Modern Stochastics: Theory and Applications

Consider the Itô stochastic differential equation where $T>0$ is a parameter, $a_{T}(x)$, $x\in \mathbb{R}$, are real-valued measurable functions such that for some constants $L_{T}>0$ and for all $x\in \mathbb{R}$ $\left|a_{T}(x)\right|\le L_{T}$, and $W_{T}=\{W_{T}(t),t\ge 0\}$, $T>0$, is a family of standard Wiener processes defined on a complete probability space $(\varOmega ,\mathrm{\Im },\operatorname{\mathsf{P}})$.

It is known from Theorem 4 in [19] that, for any $T>0$ and $x_{0}\in \mathbb{R}$, equation (1) possesses a unique strong pathwise solution $\xi _{T}=\{\xi _{T}(t),t\ge 0\}$, and this solution is a homogeneous strong Markov process.

We suppose that the drift coefficient $a_{T}(x)$ in equation (1) can have a very nonregular dependence on the parameter. For example, the drift coefficient can be of “δ”-type sequence at some points $x_{k}$ as $T\to \infty $, or it can be equal to $\sqrt{T}\sin ((x-x_{k})\sqrt{T})$, or it can have degeneracies of some other types. Such a nonregular dependence of the coefficients in equation (1) first appeared in [5] and [4], where the limit behavior of the normalized unstable solution of Itô stochastic differential equation as $t\to \infty $ was investigated. In those papers, a special dependence of the coefficients $a_{T}(x)=\sqrt{T}a(x\sqrt{T})$ on the parameter T was considered in the case where $a(x)$ is an absolutely integrable function on $\mathbb{R}$. Assume that this is the case and let $\int _{\mathbb{R}}a(x)\hspace{0.1667em}dx=\lambda $. The sufficiency of the condition $\lambda =0$ for the asymptotic equivalence of distributions $\xi _{T}$ and $W_{T}(t)$ is established in [5], and the necessity of this condition is proved in [1]. If $\lambda \ne 0$, then we can deduce from [4] that the distributions of the solution $\xi _{T}$ of equation (1) weakly converge as $T\to \infty $ to the corresponding distributions of the Markov process $\hat{\xi }(t)=l\left(\zeta (t)\right)$, where $l(x)=c_{1}x$ for $x>0$ and $l(x)=c_{2}x$ for $x\le 0$; $\zeta (t)$ is a strong solution of the Itô equation $d\zeta (t)=\bar{\sigma }(\zeta (t))\hspace{0.1667em}dW(t)$, where $\bar{\sigma }(x)=\sigma _{1}$ for $x>0$ and $\bar{\sigma }(x)=\sigma _{2}$ for $x<0$, and ${\int _{0}^{t}}P\{|\zeta (s)|=0\}\hspace{0.1667em}ds=0$. The explicit form of the transition density of the process $\hat{\xi }(t)$ is obtained. Moreover, in [10], it is proved that where $\beta (t)$ is a certain functional of $\zeta (t)$, and the necessity of the condition $\lambda \ne 0$ is established for the weak convergence as $T\to \infty $ of the solution $\xi _{T}$ of equation (1) to the process $\hat{\xi }(t)$.

Furthermore, in [5] and [4], a probabilistic method to study the “awkward” term $\sqrt{T}{\int _{0}^{t}}a(\xi _{T}(s)\sqrt{T})\hspace{0.1667em}ds$ in equation (1) is developed. This method uses a representation of this “awkward” term through a family of continuous functions $\varPhi _{T}(x)$ of $\xi _{T}(t)$ and a family of martingales ${\int _{0}^{t}}{\varPhi ^{\prime }_{T}}(\xi _{T}(s))\hspace{0.1667em}dW_{T}(s)$, with the further application of the Itô formula. After the mentioned transformations, according to this method, we can apply Skorokhod’s convergent subsequence principle for $\xi _{T_{n}}(t)$ and $W_{T_{n}}(t)$ (see [17], Chapter I, §6) in order to pass to the limit in the resulting representation.

Note that this method is also used in the present paper to study the asymptotic behavior of integral functionals.

It is known from [2], §16, that the asymptotic behavior of the solution $\xi _{T}$ of equation (1) is closely related to the asymptotic behavior of harmonic functions, that is, functions satisfying the following ordinary differential equation almost everywhere (a.e.) with respect to the Lebesgue measure:

It is obvious that the functions $f_{T}(x)$ have the form where ${c_{T}^{(1)}}$ and ${c_{T}^{(2)}}$ are some families of constants.

