Model Assumptions

The modeling decision to employ $f(\sigma W)$ rather than a time-varying correlation matrix reflects a deliberate trade-off between expressive power and analytical tractability. The function $f$ is used to capture nonlinear heteroskedastic behavior influenced by interaction between multiple stochastic systems. More Specifically:

Nonlinearity: The transformation via f permits the introduction of local, nonlinear distortion effects that are challenging to capture using purely linear correlation structures.
Parsimony: A full time-evolving correlation matrix introduces a significant number of parameters, which can lead to identifiability issues, particularly when empirical data is limited.
Interpretability: The function f offers a modular and interpretable way to model external influence on endogenous noise, aligning with methods used in stochastic volatility modeling.

The choice here is intentional and consistent with the goal of modeling systems where volatility is driven by nonlinear interaction rather than simply nonstationary correlation.

The differential form introduced as Equation (7) in the paper is given by:

$df(\mathbf{X}) = \sum_{k = 1}^{n} \frac{\partial f(\mathbf{X})}{\partial x_k} \, dx_k. \nonumber$

This expression implies that $f$ is differentiable and locally homogeneous of degree 1, satisfying:

$f(\lambda \mathbf{x}) = \lambda f(\mathbf{x}), \quad \text{for any } \lambda \in \mathbb{R}, \mathbf{x} \in \mathbb{R}^n. \nonumber$

This is not an assumption of global homogeneity, but rather a local property that ensures consistency under scalar transformation. The rationale behind this is twofold:

Stability Under Scaling: Systems influenced by proportional shocks should exhibit consistent variance scaling properties under time evolution.
Differentiability: The form of df ensures that perturbations to each dimension of X yield tractable expressions in the stochastic differential system.

This framework is particularly useful for modeling multiplicative noise processes or systems with volatility clustering.

Equation (6) as originally written,

$dx^{i,j}_t = \alpha x^{i,j}_t dt + f(\sigma W^{\{i,j\}}_t), \nonumber$

is shorthand for a more general formulation in which the driving noise $W^{\{i,j\}}_t$ is a linear combination of two Wiener processes:

$\tilde{W}_t = \lambda_1 W^i_t + \lambda_2 W^j_t, \quad \lambda_1, \lambda_2 \in \mathbb{R}. \nonumber$

The resulting system becomes:

$dx_t = \alpha x_t dt + f(\sigma \tilde{W}_t). \nonumber$

This construction acknowledges that real-world systems are rarely closed and often subject to external influences that do not respect strict orthogonality. The function $f$ absorbs these dependencies into a nonlinear transformation of noise.

Given that $f \neq 0$ , the resulting process is no longer a Levy process in the strict sense. The introduction of $f$ breaks both stationary increment and independent increment properties, depending on its form. This departure is intentional, as the goal is to model a more physically realistic, heteroskedastic process where the variance is no longer constant and memory effects may emerge. The nonlinear properties of $f(W(\boldsymbol{z}))$ described in Chapter II, which can be thought of as a type of memory function, arise from its intrinsic dependence on past values of $W_t$ . This means that for some integer-time stochastic process $\{Z_n; n \geqslant 1\}$ , our model may satisfy one of two conditions: