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_0inmathbb R, we study egin{equation*} dX_t=a(X_t),dt+sigma(X_t),dZ_t^{eta,f}, qquad X_0=x_0,quad 0leq tleq T. end{equation*} smallskipnoindent Here a:mathbb Romathbb R is the drift coefficient, sigma:mathbb Ro(0,infty) is the diffusion coefficient, and Z^{eta,f} is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance egin{equation*} operatorname{Cov}(Z_s^{eta,f},Z_t^{eta,f}) =int_0^{swedge t} f(r)q_eta(s-r,t-r),dr, qquad 0leq s,tleq T. end{equation*} smallskipnoindent Here swedge t=min{s,t}. The temporal weight f:[0,T]o[0,infty) is measurable, bounded, and positive almost everywhere, and etain(0,2) is the covariance exponent. For u,vgeq0, the kernel is q_eta(u,v)=[u^eta+v^eta-(u+v)^eta]/(1-eta) when etane1. Its continuous extension at eta=1 is q_1(u,v)=(u+v)log(u+v)-ulog u-vlog v, with 0log0=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.

Source: arXiv cs.LG | 2026-10-02