Hawkes Jump-Diffusions
- Hawkes jump-diffusions are stochastic models that combine a diffusion component with self-exciting jumps to capture clustered and endogenous jump activity.
- They leverage tractability methods such as affine, Volterra, and Markov approximations to model complex phenomena like volatility clustering and derivative pricing.
- These models support robust statistical inference and efficient hedging strategies by integrating advanced feedback mechanisms and jump dynamics into asset pricing.
to=arxiv_search.search _人人碰 天天中奖彩票_queried? ＿色(json) {"20query20 jump-diffusion\" OR 20all:\20 jump diffusions\"20 OR ti:\20"Hawkes\" AND cat:q-fin.MF20"," to=arxiv_search.search 彩神争霸能json {"20query20 AND cat:q-fin.MF20"," to=arxiv_search.search 全民彩票天天送钱json {"20query20 rough Hawkes Heston stochastic volatility model\"20 OR ti:\20"A new self-exciting jump-diffusion process for option pricing\"","20max_results20 AND cat:q-fin.MF20"," Hawkes jump-diffusions are stochastic models in which a diffusion component is coupled to jump activity that is self-exciting, clustered, or endogenously feedback-driven. In the literature, this label covers one-dimensional diffusions with Hawkes-driven jumps, affine stochastic-volatility systems in which variance is directly tied to jump intensity, marked models with separate positive and negative jump channels, and even infinite-activity specifications in which a predictable activity scale is excited by realized jumps rather than by event counts (&&&20submittedDate20&&&, &&&20query20&&&, &&&20 OR all:\20&&&). Relative to constant-intensity jump-diffusions, these models are used to encode clustering, contagion, leverage, asymmetric skew dynamics, and persistent jump-risk effects.
20all:\20. Model classes and formal scope
A canonical finite-activity specification is the one-dimensional jump-diffusion
PRESERVED_PLACEHOLDER_20query20^
where PRESERVED_PLACEHOLDER_20all:\20^ is a multivariate Hawkes process, PRESERVED_PLACEHOLDER_20 OR all:\20^ is independent of PRESERVED_PLACEHOLDER_20 OR ti:\20, and the jump amplitude is state-dependent through PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20^ (&&&20submittedDate20&&&). This formulation underlies both nonparametric inference problems and ergodicity analysis for Hawkes-driven diffusions.
A second classical specification is a Cox–Ingersoll–Ross state with Hawkes jumps,
PRESERVED_PLACEHOLDER_20max_results20^
which generalizes both the CIR process and the classical Hawkes process with exponential exciting function (&&&20all:\20all:\20&&&). In this model the state PRESERVED_PLACEHOLDER_20sort_by20^ is itself the feedback channel through which jump intensity becomes endogenous.
A third line of work embeds Hawkes activity into stochastic volatility. In the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models, and the jump component enters both variance and returns through common jumps with opposite signs (&&&20query20&&&).
Other constructions broaden the class without abandoning tractability. The Heston–Queue–Hawkes model combines Heston volatility with a Queue–Hawkes jump process whose memory expires at random times, while preserving a closed-form characteristic function for the jump component (&&&20all:\20&&&). The compound CARMAPRESERVED_PLACEHOLDER_20submittedDate20-Hawkes model replaces the standard Ornstein–Uhlenbeck Hawkes intensity by a CARMA state process driven by the counting process itself, producing a richer kernel PRESERVED_PLACEHOLDER_20sort_order20^ while retaining log-affine characteristic functions (&&&20 AND cat:q-fin.MF20&&&). A more recent affine model dispenses with finite activity altogether: jump sizes follow a normalized asymmetric tempered-stable Lévy shape, while a bounded excitation function PRESERVED_PLACEHOLDER_20descending20^ lets the asset’s own realized jumps drive future activity (&&&20 OR all:\20&&&).
