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→Koopman算子定义
pll：“伊藤公式是随机微积分里的 “链式法则”—— 专门用来算 “随布朗运动（一种随机过程）变化的函数” 的微分。”
pll：“伊藤公式是随机微积分里的 “链式法则”—— 专门用来算 “随布朗运动（一种随机过程）变化的函数” 的微分。”
<math> \begin{align*}\mathcal{L}_{\text{stoch}} g(x) &= \lim_{t \to 0} \frac{\mathbb{E}[g(x(t)) \mid x(0)=x] - g(x)}{t} \quad (\text{生成元定义}) \\
&= \lim_{t \to 0} \frac{\mathbb{E}\left[ g(x) + \int_0^t dg(x(s)) \mid x(0)=x \right] - g(x)}{t} \quad (\text{积分基本定理：}g(x(t)) - g(x) = \int_0^t dg(x(s))) \\
&= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \left( \sum_{i=1}^n \frac{\partial g}{\partial x_i}(x(s))dx_i(s) + \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2 g}{\partial x_i \partial x_j}(x(s))dx_i(s)dx_j(s) \right) \mid x(0)=x \right]}{t} \quad (\text{多元伊藤公式}) \\
&= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \sum_{i=1}^n \frac{\partial g}{\partial x_i}(x(s))\left( f_i(x(s))dt + \sum_{k=1}^m \sigma_{ik}(x(s))dW_k(s) \right) + \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2 g}{\partial x_i \partial x_j}(x(s))\left( \sum_{k=1}^m \sigma_{ik}(x(s))\sigma_{jk}(x(s))dt \right) ds \mid x(0)=x \right]}{t} \quad (\text{SDE分量代入+Wiener性质：}dW_k dW_l=\delta_{kl}dt, dt\cdot dt=dt\cdot dW_k=0) \\
&= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \sum_{i=1}^n \frac{\partial g}{\partial x_i}f_i(x(s))dt + \int_0^t \sum_{i=1}^n \sum_{k=1}^m \frac{\partial g}{\partial x_i}\sigma_{ik}(x(s))dW_k(s) + \int_0^t \frac{1}{2}\sum_{i,j=1}^n \sum_{k=1}^m \frac{\partial^2 g}{\partial x_i \partial x_j}\sigma_{ik}\sigma_{jk}(x(s))dt \mid x(0)=x \right]}{t} \quad (\text{积分项拆分+求和与积分交换}) \\
&= \lim_{t \to 0} \frac{\int_0^t \mathbb{E}\left[ \sum_{i=1}^n \frac{\partial g}{\partial x_i}f_i(x(s)) + \frac{1}{2}\sum_{i,j=1}^n \sum_{k=1}^m \frac{\partial^2 g}{\partial x_i \partial x_j}\sigma_{ik}\sigma_{jk}(x(s)) \mid x(0)=x \right] dt + \mathbb{E}\left[ \int_0^t \sum_{i,k} \frac{\partial g}{\partial x_i}\sigma_{ik}(x(s))dW_k(s) \mid x(0)=x \right]}{t} \quad (\text{Fubini定理：期望与积分交换}) \\
&= \lim_{t \to 0} \frac{\int_0^t \mathbb{E}\left[ \nabla g(x(s)) \cdot f(x(s)) + \frac{1}{2}\text{Tr}\left( \sigma(x(s))\sigma(x(s))^T \nabla^2 g(x(s)) \right) \mid x(0)=x \right] dt}{t} \quad (\text{向量点积定义+迹的定义；伊藤积分期望为0}) \\
&= \lim_{t \to 0} \frac{\int_0^t \left( \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \right) dt + o(t)}{t} \quad (\text{$t\to0$时$x(s)\to x$，被积函数连续+积分中值定理}) \\
&= \lim_{t \to 0} \frac{\left( \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \right) t + o(t)}{t} \quad (\text{积分结果展开}) \\
&= \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \quad (\text{约分$t$，$\lim_{t\to0}\frac{o(t)}{t}=0$})
\end{align*}
</math>
=残差DMD=
=残差DMD=