2.1. Local time

We start by recalling the definition of the local time, which plays a fundamental role in our analysis. Indeed, in many cases, it will be the limiting process of our statistics of interest.

The local time of a semi-martingale, X, is the random field $(L^{a}_{t}(X);\, a\in \mathbb{R},\,t\ge 0) $ , defined for all $t\ge 0 $ , $a\in \mathbb{R}$ as the term $L^{a}_{t}(X)$ such that the following equation holds (see [Reference Revuz and Yor30, Theorem 1.2]):

\begin{equation*} |X_t - a| = |X_0 - a| + \int_{0}^{t} \mathrm{sgn}(X_s - a) \,\mathrm{d} X_s + L^{a}_{t}(X).\end{equation*}

Here we used the convention $\mathrm{sgn}(x)=\mathbf{1}_{\{x>0\}}-\mathbf{1}_{\{x\le 0\}}$ .

For all $a\in\mathbb{R}$ , $t \mapsto L^{a}_{t}(X)$ is increasing and is fully supported on $\{t\ge 0\colon X_t = a\} $ . This means that, for all $t \ge 0$ , $\int_{0}^{t}\mathbf{1}_{\{|X_s-a| \not = 0\}} \,\mathrm{d} L^{a}_{s}(X) = 0$ .

The local time field can alternatively be defined as the following almost sure limit (see [Reference Revuz and Yor30, Corollary VI.1.9]):

\begin{equation*} \text{for all}\ t\ge 0,\ a\in \mathbb{R},\ \text{almost surely}, \quad L^{a}_{t}(X) = \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon} \int_{0}^{t}\mathbf{1}_{\{X_s \in [a,a + \varepsilon)\}}\,\mathrm{d}\langle X\rangle_s.\end{equation*}

2.2. The sticky Brownian motion

We now provide a definition of sticky Brownian motion as a general linear diffusion on the real line described by scale function and speed measure. We explain the role of the stickiness parameter without discussing further what the scale function and speed measure are. Instead, we provide other equivalent characterizations of the process that are useful in the proofs of our results.

Sticky Brownian motion of stickiness parameter $\rho>0$ is the diffusion process on $\mathbb{R}$ , on natural scale, with speed measure $m(\mathrm{d} x) = \mathrm{d}x + \rho\,\delta_{0}(\mathrm{d}x)$ (see, e.g., [Reference Borodin and Salminen8, pp. 123--124]). The parameter $\rho>0$ , called the stickiness parameter, expresses the propensity of the process to stick at 0. The higher it is, the more time the process spends on average at 0. The asymptotic case $\rho = 0$ corresponds to standard Brownian motion and $\rho=\infty$ to Brownian motion with an absorbing boundary at 0. In this paper we deal only with the case $\rho \in (0,\infty) $ . The Brownian motion case is known and the absorbing case is trivial.

We now provide some useful equivalent characterizations of sticky Brownian motion. Let $\mathcal P_x = (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0}, \mathbb{P}_x)$ be a filtered probability space. The subscript x in $\mathcal P_x$ is a notational choice to indicate the starting point of the process. The filtrations $(\mathcal F_t)_{t\ge 0}$ as well as all filtrations in this paper are assumed to satisfy the usual conditions (right-continuity and completeness).

The following hold (see, e.g., [Reference Itô20, Sections 5.1, 5.2] and [Reference Rogers and Williams31, Theorem 47.1 and Remark (ii), p. 277]):

1. (t1) Let X be a sticky Brownian motion of stickiness parameter $\rho$ , defined on $\mathcal P_x$ such that, $\mathbb{P}_x$ -almost surely (a.s.), $X_0=x$ . (In particular, X is $(\mathcal{F}_t)_{t\ge 0}$ -adapted.) There exists a Brownian motion Z, defined on an extension of $\mathcal P_x$ , such that $X=(Z_{\gamma(t)})_{t\ge 0}$ , where the time change $\gamma$ is the right-inverse of $A(t) \;:\!=\; t + \rho L^{0}_{t}(Z_{})$ , $t\ge 0$ , given by $\gamma(t)\;:\!=\;\inf\{s>0 \colon A(s)>t\}$ and where $L^{0}_{t}(Z_{})$ is the right local time at 0 of the process $Z_{}$ .

2. (t2) Let Z be a standard Brownian motion, defined on the probability space $\mathcal P_x$ , such that, $\mathbb{P}_x$ -a.s., $Z_0=x$ . Then the process $X \;:\!=\; (Z_{\gamma(t)})_{t\ge0}$ with $\gamma$ defined in (t1) is a sticky Brownian motion of stickiness parameter $\rho$ , and, $\mathbb{P}_x$ -a.s., $X_0 = x $ .

Also, the following SDE characterization holds.

1. (p1) Let X be a sticky Brownian motion of stickiness parameter $\rho$ , defined on the probability space $\mathcal P_x$ , such that, $\mathbb{P}_x$ -a.s., $X_0=x$ . There exists a Brownian motion W defined on an extension of $\mathcal P_x $ such that (X, W) solves

   (2.1)

   

2. (p2) The system (2.1) has a jointly unique weak solution [Reference Engelbert and Peskir13, Theorem 1] and it is a sticky Brownian motion of stickiness parameter $\rho$ .

A proof of the last two statements for some sticky diffusions can be found in [Reference Anagnostakis3, Proposition 4.2] for (p1) and in [Reference Anagnostakis3, Theorem 4.1] for (p2). For sticky Brownian motion these proofs are contained in the proof of [Reference Engelbert and Peskir13, Theorem 1].

For an historical overview of sticky Brownian motion, see [Reference Peskir28]. For results on this process, see, e.g., [Reference Amir2, Reference Anagnostakis3, Reference Anagnostakis and Mazzonetto5, Reference Davies and Truman12, Reference Howitt19, Reference Salins and Spiliopoulos32]. For applications, see [Reference Bou-Rabee and Holmes-Cerfon9] and references therein.

2.3. Notions of convergence

The results on local time approximation are formulated in terms of the types of convergence introduced below. For convenience, let $(A_n)_{n\ge 0}$ be a sequence of real-valued processes defined on the probability space $(\Omega, \mathcal F, \mathbb{P})$ .

Definition 2.1. We say that $(A^{n})_{n\ge1}$ converges locally uniformly in time, in probability to $A^{0}$ if, for all $t\ge 0$ ,

We denote this convergence with

The following result gives a sufficient condition for ucp convergence to occur.

For the estimation result we adopt the notion of conditional convergence in probability. The reason is that the estimators we consider are consistent conditionally on the event that the threshold of interest (assumed located at 0) is reached. For an event $\mathcal H $ with $\mathbb{P}(\mathcal H)\neq 0$ , let $\mathbb{P}^{\mathcal H}(\!\cdot\!) \;:\!=\; \mathbb{P}(\cdot\mid \mathcal H)$ be the conditional probability on $\mathcal H $ .