| Model family | Jump/intensity mechanism | Tractability device |
|---|---|---|
| Diffusion with Hawkes-driven jumps | Multivariate Hawkes counts drive state-dependent jumps | Markov lift under exponential kernels |
| CIR with Hawkes jumps | Intensity PRESERVED_PLACEHOLDER_20all:\20query20^ | Affine generator and Riccati ODEs |
| Rough Hawkes Heston | Variance proportional to Hawkes-type jump intensity | Affine Volterra Riccati–Volterra system |
| HQH | Queue–Hawkes intensity with random expiration | Closed-form jump CF and COS reduction |
| Compound CARMA-Hawkes | CARMA state-space intensity | Finite-dimensional ODE system |
| Endogenous infinite-activity model | Bounded feedback on activity scale PRESERVED_PLACEHOLDER_20all:\20all:\20^ | Affine transform and generalized Riccati |
This breadth is important conceptually. In the cited literature, “Hawkes jump-diffusion” does not denote a single canonical SDE, but a family of models whose common feature is endogenous jump activity rather than exogenous Poisson arrival.
20 OR all:\20. Intensity architectures, excitation channels, and stability
The standard Hawkes mechanism is linear self-excitation. In multivariate exponential-kernel models,
PRESERVED_PLACEHOLDER_20all:\20 OR all:\20^
so the intensity increases after each jump and then decays exponentially (&&&20submittedDate20&&&). Under this choice, the intensity process is Markovian; under the spectral-radius condition PRESERVED_PLACEHOLDER_20all:\20 OR ti:\20, with PRESERVED_PLACEHOLDER_20all:\20 AND cat:q-fin.MF20, the joint process can be positive Harris recurrent and exponentially PRESERVED_PLACEHOLDER_20all:\20max_results20-mixing (&&&20all:\20submittedDate20&&&).
Several models alter the excitation channel while keeping the Hawkes intuition. In the Heston–Queue–Hawkes model, the intensity is PRESERVED_PLACEHOLDER_20all:\20sort_by20, and
PRESERVED_PLACEHOLDER_20all:\20submittedDate20^
where PRESERVED_PLACEHOLDER_20all:\20sort_order20^ randomly erases past excitation. The paper uses the condition PRESERVED_PLACEHOLDER_20all:\20descending20^ to keep intensities in check (&&&20all:\20&&&). In the rough Hawkes Heston model, the compensated jump measure has intensity PRESERVED_PLACEHOLDER_20 OR all:\20query20, so the total jump intensity is PRESERVED_PLACEHOLDER_20 OR all:\20all:\20; when PRESERVED_PLACEHOLDER_20 OR all:\20 OR all:\20^ is a probability measure, the total intensity equals PRESERVED_PLACEHOLDER_20 OR all:\20 OR ti:\20, making volatility itself the Hawkes-type driver (&&&20query20&&&).
The infinite-activity affine model replaces event-count excitation by variation-based excitation. Its activity scale satisfies
PRESERVED_PLACEHOLDER_20 OR all:\20 AND cat:q-fin.MF20^
and mean-subcriticality is
PRESERVED_PLACEHOLDER_20 OR all:\20max_results20^
This is presented as the analogue of the Hawkes branching ratio (&&&20 OR all:\20&&&). The boundedness of PRESERVED_PLACEHOLDER_20 OR all:\20sort_by20^ ensures finite average excitation despite infinite jump activity.
Quadratic Hawkes models extend linear Hawkes feedback by allowing intensity to depend on terms linear and quadratic in past returns. In the ZHawkes specialization,
PRESERVED_PLACEHOLDER_20 OR all:\20submittedDate20^
with PRESERVED_PLACEHOLDER_20 OR all:\20sort_order20^ the usual Hawkes activity term and PRESERVED_PLACEHOLDER_20 OR all:\20descending20^ the off-diagonal quadratic feedback. For ZHawkes, PRESERVED_PLACEHOLDER_20 OR ti:\20query20^ is the non-critical stationarity condition, and the model can generate long memory without necessarily being at the critical point (&&&20max_results20&&&).
A recurrent source of confusion is that “Hawkes stability” is not a single universal inequality. In the cited models it appears as PRESERVED_PLACEHOLDER_20 OR ti:\20all:\20^ for multivariate exponential kernels, PRESERVED_PLACEHOLDER_20 OR ti:\20 OR all:\20^ for Queue–Hawkes, PRESERVED_PLACEHOLDER_20 OR ti:\20 OR ti:\20^ for bounded infinite-activity feedback, PRESERVED_PLACEHOLDER_20 OR ti:\20 AND cat:q-fin.MF20^ for quadratic Hawkes, and PRESERVED_PLACEHOLDER_20 OR ti:\20max_results20^ for the CIR-with-Hawkes-jumps model (&&&20all:\20submittedDate20&&&, &&&20all:\20&&&, &&&20 OR all:\20&&&, &&&20max_results20&&&, &&&20all:\20all:\20&&&).
