Jointly Persistent Mean & Volatility Dynamics

• Jointly Persistent Mean and Volatility Dynamics are models where both the conditional mean and variance exhibit ‘near-unit’ behavior, reflecting prolonged trends and clustering effects.
• They use autoregressive structures with nearly integrated stochastic volatility and multiscale factors to model asset price bubbles and persistent risk dynamics.
• Self-normalized inference methods and bootstrap techniques are employed to achieve volatility-robust estimates, essential for reliable decision-making in financial markets.

Jointly persistent mean and volatility dynamics characterize stochastic processes or economic systems in which both the conditional mean and the conditional variance exhibit persistence near nonstationarity or non-mean-reverting behavior, often approaching instability or even explosiveness. This dual persistence accurately models empirical features such as protracted asset price bubbles and long-memory volatility clustering in financial time series. Recent theoretical and econometric advances address estimation, inference, and control problems in the presence of such regimes, delivering asymptotic theory, volatility-robust inference, and optimal decision rules under multiscale uncertainty (Sarkar et al., 7 Dec 2025, Chan et al., 23 Jul 2025).

1. Model Architectures for Jointly Persistent Dynamics

The canonical archetype for jointly persistent mean and volatility dynamics consists of an autoregressive process coupled with nearly integrated stochastic volatility. In (Sarkar et al., 7 Dec 2025), the data-generating mechanism is given in the triangular array form: $$y_t = \rho_n y_{t-1} + u_t, \quad u_t = \sigma_t \varepsilon_t, \quad \varepsilon_t \sim \mathrm{i.i.d.}\; \mathcal{N}(0, 1)$$ where $\rho_n = 1 + c/k_n$ with $c\in\mathbb{R}$ and $k_n \to\infty$ such that $k_n = o(n)$. The log-volatility follows an “ultra-persistent” AR(1): $$\log \sigma_t^2 = \phi_n \log \sigma_{t-1}^2 + \eta_t, \quad \eta_t \sim \mathrm{i.i.d.}\; \mathcal{N}(0,\alpha^2)$$ with $\phi_n = 1 - d / \log r_n$, $d > 0$, $r_n \to \infty$ slowly.

In quantitative finance, jointly persistent mean and volatility dynamics are also modeled by systems where both the drift and volatility processes are themselves driven by multi-scale (mean-reverting and persistent) stochastic factors. (Chan et al., 23 Jul 2025) presents an asset model: $dP_t = x_t\,dt + \sigma(Y_t, Z_t)\,dB_t$ with $x_t$ an Ornstein–Uhlenbeck process and $\sigma(\cdot, \cdot)$ a function of fast $(Y_t)$ and slow $(Z_t)$ stochastic volatility factors.

Persistence in both the mean (via $\rho_n$ or $x_t$) and the variance (via $\phi_n$ or $(Y_t, Z_t)$) drives a “double local-to-unity” or multiscale regime, reflecting the data-generating mechanisms observed in markets subject to prolonged volatility regimes and trend persistence.

2. Limit Theory and Asymptotics under Joint Persistence

Jointly persistent frameworks require new asymptotics, as classical fixed-parameter or usual local-to-unity limit theory fails when (i) the AR(1) root is near one, and (ii) volatility itself is nearly nonstationary.

For the model of (Sarkar et al., 7 Dec 2025), under a double-local-to-unity regime, the following normalization emerges for consistent OLS inference: $$m_n \equiv \frac{1}{n} \sum_{t=1}^n \mathbb{E}[\sigma_t^2] = \exp\left( \frac{\alpha^2}{2(1-\phi_n^2)} \right)(1+o(1))$$ where $m_n$ grows at a poly-logarithmic rate, representing the average innovation scale.

Main results for the OLS estimator $\widehat{\rho}_n$ hinge on the regime:

• Mildly Stationary: $(\rho_n = 1 - c/k_n,\, c > 0)$. After scaling by $\sqrt{n k_n}$ and the average volatility $m_n$, the estimator is asymptotically normal:

$$\sqrt{n k_n} (\widehat{\rho}_n-\rho_n) \;\Rightarrow\; \mathcal{N}(0, 2c)$$

• Mildly Explosive: $(\rho_n = 1 + c/k_n,\, c > 0)$. With an explosive normalization $k_n\rho_n^n/(2c)$, the self-normalized statistic converges to a standard Cauchy law.

Remarkably, these limit laws are volatility-invariant: the form of $\sigma_t^2$—including its persistence and innovation scale—enters only in scaling but cancels in self-normalized ratios. This property is crucial for inference under ultra-persistent, nearly nonstationary volatility (Sarkar et al., 7 Dec 2025).