Definition 2.2. We say that the sequence of random variables $(X_{n})_{n\ge 1}$ converges to the random variable $X_{0}$ in probability, conditionally on the event $\mathcal H$ with positive probability, if $X_{n} \longrightarrow X_{0}$ in $\mathbb{P}^{\mathcal H}$ -probability. We denote this convergence with

2.4. Existing useful asymptotic results

For establishing the limit behavior of the number-of-crossings statistics we use the following existing results on sticky Brownian motion: an approximation of the local time, an approximation of the occupation time at 0, and a result on discrete martingales. We conclude the section by recalling a previously established limit in law of the number of crossings (of a precise type, type 0, described in Section 3), that we do not use in the paper because we have not found a proof.

Proposition 2.1. ([Reference Anagnostakis and Mazzonetto5, Proposition 6.3].) Let X be a sticky Brownian motion of stickiness parameter $\rho>0$ , defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0},\mathbb{P}_x)$ , such that, $\mathbb{P}_x$ -a.s., $X_0=x$ . Let $m_{\sqrt n \rho}$ be the measure defined for all $x\in \mathbb{R}$ by

(2.2) \begin{equation} m_{\sqrt n \rho}(\mathrm{d}x) = \mathrm{d}x + \sqrt n\rho\,\delta_{0}(\mathrm{d}x) \end{equation}

and $(g_{n})_{n}$ a sequence of measurable real functions such that, for all $x\in \mathbb{R}$ ,

(2.3) \begin{align} &\lim_{n \longrightarrow \infty}\bigg(\frac{g_{n}^2(\sqrt n x)}{n} + \frac{m_{\sqrt n \rho}(g^{2}_n)}{\sqrt n}\nonumber \\ &\quad + \frac{\big(\int_{-\infty}^{+\infty}|x|g_n(x)\,\mathrm{d}x\big)(1 + \log(n))g_n(\sqrt n x)}{n} + \frac{{(1 + \log(n))m_{\sqrt n \rho}(|g_n|)}}{\sqrt n}\bigg) = 0 \end{align}

and $\lim_{n\to \infty}m_{\sqrt n \rho}(g_n) = M$ . Then, for all $t \geq 0$ ,

Also, if (2.3) holds and $\sup_n\big(m_{\sqrt n \rho}(|g_n|)\big) < \infty$ then the above convergence is localy uniform in time, in $\mathbb{P}_x$ -probability.

In Proposition 2.1, as well as throughout the paper, we use the notation

(2.4) \begin{equation} m_{\sqrt n \rho}(g) = \int_{\mathbb{R}}g(x)\,m_{\sqrt n \rho}(\mathrm{d} x) = \int_{-\infty}^{+\infty}g(x)\,\mathrm{d}x + \sqrt{n}\rho g(0)\end{equation}

for any integrable function g.

Lemma 2.2. ([Reference Anagnostakis and Mazzonetto5, Lemma 3.7].) Let X be a sticky Brownian motion of stickiness parameter $\rho>0$ , defined on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P}_x)$ such that, $\mathbb{P}_x$ -a.s., $X_0=x$ . Then, for all intervals U (including singletons),

\begin{equation*} \frac{1}{n}\sum_{i=1}^{[n\cdot]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} \in U\}} \xrightarrow[n \longrightarrow \infty]{\mathbb{P}_x\text{-}\mathrm{ucp}} \int_{0}^{\cdot}\mathbf{1}_{\{X_s \in U\}}\,\mathrm{d}s. \end{equation*}

In particular, taking $U=\{0\}$ ,

\begin{equation*} \frac{1}{n}\sum_{i=1}^{[n\cdot]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} = 0\}} \xrightarrow[n \longrightarrow \infty]{\mathbb{P}_x\text{-}\mathrm{ucp}} \int_{0}^{\cdot}\mathbf{1}_{\{X_s = 0\}}\,\mathrm{d}s = \rho L^{0}_{}(X). \end{equation*}

See also [Reference Altmeyer1] for the approximation of smooth occupation time functionals.

Theorem 2.1. (Efimenko–Portenko (1989); see [Reference Portenko29].) We consider the setting of Lemma 2.2. For all $t>0$ , $\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}<0\}}$ converges in law to some discrete random variable $Z_t$ with values in $\mathbb{N}_0 $ , i.e. there exists $(b_k(t);\,k\in\mathbb{N}_0,\,t\ge 0)$ such that $\mathbb{P}_0(Z_t = k) = b_k(t)$ for all $k\in\mathbb{N}_0$ . The probabilities $(b_k(t);\, k\in\mathbb{N}_0,\,t\ge 0)$ are defined via their Laplace transforms

\begin{equation*} \int_{0}^{\infty}\lambda{\mathrm{e}}^{-\lambda t}b_k(t)\,\mathrm{d}t = \frac{A_{\lambda}}{1/\rho^{2} + A_{\lambda}}\bigg(\frac{1/\rho^{2}}{1/\rho^{2}+A_{\lambda}}\bigg)^{k} \quad {for\ all}\ k\in\mathbb{N}_0\ { and\ all }\ t\ge 0, \end{equation*}

with $A_{\lambda} = \sqrt{\lambda}(\sqrt\lambda + \sqrt{2}/\rho)$ .

3.1. Limit behavior of the statistics

The main result of this paper is the following.

We note that in the case of the standard Brownian motion W, all three statistics have the same non-trivial limit for the same normalizing sequence $\sqrt{n}$ . Indeed, from, e.g., [Reference Jacod21, Theorem 1.1],

\begin{equation*} \mathop{\mathbb{P}{\kern.5pt}\text{-}{\kern.5pt}\text{lim}}\limits_{n\longrightarrow\infty}\bigg(\frac{C^{(0)}_{n,t}(W)}{\sqrt n}\bigg) = \mathop{\mathbb{P}{\kern.5pt}\text{-}{\kern.5pt}\text{lim}}\limits_{n\longrightarrow\infty}\bigg(\frac{C^{(1)}_{n,t}(W)}{\sqrt n}\bigg) = \mathop{\mathbb{P}{\kern.5pt}\text{-}{\kern.5pt}\text{lim}}\limits_{n\longrightarrow\infty}\bigg(\frac{C^{(2)}_{n,t}(W)}{\sqrt n}\bigg) = \sqrt{\frac{2}{\pi}} L^{0}_{t}(W).\end{equation*}

Also, the rates of convergence are known. In [Reference Jacod21, Theorem 1.2], a central limit theorem is proven for this convergence, with a rate of $n^{1/4}$ .

