# Superdiffusive limits for Bessel-driven stochastic kinetics

Miha Brešar111Department of Statistics, University of Warwick, Coventry, CV4
7AL, UK. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Conrado
da Costa222Department of Mathematical Sciences, Durham University, Upper
Mountjoy, Durham, DH1 3LE, UK. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Aleksandar Mijatović11footnotemark: 1 333The Alan
Turing Institute. Andrew Wade22footnotemark: 2

###### Abstract

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic
dynamics, in which the stochastic drift is driven by an exogenous Bessel
noise, and also includes endogenous volatility which is permitted to have
arbitrary dependence with the exogenous noise. We identify the superdiffusive
scaling exponent for the model, and prove a weak convergence result on the
corresponding scale. We show how our result extends to admit, as exogenous
noise processes, not only Bessel processes but more general processes
satisfying certain asymptotic conditions.

Key words: Stochastic kinetic dynamics; anomalous diffusion; superdiffusivity;
Bessel process; additive functional.

AMS Subject Classification: 60J60 (Primary), 60K50 (Secondary).

##  1 Introduction and main results

The subject of this work is the long-term behaviour of an Itô process
$X=(X_{t})_{t\in\mathbb{R}_{+}}$ in $\mathbb{R}_{+}:=[0,\infty)$ with a
representation

$X_{t}=X_{0}+\int_{0}^{t}\frac{f(s,Y_{s})}{X_{s}}\,\mathrm{d}s+B_{t},\qquad\text{
$X_{0}>0$,}$ (1.1)

where the process $Y=(Y_{t})_{t\in\mathbb{R}_{+}}$ is adapted to the same
filtration as the Brownian motion (BM) $B=(B_{t})_{t\in\mathbb{R}_{+}}$. Under
suitable assumptions on the function
$f:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ and the exogenous
noise process $Y$, which drives the _stochastic drift_ of $X$, the process $X$
will exhibit _anomalous diffusion_ (see YouTube [8] for a short overview of
our results). Physical motivation includes dynamics of particles interacting
with an external field or medium, or with an internal relaxation mechanism;
see e.g. [6, 15, 16] and §2 below for further discussion of motivation and
related literature. Assume the function $f$ has the following asymptotic
behaviour at large times.

(Af)

Let the continuous function
$f:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfy
$f(t,y)\sim\rho t^{\gamma}y^{\alpha}$ as $t,y\to\infty$ for some constant
$\rho\in(0,\infty)$ and exponents $\alpha\in\mathbb{R}_{+}$ and
$\gamma\in(-\alpha,\infty)$, i.e. $\alpha+\gamma>0$. More precisely, for every
$\varepsilon>0$, there is some $r_{\varepsilon}\in\mathbb{R}_{+}$ such that

$\sup_{(t,y)\in[1,\infty)\times\mathbb{R}_{+}:\,t+y\geq
r_{\varepsilon}}\left|f(t,y)t^{-\gamma}y^{-\alpha}-\rho\right|\leq\varepsilon.$
(1.2)

Denote by ${\mathrm{BESQ}}^{\delta}(y)$ the law of a squared Bessel process
$Y$ with “dimension” parameter $\delta\in(0,\infty)$ started at an arbitrary
$Y_{0}=y\in\mathbb{R}_{+}$; this law can be defined as that of the solution of
the stochastic differential equation (SDE) in (1.6) below. Recall that
$\delta>0$ ensures $Y$ is not absorbed at $0$; for $\delta\in(0,2)$, the
process $Y$ is point recurrent over $\mathbb{R}_{+}$, while for $\delta\geq 2$
it is point transient (see [31, Ch. XI] for further details). We denote
convergence in distribution by ‘$\overset{\textup{d}}{\longrightarrow}$’. The
following theorem is our main result.

######  Theorem 1.1.

Let $f:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfy ((Af)) and
assume $\delta\in(0,\infty)$. Suppose the adapted process $(S,Y,B)$ consists
of an $\mathbb{R}$-valued Brownian motion $B$, a squared Bessel process $Y$
with law ${\mathrm{BESQ}}^{\delta}(y)$ (defined via SDE (1.6)) and an
$\mathbb{R}_{+}$-valued process $S=(S_{t})_{t\in\mathbb{R}_{+}}$, satisfying

$S_{t}=S_{0}+\int_{0}^{t}\big{(}2f(s,Y_{s})+1\big{)}\,\mathrm{d}s+2\int_{0}^{t}\sqrt{{S_{s}}}\,\mathrm{d}B_{s},\text{
for all }t\in\mathbb{R}_{+},$ (1.3)

started at a deterministic level $S_{0}\in\mathbb{R}_{+}$. Assume
$\int_{0}^{1}\operatorname{\mathbb{E}}f(t,Y_{t})\,\mathrm{d}t<\infty$. Then
the $\mathbb{R}_{+}$-valued process $X=(X_{t})_{t\in\mathbb{R}_{+}}$, given by
$X_{t}:=\sqrt{S_{t}}$, $t\in\mathbb{R}_{+}$, has the following asymptotic
properties.

  1. (a)

There is superdiffusive transience, namely,

$\lim_{t\to\infty}\frac{\log X_{t}}{\log t}=\frac{1+\gamma+\alpha}{2},\
\text{a.s.}$ (1.4)

  2. (b)

There is a distributional limit, namely,

$t^{-(1+\gamma+\alpha)/2}X_{t}\overset{\textup{d}}{\longrightarrow}\left(2\rho\int_{0}^{1}s^{\gamma}\widetilde{Y}_{s}^{\alpha}\,\mathrm{d}s\right)^{1/2},\text{
as }t\to\infty,$ (1.5)

where $\widetilde{Y}$ in the limit follows ${\mathrm{BESQ}}^{\delta}(0)$.

######  Remarks 1.2.

  1. (i)

Under a mild additional hypothesis, $X$ defined in Theorem 1.1 satisfies
dynamics (1.1), see Appendix A. In Theorem 1.1 it is more convenient to take
as the primitive $X^{2}=S$, because SDE (1.3) fully specifies the process for
a general continuous function $f$ (cf. Bessel process defined as a square-root
of $\mathrm{BESQ}$ in [31, Ch. XI]).

  2. (ii)

Since $\gamma+\alpha>0$ by ((Af)), the scaling exponent $(1+\gamma+\alpha)/2$
in (1.4)–(1.5) exceeds $1/2$, making the process $X$ _superdiffusive_ and
transient. Moreover, by ((Af)), the values of $f$ on any compact set play no
role. The log-scale convergence in (1.4) is a compact formulation of the
statement that for every small $\varepsilon>0$, it holds that for all
$t\in\mathbb{R}_{+}$ sufficiently large,
$t^{(1+\gamma+\alpha)/2-\varepsilon}<X_{t}<t^{(1+\gamma+\alpha)/2+\varepsilon}$;
put differently, $X_{t}=t^{(1+\gamma+\alpha)/2+o(1)}$ as $t\to\infty$, and, in
particular,
$\liminf_{t\to\infty}t^{\varepsilon}X_{t}/t^{(1+\gamma+\alpha)/2}=\infty$ for
all $\varepsilon>0$ (cf. Assumption ((AY))(b) below).

  3. (iii)

The dimension parameter $\delta\in(0,\infty)$ of the squared Bessel process
$Y$ does not appear in the scaling exponent in (1.5). The condition $\delta>0$
ensures that the Bessel process $\sqrt{Y}$ is _diffusive_ , which determines
how the parameter $\alpha$ in ((Af)) enters the scaling exponent in
(1.4)–(1.5). In particular, $X$ in Theorem 1.1 exhibits superdiffusive
transience for negative $\gamma\in(-\alpha,0)$, even if $Y$ is topologically
recurrent (i.e. $\delta\in(0,2]$).

  4. (iv)

We emphasize that the hypotheses in Theorem 1.1 permit arbitrary dependence
between the process $Y$ and the driving Brownian motion $B$ in (1.3). Since
$Y$ with law ${\mathrm{BESQ}}^{\delta}(y)$ is the unique strong solution of
the stochastic differential equation

$\,\mathrm{d}Y_{t}=\delta\,\mathrm{d}t+2|Y_{t}|^{1/2}\,\mathrm{d}W_{t},\text{
for }Y_{0}=y\in\mathbb{R}_{+},$ (1.6)

for some Brownian motion $W=(W_{t})_{t\in\mathbb{R}_{+}}$, the limit in (1.5)
requires only that the Brownian motions $W$ and $B$ are adapted to the same
filtration. In particular, $W$ and $B$ may be equal, independent or have
arbitrary stochastically evolving (adapted) covariation.

  5. (v)

In the case $\alpha=0$, so that $f(t,y)\sim\rho t^{\gamma}$ as $t\to\infty$,
the impact of the process $Y$ on the large-scale dynamics of $X$ vanishes as
$t\to\infty$. In particular, the limit in Theorem 1.1 is deterministic, and,
in fact, the convergence in distribution can in that case be strengthened to
almost sure convergence. Theorem 1.1 can thus be viewed as a generalisation of
certain results in [18] to non-polynomial time-inhomogeneous drift; see
Section 2 below for more details.

  6. (vi)

In contrast to Remark 1.2(ii), if we had $\gamma=\alpha=0$ (and hence $f$
asymptotically constant), then $X$ would be (essentially) a Bessel process of
dimension $1+2\rho>1$. In that case, the statement (1.5) is false, because the
distributional limit of $t^{-1/2}X_{t}$ is _random_ , while the right-hand
side of (1.5) is deterministic when $\alpha=0$. This suggests that the
condition $\gamma+\alpha>0$ in ((Af)) cannot be omitted, essentially because
Theorem 1.1 requires the (exogenous) $Y$-driven stochastic drift to dominate
the (endogenous) Brownian noise in the evolution of $X$.

