# Stochastic selection problem for a Stratonovich SDE with power non-linearity

Ilya Pavlyukevich and Georgiy Shevchenko  Institute of Mathematics, Friedrich
Schiller University Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany
<EMAIL_ADDRESS>Kyiv School of Economics, Mykoly Shpaka 3, Kyiv
03113, Ukraine<EMAIL_ADDRESS>
###### Abstract.

In our paper [Bernoulli 26(2), 2020, 1381–1409], we found all strong Markov
solutions that spend zero time at $0$ of the Stratonovich stochastic
differential equation $\hskip 1.29167pt\mathrm{d}X=|X|^{\alpha}\circ\hskip
1.29167pt\mathrm{d}B$, $\alpha\in(0,1)$. These solutions have the form
$X_{t}^{\theta}=F(B^{\theta}_{t})$, where
$F(x)=\frac{1}{1-\alpha}|x|^{1/(1-\alpha)}\operatorname{sign}x$ and
$B^{\theta}$ is the skew Brownian motion with skewness parameter
$\theta\in[-1,1]$ starting at $F^{-1}(X_{0})$. In this paper we show how an
addition of small external additive noise $\varepsilon W$ restores uniqueness.
In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion
corresponding to the physically symmetric case $\theta=0$.

Keywords: Stratonovich integral, non-uniqueness, singular stochastic
differential equation, skew Brownian motion, selection problem, generalized
Itô’s formula, local time, heterogeneous diffusion process.

MSC 2010 subject classification: Primary 60H10; secondary 60J55, 60J60.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and the main result
  3. 3 Analysis of the SDE (2.5) with singular drift
    1. 3.1 Weak existence. Proof of Theorem 2.3
    2. 3.2 Pathwise uniqueness. Proof of Theorem 2.4
  4. 4 Analysis of the SDE (1.1) with the bracket
    1. 4.1 SDE (1.16), strong existence. Proof of Theorem 2.6
    2. 4.2 SDE (1.16): uniqueness. Proof of Theorem 2.8
  5. 5 Stochastic selection. Proof of Theorem 2.9

##  1\. Introduction

In [4], Cherstvy et al. considered the so-called _heterogeneous_ diffusion
process defined as a solution of the Stratonovich stochastic differential
equation

(1.1) $X_{t}=x+\int_{0}^{t}|X_{s}|^{\alpha}\circ\hskip
1.29167pt\mathrm{d}B_{s}$

with $\alpha\in\mathbb{R}$ and $B$ being a standard Brownian motion. It is
always assumed that $|x|^{\alpha}=|x|^{\alpha}\cdot\mathbb{I}(x\neq 0)$, so
that $|0|^{\alpha}=0$ for any $\alpha\in\mathbb{R}$. This equation can be seen
as a Stratonovich version of the famous diffusion

(1.2) $\displaystyle X_{t}^{G}=x+\int_{0}^{t}|X_{s}^{G}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}$

firstly studied by Girsanov in [7]. It is well-known that equation (1.2) has a
unique strong solution for $\alpha\geq 1/2$ and has infinitely many solutions
for $\alpha\in(0,1/2)$. A complete analysis of equation (1.2) can be found in
Chapter 5 of Cherny and Engelbert [3].

It is clear that in the Stratonovich setting, the presence of the irregular
point $\\{0\\}$ also affects the existence and uniqueness of solutions of
(1.1).

The physicists’ approach to (1.1) exploited in the papers by Cherstvy et al.
[4] and Sandev et al. [15] is the solution by regularization. Indeed, let
$\alpha\in(0,1)$ and let $\\{\sigma_{\varepsilon}\\}$ be a family of smooth
functions with integrable derivatives such that $\sigma_{\varepsilon}(x)>0$
and $|\sigma_{\varepsilon}(x)-|x|^{\alpha}|\to 0$ uniformly on compacts as
$\varepsilon\to 0$. Consider a _regularized equation_ $\hskip
1.29167pt\mathrm{d}Z^{\varepsilon}_{t}=\sigma_{\varepsilon}(Z^{\varepsilon}_{t})\circ\hskip
1.29167pt\mathrm{d}B_{t}$ that can be equivalently written in the Itô form as

(1.3) $\displaystyle\hskip
1.29167pt\mathrm{d}Z^{\varepsilon}_{t}=\sigma_{\varepsilon}(Z^{\varepsilon}_{t})\,\hskip
1.29167pt\mathrm{d}B_{t}+\frac{1}{2}\sigma_{\varepsilon}(Z^{\varepsilon}_{t})\sigma^{\prime}_{\varepsilon}(Z^{\varepsilon}_{t})\,\hskip
1.29167pt\mathrm{d}t,\quad Z^{\varepsilon}_{0}=x.$

There exists a unique strong solution $Z^{\varepsilon}$ to (1.3) that can be
found explicitly with the help of the well-known Lamperti method. Indeed, let

(1.4) $\displaystyle F_{\varepsilon}(x):=\int_{0}^{x}\frac{\hskip
1.29167pt\mathrm{d}y}{\sigma_{\varepsilon}(y)},\quad x\in\mathbb{R}.$

The function $F_{\varepsilon}$ is monotonically increasing and $C^{1}$-smooth
with locally integrable second derivative. The Itô formula with generalized
derivatives from Krylov [8, Chapter 2, Section 10] is applicable and yields

(1.5) $\displaystyle
F_{\varepsilon}(Z^{\varepsilon}_{t})=F_{\varepsilon}(x)+B_{t}$

and hence

(1.6) $\displaystyle
Z^{\varepsilon}_{t}=F_{\varepsilon}^{-1}(B_{t}+F_{\varepsilon}(x)).$

Now one can argue that upon passing to the limit as $\varepsilon\to 0$, the
processes $Z^{\varepsilon}$ converge to the solution $X$ of the original SDE
(1.1). Unfortunately, the rigorous justification of this intuitively
transparent procedure turns out to be tricky. Let us illustrate this by an
example.

######  Example 1.1.

We assume that $\alpha\in(-1,1)$ and for brevity we set the initial value
$x=0$. We consider the family
$\\{\sigma_{\varepsilon}\\}_{\varepsilon\in(0,1]}$ of the form

(1.7)
$\displaystyle\sigma_{\varepsilon}(x)=\sqrt{|x|^{2\alpha}+\varepsilon^{2}},\quad
x\in\mathbb{R},$

see Figure 1 (left).

Figure 1. The function $\sigma(x)=|x|^{\alpha}$ and
$\sigma_{\varepsilon}(x)=\sqrt{|x|^{2\alpha}+\varepsilon^{2}}$ (left) and
$\sigma_{\varepsilon}(x)$ defined in (1.10) (right).

Then, a straightforward calculation yields that

(1.8) $\displaystyle F_{\varepsilon}(x)$ $\displaystyle\to
F_{0}(x):=\frac{1}{1-\alpha}|x|^{1-\alpha}\operatorname{sign}x,$
$\displaystyle F^{-1}_{\varepsilon}(x)$ $\displaystyle\to
F_{0}^{-1}(x):=|(1-\alpha)x|^{\frac{1}{1-\alpha}}\operatorname{sign}x$

so that the limit process

(1.9) $\displaystyle X_{t}^{0}:=F_{0}^{-1}(B_{t})$

is a candidate for a solution to (1.1)

On the other hand, let $A,B\geq 0$ and let

(1.10) $\displaystyle\sigma_{\varepsilon}(x)=\begin{cases}\varepsilon,\quad
x\in[-A\varepsilon,B\varepsilon],\\\
|x-B\varepsilon+\varepsilon^{1/\alpha}|^{\alpha},\quad x>B\varepsilon,\\\
|x+A\varepsilon-\varepsilon^{1/\alpha}|^{\alpha},\quad
x<-A\varepsilon,\end{cases}$

see Figure 1 (right). Applying the Lamperti transformation and passing to the
limit as above we find that

(1.11) $\displaystyle F^{-1}_{\varepsilon}(x)\to
F^{-1}_{A,B}(x)=\begin{cases}0,\quad x\in[A,B],\\\
-|(1-\alpha)(x+A)|^{\frac{1}{1-\alpha}},\quad x<-A,\\\
|(1-\alpha)(x-B)|^{\frac{1}{1-\alpha}},\quad x>B.\end{cases}$

Hence another candidate for the solution is the process

(1.12) $\displaystyle X^{A,B}_{t}:=F^{-1}_{A,B}(B_{t}).$

Note that such a process $X^{A,B}$ spends positive time at zero.

It turns out that the both processes $X^{0}$ and $X^{A,B}$ are indeed strong
solutions of the equation (1.1) for $\alpha\in(-1,1)$. This follows by a
straightforward verification with the help of the generalized Itô formula by
Föllmer, Protter and Shiryaev [6].

Excluding solutions spending positive time at zero as non-physical does not
always guarantee uniqueness. Indeed, in our previous paper [10], the following
theorem was proven.

######  Theorem 1.2 (Theorem 4.5, [10]).

Let $x\in\mathbb{R}$ and $\theta\in[-1,1]$, and let $B^{\theta}$ be a skew
Brownian motion that is a unique strong solution of the SDE

(1.13) $\displaystyle B^{\theta}=F^{-1}_{0}(x)+B_{t}+\theta
L_{t}(B^{\theta}),$

$L(B^{\theta})$ being the symmetric local time of $B^{\theta}$ at zero.

1\. Let $\alpha\in(0,1)$. Then $X^{\theta}_{t}=F_{0}(B^{\theta}_{t})$ is a
strong solution of (1.1) which is a homogeneous strong Markov process spending
zero time at $0$.

Moreover, $X^{\theta}$ is the unique strong solution of (1.1) which is a
homogeneous strong Markov process spending zero time at $0$ and such that

$\displaystyle\mathbf{P}(X^{\theta}_{t}\geq 0\mid
X_{0}=0)=\frac{1+\theta}{2},\quad t>0.$

2\. Let $\alpha\in(-1,0]$. Then $X^{0}_{t}=F(B_{t}^{0})$ is the unique strong
solution of (1.1) which is a homogeneous strong Markov process spending zero
time at $0$.

In other words, for $\alpha\in(0,1)$, the equation (1.1) is underdetermined
even in the class of homogeneous strong Markov process spending zero time at
$0$. In this respect, equation (1.1) can be seen as a stochastic counterpart
of the deterministic non-Lipschitzian equation $\dot{x}_{t}=|x_{t}|^{\alpha}$
that has infinitely many solutions if started at $x\leq 0$.