20 OR ti:\20. Affine, Volterra, and Markovian tractability
A major reason Hawkes jump-diffusions are analytically useful is that many of them preserve affine or near-affine structure. The CIR model with Hawkes jumps is a special case of affine point processes: conditional Laplace transforms of PRESERVED_PLACEHOLDER_20 OR ti:\20sort_by20, PRESERVED_PLACEHOLDER_20 OR ti:\20submittedDate20, and PRESERVED_PLACEHOLDER_20 OR ti:\20sort_order20^ are exponential-affine, with coefficients solving Riccati-type ODEs (&&&20all:\20all:\20&&&). This provides explicit law of large numbers, central limit theorem, and large deviations results in addition to transform formulas.
The rough Hawkes Heston model generalizes this idea to non-Markovian volatility. It belongs to the class of affine Volterra processes with jumps. The Fourier–Laplace transform of log returns and the Laplace transform of PRESERVED_PLACEHOLDER_20 OR ti:\20descending20^ are expressed through deterministic Riccati–Volterra equations, and the rough kernel can be approximated by a sum of exponentials
PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20query20^
which embeds the rough Volterra system in a higher-dimensional Markovian system (&&&20query20&&&).
The Heston–Queue–Hawkes model attains tractability by a different route. Because the characteristic function of the Q-Hawkes jump component is known in closed form, Fourier-based fast pricing algorithms such as the COS method can be fully exploited, and explicit partial integrals reduce the dimensionality of the COS method and its numerical complexity (&&&20all:\20&&&). The compound CARMAPRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20all:\20-Hawkes model likewise preserves log-affine characteristic functions; under both PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20 OR all:\20^ and PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20 OR ti:\20, the transform of PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20 AND cat:q-fin.MF20^ reduces to a finite-dimensional ODE system for coefficient functions PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20max_results20^ and PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20sort_by20^ (&&&20 AND cat:q-fin.MF20&&&).
The endogenous infinite-activity model is also affine. Its transform has the form
PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20submittedDate20^
where PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20sort_order20^ solve a generalized Riccati system. The paper proves real-axis well-posedness and forward invariance of the closed left half-plane for the Riccati vector field, which is what enables robust COS and Carr–Madan pricing (&&&20 OR all:\20&&&).
For non-exponential kernels, a general Markov approximation is available. Under minimal integrability conditions, a Hawkes kernel can be approximated by a linear combination of exponentials, implying that Hawkes jump-diffusions can be approximated with Markov jump-diffusions. The resulting augmented state admits an explicit generator and an HJB equation for control problems (&&&20 OR ti:\20&&&).
20 AND cat:q-fin.MF20. Ergodicity, filtering, and statistical inference
The long-run behavior of Hawkes jump-diffusions has been studied most explicitly in the exponential-kernel setting. For one-dimensional diffusions with jumps driven by a multivariate nonlinear Hawkes process, the finite-dimensional PDMP lift yields a joint process PRESERVED_PLACEHOLDER_20 AND cat:q-fin.MF20descending20^ whose generator can be analyzed by Lyapunov drift and local Dœblin minorization. Under PRESERVED_PLACEHOLDER_20max_results20query20, suitable drift conditions on PRESERVED_PLACEHOLDER_20max_results20all:\20, and regularity of PRESERVED_PLACEHOLDER_20max_results20 OR all:\20^ and PRESERVED_PLACEHOLDER_20max_results20 OR ti:\20, the process is positive Harris recurrent, ergodic, and exponentially PRESERVED_PLACEHOLDER_20max_results20 AND cat:q-fin.MF20-mixing (&&&20all:\20submittedDate20&&&).
