Squared Radial Ornstein-Uhlenbeck Processes

• Squared radial Ornstein-Uhlenbeck processes are non-negative diffusions obtained by squaring a multidimensional OU process, exhibiting power-law tail behavior and inherent asymmetry.
• They have a gamma stationary distribution, display dimension-dependent boundary behavior, and are interconnected with CIR and reflected OU processes for varied applications.
• Advanced parameter estimation methods, including maximum likelihood estimation and large deviations theory, enable precise inference and risk assessment in complex modeling scenarios.

A squared radial Ornstein-Uhlenbeck (OU) process is a non-negative, diffusion process obtained by squaring the Euclidean norm of a multidimensional OU process. These processes generalize classical OU dynamics by incorporating inherent asymmetry and power-law tail behavior, making them highly relevant in a range of mathematical, statistical, and financial modeling contexts. The squared radial OU process is closely related to the Cox-Ingersoll-Ross (CIR) process, Bessel and Wishart processes, and their reflected and generalized variants. Recent research has provided deep insights into their distributional properties, boundary behavior, estimation theory, links to reflection mechanisms, infinite divisibility, extreme value theory, and statistical inference.

1. Definition and Mathematical Structure

A squared radial Ornstein-Uhlenbeck process arises naturally from multidimensional OU dynamics. Let $\{U_t\}_{t\ge0}$ be a $d$-dimensional OU process satisfying

$$dU_t = -b U_t\,dt + \sigma\,dW_t,\quad U_0=u_0 \in \mathbb{R}^d,$$

where $b>0$, $\sigma>0$, and $W_t$ is $d$-dimensional standard Brownian motion. The squared radial process $X_t$ is defined as

$X_t = \|U_t\|^2 = \sum_{i=1}^d U_{i,t}^2.$

$X_t$ satisfies the stochastic differential equation

$$dX_t = \left(a - b X_t \right)\,dt + \sigma\sqrt{X_t}\,dW_t,$$

where $a = d\sigma^2$ and $W_t$ is a standard Brownian motion, possibly different from the original multidimensional process, due to the Itô correction.

This is the celebrated CIR process. The non-negativity of $X_t$ for $a\ge0$ and $X_0\ge0$ is a critical property, especially in modeling variances and energy-like quantities.

2. Distributional and Asymptotic Properties

The squared radial OU process exhibits asymmetric, heavy-tailed marginals (for $d > 1$), with a strictly non-negative support. Its stationary distribution (when $b > 0$ and $a > 0$) is a gamma distribution: $$X_t \sim \mathrm{Gamma}\left(k,\,\theta\right),\quad\text{with}\quad k = \frac{2a}{\sigma^2},\ \ \theta = \frac{\sigma^2}{2b},$$ reflecting the dimensional parameter $k$ and mean-reversion rate $b$.

The process exhibits ergodic behavior for $b > 0$, with a unique invariant measure if $a > 0$ (Chaumaray, 2014). The long-time behavior of the norm of extending multidimensional OU processes satisfies

$$\varlimsup_{t \to \infty} \frac{\|U_t\|}{\sqrt{\log t}} = \sqrt{2\lambda_1},\quad \text{almost surely,}$$

where $\lambda_1$ is the largest eigenvalue of the associated covariance matrix $\Sigma$ (Xie, 2014). For the squared process, this implies $X_t = O(\log t)$ a.s. as $t\to\infty$.

Boundary behavior is dimension-dependent:

• For $k>1$, $X_t$ avoids zero almost surely.
• For $0$X_t$ can hit zero, and the process must be defined with reflection or absorption conditions (Mishura et al., 2023).

3. Connection with Reflected OU and Local Time

Recent papers have shown deep connections between the squared radial OU (CIR) and reflected OU (ROU) processes, especially for low-dimensional or critical-parameter cases (Mishura et al., 2021, Mishura et al., 2023).

If $a = \sigma^2/4$ (the critical case for $d=1$), and one considers $Y_t = \sqrt{X_t}$, then $Y_t$ converges to a reflected Ornstein-Uhlenbeck process: $$Y_t = Y_0 - \frac{b}{2} \int_0^t Y_s\,ds + \frac{\sigma}{2} W_t + L_t,$$ where $L_t$ is a continuous, non-decreasing “reflection function” (the Skorokhod reflection) that ensures $Y_t\geq 0$ (Mishura et al., 2021).

For subcritical dimensions $0 < k < 1$, $X_t$ can be represented as a time- and scale-transformed reflected Brownian motion; the SDE for $Y_t$ involves a correction term given by a weighted local time of this reflected Brownian motion. Specifically,

$$Y(t) = \sqrt{x_0} - \frac{b}{2} \int_0^t Y(s)\,ds + \frac{\sigma}{2} W(t) + L(t),$$

with the reflection term $L(t)$ written as an explicit functional of local times (Mishura et al., 2023): $$L(t) = -\frac{1}{2}(\sigma^2/4 - a)\int_0^\infty y^{(4a/\sigma^2)-2}\left(\ell(t,y) - \ell(t,0)\right)\,dy,$$ where $\ell(t,y)$ is a scaled local time.

This provides a bridge between the “radial part” of multidimensional OU and the theory of reflected diffusions, and enables new representations of Skorokhod reflection mechanisms.

4. Parameter Estimation and Inference

The estimation of key parameters—most prominently the drift coefficient $b$ and the dimensional parameter $a$ (or equivalently, the “degrees of freedom” $k$)—has been studied using maximum likelihood and large deviations theory (Chaumaray, 2014).

