Summary
This paper looks at how to estimate certain hidden parts of random processes that evolve over time and are influenced by memory and varying randomness. The authors focus on stochastic equations driven by a specialized type of noise called weighted sub-fractional Brownian motion. They use a numerical approach called the Euler approximation and neural networks to learn the underlying rules from data observed at discrete times. Their method estimates the drift, diffusion, noise variation, and key parameters of this complex noise, and they compare their approach to other neural methods on simulated data.
What this means in practice
- •For financial modelers: Model and estimate complex market dynamics influenced by memory and weighted noise in asset prices using neural-based inference methods.
- •For signal processing engineers: Reconstruct and infer parameters of signals modeled by stochastic differential equations with complex Gaussian noise for better noise characterization.
Tested on simulated data.
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.