follows now easily by integrating the upper bound
in Lemma 4 with respect to $t\in A_{\tilde{\rho}}$. Here, one writes

$\displaystyle\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)\,dt$
$\displaystyle=\left(\int\limits_{A_{\tilde{\rho}}}+\int\limits_{A^{c}_{\tilde{\rho}}}\right)\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)\,dt,$

where the integral over $A_{\tilde{\rho}}$ is finite by Lemma 4, and the
integral over $A^{c}_{\tilde{\rho}}$ is finite due to the finiteness of the
volume of $A^{c}_{\tilde{\rho}}$, since the integrand

$\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\left|\mathbf{Cov}(\mathbbm{1}\\{X(t)>u\\},\mathbbm{1}\\{X(0)>v\\})\right|\mu(du)\mu(dv)\leq
1.$

## Acknowledgement

This research was supported by the Grant No. 39087941 of the German Research
Society.

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# The ReBB model at 8 TeV:
Odderon exchange is not a probability, but a certainty††thanks: Presented at
“Diffraction and Low-$x$ 2022”, Corigliano Calabro, Italy, Sept. 2022.

István Szanyi${}^{1,~{}2,~{}3,~{}{\dagger}}$ Tamás
Csörgő${}^{2,~{}3,~{}{\ddagger}}$ 1Eötvös University, H - 1117 Budapest,
Pázmány P. s. 1/A, Hungary;
2Wigner FK, H-1525 Budapest 114, POB 49, Hungary;
3MATE Institute of Technology, Károly Róbert Campus, H-3200 Gyöngyös, Mátrai
út 36, Hungary;
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The Real Extended Bialas-Bzdak (ReBB) model study is extended to the 8 TeV
$pp$ TOTEM elastic differential cross section data. The analysis shows that
the ReBB model describes the $pp$ and $p\bar{p}$ differential cross section
data in the limited $0.37\leq-t\leq 1.2$ GeV2 and $1.96\leq\sqrt{s}\leq 8$ TeV
kinematic region, in a statistically acceptable manner. In this kinematic
region a greater than 30 $\sigma$ model-dependent Odderon signal is observed
by comparing the $pp$ and ReBB extrapolated $p\bar{p}$ differential cross
sections. Thus, in practical terms, within the framework of the ReBB model,
Odderon exchange is not a probability, but a certainty.

##  1 Introduction

In a recent paper [1], published in July 2021, we showed that the Real
Extended $p=(q,d)$ version of the Bialas-Bzdak (ReBB) model developed in Ref.
[2] based on the original papers, Refs. [3, 4], and later improvements, Refs.
[5, 6], describes in a statistically acceptable manner the proton-proton
($pp$) and proton-antiproton ($p\bar{p}$) scattering data in the kinematic
range of $0.546\leq\sqrt{s}\leq 7$ TeV and $0.37\leq-t\leq 1.2$ GeV2. With
these results at hand, we reported an at least 7.08 $\sigma$, discovery level
Odderon effect111The ReBB model is based on R. J. Glauber’s multiple
diffraction theory, so it operates directly on the level of the elastic
scattering amplitude of $pp$ and $p\bar{p}$ collisions. We obtain the C-even
(Pomeron) and C-odd (Odderon) components of the elastic scattering amplitude
as the average and the difference of elastic proton-antiproton and proton-
proton amplitudes. For the details see Appendix C of Ref. [1]. by comparing
the $pp$ and $p\bar{p}$ differential cross sections at the same energies
utilizing model-dependent extrapolations of the differential cross-sections of
elastic $pp$ scattering to $\sqrt{s}$ $=$ 1.96 TeV and elastic $p\bar{p}$
scattering up to the lowest measured energy at LHC, 2.76 TeV. Extrapolating
the $p\bar{p}$ scattering up to 7 TeV, the statistical significance of Odderon
exchange increased to greater than 10 $\sigma$, however, in Ref. [1] this
significance was not quantified more precisely, due to numerical limitations
of CERN’s Root, MS Excel, Wolfram Mathematica and similar data analysis
software tools.

Based on our recently published paper [7], we present here the results of the
extension of the ReBB analysis to the new 8 TeV $pp$ differential cross
section data of TOTEM, Ref. [8]. In this Ref. [7] we also more precisely
quantified the high significance of the Odderon observation by introducing an
analytical approximation scheme (see the Appendix of Ref. [7]).

##  2 ReBB model and Odderon exchange at 8 TeV

As an extension to Ref. [1], in Fig. 1 we show the comparison of the $pp$
differential cross section calculated from the ReBB model — using the energy
calibration of the fit parameters done in Ref. [1] — with the final 8 TeV $pp$
differential cross section data measured by TOTEM and published recently in
Ref. [8]. One can see that the energy-calibrated model, in its validity range,
$0.37\leq-t\leq 1.2$ GeV2, describes the data in a statistically acceptable
manner, with a confidence level of 0.2 %.

The ReBB model thus describes the data at 8 TeV in a limited kinematic region
which is suitable to perform a search for Odderon exchange. As detailed and
utilized recently in Refs. [1, 9, 10] a possible difference between $pp$ and
$p\bar{p}$ measurable quantities at the TeV energy scale theoretically can be
attributed only to the effect of a $t$-channel C-odd Odderon exchange.

The comparison of the $p\bar{p}$ differential cross section calculated from
the ReBB model — using the energy calibration of the fit parameters done in
Ref. [1] — with the 8 TeV $pp$ differential cross section data measured by
TOTEM [8] is shown in Fig. 2, which indicates a difference between the $pp$
and $p\bar{p}$ differential cross sections with a probability of essentially
1, corresponding to a $CL=1-1.111\times 10^{-74}$, i.e., an Odderon
observation with a statistical significance $\geq$18.28 $\sigma$ (for the
details of the significance calculation see the Appendix of Ref. [7].).

The fits and the model-data comparisons are done by utilizing the $\chi^{2}$
definition developed by the PHENIX collaboration. This method is equivalent
with the diagonalization of the covariance matrix if the experimental errors
are separated into three different types: point-to-point fluctuating
uncorrelated statistical and systematic errors (type A), point-to-point
varying and 100% correlated systematic errors (type B), and point-independent,
overall correlated systematic uncertainties (type C). In our study, the
available experimental errors of the analysed data can be and are categorized
into these three types: horizontal and vertical $t$-dependent statistical
errors (type A), horizontal and vertical $t$-dependent systematic errors (type
B), and overall normalization uncertainties (type C).