  7. (vii)

We do not consider here the case where the process $Y$ in (1.1) is _ergodic_ ,
because we anticipate rather different phenomena in that case. For example, if
$f(s,y)=h(y)$ depends only on $y$, and if $h$ is integrable with respect to
the stationary measure of $Y$ on $\mathbb{R}_{+}$, then it seems natural to
suspect that $X$ behaves similarly to a Bessel process with drift coefficient
given via the stationary mean of $h(Y)$. While interesting, this case seems
unlikely to generate the anomalous diffusion that is the focus of the present
work; see Section 2 below for more details.

Remark 1.2(vii) suggests that for an ergodic process $Y$, the model $(S,Y)$ in
(1.3) does not exhibit superdiffusivity. But what properties of $Y$ do
guarantee anomalous diffusive behaviour in Theorem 1.1? By Theorem 1.3 below,
Assumption ((AY)) on the additive functional

$A_{t}:=\int_{1}^{t}s^{\gamma}Y_{s}^{\alpha}\,\mathrm{d}s,\text{ for
}t\in[1,\infty),$ (1.7)

ensures such behaviour. Theorem 1.1 then follows by proving ((AY)) for squared
Bessel processes and identifying the weak limit.

(AY)

Let $\alpha$ and $\gamma$ be as in ((Af)) and
$\int_{0}^{1}\operatorname{\mathbb{E}}f(s,Y_{s})\,\mathrm{d}s<\infty$. Assume
that $A_{t}$ in (1.7) satisfies:

  1. (a)

For a random variable $\widetilde{A}$ in $\mathbb{R}_{+}$,
$A_{t}/t^{1+\gamma+\alpha}\overset{\textup{d}}{\longrightarrow}\widetilde{A}$
as $t\to\infty$.

  2. (b)

For every $\varepsilon>0$, it holds that
$\lim_{t\to\infty}t^{\varepsilon}A_{t}/t^{1+\gamma+\alpha}=\infty$, a.s.

  3. (c)

It holds that
$\sup_{t\in[1,\infty)}\operatorname{\mathbb{E}}A_{t}/t^{1+\gamma+\alpha}<\infty$.

The next limit theorem requires no assumption beyond ((AY)) on the dynamics of
$Y$.

######  Theorem 1.3.

Assume that ((Af)) holds and that the adapted process $(S,Y,B)$ consists of an
$\mathbb{R}$-valued Brownian motion $B$, a process $Y$ on $\mathbb{R}_{+}$
satisfying hypothesis ((AY)), and an $\mathbb{R}_{+}$-valued process $S$
satisfying SDE (1.3). Then, for $X_{t}=\sqrt{S_{t}}$, it holds that

$\lim_{t\to\infty}\frac{\log X_{t}}{\log t}=\frac{1+\gamma+\alpha}{2},\
\text{a.s.},$ (1.8)

and, as $t\to\infty$, for $\widetilde{A}$ the random variable in hypothesis
((AY))(a),

$t^{-(1+\gamma+\alpha)/2}X_{t}\overset{\textup{d}}{\longrightarrow}\bigl{(}2\rho\widetilde{A}\bigr{)}^{1/2}.$
(1.9)

The proof of Theorem 1.3 is given in Section 3. The proof of Theorem 1.1 is
presented in Section 4, where we establish that the process $Y$ with law
${\mathrm{BESQ}}^{\delta}(y)$ satisfies Assumption ((AY)). Self-similarity of
the law ${\mathrm{BESQ}}^{\delta}(y)$ is only used in the final step of the
proof of Theorem 1.1 to identify the law of the limit $\widetilde{A}$. This
leads to a natural open problem to find and/or characterise self-similar
processes $Y$, satisfying the weak limit as in (1.5).

##  2 Motivation and literature

### An example motivated by a self-interacting random walk.

A discrete-time relative of the model in Theorem 1.1 is studied in [11],
motivated in part by a programme to study a certain self-interacting planar
random walk. The parameters in the present model that correspond,
heuristically, to the process described in [11] are $\gamma=0$, $\alpha=1/2$,
and $\delta=1$; informally for Brownian motions $B,W$ on $\mathbb{R}$ with
arbitrary dependence,
$\,\mathrm{d}X_{t}=\rho|W_{t}|/X_{t}\,\mathrm{d}t+\,\mathrm{d}B_{t}$. Note
that $|W|$ (reflected BM) has the same law as $\sqrt{Y}$ where $Y$ follows
${\mathrm{BESQ}}^{1}(0)$. By Theorem 1.1, as $t\to\infty$, $t^{-3/4}X_{t}$
converges weakly to the law of
$(2\rho\int_{0}^{1}|W_{s}|\,\mathrm{d}s)^{1/2}$, which can be expressed in
terms of Airy functions [5, p. 349]. Figure 1 plots the graphs of a path
$t\mapsto X_{t}$ and $t\mapsto t^{3/4}$.

Figure 1: Simulated realizations (Euler scheme, step size of $1/10$) of a
trajectory $t\mapsto X_{t}$ of $X$ (with $\gamma=0$, $\alpha=1/2$, i.e.
$f(t,y)=\sqrt{y}$, and $\delta=1$) and the graph $t\mapsto t^{p}$ for exponent
$p=(1+\alpha+\gamma)/2=3/4$ on the time interval $[0,T]$ with $T=10^{5}$.

### Stochastic kinetic dynamics.

Various physical systems motivate _stochastic kinetic models_ for processes
$(X,V)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ with dynamics of the form

$X_{t}=\int_{0}^{t}V_{s}\,\mathrm{d}s,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
V_{t}=\int_{0}^{t}G(V_{s})\,\mathrm{d}s+B_{t},$ (2.1)

where $B$ is $d$-dimensional Brownian motion and
$G:\mathbb{R}^{d}\to\mathbb{R}^{d}$. Here $V$ is an autonomous _velocity
process_ which feeds into the _s_ tochastic drift of the $d$-dimensional
process $X$. Associated to (2.1) is the so-called _kinetic Fokker–Planck_
equation [15, 16]. A classical example is Paul Langevin’s 1908 work on the
_confining_ case $G(v)=-v$ (see [29]), which was proposed to model physical
Brownian motion. In that case, $V$ is a $d$-dimensional Ornstein–Uhlenbeck
process, and, for some mean-zero Gaussian random variable $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu$,

$V_{t}\overset{\textup{d}}{\longrightarrow}\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu,\text{ as }t\to\infty,$
(2.2)

i.e., there is weak convergence to a stationary, isotropic velocity
distribution. Since $\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu$ is light-tailed and $V$ is rapidly mixing, then $X$ itself, defined
through the additive functional in (2.1), will look like Brownian motion on
large scales.

Renewed interest in the system (2.1), where $G(v)$ is of order $1/v$ for large
$v$ (so that $V$ is a Bessel-like process), has been stimulated by both
modelling of specific physical systems and realization that such processes can
generate a variety of scaling behaviours corresponding to _anomalous
diffusion_ ; we refer to [1, 6, 22, 24, 15, 16, 7, 17], and references
therein.

The mechanism for anomalous diffusion explored in these works is different
from ours. Indeed, these works maintain that $G$ be confining, taking
$G(v)=-\delta/(1+v)$ (or variations on this) for $\delta>0$. For appropriate
$\delta$, this leads to a _heavy-tailed_ stationary distribution $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu$ for which (2.2) holds.
In this case a competition between the mixing rate of $V$ and the index of the
domain of attraction in which $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu$ lives determines the
asymptotics of the additive functional $X$ defined through (2.1). In contrast,
our model $G$ is _positive_ and we are interested in velocity processes $V$
such that $V_{t}\to\infty$ in probability, as $t\to\infty$ (possibly
recurrent, however). This gives a quite distinct mechanism for anomalous
diffusion, and some different phenomena; for example, the range of scaling
laws in [16, Thm. 2] are on scale $t^{1/2}$ (diffusive) through to $t^{3/2}$
(super-ballistic), but our growth exponent in Theorem 1.1(a) has no upper
bound in $(1/2,\infty)$.

A variation of (2.1) is to replace the dynamics of $X$ by

$\,\mathrm{d}X_{t}=\frac{f(t,Y_{t})}{X_{t}}\,\mathrm{d}t,$ (2.3)

for example, where $Y$ is an autonomous process. Then
$\,\mathrm{d}(X_{t}^{2})=2f(t,Y_{t})\,\mathrm{d}t$, so, in some sense, the
generalization that (2.3) provides to (2.1) is only in the form of the
velocity process $V_{t}$. Our model (1.1) extends (2.3) further by introducing
_exogenous noise_ into the dynamics of $X$.

### Other stochastic drift models.

From an entirely different direction, a number of two-dimensional models of a
similar structure have been studied by Lefebvre and collaborators [25, 26, 27,
28], motivated by lifetime studies and models of wear, among other
applications. Specifically [27, 28] formulate a version of (2.3) for the model
$(X,Y)$, where $\,\mathrm{d}X_{t}=-c(Y_{t}/X_{t})\,\mathrm{d}t$ and $Y$ is
geometric Brownian motion. The focus of these papers is evaluating certain
hitting distributions, rather than examining $t\to\infty$ asymptotics.

### Long-time behaviour of stochastic interest-rate models.

Models for the instantaneous interest rate (also known as the short rate) $r$
typically use a mean-reverting diffusion as its stochastic representation of
$r$. Deelstra & Delbaen [13] (see also [12]) extend this framework to a model
where the mean-reversion level of the diffusion $r$ is itself stochastic and
has arbitrary dependence with the driving Brownian motion: the SDE for the
short rate $r$ has constant negative mean-reversion rate $\beta<0$ and a
stochastic mean-reversion level $Y$,

$\,\mathrm{d}r_{t}=(\beta
r_{t}+Y_{t})\,\mathrm{d}t+\sqrt{r_{t}}\,\mathrm{d}B_{t},\qquad\text{with
$\frac{1}{t}\int_{0}^{t}Y_{s}\,\mathrm{d}s\overset{\textup{a.s.}}{\longrightarrow}\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$ as $t\to\infty$.}$
(2.4)

The main result in [13] shows that $r$ is indeed mean-reverting,
$\frac{1}{t}\int_{0}^{t}r_{s}\,\mathrm{d}s\overset{\textup{a.s.}}{\longrightarrow}-\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu/\beta$ as $t\to\infty$,
allowing, as in our Theorem 1.3 above, arbitrary dependence between $Y$ and
the Brownian motion $B$. In contrast to our setting, the process $\sqrt{r}$ in
[13] is diffusive. Moreover, as long as $Y$ is ergodic as in (2.4) above, the
process $r$ is likely to remain diffusive even if $\beta=0$, as it resembles
the squared Bessel process of positive dimension.