Unlike the standard Brownian motion, we observe that there is a factor 4 in the limits of $C^{(1)}_{n\cdot}/\sqrt n$ and $\mathbf{Z}^{(1)}_{n,\cdot}/\sqrt n $ for sticky Brownian motion (see Theorem 3.1(ii)). This factor comes from the fact that $\mathbf{Z}^{(1)}_{n,t}/\sqrt n$ rewrites as a sum of four different terms, all converging to $\sqrt{2/\pi} L^{0}_{t}(X)$ . Indeed,

\begin{equation*} \begin{aligned} \frac{1}{\sqrt{n}\,}\mathbf{Z}^{(1)}_{n,t}(X) & = \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0;\,{X}_{{{i}}/{n}}> 0\}} + \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0;\,{X}_{{{i}}/{n}} < 0\}} \\ & \quad + \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} > 0;\,{X}_{{{i}}/{n}}= 0\}} + \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} < 0;\,{X}_{{{i}}/{n}}= 0\}} . \end{aligned}\end{equation*}

Each term converges to 0 if the process is a standard Brownian motion W. For the sticky Brownian motion, the limit is instead non-trivial and it corresponds to the case of the renormalized number-of-crossings statistic for Brownian motion: $C^{(0)}_{n,\cdot}(W)/\sqrt n$ .

Regarding the bouncing statistics, we prove the following.

The difference statistics $\mathbf{Z}^{(\cdot)}_{}$ are the leading parts of number-of-crossings and -bouncings statistics of type 1 and 2.

The proof of the above results relies, by virtue of Lemma 2.3, on the following limit behavior of the conditional version of the statistics $C^{(0)}_{}$ , $C^{(1)}_{}$ , and $C^{(2)}_{}$ . These are the statistics $\widetilde C^{(0)}_{}$ , $\widetilde C^{(1)}_{}$ , and $\widetilde C^{(2)}_{}$ defined for all n, t as

\begin{align*} \widetilde C^{(0)}_{n,t}(X) & \;:\!=\; \textstyle\sum_{i=1}^{[nt]}\mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}<0\}} \mid \mathcal{F}_{({i-1})/{n}}\big), \\ \widetilde C^{(1)}_{n,t}(X) & \;:\!=\; \textstyle\sum_{i=1}^{[nt]}\mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}\le 0\}} - \mathbf{1}_{\{{X}_{({{i-1}})/{n}}={X}_{{{i}}/{n}} = 0\}} \mid \mathcal{F}_{({i-1})/{n}}\big), \\ \widetilde C^{(2)}_{n,t}(X) & \;:\!=\; \textstyle\sum_{i=1}^{[nt]}\mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}\le 0\}} \mid \mathcal{F}_{({i-1})/{n}}\big).\end{align*}

For these, the following result holds.

The proof of Theorem 3.1(i) is done by bounding the distance between $ \widetilde C^{(0)}_{}$ and $C^{(0)}_{} $ , and showing that both the distance and the limit of $\widetilde C^{(0)}_{}(X)$ , if renormalized by $1/u_n $ , vanish in probability as $n \longrightarrow \infty$ .

Let us note that the only case, for which the limits of the expected and non-expected versions of the number-of-crossings statistics are not necessarily the same is for the type 0 crossings (strict crossings). In fact, they cannot be the same since $C^{(0)}_{n,t}(X)$ and its limit take values in $\mathbb{N} \cup (+\infty)$ , while the law of $(1/\rho) L^{0}_{t}(X) $ , the limit of $\widetilde C^{(0)}_{n,t}(X) $ , is continuous. The discrete limit law of $C^{(0)}_{n,t}(X)$ is described in [Reference Portenko29, §8], which is restated here in Theorem 2.1. We also mention that Theorem 3.1(i) can also be inferred directly from Theorem 2.1.

3.2. Estimation

In [Reference Anagnostakis3, Reference Anagnostakis and Mazzonetto5] an estimator was proposed based on a test function g and a normalizing sequence $u_n$ . More precisely, let g be a bounded integrable function that satisfies $g(0)=0$ , $(u_n)_n$ a sequence such that, as $n \longrightarrow \infty$ , $u_n \longrightarrow \infty$ and $u_n/n \longrightarrow 0$ , $\mathcal H_t$ be the event $\big\{\tau^{X}_0<t\big\} = \{\text{there exists}\ s < t\ \text{such that}\ X_s = 0\}$ , and $\varrho^{(0)}_{n}(X)$ the statistic

(3.1) \begin{equation} \varrho^{(0)}_{n}(X) \;:\!=\; \bigg(\int_{\mathbb{R}}g\,\mathrm{d}x\bigg)\frac{1}{u_n} \frac{\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0\}}}{\sum_{i=1}^{[nt]}g(u_n{X}_{({{i-1}})/{n}})}.\end{equation}

Then $\varrho^{(0)}_{n}(X) $ is a consistent estimator of $\rho $ conditionally on the event $\mathcal H_t $ , i.e. $\varrho^{(0)}_{n}(X) \longrightarrow \rho $ in $\mathbb{P}^{\mathcal H_t}_x $ -probability as $n \longrightarrow \infty $ .

We now devise a consistent stickiness parameter estimator based on the number-of-crossings statistics. This provides an alternative estimator to (3.1).

Like (3.1), this new estimator also converges conditionally on the event that 0 has been reached before time t. If the threshold is not reached, the number-of-crossings statistics are all trivially null.

Then, the following conditional convergence in probability holds.

Proposition 3.2. Consider the setting of Theorem 3.1 and let $t>0$ . The statistic

(3.2) \begin{equation} \varrho_n(X) = \bigg(4\sqrt{\frac2\pi}\bigg)\frac{1}{\sqrt{n}\,}\frac{C^{(2)}_{n,t}(X)}{C^{(1)}_{n,t}(X)} \end{equation}

is a consistent estimator of $\rho$ conditionally on the event $\mathcal H_t$ , i.e. $\varrho_n(X) \longrightarrow \rho $ in $\mathbb{P}^{\mathcal H_t}_x $ -probability as $n \longrightarrow \infty $ .

4.1. Crossings of type 0

Here, we prove all results regarding the number of type 0 crossings, namely Theorem 3.1(i) and Proposition 3.1(i). We first prove Proposition 3.1(i), relying on the sticky kernel asymptotics from Lemma 4.1 and the convergence result in Proposition 2.1. Then, using Lemma 2.3 (conditional and unconditioned versions of the statistic have the same asymptotics), we prove Theorem 3.1(i) by reducing to Proposition 3.1(i).