3. Practical Inference: Statistical and Numerical Procedures

Self-normalized test statistics provide volatility-robust inference tools:

• Calculate OLS estimate $\widehat{\rho}_n$ via $\sum y_{t-1} y_t / \sum y_{t-1}^2$.
• Form residuals $\widehat{u}_t = y_t - \widehat{\rho}_n y_{t-1}$.
• Compute the pivot:

$$T_n = \frac{\sum_{t=1}^n y_{t-1} \widehat{u}_t}{\sqrt{\sum_{t=1}^n y_{t-1}^2 \widehat{u}_t^2}}$$

Under mild stationarity, $T_n \sim \mathcal{N}(0,1)$ asymptotically; under mild explosiveness, $T_n$ converges to $\mathrm{Cauchy}(0,1)$. Testing for near-unit roots or bubble detection is thus achieved with standard critical values.

In dynamic portfolio settings (Chan et al., 23 Jul 2025), optimal control under multiscale volatility is derived from a nonlinear HJB, solved via multiscale perturbation. The explicit trading rule

$$u^*(q,l,x,y,z) = \frac{1}{K}\left[ (\lambda - A_{qq} + \lambda A_{ql})q + (A_{ql} + \lambda A_{ll})l + (A_{qx} + \lambda A_{xl})x \right] + \frac{\sqrt{\delta}}{K}(B_q + \lambda B_l) - \frac{\varepsilon}{K}q\phi(y,z)$$

reflects the leading order (constant-volatility) solution adjusted by explicit corrections from fast and slow volatility persistence.

Bootstrap methods, including wild and multiplier bootstraps on the martingale differences, are recommended for robust inference against heavy tails and additional heteroskedasticity (Sarkar et al., 7 Dec 2025).

4. Mechanisms and Effects of Joint Persistence

The coexistence of highly persistent mean and volatility components induces multiple, empirically-relevant features:

• Near-integrated volatility ($\phi_n \to 1$) produces slow, persistent variance dynamics. A plausible implication is extended volatility clustering and the breakdown of usual ergodicity and moment existence.
• Near-unit-root mean ($\rho_n \to 1$) allows for sustained trends or bubbles, necessitating careful separation of mild explosiveness vs. stationarity.
• In optimal trading frameworks (Chan et al., 23 Jul 2025), jointly persistent dynamics manifest as:
  • A signal-tracking term proportional to the drift persistence (momentum effect).
  • A fast-volatility de-leveraging correction, moderating risk exposure when current volatility exceeds long-run levels.
  • A slow-volatility correlation effect, either accelerating or decelerating risk-taking depending on long-run return–volatility correlation.

Table: Distinct Effects in Joint Persistence Regimes

Mechanism	Manifestation	Principal Impact
Drift persistence ($x_t$)	Signal-tracking in position	Momentum in optimal strategy
Fast-volatility	$O(\varepsilon)$ de-leveraging	Rapid risk adjustment to volatility spikes
Slow-volatility	$O(\sqrt{\delta})$ correlation adj.	Anticipatory/slowed response to volatility

5. Numerical and Empirical Evidence

Simulations in (Chan et al., 23 Jul 2025) indicate significant practical consequences of incorporating these corrections. In fast-factor (Heston) volatility models, adding the de-leveraging term yields profit and loss (PnL) improvements of about 53 basis points over a two-year horizon, while also reducing variance. In slow-factor (CIR) models, mean PnL improvements of 10–30 basis points, depending on the initial volatility regime, were observed. These results demonstrate economic significance and computational tractability of multiscale corrections in realistic settings.

The inference framework in (Sarkar et al., 7 Dec 2025) is constructed to be volatility-robust, ensuring reliability even when unconditional variance diverges or the volatility dynamics are nearly nonstationary—the scenario empirically observed in episodes of financial instability or asset price bubbles.

6. Extensions and Related Research Directions

Extensions of the jointly persistent framework accommodate:

• Stochastic liquidity and nonlinear price impact in trading models (Chan et al., 23 Jul 2025).
• Partial information scenarios for the drift.
• Bootstrap-based and self-normalized inference under generalized volatility evolution (Sarkar et al., 7 Dec 2025).

While volatility-invariance simplifies inference for AR(1) roots, a plausible implication is that similar statistical structures might generalize to higher-order autoregressions or multivariate systems subject to regime persistence. Modeling assumptions on the stochastic volatility (e.g., transition rates for $\phi_n$, nature of innovation process $\eta_t$) remain an open area for extension and further empirical validation.

These jointly persistent models unify moderate-deviation asymptotics, volatility-robust inference, and multiscale decision frameworks, serving as a foundation for ongoing econometric and applied financial research (Sarkar et al., 7 Dec 2025, Chan et al., 23 Jul 2025).