### Time-inhomogeneous one-dimensional diffusions.

In the case where the driving function $f(t,y)$ in (1.1) does not depend on
$y$ and is polynomial in $t$, the process $X$ falls into a class of one-
dimensional diffusions with space-and-time-dependent drifts, studied in detail
by Gradinaru & Offret [18] among a more general class of time-inhomogeneous
diffusions on $\mathbb{R}_{+}$, following earlier discrete-time work of [30];
there are also structural links to the elephant random walk [2, 3] and to
noise-reinforced Bessel process [4].

Specifically, a special case of dynamics (1.1) is an SDE
$\,\mathrm{d}X_{t}=\rho t^{\gamma}X_{t}^{-1}\,\mathrm{d}t+\,\mathrm{d}B_{t}$,
which is the model of [18] in the case $\alpha=-1$, $\beta=-\gamma$ (their
parametrization). Here, for $\gamma>0$, [18, Thm. 4.10(i)] states
$\lim_{t\to\infty}t^{-\frac{1+\gamma}{2}}X_{t}=\left(2\rho/(1+\gamma)\right)^{1/(1+\gamma)}\
\text{a.s.}$, strengthening the log-scale asymptotic in (1.4) (where our
$\alpha=0$) in this special case.

##  3 Strong law with stochastic drift

In this section we prove Theorem 1.3 under Assumptions ((Af)) and ((AY)). We
start with a simple observation about $f$ under the hypothesis ((Af)). By
(1.2), there exists $r_{1}>0$ such that
$f(t,y)\leq(\rho+1)(1+t)^{\gamma}y^{\alpha}$ for all
$(t,y)\in\mathbb{R}_{+}^{2}$ with $t+y\geq r_{1}$. Since $f$ is continuous
(and hence bounded on compacts), there exists $C_{0}\in(0,\infty)$ such that

$f(t,y)\leq C_{0}(1+t^{\gamma}y^{\alpha})\quad\text{ for all
}(t,y)\in[1,\infty)\times\mathbb{R}_{+}.$ (3.1)

We use bound (3.1) in several places in the proofs below.

Recall that the adapted process $(S,Y,B)$, taking values in
$\mathbb{R}_{+}^{2}\times\mathbb{R}$ satisfies the dynamics (1.3). In
particular, for all $t\in\mathbb{R}_{+}$, we can write

$S_{t}=S_{0}+U_{t}+t+M_{t},\quad\text{where}\quad
M_{t}:=2\int_{0}^{t}\sqrt{S_{s}}\,\mathrm{d}B_{s},\quad
U_{t}:=2\int_{0}^{t}f(s,Y_{s})\,\mathrm{d}s.$ (3.2)

Recall by ((AY)) that $\operatorname{\mathbb{E}}U_{1}<\infty$ and, for
$\alpha\in\mathbb{R}_{+}$, we have $\beta\coloneqq\alpha+\gamma>0$.

######  Proposition 3.1.

Suppose $\alpha\in\mathbb{R}_{+}$ and $\beta=\gamma+\alpha>0$. Assume ((Af))
and ((AY))(c) hold. Then $M=(M_{t})_{t\in\mathbb{R}_{+}}$in (3.2) is a
martingale and there exists a constant $C\in(0,\infty)$ such that
$\operatorname{\mathbb{E}}(M_{t}^{2})\leq Ct^{2+\beta}$ for all
$t\in[1,\infty)$. Moreover, for any $\varepsilon>0$, we have

$\lim_{t\to\infty}\sup_{0\leq s\leq t}|M_{s}|/t^{1+\beta/2+\varepsilon}=0,\
\text{a.s.}$ (3.3)

###### Proof.

For any $N\in\mathbb{R}_{+}$, set $\tau_{N}:=\inf\\{t\in\mathbb{R}_{+}\colon
S_{t}\geq N\\}$. The quadratic variation of the local martingale
$M^{\tau_{N}}=(M_{t\wedge\tau_{N}})_{t\in\mathbb{R}_{+}}$ is bounded: $\langle
M^{\tau_{N}}\rangle_{t}=\langle M\rangle_{t\wedge\tau_{N}}\leq
4\int_{0}^{t}S_{s\wedge\tau_{N}}\,\mathrm{d}s\leq 4tN$, a.s. By [31, Prop.
IV.1.23], $M^{\tau_{N}}$ is a uniformly integrable martingale started at zero.
Thus $\operatorname{\mathbb{E}}M_{t\wedge\tau_{N}}=0$ for all
$t\in\mathbb{R}_{+}$. Moreover, by definition (1.7), the bound in (3.1) and
((AY))(c), we have $\operatorname{\mathbb{E}}[U_{t}-U_{1}]\leq
2C_{0}(t+\operatorname{\mathbb{E}}A_{t})\leq C_{0}^{\prime}(1+t)^{1+\beta}$
for all $t\in[1,\infty)$ and a constant $C_{0}^{\prime}\in(0,\infty)$. Since,
as $\operatorname{\mathbb{E}}U_{1}<\infty$ and $U$ is increasing, by (3.2)
there exists a constant $C_{1}\in(0,\infty)$ such that

$\operatorname{\mathbb{E}}S_{s\wedge\tau_{N}}\leq
S_{0}+s+\operatorname{\mathbb{E}}U_{1}+\operatorname{\mathbb{E}}[U_{s}-U_{1}]\leq
C_{1}(1+s)^{1+\beta},\text{ for all $s,N\in\mathbb{R}_{+}$.}$

Hence $\operatorname{\mathbb{E}}\langle M\rangle_{t\wedge\tau_{N}}\leq
4\int_{0}^{t}\operatorname{\mathbb{E}}S_{s\wedge\tau_{N}}\,\mathrm{d}s\leq
2C_{1}(1+t)^{2+\beta}$ for $t,N\in\mathbb{R}_{+}$. Since $\langle
M\rangle_{t\wedge\tau_{N}}\uparrow\langle M\rangle_{t}$ as $N\to\infty$,
monotone convergence implies $\operatorname{\mathbb{E}}\langle
M\rangle_{t}\leq Ct^{2+\beta}$ for $t\in[1,\infty)$ and $C\coloneqq
2^{3+\beta}C_{1}$. By [31, Cor. IV.1.25], $M$ is a martingale and
$\operatorname{\mathbb{E}}(M_{t}^{2})=\operatorname{\mathbb{E}}\langle
M\rangle_{t}\leq Ct^{2+\beta}$ for $t\in[1,\infty)$.

The limit in (3.3) will follow by a Borel–Cantelli argument. Doob’s maximal
inequality [31, Thm. II.1.7] and the $L^{2}$-bound
$\operatorname{\mathbb{E}}M_{t}^{2}\leq Ct^{2+\beta}$ (for all
$t\in[1,\infty)$) yield

$\operatorname{\mathbb{P}}\left(\sup_{0\leq s\leq
t}|M_{s}|>a\right)\leq\operatorname{\mathbb{E}}M_{t}^{2}/a^{2}\leq
Ct^{2+\beta}/a^{2},\text{ for all $t\geq 1$ and $a>0$.}$ (3.4)

Fix arbitrary $\varepsilon>0$. Set $t_{n}:=2^{n}$ and
$a_{n}:=t_{n}^{1+\beta/2+\varepsilon}$ for
$n\in\mathbb{Z}_{+}\coloneqq\mathbb{N}\cup\\{0\\}$. By the inequality in
(3.4), applied with $t=t_{n}$ and $a=a_{n}$, the probabilities of the events

$E_{n}:=\left\\{\sup_{0\leq s\leq t_{n}}|M_{s}|>a_{n}\right\\}\quad\text{are
summable:}\quad\sum_{n=0}^{\infty}\mathbb{P}(E_{n})\leq
C\sum_{n=0}^{\infty}t_{n}^{-2\varepsilon}<\infty.$

Hence, by the Borel–Cantelli lemma, there exists a (random)
$n_{0}\in\mathbb{N}$, a.s., such that $E_{n}$ occurs for no $n\geq n_{0}$,
i.e., $\sup_{0\leq s\leq t_{n}}|M_{s}|\leq a_{n}$ for all $n\geq n_{0}$. Let
$T:=2^{n_{0}}<\infty$ a.s. For every $t\in[1,\infty)$, there exists a unique
$k\in\mathbb{Z}_{+}$, such that $t_{k}=2^{k}\leq t<2^{k+1}=t_{k+1}$. Thus, for
all $t\geq T$, we have

$\displaystyle\sup_{0\leq s\leq t}|M_{s}|$ $\displaystyle\leq\sup_{0\leq s\leq
t_{k+1}}|M_{s}|\leq
t_{k+1}^{1+\beta/2+\varepsilon}=2^{1+\beta/2+\varepsilon}\big{(}2^{k}\big{)}^{1+\beta/2+\varepsilon}\leq
2^{1+\beta/2+\varepsilon}t^{1+\beta/2+\varepsilon}.$

Since $\varepsilon>0$ was arbitrary, the limit in (3.3) holds almost surely. ∎

######  Remark 3.2 (Squared Bessel process).

A solution $Y$ of SDE (1.6) has a decomposition analogous to (3.2),
$Y_{t}=y+\delta t+{\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu}_{t}$, with the local martingale ${\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{t}\coloneqq
2\int_{0}^{t}\sqrt{Y_{s}}\,\mathrm{d}W_{s}$. Since $\delta>0$ is a constant, a
simpler version of the argument in the first paragraph of the previous proof
implies $\operatorname{\mathbb{E}}{\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}^{2}_{t}\leq Ct^{2}$ for
$t\in[1,\infty)$ and a constant $C>0$. Doob’s maximal inequality [31, Thm.
II.1.7] yields the tail bound $\operatorname{\mathbb{P}}(\sup_{0\leq s\leq
t}|{\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{s}|>a)\leq
Ct^{2}/a^{2}$ for all $t,a>1$. As in the proof of Proposition 3.1, the
Borel–Cantelli lemma and a subsequence argument implies that, for any
$\varepsilon>0$, a.s. $\sup_{0\leq s\leq t}|{\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{s}|\leq
t^{1+\varepsilon}$ for all sufficiently large $t\in\mathbb{R}_{+}$, and thus,
by SDE (1.6),

$\lim_{t\to\infty}Y_{t}/t^{1+\varepsilon}=0,\ \text{a.s.}$ (3.5)

We now establish the following key result giving the almost-sure rate of
escape for the additive functional $U$. In particular, by (3.2), the almost-
sure behaviour (as $t\to\infty$) of the process $S$ is dominated by $U$.

