1 引言
起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L 1
当前, 随机泛函微分方程研究较热, 许多深刻且有用结论被得到, 然而, 方程里参数估计工作不多, 特别是分布依赖类 McKean-Vlasov 方程, 认知甚少, 统计问题主要集中在似然比方法估计未知参数以及讨论它的渐近行为[5 ,9 ,13 ,16 ] . 最近, Zhao 和 Zhang[16 ] 介绍了一类 α - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果.
d X ϵ t = b t ( X ϵ t , μ ϵ t , θ ) d t + ϵ d W t , t ∈ [ T ] ; X ϵ 0 = x 0 ,
(1.1)
这里 x 0 θ = ( θ 1 , ⋯ , θ p ) ′ ∈ Θ ⊂ R p μ ϵ t ( A ) = P ( X ϵ t ∈ A ) { X t } t W t σ ( Ω , F , ( F t ) t ≥ 0 , P ) d - 维布朗运动, 通常称为随机噪声, 漂移项 b t : R d × P 2 ( R d ) × Θ ↦ R d t ϵ ∈ ( 0 , 1 ] 16 ]启发, 本文采用最小距离法来估计参数 θ .
设二元函数 ρ ( ⋅ , ⋅ ) ξ 1 ξ 2 , ξ 3 ∈ D ( [ T ] , R d )
ρ ( ξ 1 , ξ 2 ) = ρ ( ξ 1 + ξ 3 , ξ 2 + ξ 3 )
(1.2)
ρ ( ξ 1 , ξ 2 ) ≤ c ρ sup 0 ≤ t ≤ T | ξ 1 ( t ) − ξ 2 ( t ) | ,
(1.3)
其中 c ρ ρ
ρ ( ξ 1 , ξ 2 ) = { ∫ T 0 | ξ 1 ( t ) − ξ 2 ( t ) | r ν ( d t ) , 0 < r < 1 ( ∫ T 0 | ξ 1 ( t ) − ξ 2 ( t ) | r ν ( d t ) ) 1 r , r ≥ 1 sup 0 ≤ t ≤ T | ξ 1 ( t ) − ξ 2 ( t ) | ,
就满足 (1.2) 和 (1.3) 式, 这里 ν ( ⋅ ) [ T ]
d X 0 t = b t ( X 0 t , δ X 0 t ( ⋅ ) , θ ) d t , t ∈ [ T ] ;
(1.4)
其中 δ x ( ⋅ ) x ϵ → 0 t ∈ [ T ] X ϵ t → X 0 t .
基于距离函数 ρ ( ⋅ , ⋅ ) θ
ˆ θ ϵ := arginf θ ∈ Θ ρ ( X ϵ t , X 0 t ) .
(1.5)
许多统计问题可以归结为模型 (1.5), 若假定 θ 0 ∈ Θ θ ˆ θ ϵ [6 ,16 ] . 一旦漂移项与解分布有依赖关联性时, 渐近分布不易找到, X t ˆ θ ϵ − θ ϵ − 1 ( ˆ θ ϵ − θ 0 ) ϵ → 0 8 ,11 ]启发, 假设 θ L γ θ
2 预备知识
为了方便论述, 我们介绍一些符号和定义: 对任意 x , y ∈ R d ⟨ x , y ⟩ = d ∑ i = 1 x i y i . 使用 | ⋅ | | y | = √ y ′ y = √ d ∑ i = 1 ⟨ y i , y i ⟩ . 后文经常也用``⋅ B ‖ \mathbf{B} \mathbb{R}^d \|f\|=\sup\limits_{x\in\mathbb{R}^d}|f(x)| . 为了符号简便, 用 \mathbf{P}_\theta \theta \{X_t(\theta)\} \theta \theta X_t^\epsilon=X_t^\epsilon(\theta) . 下面介绍 Wasserstein 距离: 设 \mathcal{P}(\mathscr{C}) \mathscr{C} \mu\in\mathcal{P}(\mathscr{C}) \mu \mu(|\cdot|^2)=\int_{\mathscr{C}}|x|^2\mu(dx)<\infty . 这类概率测度空间定义为 \mathcal{P}_2(\mathscr{C}) . 对于任意 \mu,\nu \in\mathcal{P}_2(\mathscr{C}) \mathcal{W}^{(2)}(\mu,\nu)
\begin{equation*} \begin{aligned} \mathcal{W}^{(2)}(\mu,\nu)=\inf \bigg\{\Big(\int_{\mathscr{C}\times \mathscr{C}}d( x,y)^2 \pi(dx,dy)\Big)^{\frac{1}{2}}:\pi\in \mathcal{P}(\mathscr{C}\times \mathscr{C})\bigg\}, \end{aligned} \end{equation*}
这里 \mathcal{P}(\mathscr{C}\times \mathscr{C}) \mathcal{P}(\mathscr{C}) \mathcal{P}(\mathscr{C}) \pi(\cdot,\mathscr{C})=\mu(\cdot) \pi(\mathscr{C},\cdot)=\nu(\cdot) \mathcal{P}_2(\mathscr{C}) \mathcal{P}_r(\mathscr{C}) E r 3 ]知, 对任意测度 \delta_{x_0} \mu\in \mathscr{C}
\begin{equation} \mathcal{W}^{(2)}(\mu,\delta_{x_0})=\Big(\int_{\mathscr{C}}(y-x_0)^2\mu(dy)\Big)^{\frac{1}{2}}. \end{equation}
(2.1)
定义 2.1 设函数 f:\mathcal{P}(\mathbb{R}^d)\rightarrow\mathbb{R} .
(1) 泛函 \phi L^2(\mathbb{R}^d\rightarrow\mathbb{R}^d,\mu)\ni\phi\mapsto f(\mu\circ(Id+\phi)^{-1}) \gamma\in L^2(\mathbb{R}^d\rightarrow\mathbb{R}^d,\mu)
\begin{equation} \lim_{\mu(|\phi|^2)\rightarrow0}\frac{\big|f(\mu\circ(Id+\phi)^{-1})-f(\mu)-\mu(\langle\phi,\gamma\rangle)\big|}{\sqrt{\mu(|\phi|^2)}}=0, \end{equation}
(2.2)
这里 \mu(\langle\phi,\gamma\rangle):=\int_{\mathbb{R}^d}\langle\phi(\xi),\gamma(\xi)\rangle\mu(d\xi) . 则称 f \mu\in\mathcal{P}_2(\mathbb{R}^d) L - 可导, 其导数定义为 \partial^Lf(\mu)=\gamma .
(2) 若对任意 \mu\in\mathcal{P}_2(\mathbb{R}^d) L \partial^Lf(\mu) f L - 可微的. 特别, 如果对每一个 \mu\in\mathcal{P}_2(\mathbb{R}^d) \mu - 版本 \partial^Lf(\mu)(\cdot) \partial^Lf(\mu)(x) (x,\mu)\in\mathbb{R}^d\times\mathcal{P}_2(\mathbb{R}^d) f\in C^{1,0}(1,0)(\mathcal{P}_2(\mathbb{R}^d))
引理 2.1 设 (\Omega,\mathscr{F},\mathbf{P}) X,Y\in L^2(\Omega\mapsto\mathbb{R}^d,\mathbf{P}) X \mu(\cdot) f \mu L f\in C^{1,0}(\mathcal{P}_2(\mathbb{R}^d)) X Y
\begin{equation} \lim_{\epsilon\rightarrow0}\frac{f(\mu_{X+\epsilon Y})-f(\mu)}{\epsilon}=\mathbf{E}\langle \partial^Lf(\mu)(X),Y\rangle. \end{equation}
(2.3)
\Big|\lim_{\epsilon\rightarrow0}\frac{f(\mu_{X+\epsilon Y})-f(\mu)}{\epsilon}\Big|=|\mathbf{E}\langle \partial^Lf(\mu)(X),Y\rangle|\leq \mathbf{E}\|\partial^Lf(\mu)\|\sqrt{\mathbf{E}|Y|^2}.