The latter functions possess the continuous derivatives ${f^{\prime }_{T}}(x)$, and their second derivatives ${f^{\prime\prime }_{T}}(x)$ exist almost everywhere with respect to the Lebesgue measure and are locally integrable. Note that ${c_{T}^{(1)}}$ are normalizing constants and ${c_{T}^{(2)}}$ are centralizing constants in the limit theorems (see [18], §6). Further, for simplicity, we assume that in (2), ${c_{T}^{(1)}}\equiv 1$ and ${c_{T}^{(2)}}\equiv 0$.

In this paper, we assume for the coefficient $a_{T}(x)$ of equation (1) that there exists a family of functions $G_{T}(x)$, $x\in \mathbb{R}$, with continuous derivatives ${G^{\prime }_{T}}(x)$ and locally integrable second derivatives ${G^{\prime\prime }_{T}}(x)$ a.e. with respect to the Lebesgue measure such that, for all $T>0$ and $x\in \mathbb{R}$, the following inequalities hold:

Suppose additionally that the functions $G_{T}(x)$, $x\in \mathbb{R}$, introduced by condition $(A_{1})$ satisfy the following assumptions:

Let $\left\{G_{T}\right\}$ be the class of the functions $G_{T}(x)$, $x\in \mathbb{R}$, satisfying conditions $(A_{1})$ and (i)–(ii). The class of equations of the form (1) whose coefficients $a_{T}(x)$ admit $G_{T}(x)$, $x\in \mathbb{R}$, from the class $\left\{G_{T}\right\}$ will be denoted by $K\left(G_{T}\right)$. It is easy to understand that class $K\left(G_{T}\right)$ does not depend on the constants ${c_{T}^{(1)}}$ and ${c_{T}^{(2)}}$ in representation (2).

It is clear that if there exist constants $\delta >0$ and $C>0$ such that $0<\delta \le {f^{\prime }_{T}}(x)\le C$ for all $x\in \mathbb{R}$, $T>0$, then the corresponding equations (1) belong to the class $K\left(G_{T}\right)$ for $G_{T}(x)=f_{T}(x)$. We denote this subclass as $K_{1}$. Note that the class $K\left(G_{T}\right)$ contains in particular the equations for which, at some points $x_{k}$, we have the convergence ${f^{\prime }_{T}}(x_{k})\to \infty $ or the convergence ${f^{\prime }_{T}}(x_{k})\to 0$ as $T\to \infty $. For example, consider equation (1) with $a_{T}(x)=\frac{c_{0}Tx}{1+{x}^{2}T}$. It is easy to obtain that ${f^{\prime }_{T}}(x)=\frac{1}{{\left(1+{x}^{2}T\right)}^{c_{0}}}$, and if $c_{0}>-\frac{1}{2}$, then such equations belong to the class $K\left(G_{T}\right)$ with $G_{T}(x)={x}^{2}$ (here, at points $x\ne 0$, we have ${f^{\prime }_{T}}(x)\to 0$ for $c_{0}>0$, ${f^{\prime }_{T}}(x)\to \infty $ for $-\frac{1}{2}<c_{0}<0$, and ${f^{\prime }_{T}}(x)\equiv 1$ for $c_{0}=0$).

For the class of equations $K\left(G_{T}\right)$, we study the asymptotic behavior as $T\to \infty $ of the distributions of the following functionals: where the processes $\xi _{T}(t)$, $W_{T}(t)$ are related via equation (1), $g_{T}(x)$ is a family of measurable locally bounded real-valued functions, and $F_{T}(x)$ is a family of continuous real-valued functions.

This paper is a continuation of [13–15]. Note that the behavior of the distributions of functionals ${\beta _{T}^{(1)}}(t)$, ${\beta _{T}^{(2)}}(t)$ for the solutions $\xi _{T}$ of equations (1) from the class $K_{1}$ is studied in [6] and [9]. The case where $W_{T}(t)$ is replaced with $\eta _{T}(t)$, where $\eta _{T}(t)$ is a family of continuous martingales with the characteristics $\langle \eta _{T}\rangle (t)\to t$ as $T\to \infty $, was studied in [8]. Paper [7] was devoted to a discrete analogue of the results from [8]. A similar problem for the functionals $I_{T}(t)$ in the case of equation (1) with $a_{T}(x)\equiv 0$ was considered in [18] and in [11], for the class $K_{1}$. In [13–15], the behavior of the distributions of the functionals ${\beta _{T}^{(1)}}(t)$, ${\beta _{T}^{(2)}}(t)$, and $I_{T}(t)$ with a special dependence of the drift coefficients $a_{T}(x)=\sqrt{T}a(x\sqrt{T})$ on the parameter T is considered, mainly in the case where $\left|xa(x)\right|\le C$ for all $x\in \mathbb{R}$. The behavior of the distributions of functionals ${\beta _{T}^{(1)}}(t)$ was studied in [13], ${\beta _{T}^{(2)}}(t)$ was studied in [14], and $I_{T}(t)$ was investigated in [15]. A more detailed review of the known results in this area is presented in [13–15]. Note that the functionals ${\beta _{T}^{(1)}}(t)$, ${\beta _{T}^{(2)}}(t)$, and $\beta _{T}(t)$ are particular cases of the functional $I_{T}(t)$ (see [15], Lemma 4.1).

