Quadratic Jump Process Overview

• Quadratic jump processes are stochastic models characterized by discontinuous jumps and quadratic feedback, offering a robust framework for price and volatility estimation.
• They integrate pathwise quadratic variation, Lévy-driven SDEs, and QHawkes dynamics to capture fat-tails, volatility clustering, and time-reversal asymmetry in high-frequency data.
• These models enable empirical calibration and spectral inference in finance, underpinning methods for robust volatility estimation and principal component analysis.

A quadratic jump process is a stochastic process characterized by its second-order variation structure, which incorporates discontinuities (jumps) and quadratic feedback mechanisms. This class includes model-free processes admitting a pathwise quadratic variation, Lévy-driven stochastic differential equations (SDEs) with jumps, realized quadratic covariations in high-frequency pure-jump semimartingales, and specific classes of feedback-based intensities such as quadratic Hawkes (QHawkes) models. The quadratic jump process paradigm underpins robust estimation, inference, and modeling in contexts ranging from financial price processes to jump-diffusion volatility estimation and principal component analysis of high-frequency data.

1. Pathwise Quadratic Variation and Model-Free Frameworks

Quadratic variation for càdlàg paths with jumps is defined via vanishing-oscillation stopping-time partitions. Let $X:[0,T]\to\mathbb{R}^d$ be a càdlàg path, and $\Tau = \{T_k\}$ a sequence of stopping times. The discrete quadratic variation is: $$Q^{ij,\Tau}_t(X) = \sum_{k=0}^\infty \Delta X_i(T_k\wedge t) \Delta X_j(T_k\wedge t)$$ with increments $$\Delta X_i(T_k\wedge t) = X_i(T_{k+1}\wedge t) - X_i(T_k\wedge t)$$. The process $X$ has quadratic variation $[X_i, X_j]_t$ along a sequence $\Tau^n$ if $Q^{ij,\Tau^n}_t(X)$ converges in Vovk’s outer measure, uniformly in $t$, independent of the chosen partition as long as the oscillation vanishes: $\Osc(X,\Tau^n) := \max_k \sup_{s,u\in[T^n_k,T^n_{k+1})}|X(s)-X(u)| \xrightarrow[n\to\infty]{} 0$ Increments can be decomposed into a pure jump part $\sum_{s\le t}|\Delta X(s)|^2$ and a continuous part $[X]^{\rm cont}_t$, which is recovered by partitioning or via truncated variation approximations. For a truncation level $c>0$, the truncated variation is: $\TV^c(f,[0,t]) := \sup_{0 = t_0 < \dots < t_n = t} \sum_{k=1}^n \max\{|f(t_k)-f(t_{k-1})|-c,\,0\}$ As $c \downarrow 0$, $c\,\TV^c(X_i,[0,t]) \to [X_i, X_i]^{\rm cont}_t$, and similar polarization yields cross-variation.

This general viewpoint asserts that price paths with mildly restricted downward jumps possess a partition-independent quadratic variation, allowing a unique pathwise decomposition of their second-order structure (Galane et al., 2017).

2. Lévy-Type Quadratic Variation and High-Frequency Limit Theory

In the context of high-frequency observations, for a $d$-dimensional pure-jump semimartingale $(X_t)_{t\ge0}$ observed at grid points $t_{i,n} = i\Delta_n$, the realized quadratic variation matrix is: $$V_n(t) = \sum_{i=1}^{\lfloor t/\Delta_n\rfloor} \Delta X_{i,n}\, \Delta X_{i,n}^T$$ where $\Delta X_{i,n} = X_{t_{i,n}} - X_{t_{i-1,n}}$. For symmetric $\beta$-stable Lévy drivers ($\beta \in (0,2)$), after normalization $a_n = (\Delta_n\log(1/\Delta_n))^{-1/\beta}$,

$$a_n (V_n(t) - [L]_t) \xrightarrow{\mathcal{L}-s} (U_t)_{t\in[0,T]}$$

with $U_t$ a matrix-valued $\beta$-stable Lévy process and $[L]_t = \sum_{s\le t} \Delta L_s\, \Delta L_s^T$. The spectral asymptotics show that eigenvalues and eigenvectors of $V_n$ converge (after normalization) to stable distributions given by first-order perturbations of the limiting quadratic variation matrix. This structure fundamentally precludes the use of Gaussian PCA and motivates robust nonparametric confidence interval construction (Heiny et al., 2020).

A two-scale consistent subsampling method is available for estimating the limiting law of $U_1$, involving empirical QV differences across two time scales within blocks and blockwise estimation of the stability index $\beta$.