定义 2.2 设 H \hat{H} \phi:H\times H\rightarrow\hat{H} \mathscr{L}(H,\hat{H}) H \hat{H} h\in H x, y\in H \partial_x\phi(x,y)\in\mathscr{L}(H,\hat{H})
\begin{equation*} \underset{\tau\rightarrow 0}{\lim}\frac{\phi(x+\tau h,y)-\phi(x,y)}{\tau}=\partial_x\phi(x,y).h. \end{equation*}
则称 \phi x
\begin{equation*} \underset{\|h\|\rightarrow0}{\lim}\frac{\|\phi(x+h,y)-\phi(x,y)-\partial_x\phi(x,y).h\|}{\| h\|}=0, \end{equation*}
则称 \phi x
定义 2.1, 2.2 和引理 2.1 来自文献[12 ], 称为 L - 导数公式, 类似微积分公式, 泛函项 b_t(x,\mu_t,\theta) i \partial_ib_t(x,\mu_t,\theta) \theta \partial_3b_t(x,\mu_t,\theta) . 为了给出结论, 我们需要如下条件.
(A1) X^\epsilon_0\in L^p(\mathscr{F}_0,\mathbb{R}^d,\mathbf{P}) \{W_t\}
(A2) 函数 b_t (\mathbb{R}^d,\mathcal{P}_2(\mathbb{R}^d),\Theta)\mapsto\mathbb{R}^d
\begin{equation*} \mathbf{E}\Big[\int_0^T \big|b_t(X^\epsilon_0,\delta_0,\theta)\big|^p{\rm d}t\Big]<\infty. \end{equation*}
(ii) 存在一个递增函数 K(t)\geq0 t \geq0, x,y\in\mathbb{R}, \mu,\nu\in \mathcal{P}_2(\mathbb{R})
\begin{equation} \begin{aligned} \big|b_t(x,\mu,\theta_1)-b_t(y,\nu,\theta_2)\big|\leq K(t)\big( |x-y|+\mathcal{W}^{(2)}(\mu,\nu)+|\theta_1-\theta_2| \big). \end{aligned} \end{equation}
(2.4)
\begin{equation*} \begin{split} &\underset{x\in \mathbb{R}^d,\delta_x(\cdot)\in \mathcal{P}_2(\mathbb{R}^d),\theta\in \Theta}{\sup}\big|\partial_1 b_t(x,\delta_x,\theta)\big|<\infty,\\ &\underset{x\in \mathbb{R}^d,\delta_x(\cdot)\in \mathcal{P}_2(\mathbb{R}^d),\theta\in \Theta}{\sup}\big|\partial^L_2 b_t(x,\delta_x,\theta)\big|<\infty,\\ \end{split} \end{equation*}
\begin{equation*} \underset{x\in \mathbb{R}^d,\delta_x(\cdot)\in \mathcal{P}_2(\mathbb{R}^d),\theta\in \Theta}{\sup}\big|\partial_3 b_t(x,\delta_x,\theta)\big|<\infty. \end{equation*}
(iv) 存在一个固定函数 K(t)>0 \beta_1,\beta_2 x,y\in\mathbb{R}^d \mu, \nu\in\mathcal{P}_2(\mathbb{R}^d) \theta_1,\theta_2\in\Theta
\begin{equation} \begin{aligned} \big|\partial_1b_t(x,\mu,\theta_1)-\partial_1b_t(y,\nu,\theta_2)\big|\leq K(t)\big( |x-y|^{\beta_1}+\mathcal{W}^{(2)}(\mu,\nu)+|\theta_1-\theta_2|^{\beta_1} \big) \end{aligned} \end{equation}
(2.5)
\begin{equation} \begin{aligned} \big|\partial_2^Lb_t(x,\mu,\theta_1)-\partial_2^Lb_t(y,\nu,\theta_2)\big|\leq K(t)\big( |x-y|^{\beta_2}+\mathcal{W}^{(2)}(\mu,\nu)+|\theta_1-\theta_2|^{\beta_2} \big). \end{aligned} \end{equation}
(2.6)
对任意 x\in\mathbb{R}^d \mu\in\mathcal{P}_2(\mathbb{R}^d) \theta\in\Theta
\begin{equation} \begin{aligned} \max\Big\{\big|\partial_1b_t(x,\mu,\theta)\big|,\ \big|\partial_2^Lb_t(x,\mu,\theta)\big|\Big\}\leq K(t). \end{aligned} \end{equation}
(2.7)
注 2.1 如果函数项 b_t x L(t)>0 1<q\in \mathbb{N} x,y\in \mathbb{R}^d \mu \in \mathcal{P}_2(\mathbb{R}^d)
\begin{equation} \big|b_t(x,\mu,\theta)-b_t(y,\mu,\theta)\big|\leq L(t)(1+|x|^{q-1}+|y|^{q-1}) |x-y|. \end{equation}
(2.8)
且其它位置也是同样结论, 易知函数项 b_t [4 ] .
3 极限分布与证明
为了讨论极限过程, 我们需要引入一个辅助过程, 设 X^{(1)}=\{X_t^{(1)}(\theta),0\leq t\leq T\}
\begin{equation} {\rm d}X^{(1)}_t=\partial_{1}b_t\big(X_t^0(\theta),\delta_{X^0_t(\theta)},\theta\big)\cdot X^{(1)}_t{\rm d}t +\mathbf{E}\langle\partial_2^L b_t\big(X_t^0(\theta),\delta_{X^0_t(\theta)},\theta\big),X^{(1)}_t\rangle {\rm d}t +{\rm d}W_t \end{equation}
(3.1)
的一个解, 为简便叙述, 记 X_t^0 \theta
\dot{X}^0_t(\theta_0)=\Big(\frac{\partial}{\partial\theta_1}X_t^0(\theta),\cdots,\frac{\partial}{\partial\theta_p}X_t^0(\theta)\Big)^\prime\Big|_{\theta=\theta_0}.
这是一个 p\times d
\begin{matrix} \dot{X}^0_t(\theta)=& \int_0^t\partial_{1}b_s\big(X_s^0(\theta),\delta_{X^0_s(\theta)},\theta\big)\cdot\dot{X}^{(0)}_s(\theta){\rm d}t \\ & +\int_0^t\mathbf{E}\langle\partial_2^L b_s\big(X_s^0(\theta),\delta_{X^0_s(\theta)},\theta\big),\dot{X}^{0}_s(\theta)\rangle {\rm d}s+\int_0^t\partial_3b_s\big(X_s^0(\theta),\delta_{X^0_s(\theta)},\theta\big){\rm d}s. \end{matrix}
(3.2)
本文目的是证明 \epsilon\rightarrow0
\begin{equation} \frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}\longrightarrow_d\xi_T, \end{equation}
(3.3)
这里" \rightarrow_d \xi_T p
\begin{equation} \xi_T:=\underset{u\in \mathbb{R}^d}{\text{arginf}} \, \Big\{\rho\big(X^{(1)}_t(\theta_0),\dot{X}^0_t(\theta_0)\cdot u\big)\Big\}. \end{equation}
(3.4)
在后文中, 类似(3.4)式里记号 t
引理 3.1 对任意 \alpha>0 g(\alpha)=\underset{|\theta-\theta_0|>\alpha}{\inf}\rho(X^0(\theta),X^0(\theta_0))>0 . 在条件 (A1) 和 (A2) 下, 有
\begin{equation*} \underset{\epsilon\rightarrow0}{\lim}\,\mathbf{P}_{\theta_0}(|\hat{\theta}_\epsilon-\theta_0|> \alpha)=0. \end{equation*}
\begin{matrix} \mathbf{P}_{\theta_0}\left(|\hat{\theta}_\epsilon-\theta_0|>\alpha\right)&=\mathbf{P}_{\theta_0}\left(\underset{|\theta-\theta_0|\leq\alpha}{\inf} \rho(X^\epsilon(\theta),X^0(\theta))>\underset{|\theta-\theta_0|>\alpha}{\inf}\rho(X^\epsilon(\theta),X^0(\theta))\right) \\ & \leq \mathbf{P}_{\theta_0}\bigg(\underset{|\theta-\theta_0|\leq\alpha}{\inf}\Big\{\rho(X^\epsilon(\theta),X^0(\theta_0))+\rho(X^0(\theta),X^0(\theta_0))\Big\} \\ & >\underset{|\theta-\theta_0|>\alpha}{\inf}\Big\{\rho(X^0(\theta),X^0(\theta_0))-\rho(X^\epsilon(\theta),X^0(\theta_0))\Big\}\bigg) \\ & =\mathbf{P}_{\theta_0}\left(2\rho(X^\epsilon(\theta_0),X^0(\theta_0))>\underset{|\theta-\theta_0|>\alpha}{\inf}\rho(X^0(\theta),X^0(\theta_0))\right) \\ & =\mathbf{P}_{\theta_0}\left(\rho(X^\epsilon(\theta_0),X^0(\theta_0))>\frac{g(\alpha)}{2}\right), \end{matrix}
(3.5)
\begin{equation*} \underset{|\theta-\theta_0|\leq\alpha}{\inf}\,\rho(X^0(\theta),X^0(\theta_0))=0. \end{equation*}
\begin{matrix} \big|X^\epsilon_t(\theta)-X^0_t(\theta)\big|&\leq\Big|\int^t_0b_s(X^\epsilon_s(\theta),\mu_s^\epsilon,\theta)-b_s(X^0_s(\theta),\delta_{X^0_s(\theta)},\theta){\rm d}s\Big|+\epsilon|W_t| \\ & \leq\int^t_0K(s)\big(|X^\epsilon_s(\theta)-X^0_s(\theta)|+\mathcal{W}^{(2)}(\mu_s^\epsilon,\delta_{X^0_s(\theta)})\big){\rm d}s+\epsilon|W_t|. \end{matrix}
(3.6)
利用(2.1)式, 易知 \mathcal{W}^{(2)}(\mu_t^\epsilon,\delta_{X^0_t(\theta)})\leq\sup\limits_{0\leq t\leq T}|X_t^\epsilon(\theta)-X_t^0(\theta)| .
