Modern Stochastics: Theory and Applications

Consider the stochastic differential equation where a is a locally integrable function.

The aim of this paper is to study convergence in distribution of the sequence of processes $\{X_{}(nt)/\sqrt{n},\hspace{2.5pt}t\geqslant 0\}$ as $n\to \infty $.

Observe that where $W_{n}(t)=W(nt)/\sqrt{n}$, $t\geqslant 0$ is a Wiener process, and $X_{n}(t)=X_{}(nt)/\sqrt{n}$, $t\geqslant 0$.

Hence, to study the sequence $\{X_{}(nt)/\sqrt{n}\}$, it suffices to investigate the SDEs where $a_{n}(x)=na(nx)$.

If $a\in L_{1}(\mathbb{R})$, then $a_{n}$ converges in generalized sense to $\alpha \delta _{0}$, where $\delta _{0}$ is the Dirac delta function at zero, where $\alpha =\int _{\mathbb{R}}a(x)\hspace{0.1667em}dx$. It is well known that in this case the sequence $\{X_{n}\}$ converges weakly to a skew Brownian motion with parameter $\gamma =(\alpha )=\frac{{e}^{\alpha }-{e}^{-\alpha }}{{e}^{\alpha }+{e}^{-\alpha }}$; see, for example, [14, 10]. Recall that [5, 10] the skew Brownian motion $W_{\gamma }(t)$ with parameter γ, $|\gamma |\leqslant 1$, is a unique (strong) solution to the SDE where ${L_{W_{\gamma }}^{0}}(t)=\lim _{\varepsilon \to 0+}{(2\varepsilon )}^{-1}{\int _{0}^{t}}\mathbb{1}_{|W_{\gamma }(s)|\leqslant \varepsilon }\hspace{0.1667em}ds$ is the local time of the process $W_{\gamma }$ at 0. The process $W_{\gamma }$ is a continuous Markov process with transition probability density function $p_{t}(x,y)=\varphi _{t}(x-y)+\gamma (y)\hspace{0.1667em}\varphi _{t}(|x|+|y|)$, $x,y\in \mathbb{R}$, where $\varphi _{t}(x)=\frac{1}{\sqrt{2\pi t}}{e}^{-{x}^{2}/2t}$ is the density of the normal distribution $N(0,t)$. Note also that $W_{\gamma }$ can be obtained from excursions of a Wiener process pointing them (independently of each other) up and down with probabilities $p=(1+\gamma )/2$ and $q=(1-\gamma )/2$, respectively.

Kulinich et al. [8, 7] considered limit theorems in the case where a is nonintegrable function such that where $c_{\pm }>-1/2$ are constants. In this case, $a_{n}(x)$ converges in some sense to $c_{-}\mathbb{1}_{x<0}+c_{+}\mathbb{1}_{x\geqslant 0}$ as $n\to \infty $.

For instance, if $a(x)=c_{\pm }/x\hspace{2.5pt}\text{for}\hspace{2.5pt}\pm x>x_{0}$, then, for $c_{-}<1/2<c_{+}$, the sequence $X_{n}$ converges weakly to a Bessel process. If $c_{-}=c_{+}>-1/2$, then $|X_{n}|$ also converges weakly to a Bessel process. The problem of weak convergence of $X_{n}$ for (e.g.) $c_{-}=c_{+}>-1/2$ or $c_{-}<c_{+}\leqslant 1/2$ was not considered.

In this paper, we generalize the results of [14, 8] to the case where $\widetilde{a}$ is integrable on $(-\infty ;\infty )$, and

We consider all possible limit processes (depending on $c_{+}$ and $c_{-}$). In particular, we show that, for $c_{+}=c_{-}<1/2$, the limit process is a skew Bessel process (see Section 2).

We recall the definition and some properties of Bessel processes.

Let $\delta \geqslant 0$ and $x_{0}\in \mathbb{R}$. Consider the SDE where W is a Wiener process.

It is known (see [15], XI.1, (1.1)), that there exists a unique strong solution $Z({x_{0}^{2}},\cdot )$ of (3). This solution is called the squared δ-dimensional Bessel process. The process $Z({x_{0}^{2}},\cdot )$ is nonnegative a.s.

Recall the following properties of the Bessel process (see [15, Chap. XI]).

