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

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 $T&gt;0$ 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 $Z<sup>{β,f}$ 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 $q_β(u,v)=[u<sup>β+v<sup>β-(u+v)<sup>β]/(1-β)$ 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.