######  Proposition 3.3.

Suppose that $\alpha\geq 0$ and $\beta=\gamma+\alpha>0$, and assume that
((Af)) and ((AY))(b), ((AY))(c) hold. Then, for every $\varepsilon>0$, the
process $U$ in (3.2) a.s. satisfies

$t^{-\varepsilon}<U_{t}/t^{1+\beta}<t^{\varepsilon}\text{ for all large
$t\in\mathbb{R}_{+}$.}$ (3.6)

###### Proof.

By (1.2) in ((Af)), there exists a large constant $r_{\rho}>1$ such that
$f(t,y)\geq(\rho/2)t^{\gamma}y^{\alpha}$ for all
$(t,y)\in[r_{\rho},\infty)\times\mathbb{R}_{+}$. By the definition of $U_{t}$
in (3.2), for all $t>r_{\rho}$, we have

$\displaystyle\rho(A_{t}-A_{r_{\rho}})=\rho\int_{r_{\rho}}^{t}s^{\gamma}Y_{s}^{\alpha}\,\mathrm{d}s\leq
2\int_{0}^{t}f(s,Y_{s})\,\mathrm{d}s=U_{t},$

where $A_{t}$ is defined in (1.7). Hypothesis ((AY))(b) on $A_{t}$ implies
$\lim_{t\to\infty}t^{\varepsilon-(1+\beta)}A_{t}=\infty\ \text{a.s.}$ for
every $\varepsilon>0$, yielding the lower bound in (3.6).

To prove the upper bound in (3.6), note that the definition of $U_{t}$ in
(3.2) and the upper bound on $f$ in (3.1) imply $U_{t}-U_{1}\leq
2C_{0}(t+A_{t})$ a.s. Hence, by ((AY))(c), for a constant $C_{2}<\infty$,

$\operatorname{\mathbb{E}}U_{t}=\operatorname{\mathbb{E}}[U_{t}-U_{1}]+\operatorname{\mathbb{E}}U_{1}\leq
2C_{0}(t+\operatorname{\mathbb{E}}A_{t})+\operatorname{\mathbb{E}}U_{1}\leq
C_{2}t^{1+\beta}\quad\text{for all $t\in[1,\infty)$.}$

Pick any $\varepsilon>0$. The bound in the previous display and Markov’s
inequality yield

$\operatorname{\mathbb{P}}(U_{t}\geq
t^{1+\beta+\varepsilon})\leq\operatorname{\mathbb{E}}U_{t}/t^{1+\beta+\varepsilon}\leq
C_{2}/t^{\varepsilon}.$

Since $\sum_{n\in\mathbb{Z}_{+}}1/t_{n}^{\varepsilon}<\infty$, where
$t_{n}\coloneqq 2^{n}$, the Borel–Cantelli lemma implies $U_{t_{n}}\leq
t_{n}^{1+\beta+\varepsilon}$ for all but finitely many $n\in\mathbb{Z}_{+}$,
a.s. As in the proof of Proposition 3.1, for every $t\in[1,\infty)$, there
exists a unique $k\in\mathbb{Z}_{+}$, such that $t_{k}=2^{k}\leq
t<2^{k+1}=t_{k+1}$. As $U$ is increasing by ((Af)),

$U_{t}\leq
U_{t_{k+1}}\leq(2^{k+1})^{1+\beta+\varepsilon}=2^{1+\beta+\varepsilon}\cdot
t_{k}^{1+\beta+\varepsilon}\leq
2^{1+\beta+\varepsilon}t^{1+\beta+\varepsilon}.$

Since $\varepsilon>0$ is arbitrary, the upper bound in (3.6) holds for all
large $t\in\mathbb{R}_{+}$. ∎

The next result establishes weak convergence of $U_{t}$, defined in (3.2) and
suitably scaled, to a random variable proportional to the limit
$\widetilde{A}$ in hypothesis ((AY))(a).

######  Lemma 3.4.

Let $\alpha\geq 0$ and $\beta=\gamma+\alpha>0$. Assume ((Af)) and ((AY))(a)
hold. Then,

$U_{t}/t^{1+\beta}\overset{\textup{d}}{\longrightarrow}2\rho\widetilde{A},\text{
as }t\to\infty.$

###### Proof.

Recall the definitions of $A_{t}$ in (1.7) and of $U_{t}$ in (3.2). Define
also

$\displaystyle D_{t}:=U_{t}/2-\rho
A_{t}=U_{1}/2+\int_{1}^{t}f(s,Y_{s})\,\mathrm{d}s-\rho A_{t}\quad\text{ for
any $t\in[1,\infty)$.}$

Since $U_{t}/t^{1+\beta}=2(D_{t}+\rho A_{t})/t^{1+\beta}$, by Slutsky’s lemma
(see e.g. [14, p. 105]) and the weak limit in ((AY))(a), it is sufficient to
prove that $D_{t}/t^{1+\beta}$ converges to zero in probability. In
particular, it suffices to show that for every
$\varepsilon^{\prime},\varepsilon^{\prime\prime}>0$ we have

$\limsup_{t\to\infty}\operatorname{\mathbb{P}}\bigl{(}|D_{t}|/t^{1+\beta}>2\varepsilon^{\prime}\bigr{)}\leq\varepsilon^{\prime\prime}.$
(3.7)

For every $\varepsilon>0$, by (1.2), there exists
$r_{\varepsilon}\in(1,\infty)$ such that, for every
$s\in[r_{\varepsilon},\infty)$ and $y\in\mathbb{R}_{+}$, we have
$\left|f(s,y)-\rho s^{\gamma}y^{\alpha}\right|\leq\varepsilon
s^{\gamma}y^{\alpha}$. Define
$Z_{\varepsilon}\coloneqq\left|U_{1}/2+\int_{1}^{r_{\varepsilon}}f(s,Y_{s})\,\mathrm{d}s+\rho
A_{r_{\varepsilon}}\right|$ and note that, for any $t>r_{\varepsilon}$, we
have

$\displaystyle|D_{t}|\leq
Z_{\varepsilon}+\int_{r_{\varepsilon}}^{t}\left|f(s,Y_{s})-\rho
s^{\gamma}Y_{s}^{\alpha}\right|\,\mathrm{d}s\leq
Z_{\varepsilon}+\varepsilon(A_{t}-A_{r_{\varepsilon}})\leq
Z_{\varepsilon}+\varepsilon A_{t}.$ (3.8)

Pick any $\varepsilon^{\prime},\varepsilon^{\prime\prime}>0$. Fix small
$\varepsilon>0$, such that
$\lim_{t\to\infty}\operatorname{\mathbb{P}}(\varepsilon
A_{t}/t^{1+\beta}>\varepsilon^{\prime})=\operatorname{\mathbb{P}}(\widetilde{A}>\varepsilon^{\prime}/\varepsilon)$
(by ((AY))(a), limit holds for all but countably many $\varepsilon$) and
$\operatorname{\mathbb{P}}(\widetilde{A}>\varepsilon^{\prime}/\varepsilon)<\varepsilon^{\prime\prime}/2$.
Thus, by (3.8),

$\displaystyle\operatorname{\mathbb{P}}\bigl{(}|D_{t}|/t^{1+\beta}>2\varepsilon^{\prime}\bigr{)}\leq\operatorname{\mathbb{P}}\left(Z_{\varepsilon}/t^{1+\beta}>\varepsilon^{\prime}\right)+\operatorname{\mathbb{P}}(A_{t}/t^{1+\beta}>\varepsilon^{\prime}/\varepsilon)\leq\varepsilon^{\prime\prime}/2+\varepsilon^{\prime\prime}/2$

for all large $t\in\mathbb{R}_{+}$. Since
$\varepsilon^{\prime},\varepsilon^{\prime\prime}>0$ were arbitrary, (3.7)
holds and lemma follows. ∎

We can now complete the proof of Theorem 1.3.