The PHENIX method is validated by evaluating the $\chi^{2}$ from a full
covariance matrix fit of the $\sqrt{s}$ = 13 TeV TOTEM differential cross-
section data using the Lévy expansion method of Ref. [11]. The PHENIX method
and the fit with the full covariance matrix result in the same minimum within
one standard deviation of the fit parameters. Thus the PHENIX method is a
reasonable choice at energies, where the full covariance matrixes are not
published. The exact form of the $\chi^{2}$ definition 222The $\chi^{2}$
parameters $\epsilon_{B}$ and $\epsilon_{C}$ were considered as fit parameters
in Ref. [1], decreasing NDF. However $\epsilon_{B}$ and $\epsilon_{C}$, in
fact, have a known central value (0) and a known standard deviation (1), hence
they must be considered not only as fit parameters, but also new data points.
Thus in the end they are not effecting the NDF. This was done in Ref. [7], but
this correction does not effect the conclusions drawn in Ref. [1]. used in
this analysis with correlation parameters, $\epsilon_{B}$ and $\epsilon_{C}$
resulting from such a classification of measurement errors can be found in
Ref. [1].

Figure 1:  Comparison of the $pp$ differential cross sections from Ref. [7].
The ReBB model calculations for $pp$ are based on Ref. [1] and they agree, at
0.2 % CL, with the recently published TOTEM $pp$ data at $\sqrt{s}=$ 8 TeV
[8].

Figure 2: Comparison of the $p\bar{p}$ differential cross sections from Ref. [7]. The ReBB model calculations for $p\bar{p}$ are based on Ref. [1] and they disagree, at $1.1\times 10^{-72}$ % CL, with the recently published TOTEM $pp$ data at $\sqrt{s}=$ 8 TeV [8].  $\sqrt{s}$ (TeV) | $\chi^{2}$ | $NDF$ | CL | significance ($\sigma$)
---|---|---|---|---
1.96 | 24.283 | 14 | 0.0423 | 2.0
2.76 | 100.347 | 22 | 5.6093 $\times 10^{-12}$ | 6.8
7 | 2811.46 | 58 |  $\textless$ 7.2853$\times 10^{-312}$ |  $\textgreater$37.7
8 | 426.553 | 25 | 1.1111$\times 10^{-74}$ |  $\geq$18.2
Table 1: Summary on Odderon signal observation significances in the ReBB model
analysis from Ref. [7]. The significances higher than 8 $\sigma$ were
calculated by utilizing an analytical approximation schema, detailed in the
Appendix of the same paper [7].

##  3 ReBB model and Odderon at the TeV energy range

Table 1 summarises all the Odderon signal observation significances in our
ReBB model analysis. The dataset at 7 TeV carries the largest, dominant
Odderon signal, greater than 37.75 $\sigma$. The existence of a significant
Odderon signal is confirmed with the new TOTEM data at 8 TeV, which provides
an also clear-cut, greater than 18.28 $\sigma$ Odderon signal. The
significance of the Odderon signal in the $\sqrt{s}=2.76$ TeV TOTEM data is
$6.8$ $\sigma$. Within the framework of the ReBB model, no statistically
significant Odderon signal is observed from the comparison of the
$\sqrt{s}=1.96$ TeV D0 data with ReBB model extrapolated elastic $pp$
differential cross-sections.

Given that the datasets are independent measurements, we can evaluate their
combined significances step by step, by adding the individual $\chi^{2}$ and
the individual $NDF$ values. Another option for combining the significances is
Stouffer’s method ($i.e.$ by summing the significances and dividing the sum by
the square root of the number of summed significances) as used by TOTEM in
Ref. [10]. As it is detailed in Ref. [7] in Table 2, independently which
method is used, the combination of the results at the two lowest energies,
$i.e.$ 1.96 and 2.76 TeV, gives greater than 6 $\sigma$ significance for the
Odderon exchange, while the combination of the results at $\sqrt{s}=$ 1.96,
2.76, 7 and 8 TeV gives a greater than 30 $\sigma$ significance.

Fig. 3 shows the total cross-section (with systematic error band) obtained
from the optical theorem using the ReBB model amplitude of Odderon exchange,
as evaluated from the log-linear excitation functions of the model from Ref.
[1]. The result indicates that the total cross-section of the Odderon exchange
is sharply increasing in the few TeV energy range, but it is two orders of
magnitude smaller than the contribution of the Pomeron exchange that is
dominant at the same energy scale, as detailed in Ref. [1].

Figure 3:  The Odderon total cross section, determined from the ReBB model in
Ref. [1], indicates a threshold effect for Odderon exchange. The Odderon
contribution to the total cross-section starts to be statistically significant
around 1 TeV.

##  4 Summary

The Real Extended Bialas-Bzdak (ReBB) model describes all the available $pp$
and $p\bar{p}$ differential cross-section data in the kinematic range of
$0.546\leq\sqrt{s}\leq 7$ TeV and $0.37\leq-t\leq 1.2$ GeV2 in a statistically
acceptable manner. The statistical significance of Odderon exchange is greater
than 30 $\sigma$ when the results obtained from $\sqrt{s}=$1.96, 2.76, 7, and
8 TeV are combined. Thus. within the framework of the ReBB model, Odderon
exchange is not a probability, but a certainty at the TeV energy scale.

##  5 Acknowledgments

We thank A. Papa and his team for their kind hospitality and for organizing an
inspiring and useful meeting. Our research has been supported by the Hungarian
NKFIH grant K133046 and the ÚNKP-22-3 New National Excellence Program.

## References

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# Long-time behaviors of some stochastic differential equations driven by Lévy
noise

I. Orlovskyi  F. Proske  O. Tymoshenko

###### Abstract

Using key tools such as Itô formula for general semi-martingales, moments
estimates for Lévy-type stochastic integrals and properties of regular varying
functions we find conditions under which solutions of stochastic differential
equation with jumps are almost sure asymptotically equivalent nonrandom
function with $t\to\infty$.

###### keywords:

Stochastic differential equation , Lévy noise , Poisson random measure ,
Brownian motion; asymptotic behaviour.

††journal: Journal of Differential Equations

[label1]organization=Department of Mathematical Analysis and Probability
Theory, National University of Ukraine ”Igor Sikorsky Kyiv Polytechnic
Institute”, addressline=Beresteyskyi Ave, 37, city=Kyiv, postcode=03056,
country=Ukraine

[label2]organization=Department of Mathematics, University of Oslo,
addressline=Blindern, city=Oslo, postcode=0316, country=Norway

##  1 Introduction

The study of asymptotic properties of stochastic differential equations (SDEs)
has a special place in the modern theory of random processes. Gikhman and
Skorokhod [4], Keller et al. [6], and later Buldygin et al. [7] considered the
exact order of growth of solutions of homogeneous SDEs and found conditions,
under which these solutions are asymptotically equivalent, as $t\to\infty$, to
solutions of ordinary differential equations (ODEs). Closer investigations for
multidimentional SDE are done by Samoyǐlenko and Stanzhitskiy [9]. Almost-sure
(a.s.) asymptotic properties of solutions for some classes of non-homogeneous
SDEs were considered in [10] and [11].