\begin{equation} \begin{aligned} \sup\limits_{0\leq t\leq T}|X^\epsilon_t(\theta)-X^0_t(\theta)|\leq\epsilon\, {\rm e}^{2\int^T_0K(s){\rm d}s}\sup\limits_{0\leq t\leq T}|W_t|. \end{aligned} \end{equation}
(3.7)
注意到距离 \rho(\cdot,\cdot)
\begin{equation} \rho\big(X^\epsilon(\theta),\ X^0(\theta_0)\big)\leq c_{\rho}\sup\limits_{0\leq t\leq T}|X^\epsilon(\theta)-X^0(\theta_0)| \end{equation}
(3.8)
和关于 d - 维布朗运动 W_t \alpha>0
\begin{equation} \mathbf{P}\{\sup\limits_{0\leq t\leq T}|W_t|>\alpha\}\leq 4d\exp\Big\{-\frac{\alpha^2}{2dT}\Big\}. \end{equation}
(3.9)
\begin{align*} \mathbf{P}_{\theta_0}\big(\rho(X^\epsilon(\theta), X^0(\theta_0))>\frac{g(\alpha)}{2}\big) &\leq \mathbf{P}_{\theta_0}\Big(\sup\limits_{0\leq t\leq T}|W_t| > \frac{g(\alpha)}{2\epsilon c_\rho {\rm e}^{\int_0^TK(t){\rm d}t}}\Big)\\ &\leq 4d\exp\Big\{-\frac{g(\alpha)^2}{8dT\epsilon^2 c_{\rho}^2{\rm e}^{2\int^T_0K(s){\rm d}s}}\Big\}. \end{align*}
因此, 结合 (3.5) 式, 当 \epsilon\rightarrow 0 \mathbf{P}_{\theta_0}\big(|\hat{\theta}_\epsilon-\theta_0|>\alpha\big)\rightarrow0.
引理 3.2 对任意 u\in \mathbb{R}^d \epsilon \beta_1 \beta_2 T K(\cdot) C(T,\beta_1,\beta_2,K(\cdot))>0
\begin{equation*} \begin{split} \underset{0\leq t\leq T}{\sup}\Big|\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}-\dot{X}^0(\theta_0)u\Big| \leq C(T,\beta_1,\beta_2,K(\cdot))\epsilon^{\min\{1,\beta_1,\beta_2\}+1}|u|^{\max\{\beta_1,\beta_2\}+1}. \end{split} \end{equation*}
证 首先, 对足够小的 \epsilon>0 \theta_0+\epsilon u\in \Theta . 于是
\begin{matrix} &X^0_t(\theta_0+\epsilon u)-X^0_t(\theta_0) \\ =& \int^t_0\Big(b_s(X^0_s(\theta_0+\epsilon u),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)-b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0)\Big){\rm d}s \\ =& \int^t_0\Big(b_s(X^0_s(\theta_0+\epsilon u),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)-b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)\Big){\rm d}s \\ & +\int^t_0\Big(b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)-b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0+\epsilon u)\Big){\rm d}s \\ & +\int^t_0\Big(b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0+\epsilon u)-b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0) \Big){\rm d}s. \end{matrix}
(3.10)
\begin{matrix} \big|X^0_t(\theta_0+\epsilon u)-X^0_t(\theta_0)\big| \leq& \int^t_0K(s)\big|X^0_s(\theta_0+\epsilon u)-X^0_s(\theta_0)\big|{\rm d}s \\ &+\int^t_0K(s)\mathcal{W}^{(2)}(\delta_{X^\epsilon_s(\theta_0+\epsilon u)},\delta_{X^0_s(\theta_0)}){\rm d}s +\epsilon\int^T_0K(s)|u|{\rm d}s. \end{matrix}
(3.11)
\begin{equation} \mathcal{W}^{(2)}(\delta_{X^\epsilon_s(\theta_0+\epsilon u)},\delta_{X^0_s(\theta_0)})\leq \sup\limits_{0\leq s\leq t}|X^\epsilon_s(\theta_0+\epsilon u)-X^\epsilon_s(\theta_0)|. \end{equation}
(3.12)
\begin{equation*} \sup\limits_{0\leq s\leq t}\big|X^0_s(\theta_0+\epsilon u)-X^0_s(\theta_0)\big| \leq2\int_0^tK(s)\sup\limits_{0\leq \upsilon\leq s}\big|X^0_\upsilon(\theta_0+\epsilon u)-X^0_\upsilon(\theta_0)\big|{\rm d}s+\epsilon|u|\int_0^TK(s){\rm d}s.\end{equation*}
应用 Gronwall 不等式, 对任意 0\leq t\leq T
\begin{equation} \begin{aligned} \underset{0\leq s\leq t}{\sup} \big|X^0_s(\theta_0+\epsilon u)-X^0_s(\theta_0)\big| \leq \epsilon|u|\int_0^tK(s)\,{\rm e}^{2\int_0^tK(s){\rm d}s}{\rm d}s. \end{aligned} \end{equation}
(3.13)
\begin{equation} Z_t^\epsilon(\theta_0)=\frac{X^0_t(\theta_0+\epsilon u)-X^0_t(\theta_0)}{\epsilon}. \end{equation}
(3.14)
\begin{align*} &Z_t^\epsilon(\theta_0)-\dot{X}^0_t(\theta_0)\cdot u\\ =&\int^t_0\frac{1}{\epsilon}\Big(b_s(X^0_s(\theta_0+\epsilon u),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u))-b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)\\ &\ \ \ \ \ \ \ \ \ \ -\epsilon \partial _1b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0)\cdot\dot{X}^0_s(\theta_0)\cdot u\Big){\rm d}s\\ &+\int^t_0\frac{1}{\epsilon}\Big(b(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)-b(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0+\epsilon u)\\ &\ \ \ \ \ \ \ \ \ \ -\epsilon\mathbf{E}\langle \partial _2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0),\ Z_s^\epsilon(\theta_0)\rangle\Big){\rm d}s\\ & +\int^t_0\frac{1}{\epsilon}\Big(b(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0+\epsilon u)-b(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0)\\ &\ \ \ \ \ \ \ \ \ \ -\epsilon \partial _3 b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0)\cdot u\Big){\rm d}s\\ &+\int_0^t\Big(\mathbf{E}\langle\partial _2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0),Z_s^\epsilon(\theta_0)\rangle\\ &\ \ \ \ \ \ \ \ \ \ -\mathbf{E}\langle\partial _2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0),\dot{X}_s^{0}(\theta_0)\cdot u\rangle {\rm d}s\\ =&:I_1(t)+I_2(t)+I_3(t)+I_4(t). \end{align*}