###### Proof of Theorem 1.3.

Since $S_{t}=S_{0}+U_{t}+t+M_{t}$ and
$(S_{0}+t+M_{t})/t^{1+\beta}\overset{\textup{a.s.}}{\longrightarrow}0$ as
$t\to 0$ by (3.3) of Prop. 3.1, the weak limit in Lemma 3.4 and Slutsky’s
lemma (see [14, p. 105]) yield
$S_{t}/t^{1+\beta}\overset{\textup{d}}{\longrightarrow}2\rho\widetilde{A}$ as
$t\to\infty$. Then (1.9) follows by continuous mapping. By (3.6) in Prop. 3.3,
for any $\varepsilon>0$ a.s. $\log S_{t}/\log
t=1+\beta+\log\left((S_{0}+t+M_{t})/t^{1+\beta}+U_{t}/t^{1+\beta}\right)/\log
t\leq 1+\beta+\varepsilon$ for all large $t$. Similarly, for any
$\varepsilon\in(0,\beta/2)$, (3.3) of Proposition 3.1 and the lower bound in
(3.6) imply $1+\beta-\varepsilon\leq\log S_{t}/\log t$ and limit (1.8)
follows. ∎

##  4 Asymptotics of certain squared-Bessel functionals

The aim of this section is to verify Assumption ((AY)) for the squared Bessel
process $Y$ (with law ${\mathrm{BESQ}}^{\delta}(y)$) of positive “dimension”
$\delta>0$. This will, by Theorem 1.3, imply our main result Theorem 1.1. The
focus in this section is the additive functional $A_{t}$, defined in (1.7),
when $Y$ follows SDE (1.6).

######  Theorem 4.1.

Suppose that $Y$ has law ${\mathrm{BESQ}}^{\delta}(y)$ with parameter
$\delta>0$, started at arbitrary $y\in\mathbb{R}_{+}$. Let
$\alpha\in\mathbb{R}_{+}$ and $\gamma\in(-\alpha,\infty)$. Recall
$A_{t}=\int_{1}^{t}s^{\gamma}Y_{s}^{\alpha}\,\mathrm{d}s$, $t\in[1,\infty)$,
defined in (1.7).

  1. (a)

The following limit holds,

$\lim_{t\to\infty}\frac{\log A_{t}}{\log t}=1+\gamma+\alpha,\ \text{a.s.}$
(4.1)

  2. (b)

Let $\widetilde{Y}$ follow ${\mathrm{BESQ}}^{\delta}(0)$. Then, as
$t\to\infty$,

$A_{t}/t^{1+\gamma+\alpha}\overset{\textup{d}}{\longrightarrow}\int_{0}^{1}s^{\gamma}\widetilde{Y}_{s}^{\alpha}\,\mathrm{d}s.$
(4.2)

  3. (c)

The growth of the expectation $\operatorname{\mathbb{E}}A_{t}$ is at most
polynomial, $\sup_{t\geq
1}\operatorname{\mathbb{E}}A_{t}/t^{1+\gamma+\alpha}<\infty$.

Note that, since $\alpha\geq 0$, for all $t>1$ and $Y_{0}=y\in\mathbb{R}_{+}$
we have $0<A_{t}<\infty$. The inequality $\limsup_{t\to\infty}\log A_{t}/\log
t\leq 1+\gamma+\alpha$ in (4.1) of Theorem 4.1(a) follows directly from the
a.s upper bound (3.5) above for the squared Bessel process $Y$. Assumption
((AY))(b) requires only $\liminf_{t\to\infty}\log A_{t}/\log t\geq
1+\gamma+\alpha$ in (4.1), the most involved part of Theorem 4.1. Proposition
4.2 below states this as a stand-alone result, followed by an elementary,
essentially self-contained proof. From the perspective of our main result
(Theorem 1.1), the case $\delta\in(0,2]$ is also the most interesting as it
corresponds to the non-ergodic recurrent stochastic drift in (1.1), which
nevertheless induces anomalous diffusive behaviour of $X=\sqrt{S}$. In the
Brownian case ($\delta=1$), the log-limit statement (4.1) can also be deduced,
with a little work, from results of [23].

Parts (b)&(c) of Theorem 4.1 follow directly from self-similarity of the
Bessel processes.

######  Proposition 4.2.

Suppose that $\delta>0$ and $\alpha\in\mathbb{R}_{+}$, $\gamma>-\alpha$. Then,
for every $\varepsilon>0$, a.s.

$A_{t}>t^{1+\gamma+\alpha-\varepsilon}\qquad\text{for all large
$t\in\mathbb{R}_{+}$.}$ (4.3)

######  Remarks 4.3.

  1. (i)

The intuition behind (4.3) is that, typically, $Y_{t}\approx t$, by Bessel
scaling. However, the proof below reveals that the $\delta$-independent
exponent in (4.3) emerges from a rather delicate balance between the frequency
and spatial extent of excursions, which both depend critically on $\delta$; in
[9] bounds of a similar type yield lower bounds on the rates of convergence of
ergodic reflected Brownian motion. A related approach in discrete time can be
found in [19].

  2. (ii)

In the case $\gamma=0$, the quantity $A_{\tau_{\ell}}$ is studied in detail by
Çetin in [10], where $Y$ follows ${\mathrm{BESQ}}^{\delta}(y)$ and $\delta>0$
(the hitting time $\tau_{\ell}$ is defined below). The approach in [10]
appears more involved than ours, without addressing directly the a.s.
asymptotics for $A_{t}$.

Throughout this section, we write $\operatorname{\mathbb{P}}_{y}$ for the
probability measure inducing the law ${\mathrm{BESQ}}^{\delta}(y)$, and
$\operatorname{\mathbb{E}}_{y}$ for the corresponding expectation. Let
$\tau_{x}:=\inf\\{t\in\mathbb{R}_{+}:Y_{t}=x\\}$ (with convention
$\inf\emptyset=\infty$) for $x\in\mathbb{R}_{+}$. The next lemma says the
squared Bessel process started at $y$ will typically spend time at least of
order $y$ at levels of order $y$.

######  Lemma 4.4.

Suppose that $\delta>0$. Then there exists $\varepsilon>0$ such that,

$\operatorname{\mathbb{P}}_{y}(\tau_{y/2}\geq\varepsilon y)\geq 1/2,\text{ for
all }y\in\mathbb{R}_{+}.$

###### Proof.

It suffices to suppose that $Y_{0}=y\in(0,\infty)$. From (1.6), we obtain, for
all $t\in\mathbb{R}_{+}$,

$Y_{t\wedge\tau_{2y}}=y+\delta(t\wedge\tau_{2y})+M_{t\wedge\tau_{2y}},\text{
where }M_{t}:=2\int_{0}^{t}|Y_{s}|^{1/2}\,\mathrm{d}W_{s}.$ (4.4)

The process $(M_{t\wedge\tau_{2y}})_{t\in\mathbb{R}_{+}}$ is a martingale
since its quadratic variation is a.s. bounded at any finite time $t$. Hence,
by (4.4), we get $\operatorname{\mathbb{E}}_{y}(Y_{t\wedge\tau_{2y}})\leq
y+\delta t$. By [31, Prop. IV.1.23],

$\operatorname{\mathbb{E}}_{y}M_{t\wedge\tau_{2y}}^{2}=\operatorname{\mathbb{E}}_{y}\langle
M\rangle_{t\wedge\tau_{2y}}=4\int_{0}^{t}\operatorname{\mathbb{E}}_{y}Y_{s\wedge\tau_{2y}}\,\mathrm{d}s\leq
4yt+2\delta t^{2},\text{ for all }t\in\mathbb{R}_{+}.$

Consider the event $E_{y,t}:=\\{\sup_{0\leq s\leq t}|M_{s\wedge\tau_{2y}}|\geq
y/3\\}$. Then, by Doob’s maximal inequality [31, Thm. II.1.7] applied to the
non-negative submartingale $(M_{t\wedge\tau_{2y}}^{2})_{t\in\mathbb{R}_{+}}$,

$\operatorname{\mathbb{P}}_{y}(E_{y,t})\leq(9/y^{2})\operatorname{\mathbb{E}}_{y}M_{t\wedge\tau_{2y}}^{2}\leq
36(t/y)+18\delta(t/y)^{2}.$ (4.5)

If $t<y/(2\delta)$, we have $E^{\textup{c}}_{y,t}\subset\\{\tau_{2y}>t\\}$
$\operatorname{\mathbb{P}}_{y}$-a.s., implying
$E^{\textup{c}}_{y,t}=\\{\sup_{0\leq s\leq t}|M_{s}|<y/3\\}$. By (4.4) we thus
get $E^{\textup{c}}_{y,t}\subset\\{\inf_{0\leq s\leq
t}Y_{s}>y/2\\}=\\{\tau_{y/2}>t\\}$ $\operatorname{\mathbb{P}}_{y}$-a.s.,
yielding (with (4.5))

$\operatorname{\mathbb{P}}_{y}\left(\tau_{y/2}>t\right)\geq\operatorname{\mathbb{P}}_{y}(E^{\textup{c}}_{y,t})=1-\operatorname{\mathbb{P}}_{y}(E_{y,t})\geq
1-36(t/y)-18\delta(t/y)^{2}.$ (4.6)

For any $\varepsilon\in(0,(2\delta)^{-1})$, define $t\coloneqq\varepsilon
y<y/(2\delta)$, and $\operatorname{\mathbb{P}}_{y}(\tau_{y/2}\geq\varepsilon
y)\geq 1-36\varepsilon-18\delta\varepsilon^{2}$ by (4.6). Choose
$\varepsilon\in(0,(2\delta)^{-1})$, such that
$1-36\varepsilon-18\delta\varepsilon^{2}\geq 1/2$, to finish the proof. ∎

We now establish a lower bound on the tail of a functional of an excursion of
$Y$.