The idea of determining asymptotic properties of solutions of equations
excited by some random process with the solution of an ODE arises naturally.
Stochastic models approximate the real processes much better than
deterministic ones, but are more complicated in the investigation of issues
related to asymptotic. By combining powerful methodological tools from two
worlds, i.e., the field of ODEs on the one hand and the field of stochastic on
the other, researchers solve essential mathematical problems.

Works [4], [6], [7], [9], [10] [11] only focused on the case of SDEs driven by
a Brownian motion. Unfortunately, Brownian motion cannot accurately describe
stochastic disturbances in certain real systems, such as fluctuations in
financial markets, because it is a continuous stochastic process. It is
recognized that SDEs with Lévy noise are more suitable for describing these
discontinuous systems, leading to increased interest in SDEs driven by noise
with discontinuous jumps. The case is particularly interesting when the noise
is obtained from the Lévy process using the Lévy-Itô decomposition into a
continuous component perturbed by a Wiener process and a jump-shaped part with
a Poisson random measure. Since the long-time asymptotic behaviour of
solutions to SDEs is very important, in this paper we study some asymptotic
properties of equations where driving noise has jumps. In this paper we focus
on SDEs driven by Lévy noise of the form

$dX(t)=g(X(t-))\phi(t-)dt+\sigma(X(t-))\theta(t-)dW(t)+\int_{\mathbb{R}}c(X(t-))\gamma(u)\tilde{N}(dt,du),$
(1)

and we will search for conditions under which the solution of this equation is
asymptotically defined by a non-random function.

D. Applebaum, M. Siakally [2] established asymptotic stability of SDEs class
(1) with $\phi=\theta=1$ and $c(x)\gamma(t)=H(x,t).$ The similar problem for
partial case of SDE (1) with numerical approximation was considered in [15].
In [14], was investigated the conditions under which the solution $X$ of the
equation from [2] has the same asymptotic as corresponds ordinary differential
equation. The author proved that in the case when a drift coefficient is
equivalent to some non-random positive degree function then
$X(t)\sim((1-\alpha)At)^{\frac{1}{1-\alpha}}$ a.s. as $t\to\infty$ for
$\alpha\in[0;1)$, $A>0.$

In the multidimensional case the problem on identification of a limit of an
ODE with discontinuous drift that perturbed by a zero-noise is considered in
[13]. F. Proske and A. Pylypenko [12] study zero-noise limits of
$\alpha-$stable noise perturbed ODE’s which are driven by an irregular vector
field $A$ with asymptotics
$A(x)\sim\overline{a}\left(\frac{x}{\left|x\right|}\right)\left|x\right|^{\beta-1}$
at zero, where $\overline{a}>0$ is a continuous function and $\beta\in(0,1)$.
Thus, the issue considered in this paper is a classical subject of stochastic
analysis. However the asymptotic long-time properties solution equation (1)
wasn’t investigated yet.

The rest of this paper is organized as follows. In Section 2, we introduce the
model, notations and several necessary hypotheses. In Section 3, we formulate
Itô’s formula and Dynkin’s formula for special case SDE with jumps. In Section
4, we present sufficient condition for a.s. asymptotic equivalence solution of
SDEs driven by Lévy noise and solution ODE. Finally, in Section 5, we conclude
this paper with some results with regard to unboundedness of solutions SDE
driven by Lévy noise.

##  2 Preliminaries

Let $(\Omega,\mathfrak{F},(\mathfrak{F}_{t},t\geq 0),\mathsf{P})$ be a
filtered probability space that satisfies the usual hypotheses of completeness
and right continuity. Assume that we are given $\mathfrak{F}_{t}$-adapted
Wiener process $W=(W(t),t\geq 0)$, and $\mathfrak{F}_{t}$-adapted Poisson
random measure $N$ defined on
$\mathbb{R}^{+}\times(\mathbb{R}\setminus\\{0\\})$. Note that the process
$(W(t),t\geq 0)$ and $(N(t),t\geq 0)$ are independent.

We set

$\widetilde{N}(dt,du)=N(dt,du)-dt\nu(du),$

where $\widetilde{N}$ is the compensated Poisson random measure, $\nu$ is an
intensity measure such that
$\displaystyle\int_{\mathbb{R}\setminus\\{0\\}}u^{2}\nu(du)<\infty.$

Denote by $\mathbf{C}$ ($\mathbf{C}^{+}$ ) the class of all continuous (and
positive) functions and by $\mathbf{C}^{1}$ ($\mathbf{C}^{1,+}$ ) the class of
all continuously differentiable (and positive) functions.

Consider SDEs driven by Lévy noise of the form

$dX(t)=g(X(t))\phi(t)dt+\sigma(X(t))\theta(t)dW(t)+\int_{|u|<\mathcal{K}}c(X(t-))\gamma(u)\tilde{N}(dt,du),$
(2)

on $t\geq 0$ with initial value $X(0)=X_{0}$ and we also assume that
$\underset{t\to\infty}{\lim}X(t)=\infty$ a.s. Sufficient conditions of
unboundedness of $X(t)$ is considered in Section 5.

Introduce next assumtions

  * I.

Functions $g(\cdot)\in\mathbf{C}^{+}$, $\sigma(\cdot)\in\mathbf{C}^{+}$,
$\phi(\cdot)\in\mathbf{C}^{+}$, and $\theta(\cdot)\in\mathbf{C}$,
$c(\cdot)\in\mathbf{C}$, $\gamma(\cdot)\in\mathbf{C}$ are such that:

    * (i)

(Lipschitz condition) for some constants $L_{i}>0$, $i=1,2,3,$ and all
$t\in[0,T]$ and $x,y\in\mathbb{R}$

$|g(x)-g(y)|\leq L_{1}|x-y|,\,\,\,|\sigma(x)-\sigma(y)|\leq L_{2}|x-y|,$
$|c(x)-c(y)|\leq L_{3}|x-y|;$

    * (ii)

$\displaystyle\int_{\mathbb{R}}\gamma^{2}(u)\nu(du)<\infty;$

    * (iii)

$c(\cdot)$ is bounded.

######  Remark 2.1.

Note that, if functions $g$, $\sigma$ and $c$ are continuous and satisfy
Lipschitz condition, then they will also satisfy growth condition: for some
constant $K_{i}>0,\ i=1,2,3$, and all $t\in[0,T]$ and $x\in\mathbb{R}$

$|g(x)|\leq K_{1}(1+|x|),\ |\sigma(x)|\leq K_{2}(1+|x|),\ |c(x)|\leq
K_{3}(1+|x|).$ (3)

It means that if condition I is fulfilled, then the solution of the equation
(2) exists and unique (see, for example [1]).

######  Remark 2.2.

From I.(ii) and I.(iii) it follows that

$\displaystyle\mathsf{E}\left\\{\int_{0}^{t}\int_{|u|<\mathcal{K}}|c(X(r-))\gamma(u)|\nu(du)dr\right\\}<\infty.$

Then

$\displaystyle\int_{0}^{t}\int_{|u|<\mathcal{K}}c(X(t-))\gamma(u)\tilde{N}(dt,du)=$
$\displaystyle=\int_{0}^{t}\int_{|u|<\mathcal{K}}c(X(t-))\gamma(u)N(dt,du)-\int_{0}^{t}\int_{|u|<\mathcal{K}}c(X(r-))\gamma(u)\nu(du)dr$

Introduce also the following assumptions:

  * II.