设 k_0(s)=\int^1_0\partial_1\, b_s\big(X^0_s(\theta_0)+\upsilon Z_s^\epsilon(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u\big){\rm d}v
\begin{matrix} |I_1(t)| \leq\ &\Big|\int^t_0k_0(s)\big(Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u \big){\rm d}s\Big| \\ & +\Big|\int^t_0\Big(k_0(s)-\partial_1 b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u\Big)\cdot u\cdot\dot{X}^0_s(\theta_0){\rm d}s\Big| \\ & +\Big|\int^t_0\Big(\partial_1 b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u) \\ & -\partial_1 b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0)\Big)\cdot u\cdot\dot{X}^0_s(\theta_0){\rm d}s\Big|. \end{matrix}
(3.15)
利用(2.5)式和假设 (iii), 存在常数 C_1>0 \sup\limits_{0\leq s\leq T}|k_0(s)|\leq C_1
\begin{matrix} |I_1(t)|\leq\ & C_1\int^t_0\big|\big(Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u \big)\big|{\rm d}s+\epsilon^{\beta_1}\int_0^tK(s)|\dot{X}^0_s(\theta_0)||u||Z_s^\epsilon(\theta_0)|^{\beta_1}{\rm d}s \\ & +\int_0^tK(s)\Big(\mathcal{W}^{(2)}\big(\delta_{X^0_s(\theta_0+\epsilon u)},\delta_{X^0_s(\theta_0)}\big)+\epsilon^{\beta_1}|u|^{\beta_1}\Big)|u||\dot{X}^0_s(\theta_0)|{\rm d}s \\ \leq\ & C_1\int^t_0\big|\big(Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u \big)\big|{\rm d}s+\epsilon^{\beta_1}\int_0^tK(s)|\dot{X}^0_s(\theta_0)||u|^{1+\beta_1}C_T^{\beta_1}{\rm d}s \\ & +\int_0^tK(s)\Big(\epsilon|u|C_T+\epsilon^{\beta_1}|u|^{\beta_1}\Big)|u||\dot{X}^0_s(\theta_0)|{\rm d}s \end{matrix}
(3.16)
这里 C_T=\int_0^TK(s)\,{\rm e}^{2\int_0^TK(s){\rm d}s}{\rm d}s . 最后一个不等式使用了(3.13)式.
设 k_1(s):=\int^1_0\partial_2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)+v(X^0_s(\theta_0+\epsilon u)-X^0_s(\theta_0))},\theta_0+\epsilon u){\rm d}v
\begin{align*} |I_2(t)| &\leq\Big|\int^t_0\mathbf{E}\big\langle k_1(s),\ Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u\big\rangle {\rm d}s\Big|\\ & \ +\Big|\int^t_0\mathbf{E}\big\langle k_1(s)-\partial_2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u),\ \dot{X}^0_s(\theta_0)\cdot u\big\rangle ds\Big|\\ & \ +\Big|\int^t_0\mathbf{E}\big\langle\partial_2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0+\epsilon u)},\theta_0+\epsilon u)-\partial_2^L b_s(X^0_s(\theta_0),\delta_{X^0_s(\theta_0)},\theta_0),\ \dot{X}^0_s(\theta_0)\cdot u\big\rangle {\rm d}s\Big|. \end{align*}
利用假设(2.6)和条件 (iii), 存在常数 C_2>0 \sup\limits_{0\leq s\leq T}|k_1(s)|\leq C_2
\begin{matrix} |I_2(t)|\leq\ & C_2\int^t_0\big|\big(Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u \big)\big|{\rm d}s+\epsilon^{\beta_2}\int_0^tK(s)|\dot{X}^0_s(\theta_0)||u||Z_s^\epsilon(\theta_0)|^{\beta_2}{\rm d}s \\ & +\int_0^tK(s)\Big(\mathcal{W}^{(2)}\big(\delta_{X^0_s(\theta_0+\epsilon u)},\delta_{X^0_s(\theta_0)}\big)+\epsilon^{\beta_2}|u|^{\beta_2}\Big)|u||\dot{X}^0_s(\theta_0)|{\rm d}s \\ \leq\ & C \int^t_0\big|\big(Z_s^\epsilon(\theta_0)- \dot{X}^0_s(\theta_0)\cdot u \big)\big|{\rm d}s+\epsilon^{\beta_2}\int_0^tK(s)|\dot{X}^0_s(\theta_0)||u|^{1+\beta_2}C_T^{\beta_2}{\rm d}s \\ & +\int_0^tK(s)\Big(\epsilon|u|C_T+\epsilon^{\beta_2}|u|^{\beta_2}\Big)|u||\dot{X}^0_s(\theta_0)|{\rm d}s. \end{matrix}
(3.17)
不等式(3.13)同样在最后一个不等号证明中被用到.
对于 I_3(t) c_2>0
\begin{equation} \sup\limits_{0\leq t\leq T}|I_3(t)|\leq c_2\epsilon^2{\rm e}^{\int_0^TK(s){\rm d}s}. \end{equation}
(3.18)
结合(3.16)-(3.18)式, 注意到(2.6)式, 再次使用 Gronwall 不等式, 得到
\begin{equation*} \begin{split} \underset{0\leq t\leq T}{\sup}\Big|\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}-\dot{X}^0(\theta_0)\cdot u\Big| \leq C(T,\beta_1,\beta_2,K(\cdot))\epsilon^{\min\{1,\beta_1,\beta_2\}+1}|u|^{\max\{\beta_1,\beta_2\}+1}. \end{split} \end{equation*}
引理 3.3 在假设 (A1) 和 (A2) 下, 对任意 \alpha>0
\begin{equation} \lim_{\epsilon\rightarrow0}\mathbf{P}\Big(\sup\limits_{0\leq t\leq T}\Big|\frac{X_t^\epsilon(\theta)-X^0_t(\theta)}{\epsilon}-X^{(1)}_t(\theta)\Big|>\alpha\Big)=0. \end{equation}
(3.19)
其中 X^{(1)}_t(\theta)
\begin{matrix} & \ \mathbf{E}\big[\sup\limits_{0\leq t \leq T}\big|X^\epsilon_t(\theta)-X^0_t(\theta)\big|^2\big] \\ & \leq 2\mathbf{E}\Big|\int^T_0\big(b_s(X^\epsilon_s(\theta),\mu_s^\epsilon,\theta)-b_s(X^0_s(\theta),\delta_{X^0_s(\theta)},\theta)\big){\rm d}s\Big|^2+2\epsilon\mathbf{E}\big[\sup\limits_{0\leq t\leq T}|W_t|^2\big] \\ & \leq 4\int^t_0K(s)\mathbf{E}|X^\epsilon_s(\theta)-X^0_s(\theta)|^2{\rm d}s+4\int_0^TK(s)\mathcal{W}^{(2)}(\mu_s^\epsilon,\delta_{X^0_s(\theta)})\big)^2{\rm d}s+2\epsilon T \\ & \leq 8\int_0^TK(s)\mathbf{E}\big|X_s^\epsilon-X_s^0\big|^2{\rm d}s+2\epsilon T, \end{matrix}
(3.20)
最后一项使用了不等式 \mathcal{W}^{(2)}(\mu_s^\epsilon,\delta_{X^0_s(\theta)})^2\leq \mathbf{E}|X_s^\epsilon-X_s^0|^2 . 再次使用
\begin{equation} \mathbf{E}\big[\sup\limits_{0\leq t \leq T}\big|X^\epsilon_t(\theta)-X^0_t(\theta)\big|^2\big]\leq2\epsilon T {\rm e}^{8\int_0^TK(s){\rm d}s}. \end{equation}
(3.21)
这蕴含随着 \epsilon\rightarrow0 \{X^\epsilon_t(\theta)\} X^0_t(\theta) .