######  Lemma 4.5.

Let $\delta\in(0,2)$, $\alpha\in\mathbb{R}_{+}$. Then there exists a constant
$c_{\alpha,\delta}\in(0,\infty)$ satisfying

$\operatorname{\mathbb{P}}_{1}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq
z\right)\geq c_{\alpha,\delta}z^{-\frac{2-\delta}{2+2\alpha}},\quad\text{ for
all }z\in(1,\infty).$

###### Proof.

First observe that, since $Y_{s}\in\mathbb{R}_{+}$ for all
$s\in\mathbb{R}_{+}$, for every $y\in(0,\infty)$ and $z\in\mathbb{R}_{+}$,

$\displaystyle\operatorname{\mathbb{P}}_{1}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq
z\right)$
$\displaystyle\geq\operatorname{\mathbb{E}}_{1}\left[\mathbbm{1}_{\\{\tau_{y}<\tau_{0}\\}}\operatorname{\mathbb{P}}\left(\int_{\tau_{y}}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq
z\,\bigg{|}\,\mathcal{F}_{\tau_{y}}\right)\right]$
$\displaystyle=\operatorname{\mathbb{P}}_{1}(\tau_{y}<\tau_{0})\operatorname{\mathbb{P}}_{y}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq
z\right),$ (4.7)

where $(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}$ is the filtration generated by
the Brownian motion driving SDE (1.6). The equality in (4) follows from the
strong Markov property at $\tau_{y}$, and the facts:
$\\{\tau_{y}<\tau_{0}\\}\in\mathcal{F}_{\tau_{y}}$ and $Y_{\tau_{y}}=y$
$\operatorname{\mathbb{P}}_{1}$-a.s. By Lemma 4.4 there exists
$\varepsilon>0$, such that
$\operatorname{\mathbb{P}}_{y}(\tau_{y/2}\geq\varepsilon y)\geq 1/2$.
Moreover, on the event $\\{\tau_{y/2}\geq\varepsilon y\\}$, the inequality

$\displaystyle\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq\int_{0}^{\varepsilon
y}(y/2)^{\alpha}\,\mathrm{d}s=\varepsilon
y^{\alpha+1}/2^{\alpha}\quad\text{holds
$\operatorname{\mathbb{P}}_{y}$-a.s.,}$

implying
$\operatorname{\mathbb{P}}_{y}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq\varepsilon
y^{\alpha+1}/2^{\alpha}\right)\geq\operatorname{\mathbb{P}}_{y}(\tau_{y/2}\geq\varepsilon
y)\geq 1/2$ for all $y\in(0,\infty)$. By (4) with $z=\varepsilon
y^{\alpha+1}/2^{\alpha}$, we obtain

$\displaystyle\operatorname{\mathbb{P}}_{1}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq\varepsilon
y^{\alpha+1}/2^{\alpha}\right)\geq\frac{1}{2}\operatorname{\mathbb{P}}_{1}(\tau_{y}<\tau_{0})=\frac{1}{2y^{(2-\delta)/2}}\text{
for all }y\geq 1.$ (4.8)

The last equality follows from the optional sampling theorem since the process
$(Y^{(2-\delta)/2}_{t\wedge\tau_{0}\wedge\tau_{y}})_{t\in\mathbb{R}_{+}}$ is
(by Itô’s formula and SDE (1.6)) a bounded martingale. The following change of
variable $y=(2^{\alpha}z/\varepsilon)^{1/(1+\alpha)}$ in (4.8) yields the
stated bound. ∎

###### Proof of Proposition 4.2.

Suppose that $\delta\in(0,2)$ and $\alpha\in\mathbb{R}_{+}$. We will first
establish the case $\gamma=0$ of (4.3), then $\delta\geq 2$ and subsequently
deduce the general case $\gamma\in(-\alpha,\infty)$.

Define stopping times $\vartheta_{0}:=0$ and, for $n\in\mathbb{N}$,

$\displaystyle\varphi_{n}:=\inf\\{t\geq\vartheta_{n-1}:Y_{t}=1\\},\qquad\vartheta_{n}:=\inf\\{t\geq\varphi_{n}:Y_{t}=0\\}.$
(4.9)

Since the squared Bessel process is point-recurrent for $\delta\in(0,2)$, we
have $0<\vartheta_{n}<\varphi_{n+1}<\vartheta_{n+1}<\infty$, a.s., for all
$n\in\mathbb{N}$ (and $0=\vartheta_{0}<\varphi_{1}$ as well, unless
$Y_{0}=1$). Define

$N_{t}:=\sup\\{n\in\mathbb{Z}_{+}:\vartheta_{n}\leq t\\},\text{
$t\in\mathbb{R}_{+}$,}\qquad\text{and}\qquad
I_{\alpha,n}:=\int_{\varphi_{n}}^{\vartheta_{n}}Y_{s}^{\alpha}\,\mathrm{d}s,\text{
$n\in\mathbb{N}$.}$ (4.10)

Since $Y_{s}\geq 0$ for all $s\in\mathbb{R}_{+}$, and
$0\leq\vartheta_{N_{t}}\leq t$, we can write

$\int_{0}^{t}Y_{s}^{\alpha}\,\mathrm{d}s\geq\int_{0}^{\vartheta_{N_{t}}}Y_{s}^{\alpha}\,\mathrm{d}s\geq\sum_{n=1}^{N_{t}}I_{\alpha,n},\text{
for all }t\in\mathbb{R}_{+}.$ (4.11)

The strong Markov property and $Y_{\varphi_{n}}=1$ a.s. imply the random
variables $(I_{\alpha,n})_{n\in\mathbb{N}}$ are i.i.d. By Lemma 4.5 we have

$\operatorname{\mathbb{P}}(I_{\alpha,1}\geq
z)=\operatorname{\mathbb{P}}_{1}\left(\int_{0}^{\tau_{0}}Y_{s}^{\alpha}\,\mathrm{d}s\geq
z\right)\geq c_{\alpha,\delta}z^{-\frac{1-\delta/2}{1+\alpha}},\quad\text{for
all }z\in(1,\infty).$ (4.12)

For any $\varepsilon>0$, this bound and the Borel–Cantelli lemma yield: a.s.

$\sum_{n=1}^{N}I_{\alpha,n}\geq
N^{\frac{1+\alpha}{1-\delta/2}-\varepsilon}\quad\text{for all but finitely
many $N\in\mathbb{N}$.}$ (4.13)

Indeed, denoting $\theta\coloneqq\frac{1-\delta/2}{1+\alpha}\in(0,1)$ (recall
$0<\delta<2$ and $\alpha\in\mathbb{R}_{+}$) and picking $r>1$, by (4.12) for
all $N\geq 2$ we have $\operatorname{\mathbb{P}}(I_{\alpha,1}\geq(N/(\log
N)^{r})^{1/\theta})\geq c_{\alpha,\delta}(\log N)^{r}/N$. Since the variables
$(I_{\alpha,n})_{n\in\mathbb{N}}$ are i.i.d., the events
$E_{N}\coloneqq\\{\max_{1\leq n\leq N}I_{\alpha,n}<(N/(\log
N)^{r})^{1/\theta}\\}$ satisfy

$\displaystyle\operatorname{\mathbb{P}}(E_{N})=(1-\operatorname{\mathbb{P}}(I_{\alpha,1}\geq
N^{1/\theta}/(\log N)^{r/\theta}))^{N}\leq(1-c_{\alpha,\delta}(\log
N)^{r}/N)^{N}\leq\bar{c}/N^{r}$

for all $N\in\mathbb{N}$ and some constant $\bar{c}>0$ (the last inequality
follows by taking logarithms on both sides and applying $\log(1-x)\leq-x$ for
$x<1$). Since $r>1$, the Borel–Cantelli lemma implies a.s. $\max_{1\leq n\leq
N}I_{\alpha,n}\geq(N/(\log N)^{r})^{1/\theta}$ for all sufficiently large $N$.
As the sum of positive terms dominates the maximum and $(N/(\log
N)^{r})^{1/\theta}\geq N^{\frac{1+\alpha}{1-\delta/2}-\varepsilon}$ for every
$\varepsilon>0$ and all $N\in\mathbb{N}$ sufficiently large, (4.13) follows.

Let $\nu_{n}:=\vartheta_{n}-\varphi_{n}$, for $n\in\mathbb{N}$, be the
duration of the excursion (at the epoch $\varphi_{n}$) of $Y$ from level $1$
to level $0$. We now prove (4.14) below that controls the tail of the duration
$\nu_{n}$. By the strong Markov property at $\varphi_{n}$ we have
$\operatorname{\mathbb{P}}(\nu_{n}>t)=\operatorname{\mathbb{P}}_{1}(\tau_{0}>t)$
for all $t\in\mathbb{R}_{+}$ and $n\in\mathbb{N}$. For any
$Y_{0}=y\in\mathbb{R}_{+}$ and $t>0$, by the scaling property, the process
$(Y_{st}/t)_{s\in\mathbb{R}_{+}}$ follows ${\mathrm{BESQ}}^{\delta}(y/t)$.
Hence
$\operatorname{\mathbb{P}}_{y}(\tau_{0}>t)=\operatorname{\mathbb{P}}_{y}(\inf_{s\in[0,t]}Y_{s}>0)=\operatorname{\mathbb{P}}_{y/t}(\inf_{s\in[0,1]}Y_{s}>0)=G(y/t)$,
for a measurable function $G:\mathbb{R}_{+}\to[0,1]$ satisfying $G(0)=0$ and
$G(y)\to 1$ as $y\to\infty$.

The process $(Z_{s})_{s\in[0,t]}$,
$Z_{s}\coloneqq\operatorname{\mathbb{P}}(\tau_{0}>t\mid\mathcal{F}_{s})=\mathbbm{1}_{\\{\tau_{0}>s\\}}G(Y_{s}/(t-s))$,
is a martingale with respect to the filtration $(\mathcal{F}_{s})_{s\in[0,t]}$
generated by the Brownian motion $W$ in SDE (1.6). Assuming $G$ is twice
differentiable on $(0,\infty)$, Itô’s formula and the infinitesimal drift of
$(Z_{s})_{s\in[0,t]}$ being equal to zero imply that $G$ satisfies an ordinary
differential equation (ODE) $2G^{\prime\prime}(x)+(1+\delta/x)G^{\prime}(x)=0$
for $x\in(0,\infty)$ with boundary conditions above. The solution
$\bar{G}(x)\coloneqq\int_{0}^{x/2}u^{-\delta/2}{\mathrm{e}}^{-u}\,\mathrm{d}u/\Gamma(1+\delta/2)$
of the ODE, where $\Gamma$ denotes the gamma function, yields a martingale
$\bar{Z}_{s}\coloneqq\mathbbm{1}_{\\{\tau_{0}>s\\}}\bar{G}(Y_{s}/(t-s))$,
$s\in[0,t]$, satisfying $Z_{t}=\mathbbm{1}_{\\{\tau_{0}>t\\}}=\bar{Z}_{t}$.
Thus $Z_{s}=\bar{Z}_{s}$ for all $s\in[0,t]$ a.s. Since the support of $Y_{s}$
(for $s>0$) contains $(0,\infty)$, we have
$\bar{G}(y/t)=G(y/t)=\operatorname{\mathbb{P}}_{y}(\tau_{0}>t)$ for all
$y\in\mathbb{R}_{+}$ and $t>0$. Since, clearly, $\tau_{0}$ has a density and
${\mathrm{e}}^{-u}\leq 1$ for $u\in\mathbb{R}_{+}$, we obtain the bound
$\bar{G}(1/t)\leq c_{\delta}t^{-1+\delta/2}$ for all $t>0$, where
$c_{\delta}\coloneqq 2^{-1+\delta/2}/((1-\delta/2)\Gamma(1+\delta/2))$,
implying†††This upper bound is of the same order as the lower bound in the
special case $\alpha=0$ of the Lemma 4.5.

