On Cusp Location Estimation for Perturbed Dynamical Systems

1 Introduction

Let us consider the following problem. The observed continuous time trajectory $X^{ε} = (X_{t}, 0 \leq t \leq T)$ of the diffusion process satisfies the stochastic differential equation

d X_{t} = S (ϑ, X_{t}) d t + ε d W_{t}, X_{0} = x_{0}, 0 \leq t \leq T,

(1)

where $W_{t}, 0 \leq t \leq T$ is the standard Wiener process and the drift coefficient $S (ϑ, x)$ has a cusp-type singularity, i.e., at the vicinity of the point $ϑ$ we have $S (ϑ, x) \approx a {| x - ϑ |}^{κ} + h$ , where $κ \in (0, \frac{1}{2})$ . The parameter $ϑ$ is unknown and we have to estimate it by the observations $X^{ε}$ . We are interested in the asymptotic properties of the estimators of this parameter in the asymptotics of small noise: $ε \to 0$ .

Such stochastic models, called sometimes, dynamical systems with small noise or perturbed dynamical systems attract attention of probabilists and statisticians (see, for example, Freidlin and Wentzel [7] and Kutoyants [12]

and references therein). The interest to this stochastic models can be explained as follows. Suppose that we have a dynamical system described by the ordinary differential equation

\frac{d x_{t}}{d t} = S (ϑ, x_{t}), x_{0}, 0 \leq t \leq T .

(2)

The right hand part (rhp) of this system depends on some parameter $ϑ$ and therefore the state $x_{t}$ of the dynamical system of course depends on the value of this parameter, i.e., $x_{t} = x_{t} (ϑ)$ . If we know $ϑ$ , then we know the trajectory $X^{0} = (x_{t}, 0 \leq t \leq T)$ . For many real systems it is natural to suppose that the rhp contains some small noise (perturbations)

\frac{d X_{t}}{d t} = S (ϑ, X_{t}) + ε n_{t}, x_{0}, 0 \leq t \leq T .

(3)

The most “popular” noise $n_{t}, 0 \leq t \leq T$ considered in the corresponding literature is the so-called white Gaussian noise (WGN), i.e., $n_{t}$ is a Gaussian process with the properties $E n_{t} = 0, E n_{t} n_{s} = δ (t - s)$ . Here $δ (t)$ is the Dirac delta-function. In this case the observations $X^{ε} = (X_{t}, 0 \leq t \leq T)$ of the system (3) can be written as solution of the stochastic differential equation (1). Therefore we replaced $n_{t}$ by the derivative of the standard Wiener process. Of course, the Wiener process is not differentiable and the equation (1) is just a short-writing of the corresponding integral equation

X_{t} = x_{0} + \int_{0}^{t} S (ϑ, X_{s}) d s + ε W_{t}, 0 \leq t \leq T .

A wide class of estimation problems (parameter estimation and nonparametric estimation) were considered in [12]. The properties of estimators (maximum likelihood, Bayesian, minimum distance) are well studied in regular (smooth with respect to the unknown parameter) and non regular (change point, delay estimation) cases. The smooth case corresponds to the trend coefficient $S (ϑ, x)$ continuously differentiable w.r.t. $ϑ$ and finite Fisher information. The change-point problem can be described by the following example

d X_{t} = h (X_{t}) {1I}_{{ϑ < X_{t}}} d t + g (X_{t}) {1I}_{{ϑ \geq X_{t}}} d t + ε d W_{t}, X_{0} = x_{0}, 0 \leq t \leq T,

i.e., we have a switching diffusion process with unknown threshold $ϑ$ . Such models are called threshold diffusion processes like threshold autoregressive (TAR) time series [1] and statistical problems related to this model are singular [14]. If we have a cusp-type singularity as

where $κ \in (0, \frac{1}{2})$ , then for $κ$ close to zero we have cusp-type switching similar to change-point, but without jump. Usually the characteristics of the real systems can not “make jumps” and the cusp-type switching sometimes fits better to the real systems.