In general, these solutions spend positive time at $0$. However one can always
single out the unique _maximal_ solution spending zero time at 0. Such a non-
uniqueness feature of a ODE with continuous coefficients is known as the Peano
phenomenon. It is also known that uniqueness, or singling out of physically
meaningful solutions spending zero time at $0$, can be restored by means of
the _regularization by noise_ , studied for the first time by Bafico and Baldi
in [1] and Veretennikov [17]. In particular Theorem 5.1 in [1]) implies that
solutions of the SDE

(1.14) $\displaystyle
x_{t}^{\varepsilon}=x+\int_{0}^{t}|x^{\varepsilon}_{s}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}s+\varepsilon W_{t}$

driven by a small Brownian motion $\varepsilon W$ weakly converge to the
maximal deterministic solution

(1.15) $\displaystyle
x_{t}^{*}(x)=\begin{cases}\big{(}(1-\alpha)t+x^{1-\alpha}\big{)}^{\frac{1}{1-\alpha}},\quad
x>0,\\\
-\big{(}|x|^{1-\alpha}-(1-\alpha)t\big{)}_{+}^{\frac{1}{1-\alpha}}+\big{(}(1-\alpha)t-|x|^{1-\alpha}\big{)}_{+}^{\frac{1}{1-\alpha}},\quad
x\leq 0,\end{cases}$

that spends zero time at $0$. More results on the stochastic Peano phenomenon
can be found in the works by Delarue and Flandoli [5], Trevisan [16],
Pilipenko and Proske [12], Pilipenko and Pavlyukevich [11].

The goal of this paper is to apply the “regularization” procedure to the SDE
(1.1), i.e. to consider the perturbed SDE

(1.16) $\displaystyle
X_{t}^{\varepsilon}=x+\int_{0}^{t}|X_{s}^{\varepsilon}|^{\alpha}\circ\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon W_{t}$

in the presence of another independent Brownian motion $W$. From the physical
point of view, the Brownian motion $W$ with vanishing amplitude $\varepsilon$
represents the _ambient environmental noise_ whereas the Brownian motion $B$
is responsible for the diffusive behaviour of the “heavy” particle
$X^{\varepsilon}$ in the non-homogeneous medium determined by the non-linear
“temperature profile” $x\mapsto|x|^{\alpha}$.

The qualitative result of the present paper is that solutions of the
regularized equation (1.16) converge to the _benchmark_ solution

(1.17) $\displaystyle
X^{0}_{t}=F_{0}(B_{t})=\Big{|}(1-\alpha)B_{t}+|x|^{1-\alpha}\operatorname{sign}x\Big{|}^{\frac{1}{1-\alpha}}\operatorname{sign}\Big{(}(1-\alpha)B_{t}+|x|^{1-\alpha}\operatorname{sign}x\Big{)}$

that spends zero time at $0$ and has no “asymmetry” at the origin.

This result is a manifestation of the “selection” procedure in the SDE case.
In the limit as $\varepsilon\to 0$, the regularized solution $X^{\varepsilon}$
always “selects” the natural solution (1.17) among all possible solutions of
(1.1) that may have asymmetric behaviour at 0, stay positive time at zero, or
be non-Markovian, etc.

##  2\. Preliminaries and the main result

Throughout the article, we work on a stochastic basis
$(\Omega,\mathscr{F},\mathbb{F},\mathbf{P})$, i.e. a complete probability
space with a filtration $\mathbb{F}=(\mathscr{F}_{t})_{t\geq 0}$ satisfying
the standard assumptions. The process $(B,W)=(B_{t},W_{t})_{t\geq 0}$ is a
standard continuous two-dimensional Brownian motion on this stochastic basis.

First we briefly recall definitions related to stochastic integration. More
details may be found in Protter [13].

The main mode of convergence considered here is the uniform convergence on
compacts in probability (the u.c.p. convergence for short): a sequence
$X^{n}=(X_{t}^{n})_{t\geq 0}$, $n\geq 1$, of stochastic processes converges to
$X=(X_{t})_{t\geq 0}$ in u.c.p. if for any $t\geq 0$

(2.1)
$\displaystyle\sup_{s\in[0,t]}\left|X^{n}_{s}-X_{s}\right|\overset{\mathbf{P}}{\to}0,\quad
n\to\infty.$

Let a sequence of (deterministic) partitions
$D_{n}=\\{0=t^{n}_{0}<t_{1}^{n}<t_{2}^{n}<\cdots\\}=\\{0=t_{0}<t_{1}<t_{2}<\cdots\\}$
of $[0,\infty)$, $n\geq 1$, be such that for each $t\geq 0$ the number of
points in each interval $[0,t]$ is finite, and the mesh
$\|D_{n}\|:=\sup_{k\geq 1}\left|t_{k}^{n}-t_{k-1}^{n}\right|\to 0$ as
$n\to\infty$. A continuous stochastic process $X$ has quadratic variation
$[X]$ along the sequence $\\{D_{n}\\}_{n\geq 1}$ if the limit

$\displaystyle{}[X]_{t}:=\lim_{n\to\infty}\sum_{t_{k}\in
D_{n},t_{k}<t}|X_{t_{k+1}}-X_{t_{k}}|^{2}$

exists in the u.c.p. sense. Similarly, the quadratic covariation $[X,Y]$ of
two continuous stochastic processes $X$ and $Y$ is defined as a limit in
u.c.p.

$\displaystyle{}[X,Y]_{t}:=\lim_{n\to\infty}\sum_{t_{k}\in
D_{n},t_{k}<t}(X_{t_{k+1}}-X_{t_{k}})(Y_{t_{k+1}}-Y_{t_{k}}).$

When $X$ and $Y$ are semimartingales, the quadratic variations $[X]$, $[Y]$
and the quadratic covariation $[X,Y]$ exist, moreover, they have bounded
variation on any finite interval.

Further, we define the Itô integral as a limit in u.c.p.

$\displaystyle\int_{0}^{t}X_{s}\,\hskip
1.29167pt\mathrm{d}Y_{s}=\lim_{n\to\infty}\sum_{t_{k}\in
D_{n},t_{k}<t}X_{t_{k}}(Y_{t_{k+1}}-Y_{t_{k}})$

and the Stratonovich (symmetric) integral as a limit in u.c.p.

$\displaystyle\int_{0}^{t}X_{s}\circ\hskip
1.29167pt\mathrm{d}Y_{s}=\int_{0}^{t}X_{s}\,\hskip
1.29167pt\mathrm{d}Y_{s}+\frac{1}{2}[X,Y]_{t}$
$\displaystyle=\lim_{n\to\infty}\sum_{t_{k}\in
D_{n},t_{k}<t}\frac{1}{2}(X_{t_{k+1}}+X_{t_{k}})(Y_{t_{k+1}}-Y_{t_{k}}),$

provided that both the Itô integral and the quadratic variation exist. Again,
when both $X$ and $Y$ are continuous semimartingales, both integrals exists,
and the convergence holds in u.c.p.

In this paper, we define

(2.2) $\displaystyle\operatorname{sign}x:=\begin{cases}-1,\ x<0,\\\ 0,\
x=0,\\\ 1,\ x>0,\end{cases}$

and for any $\alpha\in\mathbb{R}$ we set

(2.3) $\displaystyle|x|^{\alpha}:=\begin{cases}|x|^{\alpha},\ x\neq 0,\\\ 0,\
x=0.\end{cases}$

We also denote

(2.4) $\displaystyle(x)^{\alpha}:=|x|^{\alpha}\operatorname{sign}x.$

Throughout the article, $C$ will be used to denote a generic constant, whose
concrete value is not important and may change between lines.

First we give the intuition that will lead to the main result.

Let us formally rewrite equation (1.16) in the Itô form according to the
formula (1.3). Thus we obtain another SDE

(2.5) $\displaystyle
Y^{\varepsilon}_{t}=x+\int_{0}^{t}|Y^{\varepsilon}_{s}|^{\alpha}\hskip
1.29167pt\mathrm{d}B_{s}+\frac{\alpha}{2}\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s+\varepsilon W_{t}.$

Note that the SDEs (1.1) and (2.5) are of different nature and do not have to
be equivalent.

Assume now that $Y^{\varepsilon}$ is the solution of (2.5). Then it is a
Markov process with the generator

(2.6) $\displaystyle
A^{\varepsilon}f(y)=\frac{|y|^{2\alpha}+\varepsilon^{2}}{2}f^{\prime\prime}(y)+\frac{\alpha}{2}(y)^{2\alpha-1}f(y),\quad
f\in C^{2}(\mathbb{R},\mathbb{R}).$

However, $A^{\varepsilon}$ is also the generator of the diffusion

(2.7) $\displaystyle Z^{\varepsilon}_{t}$
$\displaystyle=x+\int_{0}^{t}\sqrt{|Z^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}\,\hskip
1.29167pt\mathrm{d}\widetilde{B}_{s}+\frac{\alpha}{2}\int_{0}^{t}(Z^{\varepsilon}_{s})^{2\alpha-1}\hskip
1.29167pt\mathrm{d}s$
$\displaystyle=x+\int_{0}^{t}\sqrt{|Z^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}\circ\hskip
1.29167pt\mathrm{d}\widetilde{B}_{s}.$

driven by some other Brownian motion $\widetilde{B}$. The solution to equation
(2.7) has been determined explicitly in (1.6) as
$Z^{\varepsilon}_{t}=F^{-1}_{\varepsilon}(\widetilde{B}_{t}+F_{\varepsilon}(x))$,
and hence we get that

(2.8) $\displaystyle Y^{\varepsilon}_{t}\stackrel{{\scriptstyle\hskip
0.90417pt\mathrm{d}}}{{=}}F^{-1}_{\varepsilon}(\widetilde{B}_{t}+F_{\varepsilon}(x)),\quad
t\geq 0.$

It is clear that $Z^{\varepsilon}$, and also $Y^{\varepsilon}$, converges in
law to the benchmark solution $X^{0}$. The same convergence will also hold for
solutions $X^{\varepsilon}$ of the perturbed Stratonovich equation (1.1) if
one establishes an equivalence between the equations (1.1) and (2.5).

Now let us turn to equations (1.1) and (1.16). The concept of strong solution
for these equations is defined in a standard manner.