Function $G(x)=\int\limits_{b}^{x}\frac{ds}{g(s)},\ x\geq b$ is such that

$\underset{x\to\infty}{\lim}G(x)=\infty;$

  * III.

Function $\Phi(t)=\int\limits_{0}^{t}\phi(u)du>0,\ t>0,$ and

$\underset{t\to\infty}{\lim}\frac{t}{\Phi(t)}=0.$

  * IV.

Function $\Theta(t)=\int\limits_{0}^{t}\theta^{2}(s)ds,\ t\geq 0$ is such that

$\sum\limits_{k=0}^{\infty}\frac{\Theta(2^{k+1})}{\Phi^{2}(2^{k})}<\infty.$

######  Remark 2.3.

For the assumption IV, we can provide some simple sufficient conditions in
terms of power functions. It is easy to see, that if

$\underset{t\to\infty}{\lim\sup\,}\frac{\Theta(t)}{t^{\beta}}>0\,\,\,\textit{and}\,\,\,\underset{t\to\infty}{\lim\inf\,}\frac{\Phi(t)}{t^{\alpha}}>0$

are satisfied with $\beta<2\alpha$, then IV holds.

######  Example 2.4.

It is possible to represent the sufficient condition in terms of the regularly
varying functions.

Fix $d\geq 0$ and let $f:[d,\infty)\to(0,\infty)$ be a measurable function. We
say that $f$ is regularly varying of degree $\alpha\in\mathbb{R}$ if

$\underset{x\to\infty}{\lim}\frac{f(cx)}{f(x)}=c^{\alpha},$

for all $c>0$. We will denote the class of regularly varying functions of
degree $\alpha$ on $\mathbb{R}^{+}$ by $\mathscr{RV}_{\alpha}$. Elements of
the class $\mathscr{RV}_{0}$ are said to be _slowly varying_. Examples of
functions in $\mathscr{RV}_{0}$ are $log(1+x)$ and $log(log(e+x))$. Clearly
$f(\cdot)\in\mathscr{RV}_{\alpha}$ if and only if there exists
$l(\cdot)\in\mathscr{RV}_{0}$ such that $f(x)=l(x)x^{\alpha}$, for all
$x\in\mathbb{R}^{+}$. The following representation theorem for slowly varying
functions can be found in Bingham et al. ([3])

######  Remark 2.5.

Assume that $\phi(\cdot)\in\mathscr{RV}_{\alpha}$ and
$\theta(\cdot)\in\mathscr{RV}_{\beta}$. According to Karamata’s Theorem

$\Phi(\cdot)\in\mathscr{RV}_{\alpha+1},\qquad\Theta(\cdot)\in\mathscr{RV}_{2\beta+1}.$

Therefore condition IV holds if $\alpha\geq\beta$.

##  3 The Itô formula

######  Theorem 3.1.

Suppose

$\displaystyle X(t)=X_{0}+\int_{0}^{t}g(X(r))\phi(r)dr+$
$\displaystyle\int_{0}^{t}\sigma(X(r))\theta(r)dW(r)+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}c(X(r-))\gamma(u)\tilde{N}(dr,du).$
(4)

Then we have

$\displaystyle G(X(t))-G(X_{0})=$
$\displaystyle=\int_{0}^{t}\left(\phi(r)-\frac{1}{2}\frac{g^{\prime}(X(r))\sigma^{2}(X(r))}{g^{2}(X(r))}\cdot\theta^{2}(r)\right)dr+\int_{0}^{t}\frac{\sigma(X(r))}{g(X(r))}\cdot\theta(r)dW(r)+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\left\\{G(X(r-)+c(X(r-))\gamma(u))-G(X(r-))-\frac{c(X(r-))}{g(X(r-))}\cdot\gamma(u)\right\\}\nu(du)dr+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(G\left(X(r-)+c(X(r-))\gamma(u)\right)-G(X(r-))\right)\tilde{N}(dr,du).$

######  Theorem 3.2 (Dynkin formula).

$\displaystyle\mathsf{E}[G(X(t))]=$
$\displaystyle=G(X_{0})+\Phi(t)-\frac{1}{2}\mathsf{E}\left[\int_{0}^{t}\frac{g^{\prime}(X(r))\sigma^{2}(X(r))}{g(X(r))}\cdot\theta^{2}(r)dr\right]+$
$\displaystyle+\mathsf{E}\left[\int_{0}^{t}\int_{|u|<\mathcal{K}}\left\\{G(X(r-)+c(X(r-))\gamma(u))-G(X(r-))-\frac{c(X(r-))\gamma(u)}{g(X(r-))}\right\\}\nu(du)dr\right].$

###### Proof.

The proof follows from the Itô formula, Proposition 5.1 in D. Applebaum [1]. ∎

##  4 Almost-sure asymptotic equivalence solution of SDEs driven by Lévy noise
and solution ODE

Additionally assume that

  * V.

$\frac{\sigma}{g}$ is bounded function;

  * VI.

$\underset{x\in\mathbb{R}}{\inf}\,g(x)=k_{g}>0$ and
$\underset{x\to\infty}{\lim}g^{\prime}(x)=0$;

  * VII.

$\underset{t\to\infty}{\lim\sup}\,\frac{\theta^{2}(t)}{\varphi(t)}<\infty$.

Now we can formulate the main results of this section.

######  Theorem 4.1.

If conditions I – VII hold, then

$\underset{t\to\infty}{\lim}\frac{G(X(t))}{\Phi(t)}=1\ \ \ \textnormal{a.s.}$

######  Theorem 4.2.

Let $g(\cdot)\in\mathbf{C}^{1,+}$, $\sigma(\cdot)\in\mathbf{C}^{1,+}$,
$\phi(\cdot)\in\mathbf{C}^{+}$, $\theta(\cdot)\in\mathbf{C}^{+}$,
$c(\cdot)\in\mathbf{C}^{+}$, $\gamma(\cdot)\in\mathbf{C}^{+}$ be such that
equation (2) has a cádlág solution $X(\cdot)$ and
$\underset{t\to\infty}{\lim}X(t)=\infty$ a.s. If conditions I – VII hold and

  * VIII.

$\displaystyle\underset{t\to\infty}{\lim\inf}\int_{t}^{ct}\frac{du}{g(u)G(u)}>0,\,\,\textit{
for all}\,\,c>0$,

then

$\underset{t\to\infty}{\lim}\frac{X(t)}{\mu(t)}=1\,\,\text{a.s.,}$

where $\mu(\cdot)$ is a solution

$d\mu(t)=g(\mu(t))\phi(t)dt.$ (5)

###  4.1 Auxiliary assertions

To prove the theorems let us consider some auxiliary statements.