另一方面, 对于项 X^{(1)}_t
\begin{matrix} |X^{(1)}_t|\leq&\int_0^t\big|\partial_{1}b_s\big(X_s^0(\theta),\delta_{X^0_s(\theta)},\theta\big)\big|\big| X^{(1)}_s\big|{\rm d}s \\ & +\int_0^t\big|\mathbf{E}\langle\partial_2^L b_s\big(X_s^0(\theta),\delta_{X^0_s(\theta)},\theta\big),X^{(1)}_s\rangle\big|{\rm d}s +\sup\limits_{0\leq t\leq T}\big|W_t\big|. \end{matrix}
(3.22)
|X^{(1)}_t|=2\int_0^tK(s)\big| X^{(1)}_s\big|{\rm d}s+\sup\limits_{0\leq t\leq T}\big|W_t\big|.
\begin{equation} \sup\limits_{0\leq t\leq T}|X^{(1)}_t|\leq {\rm e}^{\int_0^TK(s){\rm d}s}.\sup\limits_{0\leq t\leq T}\big|W_t\big|. \end{equation}
(3.23)
从(3.9)式, 不难知 \sup\limits_{0\leq t\leq T}|X^{(1)}_t|
k_2(s)=\int^1_0\partial_1 b_s\big(X^0_s(\theta)+v(X^\epsilon _s(\theta)-X_s^0(\theta)),\delta_{X^0_s(\theta)},\theta\big){\rm d}v
和 k_3(s)=\int^1_0\partial_{2}^Lb_s\big(X_s^\epsilon(\theta),\delta_{X^0_s(\theta)+v(\delta_{X^\epsilon_s}(\theta)-\delta_{X^0_s}(\theta))},\theta\big){\rm d}v.
\begin{matrix} & \frac{X^\epsilon_t(\theta)-X^0_t(\theta)}{\epsilon}- X^{(1)}_t(\theta) \\ =& \int^t_0\frac{1}{\epsilon}\Big(b_s(X^\epsilon_s(\theta),\mu_s^\epsilon,\theta)-b_s(X^0_s(\theta),\delta_{X^0_s(\theta)},\theta) \\ & - \epsilon\partial_1 b_s(X^0_s(\theta),\delta_{X^0_s}(\theta),\theta)\cdot X^{(1)}_s(\theta)-\epsilon\mathbf{E}\big\langle\partial_2^L b_s(X^0_s(\theta),\delta_{X^0_s(\theta)},\theta),X^{(1)}_s(\theta)\big\rangle\Big){\rm d}s \\ =&\int^t_0\Big[\frac{1}{\epsilon}\big(b_s(X^\epsilon_s(\theta),\mu_s^\epsilon,\theta)-b_s(X^0_s(\theta),\mu_s^\epsilon,\theta)\big)- \partial_1 b_s(X^0_s(\theta),\mu_s^\epsilon,\theta)\cdot X^{(1)}_s(\theta)\Big]{\rm d}s \\ & +\int_0^t\Big[\partial_1 b_s(X^0_s(\theta),\mu_s^\epsilon,\theta)-\partial_1 b_s(X^0_s(\theta),\delta_{X^0_s}(\theta),\theta)\Big]\cdot X^{(1)}_s(\theta){\rm d}s \\ & +\int^t_0\Big[\frac{1}{\epsilon}\big(b_s(X^0_s(\theta),\mu_s^\epsilon,\theta)-b_s(X^0_s(\theta),\delta_{X^0_s(\theta)},\theta)\big)- \mathbf{E}\big\langle\partial_2^L b_s(X^0_s,\delta_{X^0_s}(\theta),\theta),X^{(1)}_s(\theta)\big\rangle\Big]{\rm d}s \\ :=&\ I_5(t)+I_6(t)+I_7(t). \end{matrix}
(3.24)
\begin{align*} |I_5(t)|&=\Big|\int_0^t\Big(k_2(s)\frac{X_s^\epsilon(\theta)-X_s^0(\theta)}{\epsilon}-\partial_1 b_s(X^0_s(\theta),\mu_s^\epsilon,\theta)\cdot X^{(1)}_s(\theta)\Big]{\rm d}s\Big|\\ &\leq\Big|\int^t_0k_2(s)\big(\frac{X_s^\epsilon(\theta)-X_s^0(\theta)}{\epsilon}- X^{(1)}_s(\theta)\big){\rm d}s\Big|+\Big|\int^t_0\big(k_2(s)-\partial_1 b(X^0_s,\mu_s^\epsilon,\theta)\big)X^{(1)}_s(\theta){\rm d}s\Big|\\ &\leq \int^t_0|k_2(s)|\cdot\big|\frac{X^\epsilon_s(\theta)-X^0_s(\theta)}{\epsilon}-X^{(1)}_s(\theta)\big|{\rm d}s+\int_0^tK(s)\big|X_s^\epsilon(\theta)-X_s^0(\theta)\big|^{\beta_1}{\rm d}s. \end{align*}
|I_6(t)|\leq\int_0^tK(s)\mathcal{W}^{(2)}(\mu_s^\epsilon,\delta_{X^0_s(\theta)})\big|X_s^{(1)}(s)\big|{\rm d}s.
\begin{align*} |I_7(t)| &=\Big|\int_0^t\Big(\big\langle k_3(s),\frac{X_s^\epsilon(\theta)-X_s^0(\theta)}{\epsilon}\big\rangle-\mathbf{E}\big\langle\partial_2^L b_s(X^0_s,\delta_{X^0_s}(\theta),\theta),X^{(1)}_s(\theta)\big\rangle\Big){\rm d}s\Big|\\ & \leq \Big|\int^t_0\Big\langle k_3(s),\frac{X_s^\epsilon(\theta)-X_s^0(\theta)}{\epsilon}- X^{(1)}_s(\theta)\Big\rangle {\rm d}s\Big|\\ & \ +\Big|\int^t_0\Big(\langle k_3(s),X^{(1)}_s(\theta)\rangle-\mathbf{E}\big\langle\partial_2^L b_s(X^0_s,\delta_{X^0_s}(\theta),\theta),X^{(1)}_s(\theta)\big\rangle\Big){\rm d}s\Big|. \end{align*}
利用 Hölder 不等式和 B-D-G 不等式, 有
\begin{matrix} &\mathbf{E}\Big[\sup\limits_{0\leq t\leq T}\Big|\frac{X_t^\epsilon(\theta)-X_t^0(\theta)}{\epsilon}-X_t^{(1)}(\theta)\Big|^2\Big] \\ \leq\ &3\mathbf{E}\big[\big|\sup\limits_{0\leq t\leq T}I_5(t)\big|^2\big]+3\mathbf{E}\big[\big|\sup\limits_{0\leq t\leq T}I_6(t)\big|^2\big]+3\mathbf{E}\big[\big|\sup\limits_{0\leq t\leq T}I_7(t)\big|^2\big] \\ \leq\ &24\int^T_0k^2_2(s)\mathbf{E}\Big|\frac{X^\epsilon_s(\theta)-X^0_s(\theta)}{\epsilon}-X^{(1)}_s(\theta)\Big|^2{\rm d}s+24\int_0^Tk_2^2(s)\big|X_s^\epsilon(\theta)-X_s^0(\theta)\big|^{2\beta_1}{\rm d}s \\ & +12\int_0^TK^2(s)\Big(\mathcal{W}^{(2)}(\mu_s^\epsilon,\delta_{X^0_s(\theta)})\Big)^2\mathbf{E}\big|X_s^{(1)}(s)\big|^2{\rm d}s \\ & +24\int^T_0\mathbf{E}\Big|\big\langle k_3(s),\frac{X_s^\epsilon(\theta)-X_s^0(\theta)}{\epsilon}- X^{(1)}_s(\theta)\big\rangle\Big|^2{\rm d}s \\ & +24\int^T_0\mathbf{E}\Big|\big\langle k_3(s)-\partial_2^L b_s(X^0_s,\delta_{X^0_s}(\theta),\theta),X^{(1)}_s(\theta)\big\rangle\Big|^2{\rm d}s. \end{matrix}
(3.25)
\int_0^1\mathcal{W}^{(2)}(\delta_{X^0_s(\theta)+\nu(X^\epsilon_s(\theta)-X^0_s(\theta))},\delta_{X^0_s(\theta)})d\nu\leq(\mathbf{E}|X_t^\epsilon(\theta)-X_t^0(\theta)|^2)^{\frac{1}{2}}.
