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

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, $\sigma :\mathbb{R}\to (0,\mathrm{\infty })$ 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^{β,f},Z_t^{β,f}) =\int_0^{s\wedge t} f®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 $f:[0,T]\to [0,\mathrm{\infty })$ is measurable, bounded, and positive almost everywhere, and $\beta \in (0,2)$ is the covariance exponent. For $u,v\ge 0$, the kernel is $q_{\beta }(u,v)=[u^{\beta }+v^{\beta }-(u+v{)}^{\beta }]\mathrm{/}(1-\beta )$ when $\beta \mathrm{\ne }1$. Its continuous extension at $\beta =1$ is $q_1(u,v)=(u+v)\log (u+v)-u\log u-v\log v$, with $0\log 0=0$. 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.