In the present work we are interested in the properties of these estimators when the trend coefficient has a singularity like cusp. This case is in some sense intermediate between regular case and the change-point (discontinuous drift) case. The statistical problems with the models having cusp-type singularities were studied since 1968, when Prakasa Rao [19] described the asymptotic distribution of the MLE ${^ϑ}_{n}$ in the case of i.i.d. observations with the density function $f (ϑ, x)$ having the representation $f (ϑ, x) \approx a {| x - ϑ |}^{κ} + h$ with $κ \in (0, \frac{1}{2})$ at the vicinity of the point $x = ϑ$ . It was shown that

where $c > 0$

is some constant and the random variable

^u

will be described later. Note that in this case the Fisher information does not exist and the study of estimators requires special techniques. The exhaustive treatment of singular estimation problems (including cusp-type singularity) can be found in the Chapter VI of the fundamental work by Ibragimov and Khasminskii [9]. In this work one can find the general results concerning the asymptotic behavior of the MLE and Bayesian estimators in the situations including cusp-type singularity. In particular, they described the asymptotic distribution of the MLE and BE and showed that the BE are asymptotically efficient in minimax sense. For inhomogeneous Poisson processes with the intensity functions

λ (ϑ, t)

having a cusp-type singularity

λ (ϑ, t) \approx a {| t - ϑ |}^{κ} + h

the properties of the MLE and BE were described in [3]. For ergodic diffusion processes with the drift coefficient having cusp-type singularity the similar results were obtained in [4]. The case of cusp-type singularity for the model of observations of regression model were treated in [20] and in [6]. For the model of signal in WGN, where the signal has cusp-type singularity such results were obtained in [2]. Note that the case

κ \in (- \frac{1}{2}, 0)

was considered in [8] (ergodic diffusion) and in [10]. The survey of the properties of estimators for the different models of stochastic processes with cusp-type singularities can be found in [5].

The method of the study of estimators through the properties of the normalized likelihood ratio developed in the work [9] is in some sense of universal nature. It was applied in the study of estimators for a wide class of models of observations and is applied in the present work too. In particular, we check the conditions of two general theorems (Theorem 1.10.1 and Theorem 1.10.2) in [9] concerning the behavior of estimators.

We show that the MLE ${^ϑ}_{ε}$ and Bayesian estimators ${~ ϑ}_{ε}$ are consistent, have different limit distributions

ε^{\frac{1}{κ + \frac{1}{2}}} ({^ϑ}_{ε} - ϑ) ⟹ c^u, ε^{\frac{1}{κ + \frac{1}{2}}} ({~ ϑ}_{ε} - ϑ) ⟹ c ~ u,

with the same constant $c > 0$ , the polynomial moments of these estimators converge and that the BE are asymptotically efficient. The random variables $^u$ and $~ u$ are defined in the next section.

2 Main result

We suppose that the following condition is fulfilled:
Condition $A$ . The drift coefficient

S (ϑ, x) = a {| x - ϑ |}^{κ} + h (x),

where $κ \in (0, 1 / 2)$ and $a > 0$ . The function $h (x)$ is bounded, has continuous bounded derivative w.r.t. $x$ : $| h^{'} (x) | \leq H_{1}$ and is separated from zero: $h (x) \geq b > 0$ (for all $x$ ). The parameter $ϑ \in Θ = (α, β)$ , where $α > x_{0}$ and $β < {inf}_{ϑ \in Θ} x_{T} (ϑ)$ .

The limit of $X^{ε}$ is $X^{0} = {x_{t}, 0 \leq t \leq T}$ – solution of the deterministic equation

\frac{d x_{t}}{d t} = a {| x_{t} - ϑ_{0} |}_{t}^{κ} + h (x_{t}), x_{0}, 0 \leq t \leq T .

(4)

Note that by this condition we have the estimate

| S (ϑ, x) | \leq L (1 + {| x |}^{κ})

(5)

with some $L > 0$ . Here and in the sequel we denoted $ϑ_{0}$ the true value. Let us denote

0 < S_{m} = inf x_{0} \leq x \leq {^x}_{T} S (ϑ_{0}, x), S_{M} = sup x_{0} \leq x \leq {^x}_{T} S (ϑ_{0}, x) < \infty,

where ${^x}_{T} = {sup}_{ϑ \in Θ} x_{T} (ϑ)$ .

The properties of the maximum likelihood and Bayesian estimates are described with the help of the limit likelihood ratio. Let us remind that the likelihood ratio in this problem is (see Liptser and Shiryaev [15])

L (ϑ, X^{ε}) = exp {\int_{0}^{T} \frac{S (ϑ, X_{t})}{ε^{2}} d X_{t} - \int_{0}^{T} \frac{S {(ϑ, X_{t})}_{t}^{2}}{2 ε^{2}} d t} .