By Itô formula from Theorem 3.1, we have

$\displaystyle G$
$\displaystyle(X(t))=G(X_{0})+\int_{0}^{t}\left(\phi(r)-\frac{1}{2}\frac{g^{\prime}(X(r))\sigma^{2}(X(r))}{g^{2}(X(r))}\cdot\theta^{2}(r)\right)dr+$
$\displaystyle+\int_{0}^{t}\frac{\sigma(X(r))}{g(X(r))}\cdot\theta(r)dW(r)+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\left\\{G(X(r-)+c(X(r-))\gamma(u))-G(X(r-))-\frac{c(X(r-))\gamma(u)}{g(X(r-))}\right\\}\nu(du)dr+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(G\left(X(r-)+c(X(r-))\gamma(u)\right)-G(X(r-))\right)\tilde{N}(dr,du)=$
$=G(X_{0})+\mathfrak{I}_{1}(t)+\mathfrak{I}_{2}(t)+\mathfrak{I}_{3}(t)+\mathfrak{I}_{4}(t).\hskip
170.71652pt$ (6)

######  Lemma 4.3.

Assume that functions $g\in\mathbf{C}^{+}$, $\phi(\cdot)\in\mathbf{C}^{+}$,
$c(\cdot)\in\mathbf{C}$, $\gamma(\cdot)\in\mathbf{C}$ satisfy I, III, VI. Then

$\underset{t\to\infty}{\lim\,}\frac{\mathfrak{I}_{3}(X(t))}{\Phi(t)}=0\,\,\,\textit{a.s.}$

###### Proof.

By Taylor’s formula

$G(x+c(x)\gamma(u))=G(x)+G^{\prime}(x)c(x)\gamma(u)+R(x,x+c(x)\gamma(u)),$

where

$R(x,x+c(x)\gamma(u))=\left(\int_{0}^{1}(1-\zeta)G^{\prime\prime}(x+\zeta\cdot
c(x)\gamma(u))d\zeta\right)c^{2}(x)\gamma^{2}(u).$

Then

$\displaystyle G(x+c(x)\gamma(u))-G(x)-$ $\displaystyle
G^{\prime}(x)c(x)\gamma(u)=$
$\displaystyle=c^{2}(x)\gamma^{2}(u)\int_{0}^{1}(1-\zeta)G^{\prime\prime}(x+\zeta\cdot
c(x)\gamma(u))d\zeta,$

and we can estimate the integral $\mathfrak{I}_{3}$:

$\displaystyle|\mathfrak{I}_{3}(t)|\leq$
$\displaystyle\leq\frac{1}{2}\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(\underset{0\leq\zeta\leq
1}{\sup\,}\Big{|}G^{\prime\prime}(X(r-)+\zeta\cdot
c(X(r-))\gamma(u))\Big{|}\right)c^{2}(X(r-))\gamma^{2}(u)\nu(du)dr\leq$
$\displaystyle\leq\frac{\underset{x\in\mathbb{R}}{\sup}\,|g^{\prime}(x)|}{2k_{g}^{2}}\cdot\int_{0}^{t}\int_{|u|<\mathcal{K}}c^{2}(X(r-))\gamma^{2}(u)\nu(du)dr\leq
K_{\mathfrak{I}_{3}}t,\ \text{a.s .}$ (7)

where
$K_{\mathfrak{I}_{3}}=\dfrac{1}{2k_{g}^{2}}\cdot\underset{x\in\mathbb{R}}{\sup}\,|g^{\prime}(x)|\cdot\underset{x\in\mathbb{R}}{\sup}\,c^{2}(x)\cdot\int_{\mathbb{R}}\gamma^{2}(u)\nu(du).$

Since III holds, then

$\underset{t\to\infty}{\lim\,}\frac{\mathfrak{I}_{3}(X(t))}{\Phi(t)}=0\,\,\,\textit{a.s.}$

∎

######  Lemma 4.4.

Assume that $g(\cdot)\in\mathbf{C}^{+}$, $\phi(\cdot)\in\mathbf{C}^{+}$, and
$c(\cdot)\in\mathbf{C}$, $\gamma(\cdot)\in\mathbf{C}$ satisfy I.(iii) and III.
Then

$\underset{t\to\infty}{\lim\,}\frac{\mathfrak{I}_{4}(X(t))}{\Phi(t)}=0\,\,\,\textit{a.s.}$

###### Proof.

The stochastic integral $\mathfrak{I}_{4}(\cdot)$ is a square integrable
martingale (see [5]).

Using the representation

$G(x+c(x)\gamma(u))-G(x)=c(x)\gamma(u)\int_{0}^{1}(1-\zeta)G^{\prime}(x+\zeta\cdot
c(x)\gamma(u))d\zeta$

and by Doob’s martingale inequality

$\mathsf{P}\left(\underset{k\geq
n}{\sup}\frac{|\mathfrak{I}_{4}(2^{k})|}{\Phi(2^{k})}>\epsilon\right)\leq\mathsf{P}\left(\underset{t\geq
2^{n}}{\sup}\frac{|\mathfrak{I}_{4}(t)|}{\Phi(t)}>\epsilon\right)\leq$
$\leq\frac{4}{\epsilon^{2}}\sum_{k=n}^{\infty}\left(\frac{1}{\Phi^{2}(2^{k})}\int_{0}^{2^{k+1}}\int_{|u|<\mathcal{K}}\mathsf{E}\left(\frac{c^{2}(X(r-))\gamma^{2}(u)}{\underset{0\leq\zeta\leq
1}{\inf\,}(g^{2}(X(r-)+\zeta\cdot
c(X(r-))\gamma(u)))}\right)\nu(du)dr\right)\leq$
$\leq\frac{4}{\epsilon^{2}k_{g}^{2}}\cdot\underset{x\in\mathbb{R}}{\sup}\,c(x)\cdot\int_{|u|<\mathcal{K}}\gamma^{2}(u)\nu(du)\cdot\sum_{k=n}^{\infty}\frac{2^{k+1}}{\Phi^{2}(2^{k})}<\infty.$

Since $n\to\infty$ the left-side last inequality tends to $0$.