在假设 (A2) 下, 利用(3.21)-(3.25)式, 存在依赖于 T K(\cdot) C(T,K(\cdot))
\begin{align*} &\mathbf{E}\Big[\sup\limits_{0\leq t\leq T}\Big|\frac{X_t^\epsilon(\theta)-X_t^0(\theta)}{\epsilon}-X_t^{(1)}(\theta)\Big|^2\Big]\\ \leq\ &12\int^t_0K^2(s)\mathbf{E}\big|\frac{X_t^\epsilon(\theta)-X_t^0(\theta)}{\epsilon}-X_t^{(1)}(\theta)\big|^2{\rm d}s+\epsilon C(T,K(\cdot)). \end{align*}
\begin{equation*} \begin{split} \mathbf{E}\Big[\sup\limits_{0\leq t\leq T}\Big|\frac{X_t^\epsilon(\theta)-X_t^0(\theta)}{\epsilon}-X_t^{(1)}(\theta)\Big|^2\Big] \leq\epsilon C(T,K(\cdot)){\rm e}^{12\int_0^TK^2(s){\rm d}s}. \end{split} \end{equation*}
定理 3.1 在假设 (A1)-(A2) 下, 随着 \epsilon\rightarrow0
\begin{equation} \frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}\longrightarrow_d\xi_T. \end{equation}
(3.26)
这里随机变量 \xi_T
\begin{equation*} \begin{split} K_\epsilon(u)=&\rho\Big(\frac{X^\epsilon(\theta_0)-X^0(\theta_0)}{\epsilon},\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\Big),\\ K_0(u)=&\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)u\big).\\ \end{split} \end{equation*}
据条件 (A2), 易知 K_\epsilon(u) K_0(u) \mathbb{R}^d \epsilon>0 L_\epsilon \lim_{\epsilon\rightarrow0}L_\epsilon=0
\begin{equation*} \xi_\epsilon=\text{arginf}_{|u|<L_\epsilon} K_0(u). \end{equation*}
由 \xi_T \epsilon \xi_\epsilon=\xi_T . 另一方面, 易知
\begin{equation} \frac{\widehat{\theta}_\epsilon-\theta_0}{\epsilon}=\text{arginf}_{|u|\leq L_\epsilon}\{K_\epsilon(u)\}. \end{equation}
(3.27)
\begin{equation*} A_\epsilon:=\Big\{\omega:\big|\frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}\big|<L_\epsilon\Big\}. \end{equation*}
对任意 \alpha>0
\begin{align*} \mathbf{P}_{\theta_0}(\big|\frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}-\xi_T\big|>2\alpha) &\leq \mathbf{P}_{\theta_0}(\big|\frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}-\xi_\epsilon\big|>\alpha)+\mathbf{P}_{\theta_0}(\big|\xi_T-\xi_\epsilon\big|>\alpha)\\ &\leq \mathbf{P}_{\theta_0}(\big|\frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}-\xi_\epsilon\big|>\alpha,A_\epsilon)+\mathbf{P}_{\theta_0}(A^c_\epsilon)+\mathbf{P}_{\theta_0}(\big|\xi_T-\xi_\epsilon\big|>\alpha)\\ &=I_8+I_9+I_{10}. \end{align*}
在集合 \omega\in A_\epsilon
\begin{align*} & |K_\epsilon(u)-K_0(u)|\\ =\ & \Big|\rho\big(\frac{X^\epsilon(\theta_0)-X^0(\theta_0)}{\epsilon},\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\big)-\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)u\big)\Big|\\ \leq\ & \left|\rho\big(\frac{X^\epsilon(\theta_0)-X^0(\theta_0)}{\epsilon},\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\big)-\rho\big(X^{(1)}(\theta_0),\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\big)\right|\\ & +\left|\rho\big(X^{(1)}(\theta_0),\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\big)-\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)u\big)\right|\\ \leq\ & \rho\big(\frac{X^\epsilon(\theta_0)-X^0(\theta_0)}{\epsilon},X^{(1)}(\theta_0)\big)+\rho\big(\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon},\dot{X}^0(\theta_0)u\big)\\ \leq\ & c_\rho\sup\limits_{0\leq t\leq T}\Big|\frac{X^\epsilon_t(\theta_0)-X^0_t(\theta_0)}{\epsilon}-X^{(1)}_t(\theta_0)\Big|+c_\rho\sup\limits_{0\leq t\leq T}\Big|\frac{X^0_t(\theta_0+\epsilon u)-X^0_t(\theta_0)}{\epsilon}-\dot{X}^0_t(\theta_0)u\Big|\\ \leq\ & c_\rho\sup\limits_{0\leq t\leq T}\Big|\frac{X^\epsilon_t(\theta_0)-X^0_t(\theta_0)}{\epsilon}-X^{(1)}_t(\theta_0)\Big|+C(T,\beta_1,\beta_2,K(\cdot))\epsilon^{\min\{1,\beta_1,\beta_2\}+1}|u|^{\max\{\beta_1,\beta_2\}+1}. \end{align*}
取 L_\epsilon=(\frac{1}{\epsilon C(T,\beta_1,\beta_2,K(\cdot))})^{\frac{1}{\max\{\beta_1,\beta_2\}}+1} . 则当 |u|<L_\epsilon \epsilon \omega\in A_\epsilon
\begin{equation*} \underset{|u|<L_\epsilon}{\sup} |K_\epsilon(u)-K_0(u)| \leq c_\rho\sup\limits_{0\leq t\leq T}\Big|\frac{X^\epsilon_t(\theta_0)-X^0_t(\theta_0)}{\epsilon}-X^{(1)}_t(\theta_0)\Big|+\epsilon^{\min\{1,\beta_1,\beta_2\}}. \end{equation*}
这蕴含 \sup\limits_{u\in L_\epsilon}|K_\epsilon(u)-K_0(u)| [-L_\epsilon,L_\epsilon] K_\epsilon(u) K_0(u) \epsilon\rightarrow0
\begin{equation*} I_8+I_{10}\rightarrow0\,\,\, \end{equation*}
且由引理 3.1 知 I_9\rightarrow0 . 利用凸最小值收敛原理,(3.26)式获得. 证毕.