######  Definition 2.1.

A strong solution to (1.16) is a continuous stochastic process $X$ such that

  1. (1)

$X^{\varepsilon}$ is adapted to the augmented natural filtration of $(B,W)$;

  2. (2)

for any $t\geq 0$, the integral
$\int_{0}^{t}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip 1.29167pt\mathrm{d}B_{s}$
and the quadratic covariation $[|X^{\varepsilon}|^{\alpha},\ B]_{t}$ exist;

  3. (3)

for any $t\geq 0$, equation (1.16) holds $\mathbf{P}$-a.s.

######  Definition 2.2.

A strong solution to (2.5) is a continuous stochastic process
$Y^{\varepsilon}$ such that

  1. (1)

$Y^{\varepsilon}$ is adapted to the augmented natural filtration of $(B,W)$;

  2. (2)

for any $t\geq 0$, the integrals
$\int_{0}^{t}|Y^{\varepsilon}_{s}|^{\alpha}\,\hskip 1.29167pt\mathrm{d}B_{s}$
and $\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s$ exist;

  3. (3)

for any $t\geq 0$, equation (2.5) holds $\mathbf{P}$-a.s.

The weak solutions are defined in a standard way.

The main results of this paper are given in the following theorems. The proofs
will be presented in the subsequent sections.

######  Theorem 2.3.

For any $\varepsilon>0$ and any $\alpha\in(-1,1)$, equation (2.5) has a weak
solution, where for $\alpha\in(-1,0)$, the integral
$\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\hskip 1.29167pt\mathrm{d}s$ is
understood in the v.p. sense, and for $\alpha=0$,
$\frac{\alpha}{2}\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\hskip
1.29167pt\mathrm{d}s:=0$.

######  Theorem 2.4.

For any $\varepsilon>0$ and any $\alpha\in(0,1)$, the solution to (2.5) is
pathwise unique.

Applying the Yamada–Watanabe theorem, we arrive at the following result.

######  Theorem 2.5.

For any $\varepsilon>0$ and $\alpha\in(0,1)$, equation (2.5) has a unique
strong solution.

Concerning the equivalence of (1.1) and (2.5) the following holds true.

######  Theorem 2.6.

For any $\alpha\in(0,1)$, a strong solution to the stochastic differential
equation (2.5) is also a strong solution to (1.16).

To establish strong uniqueness of equation (1.16), we show that any of its
strong solutions also solves (2.5) and appeal to Theorem (2.5). We will
establish the equivalence of semimartingale solutions. In other words, we
assume that the bracket process $[|X^{\varepsilon}|^{\alpha},B]$ has locally
bounded variation.

######  Remark 2.7.

It will follow from Lemma 4.1 that for any strong solution $Y^{\varepsilon}$
to (2.5), $\alpha\in(0,1)$, the process $[|Y^{\varepsilon}|^{\alpha},B]$ is
absolutely continuous and hence is of locally bounded variation.

######  Theorem 2.8.

Let $\varepsilon>0$ and let $X^{\varepsilon}$ be a strong solution to (1.16)
such that the bracket $[|X^{\varepsilon}|^{\alpha},B]$ is a process of locally
bounded variation. Then it is also a strong solution to (2.5). In particular,
strong existence and uniqueness holds for equation (1.16) in the class of
semimartingale solutions.

Eventually we formulate the main qualitative result about the stochastic
selection of the benchmark solution.

######  Theorem 2.9.

Let $\alpha\in(0,1)$. Let $X^{\varepsilon}$ be a strong solution of (2.5) or a
semimartingale solution of (1.16). Let $X^{0}$ be a benchmark solution (1.17).
Then

(2.9) $\displaystyle X^{\varepsilon}\to X^{0}$

in u.c.p. as $\varepsilon\to 0$.

##  3\. Analysis of the SDE (2.5) with singular drift

We first address the question of weak existence of a solution of equation
(2.5). It can be established even for $\alpha\in(-1,1)$. However, for negative
$\alpha$, the drift should be understood in the principal value (v.p.), see
Cherny [2].

The following function that has already appeared above in the Lamperti method
will be crucial in showing both weak existence and pathwise uniqueness of
solution. Let

(3.1)
$\displaystyle\sigma_{\varepsilon}(y)=\sqrt{|y|^{2\alpha}+\varepsilon^{2}}$

and

(3.2) $F_{\varepsilon}(x)=\int_{0}^{x}\frac{\hskip
1.29167pt\mathrm{d}y}{\sigma_{\varepsilon}(y)},\quad x\in\mathbb{R},\
\varepsilon>0.$

Clearly, for each $\varepsilon>0$ and $\alpha\in(-1,1)$, the transformation
$F_{\varepsilon}\colon\mathbb{R}\to\mathbb{R}$ is a bijection. In the
neighbourhood of zero we have

(3.3) $\displaystyle
F_{\varepsilon}(x)\approx\begin{cases}\varepsilon^{-1}\cdot
x,\quad\alpha\in(0,1),\\\ (1+\varepsilon^{2})^{-1/2}\cdot x,\quad\alpha=0,\\\
(1+|\alpha|)^{-1}\cdot(x)^{1+|\alpha|},\quad\alpha\in(-1,0).\end{cases}$

and

(3.4) $\displaystyle\frac{\hskip 1.29167pt\mathrm{d}}{\hskip
1.29167pt\mathrm{d}x}F_{\varepsilon}(x)=\sigma_{\varepsilon}(x)\approx\begin{cases}\varepsilon^{-1},\quad\alpha\in(0,1),\\\
(1+\varepsilon^{2})^{-1/2}\cdot x,\quad\alpha=0,\\\
|x|^{|\alpha|},\quad\alpha\in(-1,0).\end{cases}$

Hence, $F^{-1}_{\varepsilon}$ is a monotonically increasing absolutely
continuous function, $\frac{\hskip 0.90417pt\mathrm{d}}{\hskip
0.90417pt\mathrm{d}x}F^{-1}_{\varepsilon}(x)=\sigma_{\varepsilon}(F_{\varepsilon}^{-1}(x))$,
and in the neighbourhood of zero

(3.5) $\displaystyle\frac{\hskip 1.29167pt\mathrm{d}}{\hskip
1.29167pt\mathrm{d}x}F^{-1}_{\varepsilon}(x)\approx\begin{cases}\varepsilon,\quad\alpha\in(0,1),\\\
(1+\varepsilon^{2})^{1/2}\cdot x,\quad\alpha=0,\\\
(1+|\alpha|)^{\frac{2|\alpha|}{1+|\alpha|}}\cdot|x|^{\frac{2|\alpha|}{1+|\alpha|}},\quad\alpha\in(-1,0).\end{cases}$

In particular for $\alpha\in(-1,1)$, $\frac{\hskip 0.90417pt\mathrm{d}}{\hskip
0.90417pt\mathrm{d}x}F^{-1}_{\varepsilon}$ is locally square integrable, so
that the generalized Itô formula ([6, Theorem 4.1]) for the Brownian motion is
applicable to $F^{-1}_{\varepsilon}$.

For $\alpha\in(0,1)$, the second derivative

(3.6) $\displaystyle\frac{\hskip 1.29167pt\mathrm{d}^{2}}{\hskip
1.29167pt\mathrm{d}x^{2}}F_{\varepsilon}(x)=-\frac{\alpha(x)^{2\alpha-1}}{(|x|^{2\alpha}+\varepsilon^{2})^{3/2}},\quad
x\neq 0,$

is in $L^{1}_{\text{loc}}(\mathbb{R})$, so that the Itô formula with
generalized derivatives from Krylov [8, Chapter 2, Section 10] is applicable.

###  3.1. Weak existence. Proof of Theorem 2.3

In this section we assume that $\alpha\in(-1,1)$. Let $\varepsilon>0$ be
fixed.

Let $W^{1}$, $W^{2}$ be independent standard Wiener processes, and let
$\widehat{W}:=\frac{1}{\sqrt{2}}(W^{1}+W^{2})$. Define the process

(3.7)
$\widetilde{Y}_{t}^{\varepsilon}=F_{\varepsilon}^{-1}\Big{(}F_{\varepsilon}(x)+\widehat{W}_{t}\Big{)}$

and set

(3.8) $\displaystyle\widetilde{B}_{t}^{\varepsilon}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{\bigl{(}|\widetilde{Y}^{\varepsilon}_{s}|^{\alpha}+\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{1}_{s}+\bigl{(}|\widetilde{Y}^{\varepsilon}_{s}|^{\alpha}-\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{2}_{s}}{\sigma_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})}$
(3.9) $\displaystyle\widetilde{W}_{t}^{\varepsilon}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{\bigr{(}\varepsilon-|\widetilde{Y}^{\varepsilon}_{s}|^{\alpha}\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{1}_{s}+\bigl{(}|\widetilde{Y}^{\varepsilon}_{s}|^{\alpha}+\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{2}_{s}}{\sigma_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})}.$

It is straightforward to check that $\widetilde{B}$ and
$\widetilde{W}^{\varepsilon}$ are continuous martingales with
$[\widetilde{B}^{\varepsilon}]_{t}=[\widetilde{W}^{\varepsilon}]_{t}=t$,
$[\widetilde{B}^{\varepsilon},\widetilde{W}^{\varepsilon}]_{t}=0$. Then, by
the Lévy characterization, $\widetilde{B}^{\varepsilon}$ and
$\widetilde{W}^{\varepsilon}$ are independent standard Wiener processes. It is
also clear that

(3.10) $\displaystyle|\widetilde{Y}_{t}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}\widetilde{B}_{t}^{\varepsilon}+\varepsilon\hskip
1.29167pt\mathrm{d}\widetilde{W}_{t}^{\varepsilon}$
$\displaystyle=\frac{|\widetilde{Y}_{t}^{\varepsilon}|^{\alpha}}{\sqrt{2}}\frac{(|\widetilde{Y}^{\varepsilon}_{t}|^{\alpha}+\varepsilon)\,\hskip
1.29167pt\mathrm{d}W^{1}_{t}+(|\widetilde{Y}^{\varepsilon}_{t}|^{\alpha}-\varepsilon)\,\hskip
1.29167pt\mathrm{d}W^{2}_{t}}{\sigma_{\varepsilon}(\widetilde{Y}_{t}^{\varepsilon})}$
$\displaystyle+\frac{\varepsilon}{\sqrt{2}}\frac{(\varepsilon-|\widetilde{Y}_{t}^{\varepsilon}|^{\alpha})\,\hskip
1.29167pt\mathrm{d}W^{1}_{t}+(|\widetilde{Y}_{t}^{\varepsilon}|^{\alpha}+\varepsilon)\,\hskip
1.29167pt\mathrm{d}W^{2}_{t}}{\sigma_{\varepsilon}(\widetilde{Y}_{t}^{\varepsilon})}$
$\displaystyle=\sigma_{\varepsilon}(\widetilde{Y}_{t}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{t}.$