Given continuous observations over a long time interval $[0, T]$, maximum likelihood estimators (MLEs) for $(a, b)$ are consistent. More precisely, the joint MLEs satisfy a large deviations principle (LDP) with a good, explicit rate function $I_{a, b}(\alpha, \beta)$. The rate function is characterized as the minimum of two explicit forms $J_{a, b}$ and $K_{a, b}$, depending on the observed statistics (time averages and endpoint values) and the true process parameters.

Key analytical steps involve:

• Computing the normalized cumulant generating function for a quadruplet of relevant statistics drawn from the trajectory.
• Applying the Gärtner–Ellis theorem, followed by contraction principles for the parameter estimators.
• Explicit calculation of these rate functions, encoding precisely how deviations in observed statistics are exponentially unlikely (Chaumaray, 2014).

Simultaneous estimation of both $a$ and $b$ represents a more delicate problem than estimating either one with the other fixed and is essential for a complete statistical understanding of the process.

5. Extreme Value and Excursion Theory

Recent studies have extended extreme value analysis to closely related classes, including $q$-Ornstein-Uhlenbeck and positive self-similar Markov processes (Wang, 2016). Although these do not directly model the squared radial OU process, their methodological frameworks—particularly in terms of boundary behavior and limit laws for minima/maxima—are relevant.

Key findings include:

• Excursion probabilities near the lower boundary admit explicit asymptotic formulas involving a Pickands-type constant.
• Subsequently, for independent copies of such processes, time- and magnitude-rescaled minima converge weakly to semi-min-stable processes, as opposed to classical min-stable (or max-stable) laws.
• These phenomena have direct analogues in self-similar squared radial processes, especially in the context of positive, heavy-tailed, or boundary-reflected models.

This line of research enhances the understanding of tail events and rare excursions in models built from squared radial or related processes.

6. Generalizations and Related Processes

The mathematical structure of squared radial OU processes can be further generalized in several directions:

• Operator-valued and Hilbert–space extensions: Ornstein-Uhlenbeck processes with values in Hilbert spaces, possibly with non-Gaussian (e.g., Lévy-driven) stochastic volatility, enable the modeling of forward curves and random fields (Benth et al., 2015). In these contexts, square-root and squared-norm operations define extensions of the radial OU process in infinite dimensions.
• Transformed OU processes: By squaring or applying other nonlinear, strictly increasing transformations to a stationary OU process, one obtains models with heavy-tailed marginal distributions but mean-reverting dependencies (Borovkov et al., 2011). Flexible functional forms allow for independent tuning of dependency and tail behavior.
• Generalized kernel and Lévy-driven models: Generalized OU processes with non-exponential memory kernels or non-Gaussian noise (such as α-stable or Poisson drivers) extend the class of squared radial processes, sometimes yielding infinite divisibility and rich codifference structures (Stein et al., 2021).

These generalizations not only broaden the statistical and modeling scope but also introduce new challenges in estimation, simulation, and theoretical analysis.

7. Applications and Empirical Relevance

Squared radial OU processes and their generalizations have significant applications:

• Mathematical finance: Modeling stochastic variances (e.g., stochastic volatility models, interest rates), where non-negativity and mean reversion are essential.
• Physics and biology: Energy fluctuations, membrane potentials, and diffusive transport processes often exhibit similar stochastic structures, requiring accurate modeling of non-negativity and boundary phenomena.
• Statistical inference: Large deviation tools and explicit rate functions enable efficient estimation and risk quantification, particularly with long-memory and heavy-tailed data.

Empirical studies demonstrate that these processes can represent real-world phenomena more faithfully than their linear-Gaussian or univariate counterparts by accommodating asymmetry, heavy tails, and flexible dependence.

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Table: Key Mathematical Formulations and Domains

Model Component	Equation / Property	Domain of Applicability
SDE for squared radial OU	$dX_t = (a - b X_t)\,dt + \sigma \sqrt{X_t}\,dW_t$	$a\geq 0$, $b>0$, $X_0\geq0$
Stationary distribution	$X_t\sim \mathrm{Gamma}(k,\theta)$; $k=2a/\sigma^2$, $\theta = \sigma^2/2b$	$a>0$
LDP for MLE of $(a, b)$	$$I_{a, b}(\alpha, \beta) = \min\{J_{a, b}(\alpha, \beta), K_{a, b}(\alpha, \beta)\}$$	Large $T$, $a > 2$, $b < 0$
Reflection term (critical $a$)	$$Y_t = Y_0 - \frac{b}{2} \int_0^t Y_s\,ds + \frac{\sigma}{2} W_t + L_t$$	$a = \sigma^2/4$, reflected OU regime
Local time representation	$$L(t) = -\frac{1}{2}(\sigma^2/4 - a)\int_0^\infty y^{(4a/\sigma^2)-2}\big[\ell(t,y) - \ell(t,0)\big]dy$$	$0 < a < \sigma^2/4$, low-dimensional CIR, generalization

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In summary, squared radial Ornstein-Uhlenbeck processes represent a substantial and flexible class of mean-reverting, non-negative diffusions with rich boundary and tail properties, deeply interconnected with CIR, reflected OU, and related processes. Their theoretical tractability, adaptability to non-Gaussian settings, and relevance in high-dimensional and infinite-dimensional models ensure their continuing significance across applied probability, statistical physics, mathematical finance, and statistical inference.