Proof of Proposition 3.1(i). We begin by expressing the statistic $\widetilde C^{(0)}_{n,t}(X)$ in terms of the functions $(g_n)$ and $(h_n)$ from Lemma 4.1. Indeed, by the Markov property we have

\begin{equation*} \widetilde C^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]}\mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}< 0\}}\mid\mathcal{F}_{({i-1})/{n}}\big) = \sum_{i=1}^{[nt]}g_n({X}_{({{i-1}})/{n}}) = \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}h_n(\sqrt n{X}_{({{i-1}}(/{n}}). \end{equation*}

By Proposition 2.1 and Lemma 4.1(iii), we have

which completes the proof.

Proof of Theorem 3.1(i). We consider the families of $\sigma$ -algebras $(\mathcal{F}^{n}_{i})_{i,n}$ and random variables $(\chi^{n}_{i})_{i,n} $ defined for all i, n by $\mathcal{F}^{n}_{i} \;:\!=\; \mathcal{F}_{i/n}$ and $\chi^{n}_{i} \;:\!=\; ({1}/{u_n})\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}< 0\}}$ so that

\begin{equation*} \frac{1}{u_n}C^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]}\chi^{n}_{i}, \qquad \frac{1}{u_n}\widetilde C^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]}\mathbb{E}_x\big(\chi^{n}_{i} \mid \mathcal{F}^{n}_{i-1}\big). \end{equation*}

From Proposition 3.1(i),

and, since $(\chi^n_i)^2= \chi^n_i/u_n$ ,

Thus, from Lemma 2.3,

From Lemma 2.1, the convergence is also uniform in time, in probability (ucp). This completes the proof.

4.2. Crossings of type 1

Here, we prove all results regarding the number of type 1 crossings, namely Theorem 3.1(ii) and part of Proposition 3.1(ii).

The proof relies on studying the limit behavior of the following quantity which, according to our naming conventions, is referred to as the conditional difference statistic:

\begin{equation*} \widetilde{\mathbf Z}^{(1)}_{n,t}(X) \;:\!=\; \sum_{i=1}^{[nt]}\mathbb{E}_x\big( \mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}}\not= 0\}} + \mathbf{1}_{\{{X}_{({{i-1}})/{n}}\not =0,\,{X}_{{{i}}/{n}}= 0\}} \mid \mathcal{F}_{({i-1})/{n}}\big),\end{equation*}

which satisfies $\widetilde C^{(1)}_{n,t}(X) = \widetilde{\mathbf Z}^{(1)}_{n,t}(X) + \widetilde C^{(0)}_{n,t}(X)$ , involving the type 1 and type 0 conditional statistics.

We first establish the limit behavior of $\widetilde{\mathbf Z}^{(1)}_{n,t}(X)$ using Lemmas 2.2 (occupation time approximation) and 4.1 (sticky kernel asymptotics). The asymptotics of the type 1 conditional crossing statistic $\widetilde C^{(1)}_{n,t}(X) $ follows from the decomposition $\widetilde C^{(1)}_{n,t}(X) = \widetilde{\mathbf Z}^{(1)}_{n,t}(X) + \widetilde C^{(0)}_{n,t}(X) $ and Proposition 3.1(i) (asymptotics of the respective type 0 statistic). Finally, by Lemma 2.3 we deduce Theorem 3.1(ii) from its conditional version.

Proof. Regarding the first term of $\widetilde{\mathbf Z}^{(1)}_{}$ , by Lemmas 2.2 and 4.1(i) we have

(4.1)

Regarding the second term of $\widetilde{\mathbf Z}^{(1)}$ , by the Markov property we have

\begin{equation*} \begin{aligned} \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbb{E}_x\big[ \mathbf{1}_{\{{X}_{({{i-1}})/{n}}\not =0;\,{X}_{{{i}}/{n}} = 0\}} \mid \mathcal{F}_{({i-1})/{n}}\big] & = \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}\not =0\}} \mathbb{P}_{{X}_{({{i-1}})/{n}}}({X}_{{{1}}/{n}} = 0) \\ & = \frac{1}{\sqrt{n}\,}\sum_{i=1}^{[nt]}k_n(\sqrt n{X}_{({{i-1}})/{n}}), \end{aligned} \end{equation*}

where $(k_n)_n$ is the sequence of functions defined in Lemma 4.1. From Proposition 2.1 and Lemma 4.1(ii),

(4.2)

Combining the relations in (4.1) and (4.2) yields the desired result.

Proof of Proposition 3.1(ii) for $j=1$ . Essentially, it suffices to observe that

\begin{equation*} \frac{1}{\sqrt{n}\,}\widetilde C^{(1)}_{n,t}(X) = \frac{1}{\sqrt{n}\,}\widetilde{\mathbf Z}^{(1)}_{n,t}(X) + \frac{1}{\sqrt{n}\,}\widetilde C^{(0)}_{n,t}(X). \end{equation*}

Then, by Lemma 4.2 and Proposition 3.1(i),

By Lemma 2.1, the convergence is also uniform in time, in probability (ucp).

Proof of Theorem 3.1(ii). The proofs works by first proving convergence of the type 1 zero statistic $\mathbf{Z}^{(1)}_{\cdot,t}$ by reduction to its conditional version $\widetilde{\mathbf Z}^{(1)}_{\cdot,t}$ . We then conclude from previous results by re-writing $C^{(1)}_{\cdot,t}$ as $\mathbf{Z}^{(1)}_{\cdot,t} + C^{(0)}_{\cdot,t}$ .

For the reduction argument, we consider the families of $\sigma$ -algebras $(\mathcal{F}^{n}_{i})_{i,n}$ and random variables $(\chi^{n}_{i})_{i,n}$ , defined for all $i\in \mathbb{N}$ , $n >0$ , by

\begin{equation*} \mathcal{F}^{n}_{i} \;:\!=\; \mathcal{F}_{i/n}, \qquad \chi^{n}_{i} \;:\!=\; \frac{1}{\sqrt{n}\,}\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}= 0\}} - \mathbf{1}_{\{{X}_{({{i-1}})/{n}}={X}_{{{i}}/{n}} = 0\}}\big), \end{equation*}

so that

\begin{equation*} \frac{1}{\sqrt{n}\,}\mathbf{Z}^{(1)}_{n,t}(X) = \sum_{i=1}^{[nt]}\chi^{n}_{i}, \qquad \frac{1}{\sqrt{n}\,}\widetilde{\mathbf Z}^{(1)}_{n,t}(X) = \sum_{i=1}^{[nt]}\mathbb{E}_x\big(\chi^{n}_{i} \mid \mathcal{F}^{n}_{i-1}\big). \end{equation*}

From Lemma 4.2,

and, since $(\chi^n_i)^2= \chi^n_i/\sqrt n$ ,

By Lemma 2.3 we therefore have that

Combining this with Theorem 3.1(i) ensures that

From Lemma 2.1, the convergence is also uniform in time, in probability (ucp). This completes the proof.