The maximum likelihood estimator (MLE) ${^ϑ}_{ε}$ is defined as solution of the equation

L ({^ϑ}_{ε}, X^{ε}) = sup θ \in Θ L (θ, X^{ε}) .

If this equation has more than one solution, then we can take anyone as MLE. Note that we cannot use the maximum likelihood equation

˙ L (θ, X^{ε}) = 0, θ \in Θ,

where dot means derivative w.r.t. $ϑ$ because the likelihood ratio function $L (ϑ, X^{ε})$ is not differentiable.

The Bayesian estimator (BE) ${~ ϑ}_{ε}$

for the quadratic loss function and density a priori

p (θ), θ \in Θ

(continuous positive function) is defined by the expression

{~ ϑ}_{ε} = \int_{α}^{β} θ p (θ | X^{ε}) d θ = \frac{\int_{α}^{β} θ p (θ) L (θ, X^{ε}) d θ}{\int_{α}^{β} p (θ) L (θ, X^{ε}) d θ} .

We take quadratic loss function for the simplicity of exposition. The established in this work properties of the likelihood ratio allow to describe the behavior of the BE for essentially wider class of loss functions (see Theorem 1.10.2 in [9]).

The limit behavior of the MLE and BE are described with the help of two random variables $^u$ and $~ u$ defined as follows. Let us introduce the random function

Z (u) = exp {W^{H} (u) - \frac{{| u |}^{2 H}}{2}}, u \in R

(6)

and put

Z (^u) = sup u \in R Z (u), ~ u = \frac{\int_{R} u Z (u) d u}{\int_{R} Z (u) d u} .

(7)

Here $W^{H} (\cdot)$ is two-sided fractional Brownian motion with Hurst parameter $H = κ + 1 / 2$ . The random variable $^u$ is well defined [18]. We need as well the definitions

	$Γ_{ϑ}^{2}$	$= \frac{a^{2}}{h (ϑ)} \int_{- \infty}^{\infty} {({\| s - 1 \|}^{κ} - {\| s \|}^{κ})}^{2} d s, γ_{ϑ} = Γ_{ϑ}^{1 / H},$
	${^u}_{ϑ_{0}}$	$= \frac{^u}{γ_{ϑ_{0}}}, {~ u}_{ϑ_{0}} = \frac{~ u}{γ_{ϑ_{0}}},^W = sup 0 \leq t \leq T \| W_{t} \| .$

As usual in such problems, we can introduce the lower minimax bound on the risks of all estimators:

Proposition 1

Let the condition $A$ be fulfilled then for all $ϑ_{0} \in Θ$ and all estimators ${¯ ϑ}_{ε}$ we have

(8)

The proof of this proposition we discuss after the proof of the Theorem 1 below.

According to this bound we call an estimator $ϑ_{ε}^{*}$ asymptotically efficient if for all $ϑ_{0} \in Θ$ we have the equality

lim δ \to 0 lim ε \to 0 sup | ϑ - ϑ_{0} | \leq δ ε^{- \frac{4}{2 κ + 1}} E_{ϑ} {(ϑ_{ε}^{*} - ϑ)}^{2} = \frac{E {(~ u)}^{2}}{γ_{ϑ_{0}}^{2}} .

The main result of this work is the following theorem.

Theorem 1

Let the condition $A$ be fulfilled, then the MLE ${^ϑ}_{ε}$ and the BE ${~ ϑ}_{ε}$ are uniformly on compacts $K \subset Θ$ consistent, have different limit distributions

ε^{- 1 / H} ({^ϑ}_{ε} - ϑ_{0}) ⟹ {^u}_{ϑ_{0}}, ε^{- 1 / H} ({~ ϑ}_{ε} - ϑ_{0}) ⟹ {~ u}_{ϑ_{0}},

the moments converge (uniformly on compacts): for any $p > 0$

E_{ϑ_{0}} {∣ ∣ ∣ ∣ \frac{{^ϑ}_{ε} - ϑ_{0}}{ε^{1 / H}} ∣ ∣ ∣ ∣}^{p} ⟶ E_{ϑ_{0}} {∣ ∣ {^u}_{ϑ_{0}} ∣ ∣}_{ϑ_{0}}^{p}, E_{ϑ_{0}} {∣ ∣ ∣ \frac{{~ ϑ}_{ε} - ϑ_{0}}{ε^{1 / H}} ∣ ∣ ∣}^{p} ⟶ E_{ϑ_{0}} {∣ ∣ {~ u}_{ϑ_{0}} ∣ ∣}_{ϑ_{0}}^{p},

and the Bayesian estimators are asymptotically efficient.

