Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation

arXiv:2610.00793v1 Announce Type: cross Abstract: 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+\sigma(X_t)\,dZ_t^{\beta,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, $\sigma:\mathbb R\to(0,\infty)$ is the diffusion coefficient, and $Z^{\beta,f}$ is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance \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*} \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 $\beta\in(0,2)$ is the covariance exponent. For $u,v\geq0$, the kernel is $q_\beta(u,v)=[u^\beta+v^\beta-(u+v)^\beta]/(1-\beta)$ when $\beta\ne1$. Its continuous extension at $\beta=1$ 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.