Noise-Induced Uncertainty Quantification

• Noise-induced uncertainty quantification is a rigorous framework that models unpredictability in complex systems using stochastic differential equations and various noise processes.
• It details estimation techniques such as MLE, moment methods, and kernel-based transforms to reliably infer parameters from Brownian, fBM, and Lévy noise models.
• The approach is applicable in fields like finance and climate science, where it aids in simulating phenomena like asset price jumps, turbulence, and persistent fluctuations.

Noise-induced uncertainty quantification refers to the systematic mathematical characterization and estimation of the unpredictability in system states, observables, or inference tasks that is directly attributable to stochastic noise. In complex systems, noise is an intrinsic or extrinsic feature, often modeled via driving random processes (e.g., Brownian motion, fractional Brownian motion, Lévy processes). The rigorous modeling, estimation, and numerical treatment of such noise sources are essential for both quantifying uncertainty and for reliable model calibration to empirical data.

1. Stochastic Modeling of Noise in Complex Systems

The fundamental framework employs stochastic differential equations (SDEs) to model systems with noise-induced uncertainties. The canonical form used is:

$$dX_t = \mu(\theta, t, X_t)\,dt + \sigma(\vartheta, t, X_t)\,dW_t,\quad X_0 = \zeta$$

where $W_t$ can instantiate a variety of noise processes:

• Standard Brownian Motion: $W_t$ is a Wiener process; uncertainties result from temporal white noise, yielding the classic theory of SDEs.
• Fractional Brownian Motion (fBM): $B_t^H$ has covariance $$E[B_t^H B_s^H] = \frac{1}{2}(t^{2H} + s^{2H} - |t-s|^{2H})$$, where the Hurst parameter $H$ encodes memory effects (long-range dependence for $H>1/2$, negative correlation for $H<1/2$).
• α-stable Lévy Motion: $L_t^\alpha$ is characterized by heavy-tailed increments and, for $\alpha<2$, infinite variance, modeling impulsive or jump noise. The characteristic function of an $\alpha$-stable random variable is

$$\varphi(u) = \exp\Big\{-\sigma^\alpha|u|^\alpha[1 - i\,\mathrm{sgn}(u)\tan(\frac{\pi\alpha}{2})] + i\mu u\Big\},\quad \alpha\neq 1$$

The selection of a noise process is dictated by empirical knowledge of system dynamics, target observables’ statistical properties, or inferred via model comparison.

2. Parameter Estimation in SDEs with Noise

Parameter estimation for noise-driven SDEs is context-dependent:

• Brownian SDEs: Maximum likelihood estimation (MLE) or contrast functions are employed, with the Radon-Nikodym derivative expressing the data likelihood under the drift and diffusion parameters. For discretely observed processes, approximate likelihoods and generalized method of moments (GMM) are applicable.
• fBM-driven SDEs: Estimating the Hurst parameter $H$ is nontrivial due to long-range dependence.
  • R/S (Rescaled Range) Method: The expected rescaled range scales as $E[R/S(n)] \sim C_H n^H$, estimating $H$ via linear regression on log–log plots.
  • Aggregated variance, variance of residuals, periodogram: Each utilizes the scaling of variance or spectral density with block size or frequency to infer $H$.
  • Kernel-based Integral Transforms: For SDEs such as $X_t = \theta \int_0^t X_s ds + B_t^H$, one utilizes a kernel $k_H(t,s)$ and “variance function” $w_t^H$ to transform the process into a martingale, allowing the construction of an explicit MLE for drift parameters:

  $$\theta_T = \left\{\int_0^T Q^2(s)\,dw_s^H\right\}^{-1} \int_0^T Q(s)\,dZ_s$$

  where $$Q(t) = \frac{d}{dw_t^H} \Big[ \int_0^t k_H(t,s) X(s) ds \Big]$$ and $Z_t = \int_0^t k_H(t,s) dX_s$.

• Lévy SDEs:
  • Estimation of $\alpha$ (tail index):
  • Characteristic Function Methods (CFM)—linear regression applied to log-characteristic functions;
  • Quantile Methods—using sample quantiles, with improved estimators like McCulloch’s technique;
  • Extreme Value / Moment Estimators—e.g., via the variance of $\log|X|$:

  $$\hat{\alpha} = \left(\frac{L_2}{\psi_1} - \frac{1}{2}\right)^{-1/2}, \quad L_2 = \mathrm{Var}(\log|X|),\quad \psi_1 = \frac{\pi^2}{6}$$ - Estimation in Lévy-driven OU SDEs: Discretized observations enable adaptation of AR(1) estimation frameworks; the drift $\theta$ and scale $\sigma$ are recovered via statistics on sampled increments.

3. Numerical Algorithms and Simulation

The paper benchmarks numerical algorithms for parameter estimation under different noise models:

• Standard Brownian Case: Simulation of diffusions and related contrast functions elucidate asymptotic properties of parameter estimators (consistency, normality).
• fBM Case: Implementation of R/S, periodogram, variance-based estimators on simulated sample paths for various $H$ illustrates convergence and estimator variance. Performance varies with $H$, with high $H$ yielding slowly decaying correlations and more challenging estimation.
• Lévy Case: Simulations show that trajectories become more discontinuous as $\alpha$ decreases; estimation techniques are empirically benchmarked, with CFM and moment methods providing robust $\alpha$ estimation.

For SDEs with discrete time steps (e.g., sampled Lévy-driven OU processes), AR(1)-type modeling is effective. Estimators for the drift and diffusion are constructed from increments, with simulation validating theoretical accuracy.

4. Distinctive Features of fBM and Lévy Motion Estimation

Distinct challenges arise in the presence of fBM and $\alpha$-stable Lévy noise:

• fBM: Key estimator properties (bias, variance) are governed by the memory encoded by $H$. Transformations leveraging the kernel $k_H$ convert fBM noise into processes amenable to likelihood-based inference.
• Lévy Motion: The infinite-variance property (for $\alpha<2$) requires non-standard estimators and heavy-tailed statistical analysis. The robustness of moment and CFM estimators is corroborated by simulation, though quantile estimators gain relevance when distributional symmetry or scale properties are known.

5. Applications and Theoretical Implications

The methodologies developed are applicable across scientific domains where noise is complex:

• Finance: Heavy-tailed jumps and memory effects modelled via Lévy and fBM respectively capture empirical asset price dynamics.
• Physical/Geophysical Sciences: Turbulence and anomalous diffusion require non-Gaussian, memory-rich noise models.
• Climate and Weather Modeling: Subgrid processes, unresolved scales, and extreme events are effectively described using fBM for persistent fluctuations, Lévy motion for jump-like effects.

Simulation results underscore that the presented estimation procedures validate theoretical properties such as consistency and asymptotic normality. The framework anticipates future developments in joint inference for simultaneous estimation of both drift/diffusion and noise parameters, particularly when both process and noise characteristics are unknown.

6. Future Directions and Methodological Impact

Improvement and extension of noise-induced uncertainty quantification is facilitated by:

• The development of efficient algorithms for simultaneous estimation of drift/diffusion and complex noise parameters ($H$, $\alpha$, scale, skew).
• The adaptation to high-dimensional systems and sparse, irregularly sampled data.
• Further exploration of theoretical limits and identifiability in models with both colored (memory-rich) and impulsive (jump) noise sources.

The integrated treatment of noise models in SDEs, validated by systematic simulation studies, sets a baseline for robust uncertainty quantification in stochastic modeling throughout applied mathematics, physics, engineering, and quantitative social sciences.