Proof. Let us introduce the normalized likelihood ratio

Z_{ε} (u) = \frac{L (ϑ_{0} + ε^{1 / H} u, X^{ε})}{L (ϑ_{0}, X^{ε})}, u \in U_{ε} = (\frac{α - ϑ_{0}}{ε^{1 / H}}, \frac{β - ϑ_{0}}{ε^{1 / H}}) .

It has the representation

	$Z_{ε} (u)$	$= exp {\int_{0}^{T} \frac{S (ϑ_{0} + ε^{1 / H} u, X_{t}) - S (ϑ_{0}, X_{t})}{ε} d W_{t}$
		$- \int_{0}^{T} \frac{{(S (ϑ_{0} + ε^{1 / H} u, X_{t}) - S (ϑ_{0}, X_{t}))}^{2}}{2 ε^{2}} d t ⎫ ⎬ ⎭ .$

We show below that $Z_{ε} (u)$ converges in distribution to the random function $Z_{ϑ_{0}} (u) = Z (γ_{ϑ_{0}} u)$ .

The first result which we are going to prove is the uniform convergence of the random process $X^{ε}$ to the deterministic solution $X^{0} = (x_{t}, 0 \leq t \leq T)$ of the ordinary equation (4). To prove it we need the following estimate.

Lemma 1

(N.V. Krylov [11]) Let the conditions $A$ be fulfilled, then there exists a constant $L_{*} > 0$

such that with probability 1

sup 0 \leq t \leq T | X_{t} - x_{t} | \leq L_{*} (ε^{κ} {^W}^{κ} + ε^W) .

(9)

Proof. Let us denote by $F (x_{t})$ the right hand part of the equation (4). Then we can write

\frac{d x_{t}}{F (x_{t})} = d t, a n d \int_{x_{0}}^{x_{t}} \frac{d y}{F (y)} = t .

(10)

If we put $Y_{t} = X_{t} - ε W_{t}$ , then the equation

d X_{t} = a {| X_{t} - ϑ_{0} |}_{t}^{κ} d t + h (X_{t}) d t + ε d W_{t}, X_{0} = x_{0}

can be written as

d Y_{t} = a {| Y_{t} - ϑ_{0} + ε W_{t} |}_{t}^{κ} d t + h (Y_{t} + ε W_{t}) d t, Y_{0} = x_{0},

\frac{d Y_{t}}{d t} = a {| Y_{t} - ϑ_{0} + ε W_{t} |}_{t}^{κ} + h (Y_{t} + ε W_{t}), Y_{0} = x_{0} .

Using the smoothness of $h (\cdot)$ $(h (x + δ) = h (x) + δ h^{'} (~ x))$ and the elementary inequalities

{| a + b |}^{κ} \leq {| a |}^{κ} + {| b |}^{κ}, {| a + b |}^{κ} \geq {| a |}^{κ} - {| b |}^{κ},

we write two estimates

	$\frac{d Y_{t}}{d t}$	$\leq a {\| Y_{t} - ϑ_{0} \|}_{t}^{κ} + h (Y_{t}) + ε^{κ} {\| W_{t} \|}_{t}^{κ} + ε C \| W_{t} \|,$
	$\frac{d Y_{t}}{d t}$	$\geq a {\| Y_{t} - ϑ_{0} \|}_{t}^{κ} + h (Y_{t}) - ε^{κ} {\| W_{t} \|}_{t}^{κ} - ε C \| W_{t} \| .$

Hence we have

	$\frac{d Y_{t}}{d t}$	$\leq F (Y_{t}) + ε^{κ} {^W}^{κ} + ε C^W,$
	$\frac{d Y_{t}}{d t}$	$\geq F (Y_{t}) - ε^{κ} {^W}^{κ} - ε C^W$

and (remind that $F (y) \geq b$ )