Then from
$\mathsf{P}\left\\{\underset{t\to\infty}{\lim\sup}\frac{|\mathfrak{I}_{4}(t)|}{\Phi(t)}>0\right\\}=0$
it follows that
$\mathsf{P}\left\\{\underset{t\to\infty}{\lim\sup}\frac{|\mathfrak{I}_{4}(t)|}{\Phi(t)}=0\right\\}=1$,
which in fact means

$\mathsf{P}\left\\{\underset{t\to\infty}{\lim}\frac{\mathfrak{I}_{4}(t)}{\Phi(t)}=0\right\\}=1.$

∎

###  4.2 The Proofs of Theorems 4.1 and 4.2

Proof of Theorem 4.1

###### Proof.

From (6) we get

$\frac{G(X(t))}{\Phi(t)}=\frac{G(X_{0})}{\Phi(t)}+\frac{\mathfrak{I}_{1}(t)}{\Phi(t)}+\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}+\frac{\mathfrak{I}_{3}(t)}{\Phi(t)}+\frac{\mathfrak{I}_{4}(t)}{\Phi(t)}.$

Since

$\underset{t\to\infty}{\lim}\frac{1}{\Phi(t)}\left|\int_{0}^{t}\frac{g^{\prime}(X(r))\sigma^{2}(X(r))}{g^{2}(X(r))}\cdot\theta^{2}(r)dr\right|\leq\underset{t\to\infty}{\lim}\frac{L}{\Phi(t)}\int_{0}^{t}\Big{|}g^{\prime}(X(r))\Big{|}\cdot\theta^{2}(r)dr=0,$

where $L=\underset{x}{\sup}\,\frac{\sigma^{2}(x)}{g^{2}(x)}<\infty$. Then

$\underset{t\to\infty}{\lim}\frac{\mathfrak{I}_{1}(t)}{\Phi(t)}=1\,\,\textnormal{a.s.}$
(8)

Now, continue to prove the stated convergence result and to estimate
$\mathfrak{I}_{2}(t)$ we will use the Doob’s inequality for martingales for
fixed $\varepsilon>0$ and the convergent series IV

$\displaystyle\mathsf{P}\left\\{\sup_{k\geq
n}\frac{\mathfrak{I}_{2}(2^{k})}{\Phi(2^{k})}>\varepsilon\right\\}$
$\displaystyle\leq\mathsf{P}\left\\{\sup_{t\geq
2^{n}}\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}>\varepsilon\right\\}=$
$\displaystyle=\mathsf{P}\left\\{\sup_{t\geq
2^{n}}\frac{1}{\Phi(t)}\left|\int\limits_{0}^{t}\frac{\sigma(X(r))}{g(X(r))}\cdot\theta(r)dW(r)\right|>\varepsilon\right\\}\leq$
$\displaystyle\leq\sum_{k=n}^{\infty}\mathsf{P}\left\\{\sup_{t\in[2^{k};2^{k+1}]}\frac{1}{\Phi(t)}\left|\int\limits_{0}^{t}\frac{\sigma(X(r))}{g(X(r))}\cdot\theta(r)dW(r)\right|>\varepsilon\right\\}\leq$
$\displaystyle\leq\frac{4L}{\varepsilon^{2}}\sum_{k=n}^{\infty}\frac{\Theta(2^{k+1})}{\Phi^{2}(2^{n})}\longrightarrow\
0,\ \text{as}\ n\to\infty.$

Note also that from the last relation, given that the sequence
$\underset{t\geq 2^{n}}{\sup}\,\frac{\mathfrak{I}_{2}(t)}{\Phi(t)},\
n\in\mathbb{N}$, is monotonically decreasing, we also get $\sup_{t\geq
2^{n}}\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}\ \longrightarrow\ 0$, as
$n\to\infty$ a.s.

Let $t\geq 2$ and $n\in\mathbb{N}$ such that $2^{n}\leq t<2^{n+1}$. Then

$\displaystyle\left|\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}\right|\leq\left|\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}-\frac{\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}\right|+\frac{\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}\leq\sup_{t\in[2^{n};2^{n+1}]}\left|\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}-\frac{\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}\right|+\frac{\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}$
$\displaystyle\leq\sup_{t\in[2^{n};2^{n+1}]}\frac{\Phi(t)-\Phi(2^{n})}{\Phi(t)\Phi(2^{n})}\cdot\left|\mathfrak{I}_{2}(2^{n})\right|+$
$\displaystyle+\sup_{t\in[2^{n};2^{n+1}]}\frac{1}{\Phi(t)}\left|\int\limits_{2^{n}}^{t}\frac{\sigma(X(r))}{g(X(r))}\cdot\theta(r)dW(r)\right|+\frac{\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}\leq$
$\displaystyle\leq\left(\frac{\Phi(2^{n+1})}{\Phi(2^{n})}-1\right)\cdot\frac{\left|\mathfrak{I}_{2}(2^{n})\right|}{\Phi(t)}+\sup_{t\geq
2^{n}}\frac{\left|\mathfrak{I}_{2}(t)\right|}{\Phi(t)}+\frac{2\mathfrak{I}_{2}(2^{n})}{\Phi(2^{n})}\longrightarrow\
0,\ \text{as}\ n\to\infty\,\,\text{a.s}.$

So, we have

$\underset{t\to\infty}{\lim}\frac{\mathfrak{I}_{2}(t)}{\Phi(t)}=0\,\,\textnormal{a.s.}$
(9)

The result of the Theorem 4.1 follows from (8), (9) and Lemmas 4.3, 4.4. ∎

Proof of Theorem 4.2

###### Proof.

From the assumptions II and VIII and by the Theorem 2.2.8 [8] the inverse
function $G^{-1}$ for $G$ preserves the asymptotic equivalence. Furthermore

$\Phi(t)=G(\mu(t)),$

where $\mu$ is a solution $d\mu(t)=g(\mu(t))\phi(t)dt$ and

$\underset{t\to\infty}{\lim}\frac{G(X(t))}{\Phi(t)}=\underset{t\to\infty}{\lim}\frac{G^{-1}(G(X(t)))}{G^{-1}(\Phi(t))}=\underset{t\to\infty}{\lim}\frac{X(t)}{\mu(t)}=1\,\,\text{a.s.}$

∎

######  Remark 4.5.

In the case when the function $g(\cdot)\in\mathscr{RV}_{\alpha}$, with
$\alpha<1$, the condition VIII holds and theorem 4.1 and 4.2 establishe the
conditions for the asymptotic equivalence of the solutions of the SDE with
jumps and the corresponding ordinary differential equation.

##  5 Unboudedness of the solution of
non-homogeneous SDEs driven by Lévy noise

In the first part of the paper, we consider the asymptotic equivalence of the
solution to a deterministic equation and a stochastic differential equation,
in which a Wiener process and a Poisson noise are mixed. The main result is
obtained under the assumption II, which yields immediately that the system (5)
is unbounded as $\underset{t\to\infty}{\lim}\mu(t)=\infty.$ One of the basic
assumptions of this paper is that the solution of the SDE (2) a.s. increases
indefinitely and tends to infinity:
$\mathsf{P}\\{\underset{t\to\infty}{\lim}\,X(t)=\infty\\}>0.$ The conditions
of unboundedness of solutions are presented, for example, for autonomous SDRs
in [4], and for non-autonomous ones were studied in [16].

Denote $\displaystyle B(x)=\int_{0}^{x}\frac{dr}{\sigma(r)}.$ We further
assume that $\underset{x\to\infty}{\lim}\,B(x)=\infty$.