These ergodic properties support nonparametric estimation. For drift estimation in
PRESERVED_PLACEHOLDER_20max_results20max_results20^
the key step is to compensate the Hawkes jump contribution in the discrete increments. The resulting least-squares estimator on trigonometric sieve spaces achieves adaptive oracle inequalities, and under Besov regularity the rate
PRESERVED_PLACEHOLDER_20max_results20sort_by20^
matches the classical nonparametric drift rate when the Hawkes jump process is observed and compensated (&&&20submittedDate20&&&).
The corresponding problem for volatility and jump coefficients is subtler because the jump contribution enters squared increments nontrivially. A truncation-based estimator can recover PRESERVED_PLACEHOLDER_20max_results20submittedDate20, while an untruncated estimator targets
PRESERVED_PLACEHOLDER_20max_results20sort_order20^
The paper derives nonasymptotic risk bounds and adaptive oracle inequalities, and then proposes recovery of PRESERVED_PLACEHOLDER_20max_results20descending20^ through separate estimation of PRESERVED_PLACEHOLDER_20sort_by20query20, PRESERVED_PLACEHOLDER_20sort_by20all:\20, and PRESERVED_PLACEHOLDER_20sort_by20 OR all:\20^ (&&&20 OR ti:\20max_results20&&&).
For price-and-volatility systems with latent Hawkes intensities, inference has been carried out through a nonlinear state-space representation. In the bivariate jump-diffusion with Hawkes price and variance jumps, daily returns, a high-frequency price-jump indicator, a noisy jump-size proxy, and log bipower variation are linked to latent states, and Bayesian inference is conducted by a Gibbs–MH MCMC algorithm. Marginal likelihood comparisons then discriminate Hawkes intensity specifications from constant-intensity and purely state-dependent alternatives (&&&20sort_by20&&&).
20max_results20. Derivative pricing, calibration, and empirical performance
The option-pricing literature uses Hawkes jump-diffusions primarily to match joint return and volatility-surface phenomena that are difficult for classical Heston-type models. In the rough Hawkes Heston specification with power kernel and exponential jump law, calibration to SPX and VIX options as of May 20all:\20descending20, 20 OR all:\20query20all:\20submittedDate20^ uses five evolution-related parameters PRESERVED_PLACEHOLDER_20sort_by20 OR ti:\20^ and two initial-curve parameters PRESERVED_PLACEHOLDER_20sort_by20 AND cat:q-fin.MF20. The fitted value PRESERVED_PLACEHOLDER_20sort_by20max_results20^ is very close to PRESERVED_PLACEHOLDER_20sort_by20sort_by20, the model fits SPX and VIX implied-volatility smiles with high precision, and the usual upward shift of short-maturity VIX implied volatilities is explained by the very low kernel power (&&&20query20&&&).
The Heston–Queue–Hawkes model emphasizes computational efficiency. Numerical results for European and Bermudan options show that HQH offers a wider range of volatility smiles than the Bates model, while its computational burden is considerably smaller than that of the Heston–Hawkes process. On a grid of 20 AND cat:q-fin.MF20 OR all:\20query20^ European puts averaged over 20max_results20query20^ runs, the reported times are PRESERVED_PLACEHOLDER_20sort_by20submittedDate20^ s for HH, PRESERVED_PLACEHOLDER_20sort_by20sort_order20^ s for HQH, and PRESERVED_PLACEHOLDER_20sort_by20descending20^ s for Bates; the corresponding speedups versus HH are approximately PRESERVED_PLACEHOLDER_20submittedDate20query20^ and PRESERVED_PLACEHOLDER_20submittedDate20all:\20^ (&&&20all:\20&&&).
The compound CARMAPRESERVED_PLACEHOLDER_20submittedDate20 OR all:\20-Hawkes model is calibrated to GameStop option data from June 20all:\20all:\20, 20 OR all:\20query20 OR all:\20 AND cat:q-fin.MF20, with maturity August 20all:\20sort_by20, 20 OR all:\20query20 OR all:\20 AND cat:q-fin.MF20. The reported result is that CARMAPRESERVED_PLACEHOLDER_20submittedDate20 OR ti:\20-Hawkes and CARMAPRESERVED_PLACEHOLDER_20submittedDate20 AND cat:q-fin.MF20-Hawkes substantially reduce the error versus OU-Hawkes, especially for deep out-of-the-money options, highlighting the role of higher-order autoregressive and moving-average parameters in pricing (&&&20 AND cat:q-fin.MF20&&&). In the affine endogenous-jump-activity model, COS pricing yields an implied-volatility surface with pronounced short-maturity curvature and negative skew; increasing current activity PRESERVED_PLACEHOLDER_20submittedDate20max_results20^ mainly shifts near-term volatility levels, whereas stronger endogenous feedback slows the decay of left-right skew across maturities (&&&20 OR all:\20&&&).