The Bessel process $\xi (t)=B_{c}(x_{0},t)$ satisfies the SDE where $T_{0}$ is the first hitting time of 0. If $\delta \geqslant 2$ (i.e., $c\geqslant 1/2$), then the Bessel process with probability 1 does not hit 0.

If $0<\delta <2$ (i.e., $-1/2<c<1/2$), then with probability 1 the Bessel process hits 0 but spends zero time at 0. In particular, if $\delta =1$ (i.e., $c=0$), then the Bessel process is a reflecting Brownian motion.

If $\delta =0$ (i.e., $c=-1/2$), then with probability 1 the process attains 0 and remains there forever.

The scale function of the Bessel process $B_{c}$ equals that is, where $T_{y}=\inf \{t\geqslant 0:\hspace{2.5pt}B_{c}(t)=y\}$.

The transition density for $c>-1/2$, $x,y>0$, and $t>0$ equals where $I_{\nu }$ is a Bessel function of index $\nu =c-1/2$.

Let $c\in (-1/2,1/2)$, and let ${p_{t}^{0,c}}(x,y)$ be the transition density of the Bessel process $B_{c}$ killed at 0.

Set It is easy to verify that this function satisfies the Chapman–Kolmogorov equation, is nonnegative, and $\int _{\mathbb{R}}{p_{t}^{\mathit{skew}}}(x,y)\hspace{0.1667em}dy=1,\hspace{2.5pt}x\in \mathbb{R}$.

Let where $\widetilde{a}\in L_{1}(\mathbb{R})$ and

Let $X_{n}(t),t\geqslant 0$, be the solution of the SDE The existence and uniqueness of a strong solution of this SDE follows from [3, Thm. 4.53].

It follows from [9, Section 3] or [11, Section 3.7] that if A1 is satisfied, then, for any $\alpha >0$, we have the convergence where ${\tau }^{n,\alpha }=\inf \{t\geqslant 0:X_{n}(t)\leqslant \alpha \}$ and ${\tau }^{0,\alpha }=\inf \{t\geqslant 0:{B_{c_{+}}^{+}}(x_{0},t)\leqslant \alpha \}$. Since the process ${B_{c_{+}}^{+}}(x_{0},\cdot )$ does not hit 0, this yields the proof. Case A2 is considered similarly.

To prove all other items of Theorem 1, we use the method proposed in [13].

Let $\{{\xi }^{(n)},n\geqslant 0\}$ be a sequence of continuous homogeneous strong Markov processes. For $\alpha >0$, set

We denote by ${\xi _{x_{0}}^{(n)}}$ a process that has the distribution of ${\xi }^{(n)}$ conditioned by ${\xi }^{(n)}(0)=x_{0}$.

The next statement is a particular case of Theorem 2 of [13].

We apply this lemma for ${\xi }^{(n)}=X_{n},n\geqslant 1$, and ${\xi }^{(0)}=X_{\infty }$ in cases A1–A5 of the theorem. Case A6 will be considered separately.

Conditions (7) and (10) are obvious.

The convergence follows from [9, Section 3] or [11, Section 3.7]. Since convergence (13) yields the convergence of pairs (11).

Let us check condition (8). Set Observe that $\varPhi _{n}:\mathbb{R}\to \mathbb{R}$ is a bijection, $\varPhi _{n}(0)=0$, and Itô’s formula yields So

Let $|s-t|<\delta $, and let $\varDelta >0$ be fixed. Denote $f_{n}(t):={\int _{0}^{t}}A(nX_{n}(z))\hspace{0.1667em}dW(z)$.

a) Assume that $|X_{n}(z)|>\varDelta ,z\in [s,t]$. Then where $C=\| A\| _{\infty }\max (|c_{-}|,|c_{+}|)<\infty $. Hence, we have the estimate where $\omega _{f}(\delta )=\sup _{|s-t|<\delta ,\hspace{2.5pt}s,t\in [0,T]}|f(t)-f(s)|$ is the modulus of continuity.

b) Assume that $|X_{n}(z_{0})|\leqslant \varDelta $ for some $z_{0}\in [s,t]$.