In order to be able to prove main result in this section we need to formulate
the law of large number for martingale.

######  Lemma 5.1.

Let $M(t)$ be a squared integrable martingale. If the series

$\sum_{k=1}^{\infty}\frac{1}{2^{k+1}}\mathsf{E}M^{2}(2^{k+1})<\infty,$

then

$\underset{t\to\infty}{\lim}\frac{M(t)}{\sqrt{2t\ln\ln
t}}=0\,\,\textnormal{a.s.}$

###### Proof.

Consider, for any $k\geq 0$ and $\epsilon>0$, the following two events:
$B_{k}\subset C_{k},$ where $B_{k}=\left\\{\underset{2^{k}\leq t\leq
2^{k+1}}{\sup}\frac{|M(t)|}{\sqrt{2t\ln\ln t}}\geq\epsilon\right\\}$,
$C_{k}=\left\\{\underset{2^{k}\leq t\leq
2^{k+1}}{\sup}\frac{|M(t)|}{\sqrt{2^{k}\ln\ln 2^{k}}}\geq\epsilon\right\\}$.
So,

$\displaystyle\mathsf{P}\left(B_{k}\right)\leq\mathsf{P}\left(C_{k}\right)\leq\mathsf{P}\left(\underset{t\leq
2^{k+1}}{\sup}|M(t)|\geq\epsilon\sqrt{2^{k+1}\ln\ln 2^{k}}\right)$
$\displaystyle\leq$ (10)
$\displaystyle\leq\frac{4\mathsf{E}M^{2}(2^{k+1})}{\epsilon^{2}2^{k+1}\ln\ln
2^{k}}$
$\displaystyle\leq\frac{4\mathsf{E}M^{2}(2^{k+1})}{\epsilon^{2}2^{k+1}}.$

For any $m\geq 1$ and $\epsilon>0$

$\displaystyle\mathsf{P}\left(\underset{t\geq
2^{m}}{\sup}\frac{|M(t)|}{\sqrt{2t\ln\ln t}}\geq\epsilon\right)$
$\displaystyle=\mathsf{P}\left(\bigcup_{m=k}^{\infty}\,\left\\{\underset{2^{k}\leq
t\leq 2^{k+1}}{\sup}\frac{|M(t)|}{\sqrt{2t\ln\ln
t}}\geq\epsilon\right\\}\right)=$
$\displaystyle=\mathsf{P}\left(\bigcup_{m=k}^{\infty}B_{k}\right)\leq\sum_{k=m}^{\infty}\mathsf{P}\left(B_{k}\right).$

Using (10) we obtain

$\mathsf{P}\left(\underset{t\geq 2^{m}}{\sup}\frac{|M(t)|}{\sqrt{2t\ln\ln
t}}\geq\epsilon\right)\leq\frac{1}{\epsilon^{2}}\sum_{k=m}^{\infty}\frac{4}{2^{k+1}}\mathsf{E}M^{2}(2^{k+1}),$

and therefore
$\mathsf{P}\left(\underset{t\to\infty}{\lim\sup}\frac{|M(t)|}{\sqrt{2t\ln\ln
t}}>0\right)=0.$ Finally,

$\underset{t\to\infty}{\lim}\frac{|M(t)|}{\sqrt{2t\ln\ln
t}}=0\,\,\textnormal{a.s.}$

and Lemma 5.1 is proved. ∎

Consider

$\tilde{a}(t,x)=-\frac{\theta^{\prime}(t)}{\theta^{2}(t)}B(x)+\frac{g(x)\phi(t)}{\sigma(x)\theta(t)}-\frac{1}{2}\sigma^{\prime}(x)\theta(t).$

and denote

$A(t)=\int\limits_{0}^{t}\underset{x\in\mathbb{R}}{\inf}\,\tilde{a}(r,x)dr.$
(11)

In the following theorem we will give conditions that should be examined in
order to see if the perturbed equation is unstable.

######  Theorem 5.2.

Let $g(\cdot)\in\mathbf{C}^{1,+}$, $\sigma(\cdot)\in\mathbf{C}^{1,+}$,
$\phi(\cdot)\in\mathbf{C}^{+}$, $\theta(\cdot)\in\mathbf{C}^{+}$,
$c(\cdot)\in\mathbf{C}^{+}$, $\gamma(\cdot)\in\mathbf{C}^{+}$ be such that
equation (2) has a cádlág solution $X(\cdot)$. Assume that

  * (i)

$\underset{t\to\infty}{\lim\inf}\,\frac{A(t)}{\sqrt{2t\ln\ln t}}>1$;

  * (ii)

$\displaystyle\int_{|u|<\mathcal{K}}\frac{1}{\theta^{2}(u)}(B(x+c(x)\gamma(u))-B(x))^{2}\nu(du)<K|x|^{q}$
for $q<0$;

  * (iii)

$\displaystyle\int_{0}^{t}\int_{|u|<\mathcal{K}}\frac{c^{2}(X(r-))\gamma^{2}(u)}{\theta(u)}\nu(du)dr<\infty$;

  * (iv)

$\frac{\sigma^{\prime}(\cdot)}{\sigma^{2}(\cdot)}$ is bounded.

Then

$\underset{t\to\infty}{\lim}\,\frac{B(X(t))}{\theta(t)}=\infty\,\,\,\textit{a.s.}$

###### Proof.

Consider $f(t,x)=\frac{1}{\theta(t)}\cdot B(x)$. Since $B$ is a strictly
increasing function, $B^{-1}(\theta(t)x)$ is the inverse for $f(t,\cdot)$ with
respect to the argument $x$. Indeed,

$f(t,f^{-1}(t,x))=f(t,B^{-1}(\theta(t)x))=\frac{1}{\theta(t)}B(B^{-1}(\theta(t)x))=\frac{1}{\theta(t)}\cdot
x\theta(t)=x.$

On the other hand

$f^{-1}(t,f(t,x))=B^{-1}\left(\frac{1}{\theta(t)}\cdot\theta(t)\cdot
B(x)\right)=B^{-1}(B(x))=x.$

By Itô formula with

$f^{\prime}_{t}(t,x)=-\frac{\theta^{\prime}(t)}{\theta^{2}(t)}\cdot B(x),\ \ \
f^{\prime}_{x}(t,x)=\frac{1}{\theta(t)}\cdot\frac{1}{\sigma(x)},\ \ \
f^{\prime\prime}_{xx}(t,x)=\frac{1}{\theta(t)}\cdot\frac{\sigma^{\prime}}{\sigma^{2}(x)}$

we have

$\displaystyle f(t,X(t))=$
$\displaystyle=f(0,X(0))+W(t)+\int_{0}^{t}\left(-\frac{\theta^{\prime}(r)}{\theta^{2}(r)}\cdot
B(X(r))+\frac{g(X(r))\phi(t)}{\sigma(X(r))\theta(r)}-\frac{1}{2}\sigma^{\prime}(X(r))\theta(r)\right)dr+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(\frac{1}{\theta(u)}(B(X(r-)+c(X(r-))\gamma(u))-B(X(r-)))-\frac{c(X(r-))\gamma(u)}{\sigma(X(r-))\theta(u)}\right)\nu(du)dr+$
$\displaystyle+\int_{0}^{t}\int_{|u|<\mathcal{K}}\frac{1}{\theta(u)}(B(X(r-)+c(X(r-))\gamma(u))-B(X(r-)))\tilde{N}(dr,du)=$
$\displaystyle=f(0,X(0))+W(t)+\mathfrak{I}_{5}(t)+\mathfrak{I}_{6}(t)+\mathfrak{I}_{7}(t).$
(12)