4.3. Crossings of type 2

Here, we prove all results regarding the number of type 2 crossings, completing the proofs of Theorem 3.1 and Proposition 3.1. The proofs consist of combining results on type 1 crossings with the occupation time approximation (Lemma 2.2).

Specifically, we show that both the type 2 crossing statistic and its conditional version are asymptotically equivalent to the occupation time approximation from Lemma 2.2.

Proof of Theorem 3.1(iii). Recall the decomposition

\begin{equation*} C^{(2)}_{n,t}(X) = C^{(1)}_{n,t}(X) + \sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}} = 0\}}, \end{equation*}

where we can rewrite the second term as $\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}} = 0\}} = \mathbf{1}_{\{{X}_{({{i-1}})/{n}} =0\}} - \mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}}\neq 0\}}$ . Since $\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}}\neq 0\}$ corresponds to a particular case of type 1 crossing, we have $0 \le C^{(1)}_{n,t}(X) - \sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0,\,{X}_{{{i}}/{n}}\neq 0\}} \le C^{(1)}_{n,t}(X)$ . This yields the bound

\begin{equation*} \frac1n\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0\}} \le \frac{1}{n}C^{(2)}_{n,t}(X) \le \frac1n\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}=0\}} + \frac1n C^{(1)}_{n,t}(X). \end{equation*}

By item (ii) and Lemma 2.2, both bounds converge to $\rho L^{0}_{t}(X)$ . Therefore, by the squeeze theorem, for all $t\ge 0$ ,

Lemma 2.1 ensures ucp convergence, which completes the proof.

Proof of Proposition 3.1(ii) for $j=2$ . Observe that

\begin{equation*} \frac{1}{n}\widetilde C^{(2)}_{n,t}(X) = \frac{1}{n}\sum_{i=1}^{[nt]}\mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}={X}_{{{i}}/{n}}= 0\}} \mid \mathcal{F}_{({i-1})/{n}}\big) + \frac{1}{n}\widetilde C^{(1)}_{n,t}(X). \end{equation*}

With the same argument as in the proof of Theorem 3.1(iii),

\begin{equation*} \frac{1}{n}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} = 0\}} \le \frac{1}{n}\widetilde C^{(2)}_{n,t}(X) \le \frac{1}{n}\sum_{i=1}^{[nt]}\mathbf{1}_{\{{X}_{({{i-1}})/{n}} = 0\}} + \frac{1}{n}\widetilde C^{(1)}_{n,t}(X). \end{equation*}

From item (ii) for $j=1$ and Lemma 2.2, we have that, for all $t\ge 0$ ,

By Lemma 2.1, the convergence is ucp. This completes the proof.

4.4. Proof of Theorem 3.2

The proof relies essentially in linking the bouncing and crossing behaviors of type 0 via a reflection principle at 0 for sticky Brownian motion. Then, analogous to the results for type 1 and type 2 crossings, the leading terms in the number of type 1 and type 2 bouncings ( $B^{(1)}$ , $B^{(2)}$ ) are given by the corresponding difference statistics ( $\mathbf{Z}^{(1)}$ , $\mathbf{Z}^{(2)}$ ).

For the statement and the proof of the reflection principle, let $\tau^{Y}_{0}$ be the hitting time of 0 by the process Y, defined as

(4.3) \begin{equation} \tau^{Y}_{0} = \inf\{t>0\colon Y_t = 0\}.\end{equation}

We are now ready to state this intermediary result. The proof is deferred to Appendix B.

Lemma 4.3. (Reflection principle for sticky Brownian motion.) Let X be the sticky Brownian motion of stickiness parameter $\rho>0$ defined on the filtered probability space $\mathcal P_x = (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0},\mathbb{P}_x)$ such that, $\mathbb{P}_x$ -a.s., $X_0 = x$ (in particular, X is $(\mathcal F_t)_{t\ge 0}$ -adapted). Let X^′ be the process defined, for all $t\ge 0$ , by

The process X^′ defined this way is a sticky Brownian motion of stickiness parameter $\rho$ .

We are now ready to address the proof of Theorem 3.2.

Proof of Theorem 3.2(i). In this proof, let $\widetilde B^{(0)}$ be the conditional version of $B^{(0)}$ , defined for all $n,t >0$ by $\widetilde B^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]}\mathbb{E}_x\big( \mathbf{1}_{U^{n}_{i}(X)}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}>0\}} \mid \mathcal{F}_{({i-1})/{n}}\big)$ . We also introduce the notation of Markovian families used in [Reference Freedman16, Section 2.1]. Let X ^′ be a diffusion on J, an interval of $\mathbb{R}$ , defined on the probability space $\mathcal P_x$ such that, $\mathbb{P}_x$ -a.s., $X'_0 = x$ . We denote by $(Q^{X'}_y,\, y\in J)$ the canonical diffusion on the path-space $C([0,\infty),\mathbb{R})$ and by Y the coordinate process. Then, for every almost-surely-finite stopping time $\tau$ , we have

\begin{equation*} Q^{X'}_{X'_{\tau}}(h(Y_t)) = \mathbb{E}_x(h(X'_{\tau+t}) \mid \mathcal{F}_\tau). \end{equation*}

From the strong Markov property, using the above notation and (4.3), we have

\begin{align*} \mathbb{E}_x\big(\mathbf{1}_{U^{n}_{i}(X)}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}>0\}} \mid \mathcal{F}_{({i-1})/{n}}\big) & = Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{Y_0{Y}_{{{1}}/{n}}>0\}}\big) \\ & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}>0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}>0\}}\big) \\ & \quad + \mathbf{1}_{\{{X}_{({{i-1}})/{n}}<0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}<0\}}\big), \\ \mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}<0\}} \mid \mathcal{F}_{({i-1})/{n}}\big) & = Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{Y_0{Y}_{{{1}}/{n}}<0\}}\big) \\ & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}<0\}}Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{{Y}_{{{1}}/{n}}>0\}}\big) \\ & \quad + \mathbf{1}_{\{{X}_{({{i-1}})/{n}}>0\}}Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{{Y}_{{{1}}/{n}}<0\}}\big). \end{align*}