Denote $\tau :=\inf \{z\geqslant s:|X_{n}(z)|\leqslant \varDelta \}$ and $\sigma :=\sup \{z\leqslant t:|X_{n}(z)|\leqslant \varDelta \}$. Then

Thus, in any case, we have the following estimate of the modulus of continuity:

Let $\varDelta \leqslant \varepsilon /6$. Then, for $\delta \leqslant \frac{\varepsilon \varDelta }{6LC}$, we have The last convergence follows from the fact that the sequence of distributions of $\{f_{n}(\cdot )={\int _{0}^{.}}A(nX_{n}(z))\hspace{0.1667em}dW(z)\}_{n\geqslant 1}$ in the space of continuous functions is weakly relatively compact because the function A is bounded.

Let us prove (12) in cases A1–A5.

Let $|x|<\alpha $. It is easy to see that $P_{x}({\sigma }^{n,\alpha }<\infty )=1,\hspace{2.5pt}n\in \mathbb{N}\cup \{\infty \}$. Since the process $X_{n}$ is continuous, we have $|X_{n}({\sigma }^{n,\alpha })|=\alpha $ a.s.

By ${p_{x}^{n}}=P_{x}(X_{n}({\sigma }^{n,\alpha })=\alpha ),\hspace{2.5pt}n\in \mathbb{N}\cup \{\infty \}$, we denote the probability to reach α before reaching $-\alpha $ when starting from x.

Using formulas (4) and (5) for the scale of a Bessel process and a skew Bessel process, it is easy to check that where $\psi _{c}$ is given in (4).

For $n\in \mathbb{N}$, we have (see [4, Section 15] and [15]) where The function φ is increasing. It follows from the definition of a that φ is bounded from above (below) iff $c_{+}>1/2$ ($c_{-}>1/2$). The function φ has the following asymptotic behavior: where and

Condition (12) follows from (14), (15), (16), (17), (18) in cases A1–A5 (and in case A6 if $x_{n}=0,\hspace{2.5pt}n\geqslant 0$).

Set $\tau _{n}=\inf \{t\geqslant 0:|X_{n}(t)|\geqslant 1\}$.

Let us verify (19). It is known [6, Chap. 4.3] that is of the form where Green’s function $G_{n}$ equals with $\varphi _{n}$ given by formula (16), and The function $u_{n,\varepsilon }(x)$ is a generalized solution (because $a_{n}$ may be discontinuous) of the equation with boundary conditions $u_{n,\varepsilon }(\pm 1)=0$.

A direct verification of the condition $\lim _{\varepsilon \to 0+}\sup _{|x|\leqslant 1}\sup _{n}u_{n,\varepsilon }(x)=0$ is possible but cumbersome. We prove the corresponding convergence using the comparison theorem. We consider only the case where $a_{n}$ satisfies the Lipschitz condition. The general case follows by approximation.

It follows from the Itô–Tanaka formula that where $W_{n}$ is a new Wiener process, and $l_{n}$ is the local time of $X_{n}$ at zero.

Let $-1/2<c<\min (c_{-},c_{+},0)$. It follows from the arguments of [12] on comparison of reflecting SDEs that $|X_{n}(t)|\geqslant Y_{n}(t),\hspace{2.5pt}t\geqslant 0$, where $Y_{n}$ satisfies the following SDE with reflection at zero: Here $W_{n}(t)={\int _{0}^{t}}(X_{n}(s))\hspace{0.1667em}dW(s)$ is a Wiener process, $\tilde{l}_{n}$ is the local time of $Y_{n}$ at 0, $\bar{a}_{n}(x)=n\bar{a}(nx),\hspace{2.5pt}\bar{a}(x)=-(|a(x)|+|a(-x)|)-\frac{c}{x}\mathbb{1}_{|x|>1}+r(x)$, and r is any nonpositive function such that $\bar{a}$ satisfies Lipschitz condition. We will also assume that $\int _{\mathbb{R}}|r(x)|\hspace{0.1667em}dx\leqslant \int _{\mathbb{R}}|b(x)|\hspace{0.1667em}dx$. The Lipschitz property is used only for application of comparison theorem.

To prove (19), it suffices to verify that where $\bar{\tau }_{n}$ is the entry time of $Y_{n}$ into $[1,\infty )$.