The paper is organized as follows. Section 2 contains the statements of the main results. In Section 3, they are proved. Auxiliary results are collected in Section 4.

In what follows, we denote by $C,\hspace{0.1667em}L,\hspace{0.1667em}N,\hspace{0.1667em}C_{N}$ any constants that do not depend on T and x. Assume that, for certain locally bounded functions $q_{T}(x)$ and any constant $N>0$, the following condition holds: where ${f^{\prime }_{T}}(x)$ is the derivative of the function $f_{T}(x)$ defined by Eq. (2).

The next theorem principally follows from [11]; however, we provide its proof for the reader’s convenience and completeness of the results.

Proof of Theorem 2.1. Rewrite Eq. (3) as where The functions ${q_{T}^{(1)}}(x)$ satisfy the conditions of Lemma 4.2. Thus, for any $L>0$, as $T\to \infty $. It is clear that $\eta _{T}(t)$ is a family of continuous martingales with quadratic characteristics where

The functions ${q_{T}^{(2)}}(x)$ satisfy the conditions of Lemma 4.2. Thus, for any $L>0$, as $T\to \infty $.

We have that relations (4) and (5) hold for the processes $\zeta _{T}(t)$ and $\eta _{T}(t)$, and, according to (8) and (10), these relations hold for the processes ${\alpha _{T}^{(k)}}(t)$, $k=1,2$, as well. This means that we can apply Skorokhod’s convergent subsequence principle (see [17], Chapter I, §6) for the process $(\zeta _{T}(t),\eta _{T}(t),{\alpha _{T}^{(1)}}(t),{\alpha _{T}^{(2)}}(t))$: given an arbitrary sequence ${T^{\prime }_{n}}\to \infty $, we can choose a subsequence $T_{n}\to \infty $, a probability space $(\tilde{\varOmega },\tilde{\mathrm{\Im }},\tilde{\operatorname{\mathsf{P}}})$, and a stochastic process $(\tilde{\zeta }_{T_{n}}(t),\tilde{\eta }_{T_{n}}(t),{\tilde{\alpha }_{T_{n}}^{(1)}}(t),{\tilde{\alpha }_{T_{n}}^{(2)}}(t))$ defined on this space such that its finite-dimensional distributions coincide with those of the process $(\zeta _{T_{n}}(t),\eta _{T_{n}}(t),{\alpha _{T_{n}}^{(1)}}(t),{\alpha _{T_{n}}^{(2)}}(t))$ and, moreover, for all $0\le t\le L$, where $\tilde{\zeta }(t)$, $\tilde{\eta }(t)$, ${\tilde{\alpha }}^{(1)}(t)$, ${\tilde{\alpha }}^{(2)}(t)$ are some stochastic processes.

Evidently, relations (8) and (10) imply that ${\tilde{\alpha }}^{(k)}(t)\equiv 0$, $k=1,2$, a.s. According to (5), the processes $\tilde{\zeta }(t)$ and $\tilde{\eta }(t)$ are continuous. Moreover, applying Lemma 4.5 together with Eqs. (7) and (9), we obtain that where $\tilde{\zeta }_{T_{n}}(t)\stackrel{\tilde{\operatorname{\mathsf{P}}}}{\to }\tilde{\zeta }(t)$, $\tilde{\eta }_{T_{n}}(t)\stackrel{\tilde{\operatorname{\mathsf{P}}}}{\to }\tilde{\eta }(t)$, $\sup _{0\le t\le L}|{\tilde{\alpha }_{T_{n}}^{(k)}}(t)|\stackrel{\tilde{\operatorname{\mathsf{P}}}}{\to }0$, $k=1,2$, as $T_{n}\to \infty $. In addition, it is established in [12] that, for any constants $L>0$ and $\varepsilon >0$,

