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

J. H. Ramirez-Gonzalez

stat.MLSep 30, 2026

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

Authors: J. H. Ramirez-Gonzalez

Organizations: Institute of Mathematics and Statistics, University of São Paulo (IME–USP), São Paulo, Brazil

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.

Figures & tables

Figure 2: Observed trajectories for the coefficients in Eq. ( 31 ) (case 1). Colors distinguish the five initial states; line styles distinguish the five independent paths from each initial state. Every path contains 1001 observations.

Figure 3: Method-1 training in case 1: (a) Gaussian negative log likelihood per transition, excluding

\log(2\pi)/2

; (b) profiled exponent, with the generating value shown for reference; and (c) squared curvature of the dimensionless state curves defined in Eq. ( 28 ). Each entry is recorded at the end of a complete cycle, after the additional diffusion updates and any scheduled exponent–scale profile.

Figure 4: Drift, diffusion coefficient, and temporal weight recovered in case 1. Each panel shows the generating function and the three fitted curves, together with their MSEs. The footer gives the exponent estimates and squared errors. State-function MSEs use all recorded states, and the temporal MSE is the time average of the squared discrepancy between the continuous curves.

Index $j$	Drift $a_{j}(x)$	Diffusion $\sigma_{j}(x)$	Temporal weight $f_{j}(t)$
1	$-\dfrac{x}{1+x^{2}}$	$0.5+0.2\tanh^{2}x$	$1$
2	$-\tanh x$	$0.4$	$\dfrac{2}{3}(1+t/5)$
3	$0.5\sin x-0.25\tanh x$	$0.35+\dfrac{0.15}{1+x^{2}}$	$1+0.5\sin(2\pi t/5)$
4	$\dfrac{1-x}{1+(x-1)^{2}}$	$0.5+0.2\sin x$	$\dfrac{e^{-t/5}}{1-e^{-1}}$
5	$-0.8\tanh(x-1.5)+0.2\sin x$	$0.3+0.4e^{-(x-1.5)^{2}}$	$\dfrac{1+2e^{-((t-2.5)/0.7)^{2}}}{c_{5}}$

Table 1: Five test functions for each component of the generating model. The drift and diffusion are defined on

\mathbb{R}

and the temporal weights on

[0,5]

. The constant

c_{5}

is defined in Eq. ( 35 ). Each component is selected independently by its own index.

Method	$a$	$\sigma$	$f$	$\beta$
M1	50% 10/20	80% 16/20	70% 14/20	95% 19/20
M2	25% 5/20	15% 3/20	20% 4/20	5% 1/20
M3	25% 5/20	5% 1/20	10% 2/20	0% 0/20

Table 2: Frequency with which each method has the smallest componentwise error across the twenty cases. Each entry shows the percentage above the count. Function columns use MSE and the exponent column uses squared error. Temporal ISE is omitted because it is proportional to temporal MSE; there are no ties.

Regime	$a$	$\sigma$	$f$	$\beta$	Total
$\beta_{*}>1$	2/8	8/8	8/8	8/8	26/32
$\beta_{*}=1$	2/4	0/4	1/4	3/4	6/16
$\beta_{*}<1$	6/8	8/8	5/8	8/8	27/32

Table 3: Method-1 component wins by covariance regime. Function columns use MSE and the exponent column uses squared error; the total counts all four components in each case.

Component	Cases	Minimum gap	Maximum gap
$a$	10	$1.891\times 10^{-4}$	$1.352\times 10^{-2}$
$\sigma$	4	$3.132\times 10^{-4}$	$9.996\times 10^{-4}$
$f$	6	$5.499\times 10^{-5}$	$1.205\times 10^{-3}$
$\beta$	1	$1.840\times 10^{-5}$	$1.840\times 10^{-5}$

Table 4: Excess error of method 1 in cases where a comparator has smaller error, measured by Eq. ( 36 ). Only positive gaps are included.

Figure 5: Case 9,

\beta_{*}=1

. Each panel compares the generating function with the three estimates and reports their MSEs; the footer gives the exponent estimates and squared errors. The comparators estimate all three functions more accurately, while method 1 has the smallest exponent error.

Figure 6: Case 17,

\beta_{*}=0.2

. Panels show the generating and fitted functions with their MSEs; the footer gives the exponent estimates and squared errors. Method 1 has the smallest error for all four components.

(a) Generating functions; $c_{5}$ is defined in Eq. ( 35 )
Case (indices)	$a_{*}(x)$	$\sigma_{*}(x)$	$f_{*}(t)$
1 (2,4,2)	$-\tanh x$	$0.5+0.2\sin x$	$\frac{2}{3}(1+t/5)$
2 (3,3,3)	$0.5\sin x-0.25\tanh x$	$0.35+\frac{0.15}{1+x^{2}}$	$1+0.5\sin(2\pi t/5)$
3 (4,2,4)	$\frac{1-x}{1+(x-1)^{2}}$	$0.4$	$\frac{e^{-t/5}}{1-e^{-1}}$
4 (1,5,3)	$-\frac{x}{1+x^{2}}$	$0.3+0.4e^{-(x-1.5)^{2}}$	$1+0.5\sin(2\pi t/5)$

Table 5: Generating functions and estimation errors at

\beta_{*}=1.5