From Lemma 4.3, we have

\begin{align*} \mathbf{1}_{\{{X}_{({{i-1}})/{n}}>0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}>0\}}\big) & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}>0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}<0\}}\big) \\ & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}>0\}}Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{{Y}_{{{1}}/{n}}<0\}}\big), \\ \mathbf{1}_{\{{X}_{({{i-1}})/{n}}<0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}<0\}}\big) & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}<0\}}Q^{X}_{{X}_{({{i-1}})/{n}}} \big(\mathbf{1}_{\{\tau^{Y}_0<1/n\}}\mathbf{1}_{\{{Y}_{{{1}}/{n}}>0\}}\big) \\ & = \mathbf{1}_{\{{X}_{({{i-1}})/{n}}<0\}}Q^{X}_{{X}_{({{i-1}})/{n}}}\big(\mathbf{1}_{\{{Y}_{{{1}}/{n}}>0\}}\big). \end{align*}

Therefore, we obtain $\mathbb{E}_x\big(\mathbf{1}_{U^{n}_{i}(X)}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}>0\}} \mid \mathcal{F}_{({i-1})/{n}}\big) = \mathbb{E}_x\big(\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}<0\}} \big| \mathcal{F}_{({i-1})/{n}}\big)$ and thus

(4.4) \begin{equation} \widetilde B^{(0)}_{n,t} = \widetilde C^{(0)}_{n,t}. \end{equation}

Let us consider the families of $\sigma$ -algebras $(\mathcal{F}^{n}_{i})_{i,n}$ and random variables $(\chi^{n}_{i})_{i,n}$ defined, for all i, n, by $\mathcal{F}^{n}_{i} = \mathcal{F}_{i/n}$ and $\chi^{n}_{i} = \frac{1}{u_n}\mathbf{1}_{U^{n}_{i}(X)}\mathbf{1}_{\{{X}_{({{i-1}})/{n}}{X}_{{{i}}/{n}}> 0\}}$ . Then,

\begin{equation*} \frac{1}{u_n}B^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]} \chi^{n}_{i}, \qquad \frac{1}{u_n}\widetilde B^{(0)}_{n,t}(X) = \sum_{i=1}^{[nt]}\mathbb{E}_x\big(\chi^{n}_{i} \mid \mathcal{F}^{n}_{i-1}\big). \end{equation*}

From Proposition 3.1(i) and (4.4),

and, since $(\chi^{n}_i)^{2}=\chi^{n}_i/u_n$ ,

Thus, from Lemma 2.3,

From Lemma 2.1, the convergence is also uniform in time, in probability (ucp). This completes the proof of item (i).

Proof of Theorem 3.2(ii). We observe that, for $j=1,2$ , $\mathbf{Z}^{(\,j)}_{n,\cdot}(X) = B^{(\,j)}_{n,\cdot}(X) - B^{(\,j-1)}_{n,\cdot}(X)$ . Thus, from Theorem 3.1 and item (i),

From Lemma 2.1, the convergence is uniform in time, in $\mathbb{P}_x$ -probability ( $\mathbb{P}_x$ -ucp). This completes the proof.

4.5. Proof of Proposition 3.2

Proof. From Theorem 3.1,

This and the fact that $L^{0}_{t}(X)=L^{0}_{t}(X)\mathbf{1}_{\{L^{0}_{t}(X)>0\}}$ ensure that

and

Note that, $\mathbb{P}_x$ -a.s., $ \{L^{0}_{t}(X)>0\} = \mathcal H_t$ (this follows, e.g., from [Reference Kallenberg25, Corollary 29.18]) and, for every $\varepsilon >0$ ,

\begin{equation*} \mathbb{P}_x(\mathcal H_t \cap \{|\varrho_n -\rho|>\varepsilon\}\}) = \mathbb{P}_x(\{L^{0}_{t}(X)>0\} \cap \{|\varrho_n -\rho|>\varepsilon\}\}) = \mathbb{P}_x(|\varrho_n -\rho|\mathbf{1}_{\{L^{0}_{t}(X)>0\}}>\varepsilon), \end{equation*}

which converges to 0 as $n\to \infty$ , as we just showed. Since $\mathbb{P}_x(\mathcal H_t)>0$ , from Bayes’ rule, we prove the desired convergence.

5.1. Bouncings of the sticky–reflected Brownian motion

The sticky–reflected Brownian motion, also known as slowly reflected Brownian motion, was first discovered by Feller in his attempt to describe all possible ways to define Brownian motion on the positive semi-axis $[0,\infty)$ ; see [Reference Feller14]. In particular, he discovered that the operator $\mathrm{L} = \mathrm{D}^{2}_x$ with domain $\mathrm{dom}(\mathrm{L}) = \{\,f\in C([0,\infty))\cap C^{2}([0,\infty))\colon f'(0+)=(\rho/2)f''(0+)\}$ , with $\rho>0$ , is the infinitesimal generator of a diffusion that spends a positive amount of time at 0 and that behaves like a Brownian motion away from 0. Replacing the boundary condition $f'(0+)=(\rho/2)f''(0+)$ by the lateral condition $f(0+)-f(0-)=(\rho/2)\,f''(0+)$ defines the sticky Brownian motion, or two-sided sticky Brownian motion, introduced in Section 2.2. A nice historical overview can be found in [Reference Peskir28].

For simplicity, we define the sticky–reflected Brownian motion of stickiness parameter $\rho>0 $ as the unique (in law) weak solution to the system

(5.1)

where B is a standard Brownian motion (see [Reference Engelbert and Peskir13, Theorem 5]).

Let us note that the absolute value of a sticky Brownian motion is a sticky–reflected Brownian motion with the same stickiness parameter. More precisely, if Y is a sticky Brownian motion of stickiness parameter $\rho$ , from the Tanaka formula [Reference Revuz and Yor30, Theorem VI.1.2] and the SDE characterization in (p1) for sticky Brownian motion, there exists a Brownian motion B on an extension of the probability space such that

(5.2) \begin{equation} \mathrm{d}|Y_t| = \mathrm{sgn}(Y_t)\,\mathrm{d}Y_t + \mathrm{d}L^{0}_{t}(Y) = \mathrm{sgn}(Y_t)\mathbf{1}_{\{|Y_t|\not = 0\}}\,\mathrm{d}B_t + \mathrm{d}L^{0}_{t}(Y).\end{equation}

This, along with (p1), yield that $\mathrm{d}\langle Y\rangle_t = \mathrm{d}\langle|Y|\rangle_t = \mathbf{1}_{\{Y_t \not = 0\}}\,\mathrm{d}t$ and that, since Y is a martingale, from [Reference Revuz and Yor30, Corollary VI.1.9],

(5.3) \begin{align} L^{0}_{t}(|Y|) & = \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon}\int_{0}^{t}\mathbf{1}_{\{0<|Y_s|<\varepsilon\}}\,\mathrm{d}s \nonumber\\ & = \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon}\int_{0}^{t}\mathbf{1}_{\{0<Y_s<\varepsilon\}}\,\mathrm{d}s + \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon}\int_{0}^{t}\mathbf{1}_{\{0<-Y_s<\varepsilon\}}\,\mathrm{d}s \nonumber\\ & = 2L^{0}_{t}(Y) = 2\rho\int_{0}^{t}\mathbf{1}_{\{Y_s = 0\}}\,\mathrm{d}s. \end{align}

From (5.2) and (5.3), if $B'=\int_{0}^{\cdot}\mathrm{sgn}(Y_s)\,\mathrm{d}B_s$ , the pair $(|Y|,B')$ solves (5.1).