	$\int_{x_{0}}^{Y_{t}} \frac{d y}{F (y)}$	$\leq t + b^{- 1} T ε^{κ} {^W}^{κ} + b^{- 1} ε C T^W,$
	$\int_{x_{0}}^{Y_{t}} \frac{d y}{F (y)}$	$\geq t - b^{- 1} T ε^{κ} {^W}^{κ} - b^{- 1} ε C T^W .$

The equality (10) allows to write

	$\int_{x_{t}}^{Y_{t}} \frac{d y}{F (y)}$	$\leq b^{- 1} T ε^{κ} {^W}^{κ} + b^{- 1} ε C T^W,$
	$\int_{x_{t}}^{Y_{t}} \frac{d y}{F (y)}$	$\geq - b^{- 1} T ε^{κ} {^W}^{κ} - b^{- 1} ε C T^W .$

As the function $F (y)$ is continuous we have

- b^{- 1} T ε^{κ} {^W}^{κ} - b^{- 1} ε C T^W \leq \frac{(Y_{t} - x_{t})}{F (~ y)} \leq b^{- 1} T ε^{κ} {^W}^{κ} + b^{- 1} ε C T^W,

where $min (Y_{t}, x_{t}) \leq ~ y \leq max (Y_{t}, x_{t})$ . Hence

∣ ∣ ∣ \frac{Y_{t} - x_{t}}{F (~ y)} ∣ ∣ ∣ \leq b^{- 1} T ε^{κ} {^W}^{κ} + b^{- 1} ε C T^W .

Recall that $F (~ y)$ is bounded and separated from zero by a positive constant which does not depend on $ε$ . Further, there exists a constant $c_{1} > 0$ such that

∣ ∣ ∣ \frac{Y_{t} - x_{t}}{F (~ y)} ∣ ∣ ∣ \geq c_{1} | X_{t} - x_{t} + ε W_{t} | \geq c_{1} | X_{t} - x_{t} | - c ε^W .

Therefore

| X_{t} - x_{t} | \leq L_{*} (ε^{κ} {^W}^{κ} + ε^W)

where the constant $L_{*} = L_{*} (ϑ_{0}, b, c_{1}, C, T) > 0$ .

Lemma 2

Let the condition $A$ be fulfilled, then for any $κ_{1} \in (0, κ)$ there exist the constants $ν > 0$ and $c_{*} > 0$ such that

sup ϑ \in Θ P_{ϑ} {sup 0 \leq t \leq T | X_{t} - x_{t} | > ε^{κ_{1}}} \leq e^{- c_{*} ε^{- ν}}

(11)

for all $ε < ε_{0}$ with some $ε_{0} > 0$ .

Proof. Remind that for any $N > 0$

P{^W>N}=P{sup0≤t≤T|Wt|>N}≤4P{WT>N}≤4N√T2πe−N2/2T.

Hence we can write

	$Pϑ0{sup0≤t≤T\|Xt−xt\|>εκ1}≤P{L∗(εκ^Wκ+ε^W)>εκ1}$
	$=P{^Wκ+ε1−κ^W>L−1∗εκ1−κ}$
	$≤P{2^Wκ>L−1∗εκ1−κ}+P{ε1−κ^W>^Wκ}$
	$≤P{^W>(2L∗)−1/κεκ1−κκ}+P{^W>ε−1}$
	$\leq 4 ε^{\frac{κ - κ_{1}}{κ}} {(2 L_{})}_{}^{\frac{1}{κ}} \sqrt{\frac{T}{2 π}} exp ⎧ ⎪ ⎨ ⎪ ⎩ - \frac{ε^{\frac{- 2 (κ - κ_{1})}{κ}}}{2 T {(2 L_{})}_{}^{\frac{2}{κ}}} ⎫ ⎪ ⎬ ⎪ ⎭ + 4 ε \sqrt{\frac{T}{2 π}} e^{- \frac{ε^{- 2}}{2 T}} .$

The last expression allows us to take $ε_{0} > 0$ such that for all $ε \in (0, ε_{0})$ we have the estimate (11) where $ν = \frac{2 (κ - κ_{1})}{κ} > 0$ and $c_{*} = {(2^{\frac{2 κ + 1}{2}} T^{\frac{κ}{2}} L_{*})}_{*}^{- \frac{2}{} κ}$ .