$\operatorname{\mathbb{P}}(\nu_{n}\geq
t)=\operatorname{\mathbb{P}}_{1}(\tau_{0}>t)\leq
c_{\delta}t^{-1+\delta/2}\quad\text{for all $t\in(0,\infty)$ and any
$n\in\mathbb{N}$.}$ (4.14)

By (4.14), the i.i.d. sequence $(\nu_{n})_{n\in\mathbb{N}}$ satisfies
$\operatorname{\mathbb{E}}\nu_{n}^{(1-\delta/2)/(1+\varepsilon)}<\infty$ for
any $\varepsilon>0$. Since $0<1-\delta/2<1$, the Marcinkiewicz–Zygmund strong
law [21, Thm. 4.23] yields: a.s.,

$\sum_{n=1}^{N}\nu_{n}\leq N^{\frac{1}{1-\delta/2}+\varepsilon}\quad\text{for
all but finitely many $N\in\mathbb{N}$.}$ (4.15)

Define $\upsilon_{n}:=\varphi_{n}-\vartheta_{n-1}$, for $n\in\mathbb{N}$, to
be the time taken to exit the interval $[0,1]$ started from $0$, after time
$\vartheta_{n-1}$. Since $y\mapsto\operatorname{\mathbb{P}}_{y}(Y_{1}>1)$ is
positive and continuous on $[0,1]$, there exists a constant $c_{0}>0$ such
that $\operatorname{\mathbb{P}}_{y}(Y_{1}>1)>c_{0}$ for all $y\in[0,1]$. Thus,
$\operatorname{\mathbb{P}}(\upsilon_{n}\geq
t)\leq\operatorname{\mathbb{P}}_{0}(\cap_{i=1}^{\lfloor t\rfloor}\\{Y_{i}\leq
1\\})\leq(1-c_{0})^{\lfloor t\rfloor}$ for $t\geq 1$ (here $\lfloor
t\rfloor\coloneqq\max\\{i\in\mathbb{Z}_{+}:i\leq t\\}$), implying
$\operatorname{\mathbb{E}}\upsilon_{n}<\infty$ for all $n\in\mathbb{N}$. Since
the sequence $(v_{n})_{n\in\mathbb{N}}$ is i.i.d., the strong law of large
numbers says
$N^{-1}\sum_{n=1}^{N}\upsilon_{n}\to\operatorname{\mathbb{E}}\nu_{1}$ as
$N\to\infty$ a.s. Recalling $\nu_{n}=\vartheta_{n}-\varphi_{n}$, by (4.15) and
since $1<1/(1-\delta/2)$, for any $\varepsilon>0$ we obtain a.s.

$\displaystyle\vartheta_{N}=\sum_{n=1}^{N}(\vartheta_{n}-\vartheta_{n-1})=\sum_{n=1}^{N}\nu_{n}+\sum_{n=1}^{N}\upsilon_{n}\leq
N^{\frac{1}{1-\delta/2}+\varepsilon}\quad\text{for all large
$N\in\mathbb{N}$.}$ (4.16)

By (4.10), for every $t\in\mathbb{R}_{+}$ we have $N_{t}<\infty$ and
$t<\vartheta_{N_{t}+1}$ a.s. and
$N_{t}\overset{\textup{a.s.}}{\longrightarrow}\infty$ as $t\to\infty$. Thus,
by (4.16), for any $\varepsilon\in(0,1)$, a.s. we have $N_{t}\geq
t^{(1-\delta/2)/(1+\varepsilon)}\geq t^{(1-\delta/2)(1-\varepsilon)}$ for all
$t\in\mathbb{R}_{+}$ sufficiently large. This inequality and the bound in
(4.13), combined with (4.11), imply (4.3) when $\gamma=0$: for any
sufficiently small $\varepsilon>0$, a.s.,
$\int_{0}^{t}Y_{s}^{\alpha}\,\mathrm{d}s\geq\sum_{n=1}^{N_{t}}I_{\alpha,n}\geq
N_{t}^{\frac{1+\alpha}{1-\delta/2}-\varepsilon}$ and hence, for
$t\in\mathbb{R}_{+}$ sufficiently large,
$\int_{0}^{t}Y_{s}^{\alpha}\,\mathrm{d}s\geq t^{(1+\alpha)(1-2\varepsilon)}$.
Thus, for any (possibly random) $T>0$,

$\text{a.s.}\quad\int_{T}^{t}Y_{s}^{\alpha}\,\mathrm{d}s\geq
t^{(1+\alpha)(1-2\varepsilon)}\quad\text{for all sufficiently large
$t\in\mathbb{R}_{+}$.}$ (4.17)

If $\delta\geq 2$, since $Y$ with law ${\mathrm{BESQ}}^{\delta}(y)$ is a
continuous-state branching process with immigration, the process $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu+Y^{\prime}$, where
$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$ and
$Y^{\prime}$ are independent with laws ${\mathrm{BESQ}}^{1}(0)$ and
${\mathrm{BESQ}}^{\delta-1}(y)$, respectively, has the same law as $Y$ [31,
Thm. XI.1.2]. Since $Y^{\prime}_{s}\geq 0$ for $s\in\mathbb{R}_{+}$, by (4.17)
a.s. $\int_{0}^{t}(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{s}+Y^{\prime}_{s})^{\alpha}\,\mathrm{d}s\geq\int_{0}^{t}\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{s}^{\alpha}\,\mathrm{d}s\geq t^{(1+\alpha)(1-2\varepsilon)}$ for large
$t$, implying (4.17) for all $\delta>0$.

Suppose $\gamma>0$. By (3.5), for every $\varepsilon\in(0,\gamma\wedge 1)$ we
have $\lim_{t\to\infty}Y_{t}^{1-\varepsilon}/t=0$. Hence, there exists (a
random) $T\in\mathbb{R}_{+}$, such that $t^{\gamma}\geq
Y_{t}^{\gamma-\varepsilon}$ and
$\int_{0}^{t}s^{\gamma}Y_{s}^{\alpha}\,\mathrm{d}s\geq\int_{T}^{t}Y_{s}^{\gamma+\alpha-\varepsilon}\,\mathrm{d}s$
for $t\geq T$. Thus, by (4.17) (with $\alpha+\gamma-\varepsilon$ in place of
$\alpha$), (4.3) follows. If $\gamma\in(-\alpha,0)$, (4.3) remains valid by
(4.17) (with $T=1$) and $\int_{1}^{t}s^{\gamma}Y_{s}^{\alpha}\,\mathrm{d}s\geq
t^{\gamma}\int_{1}^{t}Y_{s}^{\alpha}\,\mathrm{d}s$, concluding the proof. ∎

###### Proof of Theorem 4.1.

Part (a). By (3.5), for every $\varepsilon>0$,
$\lim_{t\to\infty}Y_{t}/t^{1+\varepsilon}=0$ a.s. Hence, a.s. $Y_{t}\leq
t^{1+\varepsilon}$ for all sufficiently large $t\in\mathbb{R}_{+}$. Thus, by
(1.7), $A_{t}\leq t^{1+\gamma+\alpha+\varepsilon}$ for large $t$, implying

$\limsup_{t\to\infty}\frac{\log A_{t}}{\log t}\leq 1+\gamma+\alpha,\
\text{a.s.}$ (4.18)

The lower bound in (4.1) follows from (4.3) in Proposition 4.2, since, for any
$\varepsilon>0$,

$\liminf_{t\to\infty}\frac{\log A_{t}}{\log t}\geq
1+\alpha+\gamma-\varepsilon+\liminf_{t\to\infty}\frac{\log(t^{\varepsilon}A_{t}/t^{1+\alpha+\gamma})}{\log
t}\geq 1+\gamma+\alpha-\varepsilon\ \text{a.s.}$

Part (b). By self-similarity of the Bessel process, it follows that, for every
$t\in(0,\infty)$ and starting point $y\in\mathbb{R}_{+}$, the process
$(t^{-1}Y_{ts})_{s\in\mathbb{R}_{+}}$ follows ${\mathrm{BESQ}}^{\delta}(y/t)$,
where $Y$ has the law ${\mathrm{BESQ}}^{\delta}(y)$. Moreover, for
$\varepsilon_{t}\coloneqq y/t\geq 0$ and $\widetilde{Y}$ with law
${\mathrm{BESQ}}^{\delta}(0)$, the process
$(\widetilde{Y}_{s+\tilde{\tau}_{\varepsilon_{t}}})_{s\in\mathbb{R}_{+}}$ also
follows ${\mathrm{BESQ}}^{\delta}(y/t)$ by the strong Markov property, where
$\tilde{\tau}_{\varepsilon_{t}}=\inf\\{s\in\mathbb{R}_{+}:\widetilde{Y}_{s}=\varepsilon_{t}\\}$.
By substituting $s=tv$ in definition (1.7) of $A_{t}$, we obtain (4.2):

$\displaystyle
A_{t}/t^{1+\gamma+\alpha}=\int_{1/t}^{1}v^{\gamma}(t^{-1}Y_{tv})^{\alpha}\,\mathrm{d}v\overset{\textup{d}}{=}\int_{1/t}^{1}v^{\gamma}\widetilde{Y}_{v+\tilde{\tau}_{\varepsilon_{t}}}^{\alpha}\,\mathrm{d}v\overset{\textup{a.s.}}{\longrightarrow}\int_{0}^{1}v^{\gamma}\widetilde{Y}_{v}^{\alpha}\,\mathrm{d}v\quad\text{
as $t\to\infty$.}$ (4.19)

If $\gamma\geq 0$, the a.s. convergence in (4.19) clearly holds. If
$\gamma\in(-\alpha,0)$, and hence $\alpha>0$, time-reversal
$(\widetilde{Y}_{t})_{t>0}\overset{\textup{d}}{=}(t^{2}\widetilde{Y}_{1/t})_{t>0}$
and bound (3.5) imply a.s. $\widetilde{Y}_{v}\leq v^{1-\varepsilon}$ for small
$v>0$ and any $\varepsilon\in(0,1\wedge(1+\gamma/\alpha))$. Thus
$\int_{0}^{1}v^{\gamma}\widetilde{Y}_{v}^{\alpha}\,\mathrm{d}v<\infty$ and the
a.s. limit holds.

