[
    {
        "id": "osp-21333",
        "type": "article-journal",
        "title": "Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation",
        "author": [
            {
                "family": "Ramirez-Gonzalez",
                "given": "J. H."
            }
        ],
        "URL": "https://omanscience.com/en/articles/inference-for-stochastic-differential-equations-driven-by-weighted-sub-fractional-brownian-motion-using-neural-networks-and-the-euler-approximation",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "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+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."
    }
]