Lemma 3

Let the condition $A$ be fulfilled then the finite dimensional distributions of the stochastic process $Z_{ε} (\cdot)$ converge to the finite dimensional distributions of $Z_{ϑ_{0}} (\cdot)$ and this convergence is uniform on the compacts $K \subset Θ$ .

Proof. Consider the stochastic integral

	$I_{ε} (u, X^{0})$	$= \frac{1}{ε} \int_{0}^{T} (S (ϑ_{0} + ε^{1 / H} u, x_{t}) - S (ϑ_{0}, x_{t})) d W_{t}$
		$= \frac{a}{ε} \int_{0}^{T} ({∣ ∣ x_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {\| x_{t} - ϑ_{0} \|}_{t}^{κ}) d W_{t} .$

Note that $I_{ε} (u, x), u \in U_{ε}$ is a Gaussian process. By condition $A$ the solution $x_{t}, 0 \leq t \leq T$ is strictly increasing function. Therefore we can put $t = t (x)$ by the relation

t = \int_{x_{0}}^{x} \frac{d y}{S (ϑ_{0}, y)}, x \in [x_{0}, x_{T}] .

This provides us the equality ( $x_{1} < x_{2}$ )

E_{ϑ_{0}} {(W_{t (x_{1})} - W_{t (x_{2})})}_{t (x_{1})}^{2} = \int_{x_{1}}^{x_{2}} \frac{d y}{S (ϑ_{0}, y)} .

Hence if we put

w (x) = \int_{x_{0}}^{x} \sqrt{S (ϑ_{0}, y)} d W_{t (y)}, x_{0} \leq x \leq x_{T},

then $w (x), x_{0} \leq x \leq x_{T}$ is a Gaussian process with independent increments

E_{ϑ_{0}} {(w (x_{1}) - w (x_{2}))}^{2} = x_{2} - x_{1}

and

	$\int_{0}^{T} ({∣ ∣ x_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {\| x_{t} - ϑ_{0} \|}_{t}^{κ}) d W_{t}$
	$= \int_{x_{0}}^{x_{T}} \frac{({∣ ∣ x - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{0}^{κ} - {\| x - ϑ_{0} \|}_{0}^{κ})}{\sqrt{S (ϑ_{0}, x)}} d w (x) .$

Further, let us change the variables $x = ϑ_{0} + s ε^{1 / H}$ . Then

	$I_{ε} (u, X^{0})$	$= a \int_{\frac{x_{0} - ϑ_{0}}{ε^{1 / H}}}^{\frac{x_{T} - ϑ_{0}}{ε^{1 / H}}} \frac{({\| s - u \|}^{κ} - {\| s \|}^{κ})}{\sqrt{S (ϑ_{0}, ϑ_{0} + s ε^{1 / H})}} d W (s)$
		$= \frac{a}{\sqrt{h (ϑ_{0})}} \int_{\frac{x_{0} - ϑ_{0}}{ε^{1 / H}}}^{\frac{x_{T} - ϑ_{0}}{ε^{1 / H}}} ({\| s - u \|}^{κ} - {\| s \|}^{κ}) d W (s) (1 + o (1))$

with the corresponding two-sided Wiener process

W (s) = W_{1} (s) {1I}_{{s \geq 0}} + W_{2} (- s) {1I}_{{s \leq 0}}, s \in [\frac{x_{0} - ϑ_{0}}{ε^{1 / H}}, \frac{x_{T} - ϑ_{0}}{} ε^{1 / H}] .

Here $W_{1} (s), s \geq 0$ and $W_{2} (s), s \geq 0$ are two independent standard Wiener processes. We used here the relation

S (ϑ_{0}, ϑ_{0} + s ε^{1 / H} u) = a ε^{κ / H} {| s u |}^{κ} + h (ϑ_{0} + s ε^{1 / H} u) = h (ϑ_{0}) + o (1) .

Therefore for any fixed value $u$ we have the following representation of the limit process

It has the following properties: $E I_{0} (u) = 0$ and

E I_{0} {(u)}^{2} = \frac{a^{2}}{h (ϑ_{0})} \int_{- \infty}^{\infty} {({| s - u |}^{κ} - {| s |}^{κ})}^{2} d s = {| u |}^{2 κ + 1} Γ_{ϑ_{0}}^{2} .

