由加权次分数布朗运动驱动的随机微分方程的推断：基于神经网络与欧拉近似
Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation
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中文总结 AI 辅助
本文针对加权次分数布朗运动驱动的随机微分方程，提出基于欧拉近似和神经网络/径向基的漂移、扩散及噪声协方差估计方法，并通过二十种系数设置下的模拟比较验证其有效性。
中文摘要 AI 辅助
我们考虑从高斯过程驱动的随机微分方程的离散观测中估计漂移、扩散和噪声协方差的问题。对于固定的观测时域 $T>0$ 和已知的初始状态 $x_0\in\mathbb R$，我们研究 \begin{equation*} dX_t=a(X_t)\\,dt+\sigma(X_t)\\,dZ_t^{\beta,f}, \qquad X_0=x_0,\quad 0\leq t\leq T. \end{equation*} 其中 $a:\mathbb R\to\mathbb R$ 是漂移系数，$\sigma:\mathbb R\to(0,\infty)$ 是扩散系数，$Z^{\beta,f}$ 是来自加权次分数布朗族的一个中心高斯过程，其协方差为 \begin{equation*} \operatorname{Cov}(Z_s^{\beta,f},Z_t^{\beta,f}) =\int_0^{s\wedge t} f(r)q_\beta(s-r,t-r)\\,dr, \qquad 0\leq s,t\leq T. \end{equation*} 这里 $s\wedge t=\min\{s,t\}$。时间权重 $f:[0,T]\to[0,\infty)$ 是可测、有界且几乎处处为正的，$\beta\in(0,2)$ 是协方差指数。对于 $u,v\geq0$，当 $\beta\ne1$ 时，核函数为 $q_\beta(u,v)=[u^\beta+v^\beta-(u+v)^\beta]/(1-\beta)$。其在 $\beta=1$ 处的连续延拓为 $q_1(u,v)=(u+v)\log(u+v)-u\log u-v\log v$，其中 $0\log0=0$。利用欧拉近似，我们从观测到的转移中重构高斯驱动的增量，并利用它们的联合密度获得轨迹似然。神经网络和径向基表示对漂移、扩散和归一化时间权重进行建模，而似然剖面估计协方差指数和扩散尺度。我们将该方法与两种神经替代方法在二十种系数设置下的相同模拟轨迹上进行比较。
英文摘要
We consider the estimation of drift, diffusion, and noise covariance from discrete observations of stochastic differential equations driven by Gaussian processes. For a fixed observation horizon $T>0$ and a known initial state $x_0\in\mathbb R$, we study \begin{equation*} dX_t=a(X_t)\,dt+σ(X_t)\,dZ_t^{β,f}, \qquad X_0=x_0,\quad 0\leq t\leq T. \end{equation*} \smallskip\noindent Here $a:\mathbb R\to\mathbb R$ is the drift coefficient, $σ:\mathbb R\to(0,\infty)$ is the diffusion coefficient, and $Z^{β,f}$ is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance \begin{equation*} \operatorname{Cov}(Z_s^{β,f},Z_t^{β,f}) =\int_0^{s\wedge t} f(r)q_β(s-r,t-r)\,dr, \qquad 0\leq s,t\leq T. \end{equation*} \smallskip\noindent Here $s\wedge t=\min\{s,t\}$. The temporal weight $f:[0,T]\to[0,\infty)$ is measurable, bounded, and positive almost everywhere, and $β\in(0,2)$ is the covariance exponent. For $u,v\geq0$, the kernel is $q_β(u,v)=[u^β+v^β-(u+v)^β]/(1-β)$ when $β\ne1$. Its continuous extension at $β=1$ is $q_1(u,v)=(u+v)\log(u+v)-u\log u-v\log v$, with $0\log0=0$. Using the Euler approximation, we reconstruct the Gaussian driving increments from observed transitions and use their joint density to obtain a trajectory likelihood. Neural and radial-basis representations model the drift, diffusion, and normalized temporal weight, while a likelihood profile estimates the covariance exponent and diffusion scale. We compare the method with two neural alternatives on the same simulated trajectories in twenty coefficient settings.
发表机构
- Institute of Mathematics and Statistics, University of São Paulo（圣保罗大学数学与统计研究所）
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