Applying the generalized Itô formula (Theorem 4.1 in Föllmer et al. [6]) to
the Brownian motion $\widehat{W}$ and the function $F^{-1}_{\varepsilon}$ in
(3.7) and taking into account (3.10), we get

(3.11) $\displaystyle\widetilde{Y}_{t}^{\varepsilon}$
$\displaystyle=x+\int_{0}^{t}\sigma_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})\circ\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}$
$\displaystyle=x+\int_{0}^{t}\sigma_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}+\frac{1}{2}[\sigma_{\varepsilon}(\widetilde{Y}^{\varepsilon}),\widehat{W}^{\varepsilon}]_{t}$
$\displaystyle=x+\int_{0}^{t}|\widetilde{Y}_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}\widetilde{B}_{s}^{\varepsilon}+\frac{1}{2}[\sigma_{\varepsilon}(\widetilde{Y}^{\varepsilon}),\widehat{W}^{\varepsilon}]_{t}+\varepsilon\widetilde{W}_{s}^{\varepsilon}.$

For $\alpha\in(0,1)$, by Föllmer et al. [6, Remark 3.2 (c)],

(3.12)
$\displaystyle{}[\sigma_{\varepsilon}(\widetilde{Y}^{\varepsilon}),\widehat{W}]_{t}=\alpha\int_{0}^{t}\bigl{(}\widetilde{Y}^{\varepsilon}_{s}\bigr{)}^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s;$

for $\alpha\in(-1,0)$, by Cherny [2, Corollary 4.4],

(3.13)
$\displaystyle{}[\sigma_{\varepsilon}(\widetilde{Y}^{\varepsilon}),\widehat{W}]_{t}=\alpha\cdot\mathrm{v.p.}\int_{0}^{t}(\widetilde{Y}^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s,$

whence we arrive at the SDE (2.5) for the triple
$(\widetilde{Y}^{\varepsilon}$, $\widetilde{B}^{\varepsilon}$,
$\widetilde{W}^{\varepsilon})$.

###  3.2. Pathwise uniqueness. Proof of Theorem 2.4

Now we turn to the path-wise uniqueness of solution to (2.5), using the ideas
by LeGall [9]. In this section we assume that $\alpha\in(0,1)$.

Let $Y^{\varepsilon}$, $\widetilde{Y}^{\varepsilon}$ be two strong solutions
to (2.5) and let $U^{\varepsilon}:=F_{\varepsilon}(Y^{\varepsilon})$ and
$\widetilde{U}^{\varepsilon}:=F_{\varepsilon}(\widetilde{Y}^{\varepsilon})$.
Then, by the extension of Itô’s formula from Krylov [8, Chapter 2, Section 10]
we have

(3.14) $\displaystyle\hskip 1.29167pt\mathrm{d}U_{t}^{\varepsilon}$
$\displaystyle=\hskip
1.29167pt\mathrm{d}F_{\varepsilon}(Y^{\varepsilon}_{t})=G_{\varepsilon}(Y_{t}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}B_{t}+H_{\varepsilon}(Y^{\varepsilon}_{t})\,\hskip
1.29167pt\mathrm{d}W_{t}$
$\displaystyle=G_{\varepsilon}\bigl{(}F_{\varepsilon}^{-1}(U^{\varepsilon}_{t})\bigr{)}\,\hskip
1.29167pt\mathrm{d}B_{t}+H_{\varepsilon}\bigl{(}F_{\varepsilon}^{-1}(U^{\varepsilon}_{t})\bigr{)}\,\hskip
1.29167pt\mathrm{d}W_{t},\quad t\geq 0$

where

(3.15) $\displaystyle
G_{\varepsilon}(y)=\frac{|y|^{\alpha}}{\sigma_{\varepsilon}(y)},\quad
H_{\varepsilon}(y)=\frac{\varepsilon}{\sigma_{\varepsilon}(y)}.$

The same formula holds for the processes $\widetilde{Y}^{\varepsilon}$ and
$\widetilde{U}^{\varepsilon}$, too.

Since $G_{\varepsilon}(y)^{2}+H_{\varepsilon}(y)^{2}\equiv 1$, both
$U^{\varepsilon}$ and $\widetilde{U}^{\varepsilon}$ are standard Wiener
processes, by virtue of the Lévy characterization. This, in particular,
implies weak uniqueness for (2.5).

Denote
$R_{t}^{\varepsilon}=U_{t}^{\varepsilon}-\widetilde{U}_{t}^{\varepsilon}$.
Then by Tanaka’s formula

(3.16) $\displaystyle U^{\varepsilon}_{t}\vee\widetilde{U}^{\varepsilon}_{t}$
$\displaystyle=U^{\varepsilon}_{t}+(\widetilde{U}^{\varepsilon}_{t}-U^{\varepsilon}_{t})^{+}$
$\displaystyle=U^{\varepsilon}_{t}+\int_{0}^{t}\mathbb{I}(\widetilde{U}^{\varepsilon}_{s}>U^{\varepsilon}_{s})\,\hskip
1.29167pt\mathrm{d}(\widetilde{U}^{\varepsilon}_{s}-U^{\varepsilon}_{s})+\frac{1}{2}L^{0}_{t}(\widetilde{U}^{\varepsilon}-U^{\varepsilon})$
$\displaystyle=x+\int_{0}^{t}G_{1}\bigl{(}F_{\varepsilon}^{-1}(U^{\varepsilon}_{s}\vee\widetilde{U}^{\varepsilon}_{s})\bigr{)}\,\hskip
1.29167pt\mathrm{d}B_{s}+G_{2}\bigl{(}F_{\varepsilon}^{-1}(U^{\varepsilon}_{s}\vee\widetilde{U}^{\varepsilon}_{s})\bigr{)}\,\hskip
1.29167pt\mathrm{d}W_{s}+\frac{1}{2}L^{0}_{t}(R^{\varepsilon}),$

where for $a\in\mathbb{R}$

(3.17) $\displaystyle L^{a}_{t}(R^{\varepsilon})=\lim_{\delta\downarrow
0}\frac{1}{\varepsilon}\int_{0}^{t}\mathbb{I}(R^{\varepsilon}_{s}\in[a,a+\delta))\,\hskip
1.29167pt\mathrm{d}[R^{\varepsilon}]_{s}.$

Suppose that $L^{0}_{t}(R^{\varepsilon})=0$. Then, by appealing to the
extended Itô formula once more, we see that
$F_{\varepsilon}(U^{\varepsilon}\vee\widetilde{U}^{\varepsilon})$ is another
strong solution to (2.5), which implies that
$U^{\varepsilon}=\widetilde{U}^{\varepsilon}$ in view of weak uniqueness.

Thus, we are left to show that $L^{0}_{t}(R^{\varepsilon})=0$. To this end, it
suffices to show that for any $t\geq 0$

(3.18)
$\int_{0}^{t}\frac{\mathbb{I}(R^{\varepsilon}_{s}>0)}{R^{\varepsilon}_{s}}\,\hskip
1.29167pt\mathrm{d}[R^{\varepsilon}]_{s}<\infty\quad\text{a.s.,}$

see Lemma 1.1 in LeGall [9]. Indeed, the mapping $a\mapsto
L^{a}_{t}(R^{\varepsilon})$ is continuous, and if $L^{0}_{t}(R^{\varepsilon})$
does not vanish the occupation time formula will yield

(3.19)
$\displaystyle\int_{0}^{t}\frac{\mathbb{I}(R^{\varepsilon}_{s}>0)}{R^{\varepsilon}_{s}}\,\hskip
1.29167pt\mathrm{d}[R^{\varepsilon}]_{s}=\int_{0}^{\infty}\frac{L^{a}_{t}(R^{\varepsilon})}{a}\,\hskip
1.29167pt\mathrm{d}a=+\infty\quad\text{a.s.}$

To show (3.18) we apply Lemma 3.1 to get

(3.20)
$\displaystyle\int_{0}^{t}\frac{\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}[R^{\varepsilon}]_{s}}{R^{\varepsilon}_{s}}$
$\displaystyle=\int_{0}^{t}\frac{\bigl{|}G_{\varepsilon}(Y^{\varepsilon}_{s})-G_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})\bigr{|}^{2}+\bigl{|}H_{\varepsilon}(Y_{s}^{\varepsilon})-H_{\varepsilon}(\widetilde{Y}_{s}^{\varepsilon})\bigr{|}^{2}}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s$ $\displaystyle\leq
C_{\varepsilon}\int_{0}^{t}\frac{\bigl{|}|Y_{s}^{\varepsilon}|^{\alpha}-|\widetilde{Y}_{s}^{\varepsilon}|^{\alpha}\bigr{|}^{2}\wedge
1}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s$ $\displaystyle\leq
C_{\varepsilon}\int_{0}^{t}\frac{\bigl{|}(Y_{s}^{\varepsilon})^{\alpha}-(\widetilde{Y}_{s}^{\varepsilon})^{\alpha}\bigr{|}^{2}\wedge
1}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s$ $\displaystyle\leq
C_{\varepsilon}\int_{0}^{t}\frac{(Y_{s}^{\varepsilon})^{\alpha}-(\widetilde{Y}_{s}^{\varepsilon})^{\alpha}}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s$
$\displaystyle=C_{\varepsilon}\int_{0}^{t}\frac{h_{\varepsilon}(U_{s}^{\varepsilon})-h_{\varepsilon}(\widetilde{U}_{s}^{\varepsilon})}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s,$

where we have abbreviated
$h_{\varepsilon}(y)=\big{(}F_{\varepsilon}^{-1}(y)\big{)}^{\alpha}$. Note that
$h_{\varepsilon}(\cdot)$ is absolutely continuous with integrable derivative.
Using the relation