Using the latter convergence and (11), we conclude that, for any constants $L>0$ and $\varepsilon >0$,

Therefore, according to the well-known result of Prokhorov [16], we conclude that as $T_{n}\to \infty $. According to Lemma 4.3, we can pass to the limit in (11) and obtain where $\tilde{\eta }(t)$ is a continuous martingale with the quadratic characteristic

Now, it is well known that the latter representation provides the existence of a Wiener process $\hat{W}(t)$ such that Thus, the process $(\tilde{\zeta }(t),\hat{W}(t))$ satisfies Eq. (6), and the process $\tilde{\zeta }_{T_{n}}(t)$ weakly converges, as $T_{n}\to \infty $, to the process $\tilde{\zeta }(t)$. Since the subsequence $T_{n}\to \infty $ is arbitrary and since a solution of Eq. (6) is weakly unique, the proof of the Theorem 2.1 is complete.

Proof of Theorem 2.2. It is clear that, for all $t>0$, with probability one, where $\alpha _{T}(t)={\int _{0}^{t}}q_{T}(\xi _{T}(s))\hspace{0.1667em}ds$ and $q_{T}(x)=g_{T}(x)-g_{0}\left(G_{T}(x)\right)$.

The functions $q_{T}(x)$ satisfy the conditions of Lemma 4.2. Thus, for any $L>0$, as $T\to \infty $. Similarly to (11), we obtain the equality where $\tilde{\zeta }_{T_{n}}(t)\stackrel{\tilde{\operatorname{\mathsf{P}}}}{\to }\tilde{\zeta }(t)$ and $\sup _{0\le t\le L}|\tilde{\alpha }_{T_{n}}(t)|\stackrel{\tilde{\operatorname{\mathsf{P}}}}{\to }0$ as $T_{n}\to \infty $. The process $\tilde{\zeta }(t)$ is a solution of Eq. (12), whereas by Lemma 4.5 the finite-dimensional distributions of the stochastic process ${\beta _{T_{n}}^{(1)}}(t)$ coincide with those of the process ${\tilde{\beta }_{T_{n}}^{(1)}}(t)$.

Using Lemma 4.3 and Eq. (14), we conclude that as $T_{n}\to \infty $. Thus, the process ${\beta _{T_{n}}^{(1)}}(t)$ weakly converges, as $T_{n}\to \infty $, to the process ${\beta }^{(1)}(t)={\int _{0}^{t}}g_{0}(\zeta (s))\hspace{0.1667em}ds$, where $\zeta (t)$ is a solution of Eq. (6). Since the subsequence $T_{n}\to \infty $ is arbitrary and since a solution $\zeta (t)$ of Eq. (6) is weakly unique, the proof of Theorem 2.2 is complete.

Proof of Theorem 2.3. Consider the function Applying the Itô formula to the process $\varPhi _{T}(\xi _{T}(t))$, where $\xi _{T}(t)$ is a solution of Eq. (1), we get that where

Denote $P_{NT}=\operatorname{\mathsf{P}}\{\sup _{0\le t\le L}|\xi _{T}(t)|>N\}$. It is clear that, for any constants $\varepsilon >0$, $N>0$, and $L>0$, we have the inequalities and

The inequality $|G_{T}(x)|\ge C|x{|}^{\alpha }$, $\alpha >0$, together with convergence (4), implies that

Thus, using the conditions of Theorem 2.3, we get the convergence as $T\to \infty $.

The same arguments as we used establishing (11) yield that where as $T_{n}\to \infty $ for all $L>0$. According to (12) with $\tilde{\eta }(t)$ defined in (13), the process $(\tilde{\zeta }(t),\hat{W}(t))$ satisfies Eq. (6).

By Lemma 4.5 the finite-dimensional distributions of the stochastic process ${\beta _{T_{n}}^{(1)}}(t)$ coincide with those of the process ${\tilde{\beta }_{T_{n}}^{(1)}}(t)$. Using Lemma 4.3, we can pass to the limit as $T_{n}\to \infty $ in (16) and obtain as $T_{n}\to \infty $, where and $\tilde{\zeta }(t)$ is a solution of Eq. (6). Therefore, we have that Theorem 2.3 holds for the process ${\beta _{T_{n}}^{(1)}}(t)$ as $T_{n}\to \infty $. Since the subsequence $T_{n}\to \infty $ is arbitrary and since a solution $\zeta (t)$ of Eq. (6) is weakly unique, the proof of Theorem 2.3 is complete.