Hawkes jump-diffusions also support hedging and risk-premium extraction. Malliavin-calculus methods have been used to derive Wiener–Poisson Delta representations for European and Asian options when jump intensity itself is Hawkes; the paper’s central point is that stochastic Hawkes intensity materially alters the Delta weight through the Malliavin derivative of PRESERVED_PLACEHOLDER_20submittedDate20sort_by20^ (&&&20 AND cat:q-fin.MF20all:\20&&&). In a bivariate positive/negative jump model for BTC options, two additional parameters—positive and negative jump premia—are inferred from options data. The estimated premia have predictive power for both BTC futures cost of carry and delta-hedged option strategies (&&&20 AND cat:q-fin.MF20 OR all:\20&&&).
20sort_by20. Conceptual comparisons, applications, and limitations
Compared with classical Heston or Bates models, Hawkes jump-diffusions replace exogenous jump arrivals by endogenous jump clustering. In the rough Hawkes Heston model, this is used to address simultaneous SPX/VIX calibration, short-maturity skew behavior, and volatility clustering while preserving affine tractability in a Volterra setting (&&&20query20&&&). In the CIR-with-Hawkes-jumps model, the same principle generates a process that is simultaneously a generalization of classical CIR and of the classical Hawkes process with exponential exciting function (&&&20all:\20all:\20&&&).
Not all Hawkes-based models make the same structural choices. HQH assumes that variance PRESERVED_PLACEHOLDER_20submittedDate20submittedDate20^ and the jump intensity process PRESERVED_PLACEHOLDER_20submittedDate20sort_order20^ are independent, which aids factorization of the characteristic function and efficient simulation, but the paper notes that this may limit joint dynamics such as volatility-of-vol spikes concurrent with intensity bursts (&&&20all:\20&&&). By contrast, the rough Hawkes Heston model couples variance, jump intensity, and returns directly through common jumps and correlated Brownian drivers (&&&20query20&&&).
A second conceptual boundary concerns finite versus infinite activity. Classic Hawkes jump-diffusions are finite-activity point-process models with linear excitation kernels; the affine endogenous-jump-activity model instead has infinitely many jumps in any interval and ties feedback to jump-induced quadratic variation through a bounded response PRESERVED_PLACEHOLDER_20submittedDate20descending20. This suggests that the Hawkes idea can be interpreted either as event-count excitation or as endogenous activity feedback, depending on the construction (&&&20 OR all:\20&&&).
A third boundary concerns pure-jump versus jump-diffusive formulations. Quadratic Hawkes models are pure-jump price models, but their low-frequency limit is a two-dimensional diffusion in which the volatility noise amplitude is proportional to volatility itself. The cited paper argues that this produces multiplicative, fat-tailed volatility and time-reversal asymmetry that standard linear Hawkes or CIR/Heston-type models miss (&&&20max_results20&&&). For high-frequency execution, Hawkes jump-diffusions have also been diffusion-approximated to produce tractable stochastic-control problems for acquisition and liquidation under price caps or floors; increasing the Hawkes-induced volatility terms leads to earlier, more aggressive execution in the reported simulations (&&&20 AND cat:q-fin.MF20descending20&&&).
The main technical limitation across the literature is that tractability usually depends on either exponential kernels, affine structure, or exponential-sum approximation. General kernels, asymmetric excitation, and richer mark dependence are feasible, but typically require Markov lifts, numerical Riccati systems, or finite-dimensional approximations (&&&20 OR ti:\20&&&). A plausible implication is that “Hawkes jump-diffusion” should be understood less as a single closed-form model than as a design principle: jump activity is made endogenous, and the surrounding diffusion, volatility, or control architecture is then chosen to recover the desired level of tractability.