(3.21)
$\displaystyle\frac{h_{\varepsilon}(U_{s}^{\varepsilon})-h_{\varepsilon}(\widetilde{U}_{s}^{\varepsilon})}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}=\int_{0}^{1}h^{\prime}_{\varepsilon}\big{(}\theta
U_{s}^{\varepsilon}+(1-\theta)\widetilde{U}_{s}^{\varepsilon}\big{)}\,\hskip
1.29167pt\mathrm{d}\theta,$

we can write

(3.22)
$\displaystyle\int_{0}^{t}\frac{h_{\varepsilon}(U_{s}^{\varepsilon})-h_{\varepsilon}(\widetilde{U}_{s}^{\varepsilon})}{U^{\varepsilon}_{s}-\widetilde{U}^{\varepsilon}_{s}}\mathbb{I}(R^{\varepsilon}_{s}>0)\,\hskip
1.29167pt\mathrm{d}s\leq
C_{\varepsilon}\int_{0}^{t}\int_{0}^{1}h^{\prime}_{\varepsilon}\big{(}\theta
U_{s}^{\varepsilon}+(1-\theta)\widetilde{U}_{s}^{\varepsilon}\big{)}\,\hskip
1.29167pt\mathrm{d}\theta\hskip 1.29167pt\mathrm{d}s.$

For each $\theta\in[0,1]$, the process $R^{\varepsilon,\theta}:=\theta
U^{\varepsilon}+(1-\theta)\widetilde{U}^{\varepsilon}$ can be written as

(3.23) $\displaystyle
R^{\varepsilon,\theta}_{t}=\int_{0}^{t}\mu_{s}^{\varepsilon,\theta}\,\hskip
1.29167pt\mathrm{d}B_{s}+\int_{0}^{t}\nu_{s}^{\varepsilon,\theta}\,\hskip
1.29167pt\mathrm{d}W_{s},$

with the processes $\mu^{\varepsilon,\theta}$ and $\nu^{\varepsilon,\theta}$
given by

(3.24)
$\displaystyle\mu^{\varepsilon,\theta}_{t}=\theta\Big{(}G_{\varepsilon}(Y^{\varepsilon}_{t})+G_{\varepsilon}(\widetilde{Y}^{\varepsilon}_{t})\Big{)}\quad\text{and}\quad\nu^{\varepsilon,\theta}_{t}$
$\displaystyle=(1-\theta)\Big{(}H_{\varepsilon}(Y^{\varepsilon}_{t})+H_{\varepsilon}(\widetilde{Y}^{\varepsilon}_{t})\Big{)}.$

One easily checks that
$\frac{1}{2}\leq|\mu_{s}^{\varepsilon,\theta}|^{2}+|\nu_{s}^{\varepsilon,\theta}|^{2}\leq
1$, Therefore, $R^{\varepsilon,\theta}$ possesses a local time and for all
$\theta\in[0,1]$, $x\in\mathbb{R}$,

(3.25)
$\displaystyle\mathbf{E}L^{x}_{t}(R^{\varepsilon,\theta})\leq\sqrt{2}\mathbf{E}L^{x}_{t}(W)=2\sqrt{\frac{t}{\pi}}\mathrm{e}^{-x^{2}/2t}.$

Using Fubini’s theorem and the occupation time formula we obtain the estimate

(3.26)
$\displaystyle\mathbf{E}\int_{0}^{1}\int_{0}^{t}h^{\prime}_{\varepsilon}\big{(}\theta
U_{s}^{\varepsilon}+(1-\theta)\widetilde{U}_{s}^{\varepsilon}\big{)}\,\hskip
1.29167pt\mathrm{d}s\,\hskip 1.29167pt\mathrm{d}\theta$
$\displaystyle=\mathbf{E}\int_{0}^{1}\int_{\mathbb{R}}h^{\prime}_{\varepsilon}(x)L_{t}^{x}(R^{\varepsilon,\theta})\,\hskip
1.29167pt\mathrm{d}x\,\hskip 1.29167pt\mathrm{d}\theta$ $\displaystyle\leq
C\sqrt{t}\int_{0}^{1}\int_{\mathbb{R}}h^{\prime}_{\varepsilon}(x)\mathrm{e}^{-x^{2}/2t}\,\hskip
1.29167pt\mathrm{d}x\,\hskip 1.29167pt\mathrm{d}\theta\leq
C_{t,\varepsilon}<\infty,$

whence we get (3.18).

In the following Lemma we prove a technical estimate that has been used in
(3.20).

######  Lemma 3.1.

Let $\alpha\in(0,1)$, and let $G_{\varepsilon}$ and $H_{\varepsilon}$ be
functions defined in (3.15). Then for any $\varepsilon>0$ there is
$C_{\varepsilon}>0$ such that for all $y_{1},y_{2}\in\mathbb{R}$ we have

(3.27)
$\displaystyle\bigl{|}G_{\varepsilon}(y_{1})-G_{\varepsilon}(y_{2})\bigr{|}^{2}+\bigl{|}H_{\varepsilon}(y_{1})-H_{\varepsilon}(y_{2})\bigr{|}^{2}\leq
C_{\varepsilon}\Big{|}|y_{1}|^{\alpha}-|y_{2}|^{\alpha}\Big{|}^{2}.$

###### Proof.

Denote for some $\beta_{1},\beta_{2}\in\mathbb{R}$

(3.28) $\displaystyle G_{\varepsilon}(y_{1})=\sin\beta_{1},\quad
H_{\varepsilon}(y_{1})=\cos\beta_{1},$ $\displaystyle
G_{\varepsilon}(y_{2})=\sin\beta_{2},\quad
H_{\varepsilon}(y_{2})=\cos\beta_{2}.$

Then we have

(3.29)
$\displaystyle|\sin\beta_{1}-\sin\beta_{2}|^{2}+|\cos\beta_{1}-\cos\beta_{2}|^{2}$
$\displaystyle=2-2\sin\beta_{1}\sin\beta_{2}-2\cos\beta_{1}\cos\beta_{2}$
$\displaystyle=2(1-\cos(\beta_{1}-\beta_{2}))$
$\displaystyle\leq|\beta_{1}-\beta_{2}|^{2}$ $\displaystyle=\Big{|}\arcsin
G_{\varepsilon}(y_{1})-\arcsin G_{\varepsilon}(y_{2})\Big{|}^{2}$
$\displaystyle=\Big{|}\arcsin\Big{(}G_{\varepsilon}(y_{1})H_{\varepsilon}(y_{2})-G_{\varepsilon}(y_{2})H_{\varepsilon}(y_{1})\Big{)}\Big{|}^{2}$
$\displaystyle=\Big{|}\arcsin\frac{\varepsilon(|y_{1}|^{\alpha}-|y_{2}|^{\alpha})}{\sigma_{\varepsilon}(y_{1})\sigma_{\varepsilon}(y_{2})}\Big{|}^{2}$
$\displaystyle\leq\frac{\varepsilon^{2}||y_{1}|^{\alpha}-|y_{2}|^{\alpha}|^{2}}{\sigma_{\varepsilon}(y_{1})^{2}\sigma_{\varepsilon}(y_{2})^{2}}\wedge\frac{\pi^{2}}{4}$
$\displaystyle\leq\frac{||y_{1}|^{\alpha}-|y_{2}|^{\alpha}|^{2}}{\varepsilon^{2}}\wedge
4$
$\displaystyle\leq\frac{4\varepsilon^{2}+1}{\varepsilon^{2}}\cdot\Big{(}\Big{|}|y_{1}|^{\alpha}-|y_{2}|^{\alpha}\Big{|}^{2}\wedge
1\Big{)},$

and the inequality (3.27) follows. ∎

##  4\. Analysis of the SDE (1.1) with the bracket

In this and further sections we restrict ourselves to the case
$\alpha\in(0,1)$.

###  4.1. SDE (1.16), strong existence. Proof of Theorem 2.6

Our main goal in this section is to show that for $\alpha\in(0,1)$, the
solution $Y^{\varepsilon}$ to (2.5) also solves (1.16), i.e. that
$[|Y^{\varepsilon}|^{\alpha},B]_{t}=\alpha\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s$.

######  Lemma 4.1.

Let $\varepsilon>0$ and let $Y^{\varepsilon}$ be a strong solution to (2.5).
Then, for any $\alpha\in(0,1)$ the quadratic variation

(4.1)
$\displaystyle{}[|Y^{\varepsilon}|^{\alpha},B]_{t}=\lim_{n\to\infty}\sum_{t_{k}\in
D_{n},t_{k}<t}\big{(}|Y^{\varepsilon}_{t_{k}}|^{\alpha}-|Y^{\varepsilon}_{t_{k-1}}|^{\alpha}\big{)}(B_{t_{k}}-B_{t_{k-1}})$

exists as a limit in u.c.p. and

(4.2)
$\displaystyle{}[|Y^{\varepsilon}|^{\alpha},B]_{t}=\alpha\int_{0}^{t}(Y^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s.$

###### Proof.

Define the processes

(4.3) $\displaystyle W^{1,\varepsilon}_{t}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{(|Y_{s}^{\varepsilon}|^{\alpha}+\varepsilon)\,\hskip
1.29167pt\mathrm{d}B_{s}-(|Y_{s}^{\varepsilon}|^{\alpha}-\varepsilon)\,\hskip
1.29167pt\mathrm{d}W_{s}}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})},$ (4.4)
$\displaystyle W^{2,\varepsilon}_{t}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{(|Y_{s}^{\varepsilon}|^{\alpha}-\varepsilon)\,\hskip
1.29167pt\mathrm{d}B_{s}+(|Y_{s}^{\varepsilon}|^{\alpha}+\varepsilon)\,\hskip
1.29167pt\mathrm{d}W_{s}}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}.$

It is easy to check that $W^{1,\varepsilon}$, $W^{2,\varepsilon}$ are
independent standard Brownian motions and the original Brownian motions $B$
and $W$ satisfy the no-tilde counterparts of (3.8) and (3.9), namely, we have

(4.5) $\displaystyle B_{t}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{\bigl{(}|Y^{\varepsilon}_{s}|^{\alpha}+\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{1,\varepsilon}_{s}+\bigl{(}|Y^{\varepsilon}_{s}|^{\alpha}-\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{2,\varepsilon}_{s}}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}$
$\displaystyle W_{t}$
$\displaystyle=\frac{1}{\sqrt{2}}\int_{0}^{t}\frac{\bigr{(}\varepsilon-|Y^{\varepsilon}_{s}|^{\alpha}\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{1,\varepsilon}_{s}+\bigl{(}|Y^{\varepsilon}_{s}|^{\alpha}+\varepsilon\bigr{)}\,\hskip
1.29167pt\mathrm{d}W^{2,\varepsilon}_{s}}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}.$

The process

(4.6)
$\displaystyle\widehat{W}^{\varepsilon}_{t}=\frac{1}{\sqrt{2}}(W^{1,\varepsilon}_{t}+W^{2,\varepsilon}_{t}).$

is also a Brownian motion and we have the equality

(4.7) $\displaystyle\hskip
1.29167pt\mathrm{d}\widehat{W}_{t}^{\varepsilon}=|Y_{t}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{t}+\varepsilon\hskip 1.29167pt\mathrm{d}W_{t}.$

Literally repeating the argument of Section 3.1 we get that the process
$\widehat{Y}_{t}^{\varepsilon}=F_{\varepsilon}^{-1}\bigl{(}F_{\varepsilon}(x)+\widehat{W}_{t}^{\varepsilon}\bigr{)}$
solves (2.5). Therefore, by the uniqueness Theorem 2.5,
$\widehat{Y}_{t}^{\varepsilon}=Y_{t}^{\varepsilon}=F_{\varepsilon}^{-1}(F_{\varepsilon}(x)+\widehat{W}^{\varepsilon}_{t})$
a.s.