4 带惩罚函数项的最小距离估计法
受文献[8 ]的思想启发, 考虑惩罚函数项的距离估计, 依然假定有分布依赖方程 (1.1)和对应的常微分方程(1.4), 定义未知参数 \theta=(\theta_1,\cdots,\theta_p)^{\prime}\in\Theta
\begin{equation} \widehat{\theta}_{\epsilon,\gamma}:=\text{arginf}_{\theta\in\Theta}\Big\{\rho(X^\epsilon,X^0)+\lambda_\epsilon\sum\limits_{j=1}^p|\theta_j|^\gamma\Big\}. \end{equation}
(4.1)
这里 \gamma>0 \lambda_\epsilon \epsilon\rightarrow0 \widehat{\theta}_{\epsilon,\gamma}
定理 4.1 设 \gamma\geq1 \frac{\lambda_\epsilon}{\epsilon}\rightarrow\lambda_0\geq0 . 在假设 (A1)-(A2) 下, 随着 \epsilon\rightarrow0
\begin{equation} \frac{\hat{\theta}_{\epsilon,\gamma}-\theta_0}{\epsilon}\longrightarrow_d\xi_{T,\gamma}. \end{equation}
(4.2)
这里 \xi_{T,\gamma} p>1
\begin{equation*} \xi_{T,\gamma}=\underset{u\in \mathbb{R}^d}{\text{arginf}} \, \Big\{\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)\cdot u\big)\Big\}+\lambda_0\sum\limits_{j=1}^pu_j \text{sgn}(\theta_{0j})|\theta_{0j}|^{\gamma-1}. \end{equation*}
\begin{equation*} \xi_{T,\gamma}=\underset{u\in \mathbb{R}^d}{\text{arginf}} \, \Big\{\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)\cdot u\big)\Big\}+\lambda_0\sum\limits_{j=1}^p\Big(|u_j|\mathcal{I}_{\{\theta_{0j}=0\}}+u_j \text{sgn}(\theta_{0j})|\theta_{0j}|\mathcal{I}_{\{\theta_{0j}\neq0\}}\Big). \end{equation*}
其中 \text{sgn}(\cdot)
证 对任意 u=(u_1,\cdots,u_p)
\begin{matrix} K_\epsilon(u)=\ &\rho\Big(\frac{X^\epsilon(\theta_0)-X^0(\theta_0)}{\epsilon},\frac{X^0(\theta_0+\epsilon u)-X^0(\theta_0)}{\epsilon}\Big)+\frac{\lambda_\epsilon}{\epsilon^2}\sum\limits_{j=1}^p\big(|\theta_{0j}+\epsilon u_j|^\gamma-|\theta_{0j}|^\gamma\big) \\ :=\ & K_\epsilon^{(1)}(u)+K_\epsilon^{(2)}(u) \end{matrix}
(4.3)
\begin{equation*} \begin{aligned} K_0(u)=\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)u\big). \end{aligned} \end{equation*}
与定理 3.1 相同定义法, 定义 L_\epsilon \sup\limits_{u\in L_\epsilon}|K_\epsilon^{(1)}(u)-K_0(u)|
\begin{equation} \begin{aligned} \frac{\lambda_\epsilon}{\epsilon^2}\sum\limits_{j=1}^p\big(|\theta_{0j}+\epsilon u_j|^\gamma-|\theta_{0j}|^\gamma\big) =\frac{\lambda_\epsilon}{\epsilon^2}\sum\limits_{j=1}^pu_j^\prime \frac{|\theta_{0j}+\epsilon u_j|^\gamma-|\theta_{0j}|^\gamma}{\epsilon u_j} \longrightarrow\lambda_0\sum\limits_{j=1}^pu_j \text{sgn}(\theta_{0j})|\theta_{0j}|^{\gamma-1}. \end{aligned} \end{equation}
(4.4)
这里" \longrightarrow \epsilon \gamma=1
\begin{equation} \begin{aligned} \frac{\lambda_\epsilon}{\epsilon^2}\sum\limits_{j=1}^p\big(|\theta_{0j}+\epsilon u_j|-|\theta_{0j}|\big) \longrightarrow\lambda_0\sum\limits_{j=1}^p\Big(|u_j|\mathcal{I}_{\{\theta_{0j}=0\}}+u_j \text{sgn}(\theta_{0j})|\theta_{0j}|\mathcal{I}_{\{\theta_{0j}\neq0\}}\Big). \end{aligned} \end{equation}
(4.5)
在一致收敛拓扑下, 需要证明 K_\epsilon(u)\longrightarrow_dK_0(u) . 注意到随机函数 K_\epsilon(u)
(i) K_\epsilon(u) K_0(u)
(ii) 胎紧性: \lim_{h\rightarrow0}\limsup_{\epsilon\rightarrow0}\mathbf{E}[\varrho(K_\epsilon(u),h)\wedge1]=0
这里 \varrho(y,h)=\sup\{\rho(y(u),y(\nu)):\ |u-v|\leq h\} y(u) \mathbb{R}^d
对于 K_\epsilon^{(1)}(u) K_\epsilon^{(2)}(u) K_\epsilon(u) K_\epsilon(u) u
\mathbf{E}[\varrho(K_\epsilon(u),h)\wedge1]\leq hc_\rho\sup\limits_{0<\epsilon\leq1}\mathbf{E}\sup\limits_{u\in K}\Big|\frac{\rm d}{{\rm d}u}K_\epsilon(u)\Big|,
同时让 \epsilon\rightarrow0 h\rightarrow0
对于 0<\gamma<1 \lambda_\epsilon
定理 4 .2设 0<\gamma<1 \frac{\lambda_\epsilon}{\epsilon^{2-\eta}}\rightarrow\lambda_0\geq0 . 这里 0<\eta<1 \epsilon\rightarrow0
\begin{equation} \frac{\hat{\theta}_{\epsilon,\gamma,\eta}-\theta_0}{\epsilon}\longrightarrow_d\xi_{T,\gamma,\eta}. \end{equation}
(4.6)
这里 \xi_{T,\gamma,\eta} p>1
\begin{equation*} \xi_{T,\gamma,\eta}=\underset{u\in \mathbb{R}^d}{\text{arginf}} \, \Big\{\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)\cdot u\big)\Big\}+\lambda_0\sum\limits_{j=1}^p|u_j|^\eta\mathcal{I}_{\{\theta_{0j}=0\}}. \end{equation*}
证 根据定理 3.1 的证明, 仅仅证明惩罚函数项收敛性问题, 对任意 u=(u_1,\cdots,u_p) \theta_{0k}\neq0
\frac{\lambda_\epsilon}{\epsilon}u_k\Big(\frac{|\theta_{0k}+\epsilon u_k|^p-|\theta_{0k}|^p}{\epsilon u_k}\Big)\rightarrow0.
frac{\lambda_\epsilon}{\epsilon^2}\sum\limits_{j=1}^p\big(|\theta_{0k}+\epsilon u_k|^p-|\theta_{0k}|^p\big)\rightarrow\lambda_0\sum\limits_{j=1}^p|u_j|^\lambda\mathcal{I}_{\{\theta_{0j}=0\}}.
结合有限维分布的收敛和胎紧性, 便完成这个结论证明.
在结束本节前, 我们给出一个例子去验证定理 4.1.
例子 设 \theta=(\theta^{(1)},\theta^{(2)})^\prime\in\Theta_0:=(c_1,c_2)\times(c_3,c_4)\subset\mathbb{R}^2 c_1<c_2 c_3<c_4 . 对任意 \epsilon\in(0,1)
\begin{equation} \begin{aligned} {\rm d}X^\epsilon_t=\Big(\theta^{(1)}+\theta^{(2)}\int_{\mathbb{R}}b_t(X_t^\epsilon,\xi)\mu_t(d\xi)\Big){\rm d}t+\epsilon dB_t,\ \ \ \ \ \ \ \ t\in[T]. \end{aligned} \end{equation}
(4.7)
其初始条件为 X^\epsilon_0=x_0 . 这里 \theta \theta_0=(\theta^{(1)}_0,\theta^{(2)}_0)\in\Theta_0 \mu_t(\cdot)=\mathbf{P}(X_t^\epsilon\in\cdot) X_t^\epsilon b_t:\mathbb{R}\times\mathbb{R}\mapsto\mathbb{R} L(t)>0 \forall x,y\in \mathbb{R} z_1,z_2 \in \mathbb{R}
\begin{equation*} \begin{aligned} \big|b_t(x,z_1)-b_t(y,z_2)\big|\leq K(t)\big( |x-y|+|z_1-z_2| \big). \end{aligned} \end{equation*}
\begin{equation*} \underset{x\in \mathbb{R},z\in\mathbb{R}}{\sup}\Big\{\big|\partial_1 b_t(x,z)\big|,\ \big|\partial_2 b_t(x,z))\big|\Big\}<\infty. \end{equation*}
\begin{equation*} \widehat{\theta}_{\epsilon}:=\text{arginf}_{\theta\in\Theta}\Big\{\rho(X^\epsilon,X^0)+\epsilon|\theta_1|+\epsilon|\theta_2|\Big\}. \end{equation*}
\begin{equation*} \frac{\hat{\theta}_\epsilon-\theta_0}{\epsilon}\longrightarrow_d\xi_{\Theta_0}. \end{equation*}
\begin{align*} \xi_{\Theta_0}&=\underset{u\in \mathbb{R}^d}{\text{arginf}} \, \Big\{\rho\big(X^{(1)}(\theta_0),\dot{X}^0(\theta_0)\cdot u\big)\Big\}+|u_1|\mathcal{I}_{\{\theta_{1}=0\}}+|u_2|\mathcal{I}_{\{\theta_{2}=0\}}\\ & \ +u_1 \text{sgn}(\theta_{1})|\theta_{1}|\mathcal{I}_{\{\theta_{1}\neq0\}}+u_2 \text{sgn}(\theta_{2})|\theta_{2}|\mathcal{I}_{\{\theta_{2}\neq0\}}. \end{align*}
X^{(1)}(\theta_0) \dot{X}^0(\theta_0)
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Parameter estimation of discretely observed interacting particle systems
1
2023
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
1
2014
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
2
2018
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
... 这里 \mathcal{P}(\mathscr{C}\times \mathscr{C}) \mathcal{P}(\mathscr{C}) \mathcal{P}(\mathscr{C}) \pi(\cdot,\mathscr{C})=\mu(\cdot) \pi(\mathscr{C},\cdot)=\nu(\cdot) \mathcal{P}_2(\mathscr{C}) \mathcal{P}_r(\mathscr{C}) E r 3 ]知, 对任意测度 \delta_{x_0} \mu\in \mathscr{C}
Freidlin Wentzell LDP in path space for Mckean Vlasov equations and the functional iterated logarithm law
2
2019
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
... 且其它位置也是同样结论, 易知函数项 b_t [4 ] . ...
Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models
1
2021
... 当前, 随机泛函微分方程研究较热, 许多深刻且有用结论被得到, 然而, 方程里参数估计工作不多, 特别是分布依赖类 McKean-Vlasov 方程, 认知甚少, 统计问题主要集中在似然比方法估计未知参数以及讨论它的渐近行为[5 ,9 ,13 ,16 ] . 最近, Zhao 和 Zhang[16 ] 介绍了一类 \alpha - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果. ...
On penalized estimation for dynamical systems with small noise
1
2018
... 许多统计问题可以归结为模型 (1.5), 若假定 \theta_0\in\Theta \theta \widehat{\theta}_\epsilon [6 ,16 ] . 一旦漂移项与解分布有依赖关联性时, 渐近分布不易找到, X_t \widehat{\theta}_\epsilon-\theta \epsilon^{-1}(\widehat{\theta}_\epsilon-\theta_0) \epsilon\rightarrow0 8 ,11 ]启发, 假设 \theta L^\gamma \theta
Large deviations and a Kramers'type law for self-stabilizing diffusions
1
2008
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
Asymptotics for lasso-type estimators
2
2000
... 许多统计问题可以归结为模型 (1.5), 若假定 \theta_0\in\Theta \theta \widehat{\theta}_\epsilon [6 ,16 ] . 一旦漂移项与解分布有依赖关联性时, 渐近分布不易找到, X_t \widehat{\theta}_\epsilon-\theta \epsilon^{-1}(\widehat{\theta}_\epsilon-\theta_0) \epsilon\rightarrow0 8 ,11 ]启发, 假设 \theta L^\gamma \theta
... 受文献[8 ]的思想启发, 考虑惩罚函数项的距离估计, 依然假定有分布依赖方程 (1.1)和对应的常微分方程(1.4), 定义未知参数 \theta=(\theta_1,\cdots,\theta_p)^{\prime}\in\Theta
On minimum uniform metric estimate of parameters of diffusion-type processes
1
1994
... 当前, 随机泛函微分方程研究较热, 许多深刻且有用结论被得到, 然而, 方程里参数估计工作不多, 特别是分布依赖类 McKean-Vlasov 方程, 认知甚少, 统计问题主要集中在似然比方法估计未知参数以及讨论它的渐近行为[5 ,9 ,13 ,16 ] . 最近, Zhao 和 Zhang[16 ] 介绍了一类 \alpha - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果. ...
The LAN property for Mckean-Vlasov models in a mean-field regime
1
2023
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
Shrinkage drift parameter estimation for multi-factor Ornstein-Uhlenbeck processes
1
2010
... 许多统计问题可以归结为模型 (1.5), 若假定 \theta_0\in\Theta \theta \widehat{\theta}_\epsilon [6 ,16 ] . 一旦漂移项与解分布有依赖关联性时, 渐近分布不易找到, X_t \widehat{\theta}_\epsilon-\theta \epsilon^{-1}(\widehat{\theta}_\epsilon-\theta_0) \epsilon\rightarrow0 8 ,11 ]启发, 假设 \theta L^\gamma \theta
Bismut formula for lions derivative of distribution dependent SDEs and applications
1
2019
... 定义 2.1, 2.2 和引理 2.1 来自文献[12 ], 称为 L - 导数公式, 类似微积分公式, 泛函项 b_t(x,\mu_t,\theta) i \partial_ib_t(x,\mu_t,\theta) \theta \partial_3b_t(x,\mu_t,\theta) . 为了给出结论, 我们需要如下条件. ...
Least squeres estimation for path-distribution McKean-Vlasov SDEs via discrete-time observations
1
2019
... 当前, 随机泛函微分方程研究较热, 许多深刻且有用结论被得到, 然而, 方程里参数估计工作不多, 特别是分布依赖类 McKean-Vlasov 方程, 认知甚少, 统计问题主要集中在似然比方法估计未知参数以及讨论它的渐近行为[5 ,9 ,13 ,16 ] . 最近, Zhao 和 Zhang[16 ] 介绍了一类 \alpha - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果. ...
Online parameter estimation for the McKean-Vlasov stochastic differential equation
1
2023
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
An asymptotic expansion approach to pricing financial contingent claims
1
1999
... 起源于随机场博弈论和交互粒子系统研究, 分布依赖的随机泛函微分方程 (俗称 McKean-Vlasov SDE)[1 ,4 ,7 ,14 ] 最近吸引了许多学者关注, 该方程在金融经济、生物学、统计物理等领域[2 ,3 ,10 ,15 ] 有许多应用, 由于解缺乏马氏性, 因而这类方程比传统随机微分方程结构复杂, 如何解决方程解的分布与扩散项和漂移项共存关系是这类问题关键. 作为统计应用, 传统随机微分方程 (带跳和不带跳) 的统计推断, 如系数估计, 一直是学术界感兴趣问题, 极大似然估计和贝叶斯估计占绝大多数, 与之相比, 最小二乘和 L1
Minimum distance parameter estimation for SDEs with small \alpha
4
2019
... 当前, 随机泛函微分方程研究较热, 许多深刻且有用结论被得到, 然而, 方程里参数估计工作不多, 特别是分布依赖类 McKean-Vlasov 方程, 认知甚少, 统计问题主要集中在似然比方法估计未知参数以及讨论它的渐近行为[5 ,9 ,13 ,16 ] . 最近, Zhao 和 Zhang[16 ] 介绍了一类 \alpha - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果. ...
... [16 ]介绍了一类 \alpha - stable 过程驱动的随机微分方程的最小距离估计法, 紧跟也讨论了估计量的极限行为. 然而, 漂移项依赖于解分布这类情形, 即便是此时扩散项为常数, 或者为相应的随机泛函微分方程, 相合性证明和极限分布等统计问题都没有满意结果. ...
... 这里 x_0 \theta=(\theta_1,\cdots,\theta_p)^{\prime}\in\Theta\subset\mathbb{R}^p \mu^\epsilon_t(A)=\mathbf{P}(X^\epsilon_t\in A) \{X_t\} t W_t \sigma (\Omega,\mathscr{F},(\mathscr{F}_t)_{t\geq0},\mathbf{P}) d - 维布朗运动, 通常称为随机噪声, 漂移项 b_t:\mathbb{R}^d\times\mathcal{P}_2(\mathbb{R}^d)\times\Theta\mapsto\mathbb{R}^d t \epsilon\in(0,1] 16 ]启发, 本文采用最小距离法来估计参数 \theta . ...
... 许多统计问题可以归结为模型 (1.5), 若假定 \theta_0\in\Theta \theta \widehat{\theta}_\epsilon [6 ,16 ] . 一旦漂移项与解分布有依赖关联性时, 渐近分布不易找到, X_t \widehat{\theta}_\epsilon-\theta \epsilon^{-1}(\widehat{\theta}_\epsilon-\theta_0) \epsilon\rightarrow0 8 ,11 ]启发, 假设 \theta L^\gamma \theta