The process

W^{H} (u) = \frac{a}{\sqrt{h (ϑ_{0})} Γ_{ϑ_{0}}} \int_{- \infty}^{\infty} ({| s - u |}^{κ} - {| s |}^{κ}) d W (s), u \in R

is known as a representation of the two-sided fractional Brownian motion, because $W^{H} (\cdot)$ is a Gaussian process with the properties:

E W^{H} (u) = 0, E {[W^{H} (u)]}^{2} = {| u |}^{2 κ + 1} = {| u |}^{2 H} .

Hence using the standard arguments we obtain the convergence of the finite-dimensional distributions

(I_{ε} (u_{1}, X^{0}), \dots, I_{ε} (u_{k}, X^{0})) ⟹ (I_{0} (u_{1}), \dots, I_{0} (u_{k}))

and this convergence is uniform on the compacts $ϑ_{0} \in K \subset Θ$ .

Let us consider the ordinary integral

	$J_{ε} (u, X^{ε})$	$= \int_{0}^{T} {(\frac{S (ϑ_{0} + ε^{1 / H} u, X_{t}) - S (ϑ_{0}, X_{t})}{ε})}^{2} d t$
		$= \frac{a^{2}}{ε^{2}} \int_{0}^{T} {({∣ ∣ X_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {\| X_{t} - ϑ_{0} \|}_{t}^{κ})}^{2} d t .$

If we show the convergence in probability

J_{ε} (u, X^{ε}) - J_{ε} (u, X^{0}) ⟶ 0,

(12)

then we obtain the convergence

I_{ε} (u, X^{ε}) ⟹ I_{0} (u) = Γ_{ϑ_{0}} W^{H} (u) .

We can write

J_{ε} (u, X^{ε}) - J_{ε} (u, X^{0}) = \frac{a^{2}}{ε^{2}} \int_{0}^{T} (Δ {(u, X_{t})}_{t}^{2} - Δ {(u, x_{t})}_{t}^{2}) d t,

where we denoted

Δ (X_{t}, u) = {∣ ∣ X_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {| X_{t} - ϑ_{0} |}_{t}^{κ} .

Let us denote $ℓ_{ε} (x) = ε^{- 2} Λ_{T} (x)$ the normalized local time of the diffusion process $X_{t}$ and remind that for any function $g (\cdot) \geq 0$ we have the occupation time formula

\int_{0}^{T} g (X_{t}) d t = \int_{- \infty}^{\infty} g (x) ℓ_{ε} (x) d x .

(13)

Moreover, according to (9), we know that

	$\int_{0}^{T} g (X_{t}) d t ⟶$	$\int_{0}^{T} g (x_{t}) d t = \int_{0}^{T} \frac{g (x_{t})}{S (ϑ_{0}, x_{t})} d x_{t} = \int_{x_{0}}^{x_{T}} \frac{g (x)}{S (ϑ_{0}, x)} d x$
		$= \int_{- \infty}^{\infty} g (x) ℓ_{0} (x) d x,$		(14)

where we denoted $ℓ_{0} (x) = S {(ϑ_{0}, x)}_{0}^{- 1} {1I}_{{x_{0} \leq x \leq x_{T}}} .$ Hence for any continuous function $g (\cdot) \geq 0$ we have the convergence

	$\int_{- \infty}^{\infty} g (x) ℓ_{ε} (x) d x ⟶ \int_{x_{0}}^{x_{T}} g (x) ℓ_{0} (x) d x,$
	$\int_{- \infty}^{\infty} g (x) E_{ϑ_{0}} ℓ_{ε} (x) d x ⟶ \int_{x_{0}}^{x_{T}} g (x) ℓ_{0} (x) d x$

(see details in [13]). For example, for any small $δ > 0$ and $y \in (x_{0}, x_{T})$

E_{ϑ_{0}} \int_{0}^{T} {1I}_{{y - δ < X_{t} < y + δ}} d t = \int_{y - δ}^{y + δ} E_{ϑ_{0}} ℓ_{ε} (x) d x \to \int_{y - δ}^{y + δ} \frac{d x}{S (ϑ_{0}, x)} \approx \frac{2 δ}{S (ϑ_{0}, y)} .