Let us apply the generalized Itô formula from Föllmer et al. [6] to the
function

(4.8) $\displaystyle
h_{\varepsilon}(z):=|F_{\varepsilon}^{-1}\bigl{(}F_{\varepsilon}(x)+z\bigr{)}|^{\alpha},\quad
h^{\prime}_{\varepsilon}\in L^{2}_{\mathrm{loc}}(\mathbb{R}),$

and the Brownian motion $\widehat{W}^{\varepsilon}$:

(4.9)
$\displaystyle|Y_{t}^{\varepsilon}|^{\alpha}=h_{\varepsilon}(\widehat{W}_{t}^{\varepsilon})$
$\displaystyle=|x|^{\alpha}+\int_{0}^{t}h^{\prime}_{\varepsilon}(\widehat{W}_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}+\frac{1}{2}[h^{\prime}_{\varepsilon}(\widehat{W}^{\varepsilon}),\widehat{W}^{\varepsilon}]_{t}$
$\displaystyle=|x|^{\alpha}+\alpha\int_{0}^{t}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}+A^{\varepsilon}_{t}$

where
$A^{\varepsilon}_{t}=\frac{1}{2}[h^{\prime}_{\varepsilon}(\widehat{W}^{\varepsilon}),\widehat{W}^{\varepsilon}]_{t}$
is an adapted zero energy process [6, Theorem 3.5]. Now we decompose the
bracket’s partial sum as

(4.10) $\displaystyle\sum_{t_{k}\in
D_{n},t_{k}<t}\Big{(}|Y_{t_{k}}^{\varepsilon}|^{\alpha}$
$\displaystyle-|Y_{t_{k-1}}^{\varepsilon}|^{\alpha}\Big{)}(B_{t_{k}}-B_{t_{k-1}})$
$\displaystyle=\sum_{t_{k}\in
D_{n},t_{k}<t}\alpha\int_{t_{k-1}}^{t_{k}}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}\cdot(B_{t_{k}}-B_{t_{k-1}})$
$\displaystyle+\sum_{t_{k}\in
D_{n},t_{k}<t}\Big{(}A_{t_{k}}^{\varepsilon}-A_{t_{k-1}}^{\varepsilon}\Big{)}(B_{t_{k}}-B_{t_{k-1}})=:J_{1}^{\varepsilon}+J_{2}^{\varepsilon}.$

Since $A^{\varepsilon}$ is of zero energy, the second sum
$J_{2}^{\varepsilon}$ converges to zero in probability by the Cauchy–Schwarz
inequality.

We rewrite the first sum in terms of $W^{1,\varepsilon}$ and
$W^{2,\varepsilon}$:

(4.11) $\displaystyle J_{1}^{\varepsilon}$ $\displaystyle=\sum_{t_{k}\in
D_{n},t_{k}<t}\frac{\alpha}{2}\int_{t_{k-1}}^{t_{k}}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s})\,\hskip
1.29167pt\mathrm{d}W_{s}^{1,\varepsilon}\cdot\int_{t_{k-1}}^{t_{k}}\frac{|Y_{s}^{\varepsilon}|^{\alpha}+\varepsilon}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}\,\hskip
1.29167pt\mathrm{d}W^{1,\varepsilon}_{s}$ $\displaystyle+\sum_{t_{k}\in
D_{n},t_{k}<t}\frac{\alpha}{2}\int_{t_{k-1}}^{t_{k}}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s})\,\hskip
1.29167pt\mathrm{d}W_{s}^{1,\varepsilon}\cdot\int_{t_{k-1}}^{t_{k}}\frac{|Y_{s}^{\varepsilon}|^{\alpha}-\varepsilon}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}\,\hskip
1.29167pt\mathrm{d}W^{2,\varepsilon}_{s}$ $\displaystyle+\sum_{t_{k}\in
D_{n},t_{k}<t}\frac{\alpha}{2}\int_{t_{k-1}}^{t_{k}}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}W_{s}^{2,\varepsilon}\cdot\int_{t_{k-1}}^{t_{k}}\frac{|Y_{s}^{\varepsilon}|^{\alpha}+\varepsilon}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}\,\hskip
1.29167pt\mathrm{d}W^{1,\varepsilon}_{s}$ $\displaystyle+\sum_{t_{k}\in
D_{n},t_{k}<t}\frac{\alpha}{2}\int_{t_{k-1}}^{t_{k}}(Y_{s}^{\varepsilon})^{\alpha-1}\sigma_{\varepsilon}(Y_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}W_{s}^{2,\varepsilon}\cdot\int_{t_{k-1}}^{t_{k}}\frac{|Y_{s}^{\varepsilon}|^{\alpha}-\varepsilon}{\sigma_{\varepsilon}(Y_{s}^{\varepsilon})}\,\hskip
1.29167pt\mathrm{d}W^{2,\varepsilon}_{s}$
$\displaystyle:=\frac{\alpha}{2}\Big{(}I_{11}^{\varepsilon}+I_{12}^{\varepsilon}+I_{21}^{\varepsilon}+I_{22}^{\varepsilon}\Big{)}.$

By Theorems II.23 and II.29 from Protter [13] we obtain that as $n\to\infty$

(4.12) $\displaystyle I_{11}^{\varepsilon}$
$\displaystyle\stackrel{{\scriptstyle\mathbf{P}}}{{\to}}\int_{0}^{t}(Y_{s}^{\varepsilon})^{\alpha-1}\big{(}|Y_{s}^{\varepsilon}|^{\alpha}+\varepsilon\big{)}\,\hskip
1.29167pt\mathrm{d}s,$ $\displaystyle I_{22}^{\varepsilon}$
$\displaystyle\stackrel{{\scriptstyle\mathbf{P}}}{{\to}}\int_{0}^{t}(Y_{s}^{\varepsilon})^{\alpha-1}\big{(}|Y_{s}^{\varepsilon}|^{\alpha}-\varepsilon\big{)}\,\hskip
1.29167pt\mathrm{d}s,$ $\displaystyle I_{12}^{\varepsilon}$
$\displaystyle\stackrel{{\scriptstyle\mathbf{P}}}{{\to}}0,\quad
I_{21}^{\varepsilon}\stackrel{{\scriptstyle\mathbf{P}}}{{\to}}0,$

and the statement of the Lemma follows. ∎

###  4.2. SDE (1.16): uniqueness. Proof of Theorem 2.8

To establish strong uniqueness of equation (1.16), we can show that any of its
strong solutions also solves (2.5) and appeal to Theorem 2.5. This is done
under the additional requirement that the solution is a semimartingale.

######  Remark 4.2.

From Lemma 4.1 it follows that for any strong solution $Y^{\varepsilon}$ to
(2.5), the process $[|Y^{\varepsilon}|^{\alpha},B]$ is absolutely continuous
and hence of locally bounded variation.

Recall that $\sigma_{\varepsilon}$ and $F_{\varepsilon}$ satisfy (3.1) and
(3.2).

######  Lemma 4.3.

$F_{\varepsilon}(X^{\varepsilon})^{2}$ is a semimartingale with decomposition

(4.13) $\displaystyle
F_{\varepsilon}(X_{t}^{\varepsilon})^{2}=F_{\varepsilon}(x)^{2}+2\int_{0}^{t}F_{\varepsilon}(X_{s}^{\varepsilon})\frac{|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}}{\sqrt{|X_{s}^{\varepsilon}|^{2\alpha}+\varepsilon^{2}}}+t+L_{t}^{\varepsilon},\quad
t\geq 0.$

where $t\mapsto L_{t}^{\varepsilon}$ is a non-decreasing continuous process.
††margin:  continuous

###### Proof.

Let $T>0$ and let $D_{n}=\\{t_{k}^{n}=kT2^{-n},k=0,\dots,2^{n}\\}$ be the
dyadic partition of $[0,T]$.

The semimartingale $X$ has quadratic variation equal to

(4.14)
$\displaystyle{}[X^{\varepsilon}]_{t}=\int_{0}^{t}|X_{s}^{\varepsilon}|^{2\alpha}\,\hskip
1.29167pt\mathrm{d}s+\varepsilon\,t.$

Note also that $F_{\varepsilon}(\cdot)^{2}\in C^{2}(\mathbb{R},\mathbb{R})$
with

(4.15) $\displaystyle h_{\varepsilon}(z)$ $\displaystyle:=\frac{\hskip
1.29167pt\mathrm{d}}{\hskip
1.29167pt\mathrm{d}z}F_{\varepsilon}(z)^{2}=\frac{2F_{\varepsilon}(z)}{\sqrt{|z|^{2\alpha}+\varepsilon^{2}}}$
(4.16) $\displaystyle h^{\prime}_{\varepsilon}(z)$
$\displaystyle:=\frac{\hskip 1.29167pt\mathrm{d}^{2}}{\hskip
1.29167pt\mathrm{d}z^{2}}F_{\varepsilon}(z)^{2}=\frac{2}{|z|^{2\alpha}+\varepsilon^{2}}-\frac{2\alpha\,F_{\varepsilon}(z)(z)^{2\alpha-1}}{\sqrt{(|z|^{2\alpha}+\varepsilon^{2})^{3}}}.$

Note that the last term in (4.16) is well defined and continuous since
$F(z)\sim z/\varepsilon$, $z\to 0$, see (3.3). By virtue of the generalized
Itô formula from [6] we get