Proof of Theorem 2.4. It is clear that where

The process $\gamma _{T}(t)$ is a continuous martingale with the quadratic characteristics

According to the conditions of Theorem 2.4, the functions ${q_{T}^{2}}(x)$ satisfy the conditions of Lemma 4.2. Thus, for any $L>0$, we have the convergence $\langle \gamma _{T}\rangle (L)\stackrel{\operatorname{\mathsf{P}}}{\to }0$ as $T\to \infty $.

The following inequality holds for any constants $\varepsilon >0$ and $\delta >0$: (see [2], §3, Theorem 2), which implies the relation as $T\to \infty $.

Then, similarly to representation (11), on a certain probability space $(\tilde{\varOmega },\tilde{\mathrm{\Im }},\tilde{\operatorname{\mathsf{P}}})$, for an arbitrary subsequence $T_{n}$, we get the equality where as $T_{n}\to \infty $ for any $L>0$, where the process $(\tilde{\zeta }(t),\hat{W}(t))$ satisfies Eq. (6), $\tilde{\eta }(t)$ is defined in (13), and the processes ${\tilde{\beta }_{T_{n}}^{(2)}}(t)$ and ${\beta _{T_{n}}^{(2)}}(t)$ are stochastically equivalent.

Similarly to the proof of convergence (17), we obtain as $T_{n}\to \infty $, where

Thus, the process ${\tilde{\beta }_{T_{n}}^{(2)}}(t)$ weakly converges, as $T_{n}\to \infty $, to the process ${\tilde{\beta }}^{(2)}(t)$. Since the subsequence $T_{n}\to \infty $ is arbitrary and since the processes ${\tilde{\beta }_{T_{n}}^{(2)}}(t)$ and ${\beta _{T_{n}}^{(2)}}(t)$ are stochastically equivalent, the proof of Theorem 2.4 is complete.

Proof of Theorem 2.5. It is clear that where

Denote, as before, $P_{NT}=\operatorname{\mathsf{P}}\{\sup _{0\le t\le L}|\xi _{T}(t)|>N\}$. Since for any constants $\varepsilon >0$, $N>0$, and $L>0$, we have the inequality we can apply conditions of Theorem 2.5 and convergence (15) to get that as $T\to \infty $. The proof of the fact that, for $\gamma _{T}(t)$, an analogue of convergence (19) holds is literally the same as in the proof of Theorem 2.4. Then, we can apply Skorokhod’s convergent subsequence principle to the process $\left(\zeta _{T}(t),\eta _{T}(t),\alpha _{T}(t),\gamma _{T}(t)\right)$ and, similarly to representation (11), obtain the following equality for an arbitrary subsequence $T_{n}$ in a certain probability space $(\tilde{\varOmega },\tilde{\mathrm{\Im }},\tilde{\operatorname{\mathsf{P}}})$: where, as $T_{n}\to \infty $, for any $L>0$,

To complete the proof of Theorem 2.5, we repeat the same arguments as in the proof of Theorem 2.4.

Proof of Theorem 2.6. According to the Itô formula, the process $\zeta _{T}(t)\hspace{0.1667em}=\hspace{0.1667em}f_{T}(\xi _{T}(t))$ satisfies the equation $d\zeta _{T}(t)=\hat{\sigma }_{T}(\zeta _{T}(t))\hspace{0.1667em}dW_{T}(t)$, where $\hat{\sigma }_{T}(x)={f^{\prime }_{T}}\left(\varphi _{T}(x)\right)$, $\varphi _{T}(x)$ is the inverse function of the function $f_{T}(x)$, and $\zeta _{T}(0)=f_{T}(x_{0})\to y_{0}$ as $T\to \infty $. In addition, the following equality holds: where $\hat{F}_{T}(x)=F_{T}(\varphi _{T}(x))$ and $\hat{g}_{T}(x)=g_{T}(\varphi _{T}(x))\cdot {\hat{\sigma }_{T}^{-1}}(x)$.

It is easy to see that condition 1 of the present theorem implies that as $T\to \infty $ for all x, whereas condition 2 implies that and as $T\to \infty $ for any $N>0$.

This means that the necessary and sufficient conditions of weak convergence of the process $\left(\zeta _{T}(t),I_{T}(t)\right)$ as $T\to \infty $ to the process $\left(\zeta (t),I(t)\right)$ from [11] hold with $b_{T}={c_{T}^{(1)}}$ and $a_{T}={c_{T}^{(2)}}$.