We can write

	$J_{ε} (u, X^{ε})$
		$= a^{2} \int_{- \infty}^{\infty} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}^{2} ℓ_{ε} (ϑ_{0} + v ε^{1 / H}) d v$
		$⟶ a^{2} ℓ_{0} (ϑ_{0}) \int_{- \infty}^{\infty} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}^{2} d v$
		$= a^{2} ℓ_{0} (ϑ_{0}) {\| u \|}^{2 κ + 1} \int_{- \infty}^{\infty} {({\| s - 1 \|}^{κ} - {\| s \|}^{κ})}^{2} d s = Γ_{ϑ_{0}}^{2} {\| u \|}^{2 κ + 1},$

where we put $x = ϑ_{0} + v ε^{1 / H}$ and $v = u s$ .

For $J_{ε} (u, X^{0})$ we have the similar relations

	$J_{ε} (u, X^{0})$	$= \frac{a^{2}}{ε^{2}} \int_{0}^{T} {({∣ ∣ x_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {\| x_{t} - ϑ_{0} \|}_{t}^{κ})}^{2} d t$
		$= \frac{a^{2}}{ε^{2}} \int_{x_{0}}^{x_{T}} \frac{{({∣ ∣ x - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{0}^{κ} - {\| x - ϑ_{0} \|}_{0}^{κ})}^{2}}{S (ϑ_{0}, x)} d x$
		$= \frac{a^{2}}{ε^{2}} ε^{\frac{2 κ + 1}{H}} \int_{\frac{x_{0} - ϑ_{0}}{ε^{1 / H}}}^{\frac{x_{T} - ϑ_{0}}{ε^{1 / H}}} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}$

	$J_{ε} (u, X^{ε})$
		$= a^{2} \int_{- \infty}^{\infty} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}^{2} ℓ_{ε} (ϑ_{0} + v ε^{1 / H}) d v$
		$⟶ a^{2} ℓ_{0} (ϑ_{0}) \int_{- \infty}^{\infty} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}^{2} d v$
		$= a^{2} ℓ_{0} (ϑ_{0}) {\| u \|}^{2 κ + 1} \int_{- \infty}^{\infty} {({\| s - 1 \|}^{κ} - {\| s \|}^{κ})}^{2} d s = Γ_{ϑ_{0}}^{2} {\| u \|}^{2 κ + 1},$

	$J_{ε} (u, X^{0})$	$= \frac{a^{2}}{ε^{2}} \int_{0}^{T} {({∣ ∣ x_{t} - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{t}^{κ} - {\| x_{t} - ϑ_{0} \|}_{t}^{κ})}^{2} d t$
		$= \frac{a^{2}}{ε^{2}} \int_{x_{0}}^{x_{T}} \frac{{({∣ ∣ x - ϑ_{0} - ε^{1 / H} u ∣ ∣}_{0}^{κ} - {\| x - ϑ_{0} \|}_{0}^{κ})}^{2}}{S (ϑ_{0}, x)} d x$
		$= \frac{a^{2}}{ε^{2}} ε^{\frac{2 κ + 1}{H}} \int_{\frac{x_{0} - ϑ_{0}}{ε^{1 / H}}}^{\frac{x_{T} - ϑ_{0}}{ε^{1 / H}}} {({\| v - u \|}^{κ} - {\| v \|}^{κ})}$

On Cusp Location Estimation for Perturbed Dynamical Systems

Adaptive estimation and noise detection for an ergodic diffusion with observation noises

Inference in the stochastic Cox-Ingersol-Ross diffusion process with continuous sampling: Computational aspects and simulation

On the parameter estimation of ARMA(p,q) model by approximate Bayesian computation

Convergence of Bayesian Estimators for Diffusions in Genetics

Bayesian Estimations for Diagonalizable Bilinear SPDEs

Diffusion Means in Geometric Spaces

Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

1 Introduction

2 Main result

Proposition 1

Theorem 1

Lemma 1

Lemma 2

Lemma 3

On Cusp Location Estimation for Perturbed Dynamical Systems

Related Research

Adaptive estimation and noise detection for an ergodic diffusion with observation noises

Inference in the stochastic Cox-Ingersol-Ross diffusion process with continuous sampling: Computational aspects and simulation

On the parameter estimation of ARMA(p,q) model by approximate Bayesian computation

Convergence of Bayesian Estimators for Diffusions in Genetics

Bayesian Estimations for Diagonalizable Bilinear SPDEs

Diffusion Means in Geometric Spaces

Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

1 Introduction

2 Main result

Proposition 1

Theorem 1

Lemma 1

Lemma 2

Lemma 3