(4.17) $\displaystyle F_{\varepsilon}(X_{t}^{\varepsilon})^{2}$
$\displaystyle=F_{\varepsilon}(x)^{2}+\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}X_{s}^{\varepsilon}+t-\alpha\int_{0}^{t}\frac{F_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}}{\sqrt{|X_{s}^{\varepsilon}|^{2\alpha}+\varepsilon^{2}}}\,\hskip
1.29167pt\mathrm{d}s$
$\displaystyle=F_{\varepsilon}(x)^{2}+\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}X_{s}^{\varepsilon}+t-\frac{\alpha}{2}\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s,$

where the first integral exists as a limit in u.c.p.:

(4.18) $\displaystyle\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})\,\hskip
1.29167pt\mathrm{d}X_{s}^{\varepsilon}=\lim_{n\to\infty}\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}X_{t^{n}_{k}}^{\varepsilon}-X_{t^{n}_{k-1}}^{\varepsilon}\big{)}.$

We expand

(4.19) $\displaystyle\sum_{t^{n}_{k}\in D_{n},t^{n}_{k}<t}$ $\displaystyle
h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}X_{t^{n}_{k}}^{\varepsilon}-X_{t^{n}_{k-1}}^{\varepsilon}\big{)}$
$\displaystyle{}=\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\int_{t^{n}_{k-1}}^{t^{n}_{k}}\big{(}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip 1.29167pt\mathrm{d}W_{s}\big{)}$
$\displaystyle{}+\frac{1}{2}\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{t^{n}_{k}}^{\varepsilon}-A_{t^{n}_{k-1}}^{\varepsilon}\big{)},$

where $A^{\varepsilon}=[|X^{\varepsilon}|^{\alpha},B]$. Since
$h_{\varepsilon}(X^{\varepsilon})$ is continuous, we have

(4.20) $\displaystyle\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\int_{t^{n}_{k-1}}^{t^{n}_{k}}\big{(}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}\big{)}\to\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})\big{(}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}\big{)},\quad n\to\infty,$

in u.c.p. Further, define

(4.21) $\displaystyle\tau^{n,\varepsilon}_{k}=\min\\{s\geq t^{n}_{k-1}\colon
X_{s}^{\varepsilon}=0\\}\wedge t^{n}_{k}$

and write

(4.22) $\displaystyle\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{t^{n}_{k}}^{\varepsilon}-A_{t^{n}_{k-1}}^{\varepsilon}\big{)}$
$\displaystyle=\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{t^{n}_{k}}^{\varepsilon}-A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}\big{)}+\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}-A_{t^{n}_{k-1}}^{\varepsilon}\big{)}.$

Note that $h_{\varepsilon}(X_{\tau^{n,\varepsilon}_{k}}^{\varepsilon})=0$ if
$\tau^{n,\varepsilon}_{k}<t^{n}_{k}$ and
$A_{t^{n}_{k}}^{\varepsilon}-A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}=0$ if
$\tau^{n,\varepsilon}_{k}=t^{n}_{k}$. Therefore,

(4.23) $\displaystyle\Big{|}\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{t^{n}_{k}}-A_{\tau^{n,\varepsilon}_{k}}\big{)}\Big{|}^{2}=\Big{|}\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}\big{(}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})-h_{\varepsilon}(X_{\tau^{n}_{k}}^{\varepsilon})\big{)}\big{(}A_{t^{n,\varepsilon}_{k}}^{\varepsilon}-A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}\big{)}\Big{|}^{2}$
$\displaystyle\leq\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}\big{(}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})-h_{\varepsilon}(X_{\tau^{n,\varepsilon}_{k}}^{\varepsilon})\big{)}^{2}\sum_{t^{n}_{k}\in
D_{n},t^{n}_{k}<t}\big{(}A_{t^{n}_{k}}^{\varepsilon}-A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}\big{)}^{2}\to
0,\quad n\to\infty,$

thanks to the fact that $A^{\varepsilon}$ is of bounded variation on $[0,t]$.
On $[t^{n}_{k-1},\tau^{n,\varepsilon}_{k}]$, $X^{\varepsilon}$ satisfies a
Stratonovich SDE with smooth coefficients, therefore (see Section V.5 in [13])

(4.24) $\displaystyle
A_{u}^{\varepsilon}-A_{t^{n}_{k-1}}^{\varepsilon}=\alpha\int_{t^{n}_{k-1}}^{u}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s,\quad u\in[t^{n}_{k-1},\tau^{n,\varepsilon}_{k}].$

Fix $m\geq 1$ and consider $n>m$. Let

(4.25) $\displaystyle D_{m}^{\varepsilon}=\\{t_{j}^{m}\in
D_{m}\colon\tau^{m,\varepsilon}_{j}=t^{m}_{j}\\}$

be the points of partition $D_{m}$ such that $X^{\varepsilon}$ does not vanish
on the intervals $[t^{m}_{j-1},t^{m}_{j}]$. Let

(4.26) $\displaystyle P_{m,n}^{\varepsilon}=\bigcup_{t_{j}^{m}\in
D_{m}^{\varepsilon}}P_{m,n,j}=\bigcup_{t_{k}^{m}\in
D_{m}^{\varepsilon}}\\{t^{n}_{k}\in D_{n}\colon
t^{n}_{k}\in[t^{m}_{j-1},t^{m}_{j}]\\}$

be the corresponding points of the finer partition $D_{n}\supseteq D_{m}$.
Clearly,

(4.27) $\displaystyle\sum_{t^{n}_{k}\in
P_{m,n}^{\varepsilon},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})$
$\displaystyle\big{(}A_{\tau^{n,\varepsilon}_{k}}^{\varepsilon}-A_{t^{n}_{k-1}}^{\varepsilon}\big{)}=\sum_{t^{m}_{j}\in
D_{m}^{\varepsilon},t_{j}^{m}<t}\sum_{t_{n}^{k}\in
P_{m,n,j}^{\varepsilon}}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\cdot\alpha\int_{t^{n}_{k-1}}^{t_{k}^{n}}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s$ $\displaystyle\to\sum_{t^{n}_{j}\in
D_{m}^{\varepsilon},t_{j}^{m}<t}\alpha\int_{t^{m}_{j-1}}^{t^{m}_{j}}h_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s=:K^{m,\varepsilon}_{t},\quad n\to\infty.$

Further, using again (4.24) we get

(4.28) $\displaystyle\sum_{t^{n}_{k}\in D_{n}\setminus
P_{m,n}^{\varepsilon},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\big{(}A_{\tau^{n,\varepsilon}_{k}}-A_{t^{n}_{k-1}}^{\varepsilon}\big{)}=\sum_{t^{n}_{k}\in
D_{n}\setminus
P_{m,n}^{\varepsilon},t^{n}_{k}<t}h_{\varepsilon}(X_{t^{n}_{k-1}}^{\varepsilon})\cdot\alpha\int_{t^{n}_{k-1}}^{\tau^{n,\varepsilon}_{k}}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s$

is a non-decreasing process. As $n\to\infty$, it converges to some non-
negative non-decreasing process, say, $L^{m,\varepsilon}$, and we get

(4.29) $\displaystyle
F_{\varepsilon}(X_{t}^{\varepsilon})^{2}=F_{\varepsilon}(x)^{2}+2\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})\big{(}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}\big{)}+t+K_{t}^{m,\varepsilon}+L^{m,\varepsilon}_{t}-\frac{\alpha}{2}\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s.$

Moreover, since the set of zeros of $X^{\varepsilon}$ is closed and
$h_{\varepsilon}(z)(z)^{2\alpha-1}=0$ for $z=0$, we get

(4.30) $\displaystyle
K^{m,\varepsilon}_{t}\to\frac{\alpha}{2}\int_{0}^{t}\mathbb{I}(X_{s}^{\varepsilon}\neq
0)h_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s=\frac{\alpha}{2}\int_{0}^{t}h_{\varepsilon}(X_{s}^{\varepsilon})(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s,\quad m\to\infty.$

As a result, the processes $L^{m,\varepsilon}$ also monotonically converge as
$m\to\infty$ to some non-decreasing limit $L^{\varepsilon}$. Hence, passing to
the limit in (4.29), we get the desired statement. ∎

######  Lemma 4.4.

For any $a\in(-1,0)$ and $t\in[0,T]$, there is $C_{a,\varepsilon,T}>0$ such
that

(4.31) $\displaystyle\mathbf{E}|X_{t}^{\varepsilon}|^{a}\leq
C_{a,\varepsilon,T}.$

Moreover, $X^{\varepsilon}$ spends zero time at $0$.

###### Proof.

By Lemma 4.3, $Q_{t}^{\varepsilon}:=F_{\varepsilon}(X_{t}^{\varepsilon})^{2}$
solves the equation

(4.32) $\displaystyle
Q_{t}^{\varepsilon}=F_{\varepsilon}(x)^{2}+2\int_{0}^{t}\sqrt{Q_{s}^{\varepsilon}}\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}+t+L_{t}^{\varepsilon},$

where

(4.33)
$\displaystyle\widehat{W}_{t}^{\varepsilon}=\int_{0}^{t}\frac{\operatorname{sign}\big{(}F_{\varepsilon}(X_{s}^{\varepsilon})\big{)}\big{(}|X_{s}^{\varepsilon}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}\big{)}}{\sqrt{|X_{s}^{\varepsilon}|^{2\alpha}+\varepsilon^{2}}}$

is a standard Wiener process. Since $L^{\varepsilon}$ is non-decreasing and
continuous, a slight modification of the comparison Theorem 3.7 in Chapter IX
in Revuz and Yor [14] yields that
$Q_{t}^{\varepsilon}\geq\widehat{Q}^{\varepsilon}_{t}$, $t\geq 0$, with
probability 1, where $\widehat{Q}^{\varepsilon}$ is the unique strong solution
to the equation

(4.34)
$\displaystyle\widehat{Q}_{t}^{\varepsilon}=F_{\varepsilon}(x)^{2}+2\int_{0}^{t}\sqrt{\widehat{Q}_{s}^{\varepsilon}}\,\hskip
1.29167pt\mathrm{d}\widehat{W}_{s}^{\varepsilon}+t,$

and hence is a square of a standard Wiener process started at
$F_{\varepsilon}(x)$. Consequently, $Q^{\varepsilon}$ spends zero time at
zero. Furthermore, noting that for $|F_{\varepsilon}(x)|\leq
C_{\varepsilon}|x|$, $x\in\mathbb{R}$, for some $C_{\varepsilon}>0$ (recall
(3.3)), we obtain for $a>-1$