It is known [6] that is a (generalized) solution of the equation with boundary conditions ${u^{\prime }_{n,\varepsilon }}(0)=0,\hspace{2.5pt}u_{n,\varepsilon }(1)=0$. So where K is a constant that depends only on $\int _{\mathbb{R}}|b(x)|\hspace{0.1667em}dx$ and c (and is independent of n). By our choice, $c\in (-1/2,0)$, so the right-hand side of (21) tends to 0 as $\varepsilon \to 0+$ by the Lebesgue dominated convergence theorem.

The theorem is proved in cases A1–A5.

Consider case A6. Note that conditions (7)–(11) are satisfied for ${\xi }^{(n)}=X_{n}$, $n\geqslant 1$, and ${\xi }^{(0)}=X_{\infty }$, where $X_{\infty }$ is given in the theorem. In particular, this implies that the sequence of distributions of stochastic processes $\{X_{n}\}$ in the space of continuous functions is weakly relatively compact. Choosing an arbitrary convergent subsequence, without loss of generality, we may assume that $\{X_{n}\}$ itself converges weakly to a continuous process X. Let $\delta >0$, and let ${\sigma }^{n,\delta }=\inf \{t\geqslant 0:X_{n}(t)=\delta \},\hspace{2.5pt}{\sigma }^{\delta }=\inf \{t\geqslant 0:X(t)=\delta \}$. It follows from formulas for the scale function of the processes $\{X_{n}\}$ that $\lim _{n\to \infty }P(X_{n}({\sigma }^{n,\delta })=\delta )=p,\hspace{2.5pt}\lim _{n\to \infty }P(X_{n}({\sigma }^{n,\delta })=-\delta )=1-p$, where p is given by (6). Formulas (9) and (11) imply that the limit process exits from the interval $[-\delta ,\delta ]$ with probability 1.

Observe that, for almost all $\delta >0$, with respect to the Lebesgue measure, the distribution of $X_{n}({\sigma }^{n,\delta }+\cdot )$ converges weakly as $n\to \infty $ to the distribution of $X(\delta +\cdot )$. Indeed, by the Skorokhod theorem on a single probability space (see [16]), without loss of generality, we may assume that the sequence $\{X_{n}\}$ converges to X uniformly on compact sets with probability 1. For simplicity, we will assume that the convergence holds for all ω and that also ${\sigma }^{n,\delta },{\sigma }^{\delta }<\infty $ for all $\omega ,n,\delta >0$. So we show convergence if we prove that Convergence (23) may fail only if ${\sigma }^{\delta }$ is a point of a local maximum of X. It follows from the definition that ${\sigma }^{\delta }$ is a point of a strict local maximum of X from the left. The set of points of local maximums that are strict maximums from the left is at most countable. This yields that, for almost all ω and almost all $\delta >0$ with respect to the Lebesgue measure, we have convergence (23) and hence (22).

On the other hand, the distribution of $X_{n}({\sigma }^{n,\delta }+\hspace{0.1667em}\cdot )$ converges weakly as $n\to \infty $ to the distribution of the process $\mathbb{1}_{\varOmega _{-}}{B_{c_{-}}^{-}}(-\delta ,\cdot )+\mathbb{1}_{\varOmega _{+}}{B_{c_{+}}^{+}}(\delta ,\cdot )$, where $P(\varOmega _{-})=1-p,\hspace{2.5pt}P(\varOmega _{+})=p$, and the σ-algebra $\{\varnothing ,\varOmega _{-},\varOmega _{+},\varOmega \}$ is independent of $\sigma ({B_{c_{\pm }}^{\pm }}(\pm \delta ,t),t\geqslant 0)$.

Recall that assumptions of the theorem yield It follows from (9) that Thus, we have the almost sure convergence in $C([0,\infty ))$ The processes $\mathbb{1}_{\varOmega _{-}}{B_{c_{-}}^{-}}(-\delta ,\cdot )+\mathbb{1}_{\varOmega _{+}}{B_{c_{+}}^{+}}(\delta ,\cdot )$ converge in distribution to where the σ-algebras $\{\varnothing ,\varOmega _{-},\varOmega _{+},\varOmega \}$ and $\sigma ({B_{c_{\pm }}^{\pm }}(0,t),t\geqslant 0)$ are independent.

This completes the proof of Theorem 1.