We are now ready to state our main results regarding the number of bouncings of this process. The proof reduces to the case of non-reflected sticky Brownian motion.

Proof. Let Y be a sticky Brownian motion of stickiness parameter $\rho $ defined on the probability space $\mathcal{P}'_x = (\Omega',\mathcal{F}',(\mathcal{F}'_t)_{t\ge 0},\mathbb{P}'_x) $ such that, $\mathbb{P}'_x$ -a.s., $Y_0 = x$ and Y is $(\mathcal{F}'_t)_{t\ge 0}$ -adapted. We note that X and $|Y|$ are equal in law (see the discussion before the statement) and that

(5.4) \begin{align} B^{(0)}_{n,\cdot}(|Y|) & = B^{(0)}_{n,\cdot}(Y)+C^{(0)}_{n,\cdot}(Y), \nonumber\\ B^{(1)}_{n,\cdot}(|Y|) & = \mathbf{Z}^{(\,j)}_{n,\cdot}(Y) + B^{(0)}_{n,\cdot}(|Y|), \nonumber\\ B^{(2)}_{n,\cdot}(|Y|) & = \mathbf{Z}^{(\,j)}_{n,\cdot}(Y) + B^{(0)}_{n,\cdot}(|Y|). \end{align}

From Theorems 3.1 and 3.2, for every diverging sequence $(u_n)_n$ ,

This, and the fact that $L^{0}_{t}(Y)= L^{0}_{t}(Y) \mathbf{1}_{\{L^{0}_{t}(Y)>0\}}$ an d $L^{0}_{t}(|Y|)=2L^{0}_{t}(Y)$ , imply the convergences

and

It is known that, in the case of constant limits, convergence in probability is equivalent to convergence in law (see [Reference Billingsley7, Theorem 25.2] and the discussion thereafter). And since $|Y|= X$ in law (which entails that $(|Y|,L^{0}_{}(|Y|))=(X,L^{0}_{}(X))$ in law), the latter convergences hold with X instead of $|Y|$ and $\mathbb{P}_x$ instead of $\mathbb{P}'_x$ . Therefore, again using that ${L^{0}_{t}(X)} = {L^{0}_{t}(X)} \mathbf{1}_{\{L^{0}_{t}(X)>0\}}$ , we obtain

and

Thus, we have

(5.5)

From Lemma 2.1, the convergences are uniform in time, in $\mathbb{P}_x$ -probability ( $\mathbb{P}_x$ -ucp). This proves (i), (ii), and (iii). From (5.4) and (5.5), we infer (iv). This completes the proof.

Corollary 5.1. We consider the setting of Theorem 5.1 and let $\mathcal H_t$ be the event $\big\{\tau^{X}_0<t\big\} = \{$ there exists $s \in [0,t) \colon X_s = 0\}$ . The statistic

\begin{equation*} \varrho_n(X) = \bigg(4\sqrt{\frac2\pi}\bigg)\frac{1}{\sqrt{n}\,} \frac{\mathbf{Z}^{(2)}_{n,\cdot}(X)}{\mathbf{Z}^{(1)}_{n,\cdot}(X)} \end{equation*}

is a consistent estimator of $\rho$ , conditionally on the event $\mathcal H_t$ , i.e. $\varrho_n(X) \longrightarrow \rho$ , in $\mathbb{P}^{\mathcal H_t}_x$ -probability, as $n \longrightarrow \infty$ .

5.2. Crossings and bouncings of sticky Itô diffusions

In this section we generalize Theorems 3.1 and 3.2 to sticky Itô diffusions. The generalization procedure is analogous to the ones in, e.g., [Reference Anagnostakis3, Reference Anagnostakis and Mazzonetto5, Reference Azas6, Reference Jacod21, Reference Mazzonetto27].

We consider the system

(5.6)

with $W_{}$ a standard Brownian motion, defined on the probability space $\mathcal P_x = (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0},\mathbb{P}_x) $ such that, $\mathbb{P}_x$ -a.s., $X_0=x$ . We suppose $(\mu,\sigma) $ satisfy the following conditions:

1. (c1) The diffusion coefficient $\sigma$ is strictly positive and $C^{1}$ .

2. (c2) Weak existence and uniqueness in law holds for the system

   (5.7)

   
   on $\mathcal P_x$ such that, $\mathbb{P}_x$ -a.s., $X'_0=x$ .

3. (c3) The laws of the solutions X, X ^′ of (5.6) and (5.7) with the same initial condition ( $X_0 = X'_0 = x $ a.s.) are locally equivalent: they are mutually locally absolutely continuous in the sense of [Reference Jacod and Shiryaev24, Definition III.3.2]. Basically, this means for all $t \ge 0$ , $\mathrm{Law}\big(X|_{[0,t]}\big) \sim \mathrm{Law}\big(X'|_{[0,t]}\big)$ . Note that the processes are not necessarily defined on the same probability space.

4. (c4) The state space of X ^′ that solves (5.7) is an open interval J of $\mathbb{R}$ , i.e. X ^′ has only inaccessible boundaries (see, e.g., [Reference Kallenberg25, Section 33]).

Under conditions (c1)–(c4), the following results hold.

In the proof we reduce ourselves to the case of sticky Brownian motion. By the local equivalence in law (c3), it suffices to prove the result for (5.7). We then further reduce to the case of sticky Brownian motion by Girsanov’s lemma and a Lamperti transform.

Proof of Theorem 5.2. Let us first assume that the diffusion coefficient and its derivative satisfy the following conditions: there exists a real constant $\delta >0 $ such that, for all $x \in \mathbb{R}$ , $\delta \le \sigma(x)\le 1/\delta$ and $|\sigma'(x)| \le 1/\delta$ . We note that under these conditions, the state space J is $\mathbb{R}$ . We consider X ^′ to be the process that is the solution to (5.7), defined on $\mathcal P_x$ and with $(\mathcal{F}_t)_{t\ge 0}$ the natural filtration generated by X ^′. By Definition 2.1 of ucp convergence, we can fix $T>0$ and consider that all processes are restricted to the time interval [0,T].