Part (c). The scaling property implies $Y_{t}\overset{\textup{d}}{=}t\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{1}$, where $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$ follows
${\mathrm{BESQ}}^{\delta}(y/t)$. Thus, for $\alpha\geq 0$, there is a constant
$C_{\delta,\alpha,y}<\infty$, such that
$\operatorname{\mathbb{E}}Y_{t}^{\alpha}=t^{\alpha}\operatorname{\mathbb{E}}\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{1}^{\alpha}\leq
C_{\delta,\alpha,y}t^{\alpha}$ for all $t\in[1,\infty)$. Since
$1+\gamma+\alpha>0$, definition (1.7) implies
$\operatorname{\mathbb{E}}A_{t}=O(t^{1+\gamma+\alpha})$ as $t\to\infty$. ∎

###### Proof of Theorem 1.1.

Theorem 4.1 establishes that hypothesis ((AY)) is satisfied by $Y$ following
${\mathrm{BESQ}}^{\delta}(y)$, provided that $\delta>0$,
$\alpha\in\mathbb{R}_{+}$, and $\gamma+\alpha>0$. Indeed, Theorem 4.1(a)
implies hypothesis ((AY))(b), Theorem 4.1(b) implies ((AY))(a), with limit
$\widetilde{A}=\int_{0}^{1}s^{\gamma}\widetilde{Y}^{\alpha}_{s}\,\mathrm{d}s$,
and Theorem 4.1(c) implies ((AY))(c). Hence Theorem 1.1 is a consequence of
Theorem 1.3. ∎

##  Appendix A Existence, uniqueness and positivity for SDE (1.3)

Let $(Y,B)$ be an adapted process, on a filtered probability space, where $B$
is a scalar Brownian motion with respect to the given filtration and $Y$ is a
continuous adapted process in $\mathbb{R}_{+}$.

Existence and pathwise uniqueness. As noted in [12], the SDE in (1.3) for $S$
is a special case of the Doléans-Dade and Protter’s equation, driven by the
semimartingale $B$ and an adapted continuous process
$K=(K_{t})_{t\in\mathbb{R}_{+}}$, defined by
$K_{t}\coloneqq\int_{0}^{t}(2f(s,Y_{s})+1)\,\mathrm{d}s$ for some continuous
function $f:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$. This
equation is well-known [20] to possess a solution and satisfy pathwise
uniqueness (making every solution strong). In [12], Deelstra & Delbaen
construct the solution $S$ of SDE (1.3) (which may a priori take values in
$\mathbb{R}$ with the volatility coefficient given by $\sqrt{|S_{s}|}$) from
an Euler scheme approximation, and prove that if $S_{0}\in\mathbb{R}_{+}$ then
$S_{t}\in\mathbb{R}_{+}$ for all $t\in\mathbb{R}_{+}$ a.s. We stress that,
beyond $Y$ and $B$ being adapted to the same filtration, no assumption is made
on the dependence between the two processes (or on the dynamics of $Y$).

Positivity. Consider now the strong solution $S$ of (1.3), driven by $B$ and
$K$ and started at $S_{0}\in\mathbb{R}_{+}$. Assume in addition that
$1+2f\geq\delta_{0}$ a.s. for all $t>0$ and a constant $\delta_{0}>0$. Since
the squared Bessel SDE in (1.6) has pathwise uniqueness, we may construct a
squared Bessel process $Z$, satisfying SDE (1.6) with “dimension” parameter
$\delta_{0}$, started at $Z_{0}=0$ and driven by the Brownian motion $B$. Note
that by (1.3) and (1.6) the quadratic variation of the semimartingale $S-Z$
equals $\langle
S-Z\rangle_{t}=4\int_{0}^{t}(\sqrt{S_{s}}-\sqrt{Z_{s}})^{2}\,\mathrm{d}s$. In
particular, since $(\sqrt{y}-\sqrt{x})^{2}\leq|x-y|$ for all
$x,y\in\mathbb{R}_{+}$, we have

$\int_{0}^{t}\mathbbm{1}_{\\{Z_{s}-S_{s}>0\\}}\left|\sqrt{S_{s}}-\sqrt{Z_{s}}\right|^{-1}\,\mathrm{d}\langle
S-Z\rangle_{s}\leq 4t<\infty\quad\text{a.s. for all $t\in\mathbb{R}_{+}$.}$

By [31, Lem. IX.3.3], the local time $L^{0}(S-Z)=0$ vanishes. Since $S_{0}\geq
Z_{0}=0$, the Tanaka formula [31, Thm. VI.1.2] on $[0,\tau_{N}]$, where
$\tau_{N}\coloneqq\inf\\{s\geq 0:S_{s}\geq N\\}$ for $N>0$, yields
$\operatorname{\mathbb{E}}\max\\{0,Z_{t\wedge\tau_{N}}-S_{t\wedge\tau_{N}}\\}\leq
0$ (since $s\delta_{0}\leq K_{s}$ for all $s\in\mathbb{R}_{+}$ and $x\wedge
y\coloneqq\min\\{x,y\\}$ for $x,y\in\mathbb{R}$). Thus
$Z_{t\wedge\tau_{N}}\leq S_{t\wedge\tau_{N}}$, implying $S_{t}\geq Z_{t}$ a.s.
for all $t\in\mathbb{R}_{+}$ (since $\lim_{N\uparrow\infty}\tau_{N}=\infty$).

Bessel-type representation (1.1) of $X=\sqrt{S}$. Since $f\geq 0$, we can
always take $\delta_{0}=1$, implying that the modulus of BM (i.e. $\sqrt{Z}$
for $\delta_{0}=1$) bounds $X=\sqrt{S}$ from below, making the point $0$
instantaneously reflecting for $X$. Moreover, if we had $f\geq\varepsilon/2$
for for some $\varepsilon>0$, a Bessel process $\sqrt{Z}$ with “dimension”
$\delta_{0}=1+\varepsilon$, which satisfies the Bessel SDE and, in particular,
$\int_{0}^{t}\,\mathrm{d}s/\sqrt{Z_{s}}<\infty$ for all $t\in\mathbb{R}_{+}$,
would bound $X$ from below. Thus,

$\int_{0}^{t}(f(s,Y_{s})/X_{s})\,\mathrm{d}s\leq\int_{0}^{t}(f(s,Y_{s})/\sqrt{Z_{s}})\,\mathrm{d}s<\infty\quad\text{for
all $t\in\mathbb{R}_{+}$.}$ (A.1)

Once we know the integral in (A.1) is finite, by considering the excursions of
$X$ away from $0$, it is not hard to see that the quadratic variation at $t$
of the continuous process $X-\int_{0}^{\cdot}(f(s,Y_{s})/X_{s})\,\mathrm{d}s$
equals $t$, making it, by Lévy’s characterisation, a Brownian motion and thus
implying SDE (1.1) for $X$. However Assumption ((Af)) is not consistent with
$f\geq\varepsilon/2$ when $\alpha>0$ (at large $t$, by (1.2), $f(t,y)\to 0$ as
$y\downarrow 0$). But we may assume $f>0$ on $(0,\infty)\times(0,\infty)$,
implying (A.1) on the stochastic interval $[0,\tau^{(f)}_{\varepsilon})$,
where
$\tau^{(f)}_{\varepsilon}\coloneqq\\{t\in\mathbb{R}_{+}:f(t,Y_{t})=\varepsilon/2\\}$
for small $\varepsilon>0$. Since, for $\delta\geq 2$ and $y>0$, the point $0$
is polar for $Y\sim{\mathrm{BESQ}}^{\delta}(y)$, we have
$\tau^{(f)}_{\varepsilon}\to\infty$ as $\varepsilon\downarrow 0$ implying SDE
(1.1) for $X$ in this case.

If $\delta\in(0,2)$, $Y$ hits zero a.s., bringing SDE (1.1) into the realm of
the Bessel process with $\delta=1$ (which does not satisfy the corresponding
SDE) at times $Y_{t}=0$. This case would require the analysis of the joint
zeros $S_{t}=Y_{t}=0$, where the dependence in $(Y,B)$ clearly matters.

## Acknowledgements

The authors are grateful to Mikhail Menshikov and Vadim Shcherbakov for useful
conversations. CdC and AW are supported by EPSRC grant EP/W00657X/1. MB is
supported by EPSRC grant EP/V009478/1. AM is supported by EPSRC grants
EP/V009478/1 and EP/W006227/1 and, through The Alan Turing Institute, by
EP/X03870X/1.

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