3. Quadratic Hawkes and Endogenous Feedback Processes

Quadratic Hawkes (QHawkes) processes extend the linear Hawkes paradigm by combining linear and quadratic feedback in the jump intensity of a pure-jump process $P_t$: $$\lambda_t = \lambda_\infty + \frac{1}{\psi}\int_{-\infty}^{t} L(t-s)\,dP_s + \frac{1}{\psi^2}\iint_{-\infty}^{t} K(t-s, t-u)\,dP_s\,dP_u$$ with $\lambda_\infty$ the baseline rate, $L$ a leverage kernel, and $K$ a quadratic kernel, often represented in terms of a time-diagonal "Hawkes" part and a rank-one ("trend" or Zumbach effect) part: $K(t,s) = \phi(t)\delta_{t-s} + k(t)k(s)$ yielding

$\lambda_t = \lambda_\infty + H_t + Z_t^2$

where $H_t = \int_{-\infty}^t \phi(t-s)\,dN_s$ ("Hawkes term"), $Z_t = \psi^{-1}\int_{-\infty}^t k(t-s)\,dP_s$ ("Zumbach term"), and $dN_s = (dP_s)^2/\psi^2$.

The stationarity condition is

$$\int_0^\infty \phi(u)\,du + \int_0^\infty k(u)^2\,du = n_H + n_Z < 1$$

Quadratic feedback induces key empirical features: time-reversal asymmetry, multiplicative fat-tailed volatility, and spontaneous long memory without criticality. The Markovian exponential-kernel limit connects the process to Pearson diffusions, enabling analytical computation of tail exponents (Blanc et al., 2015, Aubrun et al., 2022).

In multivariate settings, the Multivariate Quadratic Hawkes process (MQHawkes) involves cross-dependencies in multiple assets, with the intensity for each asset influenced by both linear and quadratic kernels across all assets. Yule–Walker-type equations relate the feedback kernels to empirical correlation structures, making the process amenable to nonparametric calibration (Aubrun et al., 2022).

4. Volatility Estimation and Quadratic Variations in Jump Diffusions

For Lévy-driven SDEs as in

$$dX_t = b_t\,dt + \sigma_t\,dW_t + \int_{\mathbb{R}} \kappa(t, z)\,(\mu - \nu)(dt,dz)$$

quadratic variation estimators are constructed using observed increments and a truncation threshold. Classical truncated quadratic variation is: $$\mathrm{QV}^{(n,\beta)}_T = \sum_{i=1}^n (\Delta_i^n X)^21_{|\Delta_i^nX| \leq u_n}$$ This estimator retains a bias $B_n$ due to small jumps: $$B_n = n^{1 - \beta(2-\alpha)} C(\alpha) \int_0^T \lambda_s\,ds$$ with $\alpha$ the Blumenthal–Getoor index. A bias-corrected estimator subtracts $\widehat{B}_n$ using local jump activity estimates: $$\mathrm{QV}^{\mathrm{unb}}_T = \mathrm{QV}^{(n,\beta)}_T - \widehat{B}_n$$ This yields an efficient, unbiased estimator of integrated volatility with standard central limit theorem properties for all $\alpha$ and $\beta$, resolving classical restrictions on the jump activity and convergence rate (Amorino et al., 2019).

5. Empirical Features and Model Calibration

Quadratic jump processes, particularly in high-frequency financial contexts, have been shown by nonparametric estimation and maximum likelihood fitting on large datasets (e.g., 133 NYSE stocks) to capture prevailing stylized facts:

• Correct magnitude of fat-tails in returns and volatility distributions;
• Robust time-reversal asymmetry matching empirical time evolution;
• Slow decay in return squared autocorrelations (volatility clustering);
• Stationarity with a small quadratic kernel norm reproduces broad market features (Blanc et al., 2015).

Empirical calibration leverages the generalized method of moments to estimate linear and quadratic feedback kernels, followed by refinement using Student-$t$ likelihoods. In QARCH representations, the estimated quadratic matrix typically displays both a diagonal Hawkes part and a low-rank off-diagonal trend structure.

6. Applications and Spectral Inference

Quadratic jump processes form the backbone of model-free realized volatility calculations and robust principal component analysis in pure-jump settings. In high-frequency jump data, the eigenvalues and eigenvectors of the realized QV matrix converge in law to matrix-stable distributions, necessitating tailored inference procedures—such as blockwise subsampling—to obtain valid confidence intervals around principal components and related low-dimensional spectral functionals (Heiny et al., 2020).

The partition-independence and truncated-variation tools make the quadratic variation and jump decomposition robust to path discretization, supporting pathwise Itô calculus in non-probabilistic settings, directly applicable to financial hedging, calibration, and model validation (Galane et al., 2017).

7. Summary Table: Quadratic Jump Process Features

Feature	Representative Model/Framework	Key Mathematical Object
Pathwise quadratic variation	Model-free càdlàg process	$[X]_t = \text{jumps} + [X]^{\rm cont}_t$
Lévy jump variation	Stable Lévy process, high-frequency QV	$V_n(t),\, [L]_t$
Feedback dynamics (QHawkes)	Quadratic Hawkes, MQHawkes	$\lambda_t,\, K(t,s),\, Z_t^2$
Truncated estimation	Jump-diffusion SDE	$$\mathrm{QV}^{(n,\beta)}_T,\, \mathrm{QV}^{\mathrm{unb}}_T$$
Spectral/statistical inference	Multivariate stable limit, PCA	Eigenpairs, blockwise estimates

Quadratic jump processes unify robust pathwise variation, endogenous feedback in intensities, and high-frequency estimation under a shared mathematical structure, supporting theoretical results and empirical calibration across stochastic analysis and quantitative finance.