(4.35) $\displaystyle\mathbf{E}|X_{t}^{\varepsilon}|^{a}\leq
C_{\varepsilon}^{a}\mathbf{E}|F_{\varepsilon}(X_{t}^{\varepsilon})|^{a}=C_{\varepsilon}^{a}\mathbf{E}|Q_{t}^{\varepsilon}|^{a/2}\leq
C_{\varepsilon}^{a}\mathbf{E}|\widehat{Q}_{t}^{\varepsilon}|^{a/2}=C_{\varepsilon}^{a}\mathbf{E}|F_{\varepsilon}(x)+\widehat{W}_{t}^{\varepsilon}|^{a}\leq
C_{\varepsilon,a,T}.$

∎

######  Lemma 4.5.

Let $X^{\varepsilon}$ be a semimartingale solution of the equation (1.16).
Then

(4.36)
$\displaystyle{}[|X^{\varepsilon}|^{\alpha},B]_{t}=\alpha\int_{0}^{t}(X^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s.$

###### Proof.

Denote $A_{t}^{\varepsilon}=[|X^{\varepsilon}|^{\alpha},B]_{t}$. Without loss
of generality assume that $x=0$. For each $n\geq 1$ consider the stopping
times

(4.37) $\displaystyle\sigma^{n}_{0}$ $\displaystyle=0,$
$\displaystyle\tau^{n,\varepsilon}_{k}$
$\displaystyle=\inf\Big{\\{}t>\theta^{n,\varepsilon}_{k-1}\colon|X^{\varepsilon}_{t}|=\frac{1}{n}\Big{\\}},$
$\displaystyle\sigma^{n,\varepsilon}_{k}$
$\displaystyle=\inf\\{t>\tau^{n,\varepsilon}_{k}\colon
X^{\varepsilon}_{t}=0\\},\quad k\geq 1.$

Denote

(4.38)
$\displaystyle\theta^{n,\varepsilon}_{t}:=\sum_{k}\mathbb{I}_{(\sigma_{k-1}^{n,\varepsilon},\tau_{k}^{n,\varepsilon}]}(t).$

Then by Lemma 4.4, since $X^{\varepsilon}$ spends zero time at $0$, we get

(4.39)
$\displaystyle\theta^{n,\varepsilon}_{t}\leq\int_{0}^{t}\mathbb{I}(|X_{s}^{\varepsilon}|\leq
1/n)\,\hskip 1.29167pt\mathrm{d}s\to 0,\quad n\to\infty.$

Hence

(4.40) $\displaystyle A_{t}^{\varepsilon}$
$\displaystyle=\sum_{k}\alpha\int_{\tau^{n,\varepsilon}_{k}\wedge
t}^{\sigma^{n,\varepsilon}_{k}\wedge
t}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s+\sum_{k}\Big{(}A_{\tau^{n,\varepsilon}_{k}\wedge
t}^{\varepsilon}-A_{\sigma^{n,\varepsilon}_{k-1}\wedge
t}^{\varepsilon}\Big{)}$
$\displaystyle=\alpha\int_{0}^{t}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s-\sum_{k}\alpha\int_{\sigma^{n,\varepsilon}_{k}}^{\tau^{n,\varepsilon}_{k}}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s+\int_{0}^{t}\theta^{n,\varepsilon}_{s}\,\hskip
1.29167pt\mathrm{d}A_{s}^{\varepsilon}$
$\displaystyle=\alpha\int_{0}^{t}(X_{s}^{\varepsilon})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s-S_{1}^{n,\varepsilon}(t)+S_{2}^{n,\varepsilon}(t).$

Clearly, by Lemma 4.4 and the dominated convergence theorem

(4.41)
$\displaystyle\mathbf{E}|S_{1}^{n,\varepsilon}(t)|\leq\alpha\mathbf{E}\int_{0}^{t}|X_{s}^{\varepsilon}|^{2\alpha-1}\mathbb{I}(|X_{s}^{\varepsilon}|\leq
1/n)\,\hskip 1.29167pt\mathrm{d}s\to 0$

and

(4.42)
$\displaystyle|S_{2}^{n,\varepsilon}(t)|\leq\int_{0}^{t}\theta^{n,\varepsilon}_{s}\,\hskip
1.29167pt\mathrm{d}|A^{\varepsilon}|_{s}\to 0\quad\text{a.s.}$

in the limit as $n\to\infty$. ∎

##  5\. Stochastic selection. Proof of Theorem 2.9

Recall that

(5.1) $\displaystyle
X^{0}_{t}:=\Big{(}(1-\alpha)B_{t}-(x)^{1-\alpha}\Big{)}^{\frac{1}{1-\alpha}},\quad
t\geq 0,$

is the strong solution of the equation (1.1) that spends zero time at zero and
has no skew behaviour at zero. Let $X^{\varepsilon}$ be the semimartingale
solution of (1.16) or (2.5). We apply the generalized Itô formula from Krylov
[8] to get

(5.2) $\displaystyle F_{\varepsilon}(X^{\varepsilon}_{t})$
$\displaystyle=F_{\varepsilon}(x)+\int_{0}^{t}\frac{|X^{\varepsilon}_{s}|^{\alpha}\,\hskip
1.29167pt\mathrm{d}B_{s}+\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}}{\sqrt{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}}$
$\displaystyle+\frac{\alpha}{2}\int_{0}^{t}\frac{(X^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s}{\sqrt{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}}-\frac{\alpha}{2}\int_{0}^{t}\frac{|X^{\varepsilon}_{s}|^{2\alpha}(X^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s}{(|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2})^{3/2}}-\frac{\alpha}{2}\int_{0}^{t}\frac{\varepsilon^{2}(X^{\varepsilon}_{s})^{2\alpha-1}\,\hskip
1.29167pt\mathrm{d}s}{(|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2})^{3/2}}$
$\displaystyle=B_{t}-\int_{0}^{t}\frac{\varepsilon\,\hskip
1.29167pt\mathrm{d}B_{s}}{\sqrt{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}}+\int_{0}^{t}\frac{\varepsilon\,\hskip
1.29167pt\mathrm{d}W_{s}}{\sqrt{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}}$

Hence, by the Doob inequality, for any $t>0$

(5.3)
$\displaystyle\mathbf{E}\sup_{[0,t]}\Big{|}\int_{0}^{t}\frac{\varepsilon\,\hskip
1.29167pt\mathrm{d}B_{s}}{\sqrt{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}}\Big{|}^{2}\leq
4\mathbf{E}\int_{0}^{t}\frac{\varepsilon^{2}\,\hskip
1.29167pt\mathrm{d}s}{|X^{\varepsilon}_{s}|^{2\alpha}+\varepsilon^{2}}$

Taking into account that $X^{\varepsilon}\stackrel{{\scriptstyle\hskip
0.90417pt\mathrm{d}}}{{=}}F^{-1}_{\varepsilon}(F_{\varepsilon}(x)+\widehat{W}))$
for some Brownian motion $\widehat{W}$ we have to show that

(5.4) $\displaystyle\lim_{\varepsilon\to
0}\mathbf{E}\int_{0}^{t}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(F_{\varepsilon}(x)+\widehat{W}_{s}))|^{2\alpha}+\varepsilon^{2}}\,\hskip
1.29167pt\mathrm{d}s=0.$

We have

(5.5)
$\displaystyle\mathbf{E}\int_{0}^{t}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(F_{\varepsilon}(x)+\widehat{W}_{s}))|^{2\alpha}+\varepsilon^{2}}\,\hskip
1.29167pt\mathrm{d}s=\int_{0}^{t}\frac{1}{\sqrt{2\pi}s}\int_{-\infty}^{\infty}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(y)|^{2\alpha}+\varepsilon^{2}}\mathrm{e}^{-\frac{(y-F_{\varepsilon}(x))^{2}}{2s}}\,\hskip
1.29167pt\mathrm{d}y\,\hskip 1.29167pt\mathrm{d}s.$

Since for $y>0$ and $\varepsilon>0$

(5.6) $\displaystyle F_{\varepsilon}(y)\leq\int_{0}^{y}\frac{\hskip
1.29167pt\mathrm{d}z}{z^{\alpha}}=\frac{y^{1-\alpha}}{1-\alpha}=F_{0}(y)$

we have

(5.7) $\displaystyle F_{\varepsilon}^{-1}(y)\geq
F_{0}^{-1}(y)=\Big{(}(1-\alpha)y\Big{)}^{\frac{1}{1-\alpha}}.$

Thus for $x=0$, all $y\neq 0$ and $t\in(0,T]$ we have

(5.8)
$\displaystyle\frac{1}{\sqrt{2\pi}t}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(y)|^{2\alpha}+\varepsilon^{2}}\mathrm{e}^{-\frac{(y-F_{\varepsilon}(x))^{2}}{2t}}\leq\frac{1}{\sqrt{2\pi}t}\frac{\varepsilon^{2}}{|(1-\alpha)y|^{2\alpha/(1-\alpha)}+\varepsilon^{2}}\to
0$

and (5.4) follows by the dominated convergence theorem.

Let us determine the dominating function if $x>0$. For $\varepsilon\in(0,1]$
small enough we have $\frac{1}{2}F_{0}(x)\leq F_{\varepsilon}(x)\leq
F_{0}(x)$. Hence, for $y\geq 0$ we have

(5.9) $\displaystyle\frac{1}{\sqrt{2\pi
t}}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(y)|^{2\alpha}+\varepsilon^{2}}\mathrm{e}^{-\frac{(y-F_{\varepsilon}(x))^{2}}{2t}}\leq\frac{1}{\sqrt{2\pi
t}}\Big{(}\mathbb{I}_{[0,F(x))}(y)+\mathrm{e}^{-\frac{(y-F(x))^{2}}{2t}}\mathbb{I}_{[F(x),\infty)}(y)\Big{)}$

whereas for $y\leq 0$

(5.10) $\displaystyle\frac{1}{\sqrt{2\pi
t}}\frac{\varepsilon^{2}}{|F^{-1}_{\varepsilon}(y)|^{2\alpha}+\varepsilon^{2}}\mathrm{e}^{-\frac{(y-F_{\varepsilon}(x))^{2}}{2t}}\leq\frac{1}{\sqrt{2\pi
t}}\mathrm{e}^{-\frac{(y-F(x)/2)^{2}}{2t}}$

The right hand sides of (5.9) and (5.10) are obviously integrable on
$(0,T]\times\mathbb{R}$ and (5.4) follows again by the dominated convergence
theorem.

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