Let $\mathcal{E}(\theta)$ , $\theta$ be the processes defined by

\begin{equation*} \mathcal{E}_t(\theta) = \exp\bigg(\int_{0}^{t}\theta_s\,\mathrm{d}W_s - \frac{1}{2}\int_{0}^{t}\theta^2_s\,\mathrm{d}s\bigg), \qquad \theta_t = \frac12\sigma'(X_t), \quad t \ge 0. \end{equation*}

Since $\sigma'$ is bounded, from Girsanov’s theorem [Reference Anagnostakis3, Lemma 4.4] there exists a measure $\mathbb{Q}_x$ such that the process $(X'_t, W^{\theta}_{t})_{t\in [0,T]}$ jointly solves a drifted version of (5.7),

(5.8)

where $(W^{\theta}_{t})_{t\in [0,T]}$ is the $\mathbb{Q}_x$ -Brownian motion defined by $W^{\theta} = W_{} - \int_{0}^{\cdot}\theta_s\,\mathrm{d}s$ . Moreover, the Radon–Nikodym derivative $\mathrm{d}\mathbb{Q}_x/\mathrm{d}\mathbb{P}_x|_{\mathcal{F}_T} = \mathcal{E}_T(\theta)$ .

Let S be the function defined by

\begin{equation*} S(y) = \int_{0}^{y} \frac{1}{\sigma(u)}\,\mathrm{d}u, \quad y \in \mathbb R. \end{equation*}

The function S is strictly increasing, $S(0)=0$ , $S \in C^{2}(\mathbb{R})$ , $S'(y) = (\sigma(y))^{-1}$ , and $S''(y) = -\sigma'(y)(\sigma(y))^{-2}$ , $y \in \mathbb R$ . Let $Y'_{} \;:\!=\; S(X')$ , Then, from the Itô–Tanaka formula (see, e.g., [Reference Anagnostakis3, Lemma 4.5]), (5.8), and the fact that $S(0)=0$ , $(Y'_{}, W^{\theta}_{})$ solves

From characterization (p2), the process Y ^′ is a sticky Brownian motion under $\mathbb{Q}_x$ of stickiness parameter $\rho\sigma(0)$ and Theorems 3.1 and 3.2 do apply to Y ^′ for the probability $\mathbb{Q}_x$ . In particular:

1. (i) ,

2. (ii) ,

3. (iii) .

Since S is strictly increasing and $S(0)=0$ ,

• • for all n, t and $j\in\{0,1,2\}$ , $C^{(\,j)}_{n,t}(Y'_{}) = C^{(\,j)}_{n,t}(X')$ , $B^{(\,j)}_{n,t}(Y'_{}) = B^{(\,j)}_{n,t}(X')$ ,

• • for all n, t and $j\in \{1,2\} $ , $\mathbf{Z}^{(\,j)}_{n,t}(Y'_{}) = \mathbf{Z}^{(\,j)}_{n,t}(X') $ ,

• • for all i, n, $U^{n}_{i}(Y'_{})=U^{n}_{i}(X') $ ,

and, $\mathbb{Q}_x $ -a.s., $L^{0}_{t}(X') = \sigma(0)L^{0}_{t}(Y'_{})$ . From the equivalence of measure $\mathbb{P}_x \sim \mathbb{Q}_x$ when restricted to $\mathcal{F}_T$ , and the fact that $\mathcal E(\theta)_T $ is an exponential martingale (even square integrable), we deduce that the previous convergences also hold under $\mathbb{P}_x$ . The proof is thus completed for (5.7) under the boundedness assumptions on the diffusion coefficient and its derivative.

Let us consider a standard localization argument to go from bounded to unbounded $\sigma$ , $1/\sigma$ , and $\sigma'$ by considering $X'^{\tau'_m} = \big(X'_{t\wedge \tau'_m}\big)_{t\ge 0}$ . More precisely, let $(\mathcal{F}_t)_{t\ge 0}$ be the filtration generated by X ^′, let $(K_m)_m $ be an increasing sequence of compacts of J such that $\bigcup_m K_m = J $ , and let $(\tau'_m)_m $ be the sequence of stopping times defined by $\tau'_m = \inf\{t\ge 0\colon X'_t \not\in K_m \}$ and $X'^{\tau'_m} $ be the stopped process, defined by $X'^{\tau'_m} = (X'_{t\wedge \tau'_m})_{t\ge 0} $ . Condition (c4) ensures that

(5.9) \begin{equation} \lim_{m \rightarrow \infty} \mathbb{P}_x(t > \tau'_m) = 0 \end{equation}

for all $t\geq 0$ . For a process Z and all $t\geq 0$ , let $(\,f_n(Z,t))_n$ be a sequence of statistics such that $f_n(Z,t)$ is $\mathcal{F}_t $ -measurable. These quantities represent the difference between the statistics of interest and their limits. We also observe that

\begin{equation*} \begin{aligned} \mathbb{P}_x (|\,f_n(X',t)|>\varepsilon) & \le \mathbb{P}_x(|\,f_n(X',t)|>\varepsilon;\, t\le \tau'_m) + \mathbb{P}_x(t > \tau'_m) \\ & = \mathbb{P}_x(|\,f_n(X',t \wedge \tau'_m)|>\varepsilon;\, t\le \tau'_m) + \mathbb{P}_x(t > \tau'_m). \end{aligned} \end{equation*}

We use the previous part to pass to the limit as $n\to \infty$ . Next, we let $m \longrightarrow \infty$ . From (5.9), this proves the result for X ^′.

From (c3), these convergences also hold in $\mathbb{P}_x$ -probability for the solution to (5.6).

From Lemma 2.1, the convergence is also uniform in time, in $\mathbb{P}_x$ -probability ( $\mathbb{P}_x$ -ucp). This completes the proof.

Corollary 5.2. Let $\mathcal P_x =(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0}, \mathbb{P}_x)$ be a filtered probability space and let X be an $(\mathcal{F}_t)_{t\ge 0}$ -adapted process solution to (5.6) on $\mathcal P_x$ . The statistic defined by

\begin{equation*} \varrho_n(X) = \bigg(\frac{4}{\sigma(0)}\sqrt{\frac2\pi}\bigg)\frac{1}{\sqrt{n}\,} \frac{C^{(2)}_{n,t}(X)}{C^{(1)}_{n,t}(X)}, \quad n \in \mathbb{N}, \end{equation*}

is a consistent estimator of $\rho$ , conditionally on the event $\mathcal H_t = \{\tau^{X}_0 < t\} = \{$ there exists $s \in [0,t) \colon X_s = 0\}$ . This means that (see Definition 2.2), $\varrho_n(X) \longrightarrow \rho$ , in $\mathbb{P}^{\mathcal H_t}_x $ -probability, as $n \longrightarrow \infty$ .