By assumption (ii) of the Theorem we obtain

$\displaystyle\mathsf{E}\mathfrak{I}_{7}^{2}(2^{k+1})$ $\displaystyle=$
$\displaystyle=\mathsf{E}\int_{0}^{2^{k}}\int_{|u|<K}\frac{1}{\theta^{2}(u)}(B(X(r-)+c(X(r-))\gamma(u))-B(X(r-)))^{2}\nu(du)dr\leq$
$\displaystyle\leq\int_{0}^{2^{k}}K|r|^{q}dr=\frac{K2^{k(q+1)}}{q+1}.$

Using Lemma 5.1 for $M(t)=\mathfrak{I}_{7}(t)$,

$\sum_{k=1}^{\infty}\frac{1}{2^{k+1}}\mathsf{E}\mathfrak{I}_{7}^{2}(2^{k+1})=\frac{K}{2(q+1)}\sum_{k=1}^{\infty}2^{kq}<\infty,$

as $q<0$, we get

$\underset{t\to\infty}{\lim}\frac{\mathfrak{I}_{7}(t)}{\sqrt{2t\ln\ln
t}}=0\,\,\textnormal{a.s.}$

By assumptions (iii) and (iv),

$\displaystyle\mathfrak{I}_{6}(t)=$
$\displaystyle\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(\frac{(B(X(r-)+c(X(r-))\gamma(u))-B(X(r-)))}{\theta(u)}-\frac{c(X(r-))\gamma(u)}{\sigma(X(r-))\theta(u)}\right)\nu(du)dr\leq$
$\displaystyle\leq\frac{1}{2}\int_{0}^{t}\int_{|u|<\mathcal{K}}\left(\underset{0\leq\zeta\leq
1}{\sup\,}\Big{|}B^{\prime\prime}(X(r-)+\zeta\cdot
c(X(r-))\gamma(u))\Big{|}\right)\frac{c^{2}(X(r-))\gamma^{2}(u)}{\theta(u)}\nu(du)dr\leq$
$\displaystyle\leq\frac{1}{2}\underset{x\in\mathbb{R}}{\sup\,\,}\underset{0<|u|\leq\mathcal{K}}{\sup\,}|B^{\prime\prime}(x,u)|\int_{0}^{t}\int_{|u|<\mathcal{K}}\frac{c^{2}(X(r-))\gamma^{2}(u)}{\theta(u)}\nu(du)dr<\infty.$

Therefore

$\underset{t\to\infty}{\lim}\frac{\mathfrak{I}_{6}(t)}{\sqrt{2t\ln\ln
t}}=0\,\,\textnormal{a.s.}$

Let $\varepsilon(\cdot)$ be some positive function, such that
$\int_{0}^{\infty}\varepsilon(r)dr<\infty$. Since

$\underset{x\in\mathbb{R}}{\inf}\,\tilde{a}(t,x)-\varepsilon(t)<\tilde{a}(t,x),$

then
$f(t,X(t))>f(0,X(0))+W(t)+\int_{0}^{t}(\underset{x\in\mathbb{R}}{\inf}\,\tilde{a}(r,x)-\varepsilon(r))dr+\mathfrak{I}_{6}(t)+\mathfrak{I}_{7}(t)$.

By the low of large number for Wiener process we have

$\displaystyle\underset{t\to\infty}{\lim\inf}\frac{f(t,X(t))}{\sqrt{t\ln\ln
t}}>\underset{t\to\infty}{\lim\inf}\frac{f(0,X(0))}{\sqrt{t\ln\ln t}}$
$\displaystyle+\underset{t\to\infty}{\lim\inf}\frac{W(t)}{\sqrt{2t\ln\ln t}}+$
$\displaystyle+\underset{t\to\infty}{\lim\inf}\frac{1}{\sqrt{2t\ln\ln
t}}\left(A(t)-\int_{0}^{t}\varepsilon(r)dr\right)+$
$\displaystyle+\underset{t\to\infty}{\lim\inf}\frac{\mathfrak{I}_{6}(t)}{\sqrt{2t\ln\ln
t}}+\underset{t\to\infty}{\lim\inf}\frac{\mathfrak{I}_{7}(t)}{\sqrt{2t\ln\ln
t}}.$

From the obtained inequality follows the statement of the Theorem 5.2. ∎

######  Theorem 5.3.

If all conditions of Theorem 5.2 hold and
$\underset{t\to\infty}{\lim}\theta(t)=\infty$, then

$\underset{t\to\infty}{\lim}X(t)=\infty\ \text{a.s.}$

###### Proof.

The result of the Theorem follows from the Theorem 5.2 ∎

##  6 Conclusion

To conclude, the purpose of this paper is to investigate the asymptotic
properties of SDEs perturbed by a Wiener process and a jump-shaped part with a
Poisson random measure. We have focused on SDEs with coefficients expressed as
the product of two functions: one non-random and depending on the time, the
other depending on a random process. We have:

  * 1.

Proposed a method for studying the asymptotic properties of a solution of a
solution to an SDE by comparing it with a solution to an ODE obtained by
removing the stochastic part;

  * 2.

Proved that under certain conditions the asymptotic properties of SDEs
solutions are determined by a nonrandom function.

  * 3.

Provided results regarding the unboundedness of solutions for SDEs with time-
dependent coefficients as $t\to\infty$.

We plan to further our research in order to extend our investigation of the
asymptotic behavior of solutions of SDEs to the multidimensional case.

## Statements and Declarations

  * 1.

Funding. The article was written within the project ”Robustness, Asymptotic
Behavior and Stability of Models Based on Stochastic Differential Equations”.
This research was supported by the MSC4Ukraine grant (AvH ID:1233636), which
is funded by the European Union.

We would like to thank the Alexander von Humboldt Foundation for supporting
Ukrainian scientists and their research.

  * 2.

Conflict of interest. The authors have no competing interests or other
interests that might be perceived to influence the results and/or discussion
reported in this paper.

  * 3.

Authors’ contributions. All authors contributed to of the research concept.
Material preparation and analysis were performed by all authors. The first
draft of the manuscript was written by O. Tymoshenko. All authors read and
approved the final manuscript.

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