Hawkes Jump-Diffusion Process

• Hawkes jump-diffusion process is a stochastic model that blends continuous, mean-reverting diffusion with self-exciting jump components to capture clustered events.
• Its affine structure yields closed-form characteristic functions and limit theorems (LLN, CLT, LDP), enabling efficient computation of transition laws and risk quantification.
• The model’s applications span interest rate dynamics, credit risk analysis, and option pricing, demonstrating its versatility in capturing heavy tails and feedback effects.

A Hawkes jump-diffusion process integrates continuous stochastic evolution typical of diffusion processes with discontinuous, clustered jumps triggered by a self-exciting point process. This construction captures dynamics where event clustering and feedback effects—often seen in finance, neuroscience, and other domains—are empirically observed, but are not adequately represented by diffusions or Poisson-type jump diffusions alone. The formalism allows for mean-reverting diffusive dynamics, as in the Cox–Ingersoll–Ross (CIR) model, coupled with a Hawkes process where each jump raises the likelihood of subsequent jumps, leading to endogenous clustering, heavy tails, and realistic long-run behavior.

1. Mathematical Construction and Model Properties

The canonical Hawkes jump-diffusion process augments a (possibly mean-reverting) diffusion SDE with jump increments occurring at random times, the inter-arrival times being governed by a Hawkes process whose intensity depends on both exogenous and endogenous factors. A representative example is the CIR process with Hawkes jumps: $$dr_t = b(c - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t + a\,dN_t,$$ with the intensity

$I_t = \alpha + B r_t, \qquad \alpha, B > 0.$

Here, $N_t$ is a simple point process where past jumps increase current intensity, yielding clusters of jumps and autocorrelation in jump arrivals. In the limiting cases ($a = 0$ or $B = 0$), the model reduces to the classical CIR process. If $c = 0$ and $\sigma = 0$ with $a, \alpha > 0$, it reduces to a linear Hawkes process with an exponential kernel.

This architecture generalizes both the CIR and linear Hawkes processes, embedding state-dependency and self-excitation that are essential for modeling phenomena such as contagion in credit risk or clusters of neural spikes.

2. Affine Point Process Framework

The process belongs to the class of affine point processes, where the logarithm of characteristic/laplace functionals is affine in the state. Specifically, the Laplace transform

$$\mathbb{E}[e^{-\theta r_t}] = \exp\{ A(t) r + B(t) \}$$

with $(A(t), B(t))$ satisfying coupled ODEs: $$\begin{aligned} A'(t) &= -bA(t) + \frac{\sigma^2}{2}A^2(t) + B(e^{aA(t)} - 1),\ B'(t) &= b c A(t) + a (e^{aA(t)} - 1),\ A(0) &= -\theta,\quad B(0) = 0. \end{aligned}$$ This exponential-affine representation is shared by a broad class of tractable models, making computation of transition probabilities and expectations of functionals feasible. Affineness enables semi-closed-form solutions for derivative pricing, risk assessment, and facilitates statistical inference for parameter estimation within these models.

3. Limit Theorems and Long-Term Behavior

The Hawkes jump-diffusion process exhibits robust asymptotic properties:

• Law of Large Numbers (LLN):

$\frac{1}{t} \int_0^t r_s ds \longrightarrow \frac{b c + a \alpha}{b - a B}, \quad \text{in %%%%7%%%% and a.s. (under ergodicity)}.$

This ensures that time-averaged quantities converge to deterministic values governed by model parameters, provided $b > aB$ ensures stationarity and the Feller condition $2bc > \sigma^2$ ensures positivity.

• Central Limit Theorem (CLT):

The fluctuations about LLN limits are Gaussian. For instance,

$$\frac{1}{\sqrt{t}}\left( \int_0^t r_s ds - t \frac{b c + a \alpha}{b - a B} \right) \xrightarrow{d} N(0, V),$$

with explicit formulae for $V$ in terms of $a, B, \sigma$.

• Large Deviations Principle (LDP):

The empirical measures of the process satisfy an LDP, with rate function for $r_t$ given by

$$I(x) = \sup_{\theta > 0}\{ \theta x - b c y(\theta) - a (e^{a y(\theta)} - 1)\},$$

where $y(\theta)$ solves a transcendental equation reflecting jump effects. This quantifies the probability of large deviations of empirical means, critical for risk control.

4. Applications and Model Implications

The coupling of mean-reverting diffusion with self-exciting jumps makes the Hawkes jump-diffusion process particularly suited to several applications:

• Interest Rate Modeling: The process is a natural candidate for short-term rate models, extending CIR properties with clustered jumps. Analytical tractability via Feynman–Kac arguments permits derivation of explicit or semi-closed pricing formulas for bonds.
• Credit and Default Risk: The self-excitation in the Hawkes component models default clustering, a phenomenon observed empirically (e.g., during financial crises). The capacity of the model to endogenously capture feedback effects and mutual excitation is pivotal for portfolio credit derivatives and risk management applications.
• Stochastic Volatility and Option Pricing: The jump component produces heavy-tails and skewness in distributions, improving the fit to observed option prices and tail risk. The LDP and CLT results are essential for quantifying risk measures, including Value-at-Risk and ruin probabilities.
• Extensions: Generalizations include random jump sizes and nonlinear (state or history-dependent) Hawkes intensities, allowing richer feedback structures and even nonlinearity in excitation.

5. Structural and Computational Considerations

The analytical tractability of the Hawkes jump-diffusion process is a direct consequence of its affine structure. Core performance guarantees and theoretical results hold when $b > aB$ (to prevent explosion/clustering from becoming dominant) and the Feller non-negativity condition is satisfied. The ODE system for the Laplace transform allows efficient computation of moments, transition laws, and pricing formulas.

Resource requirements for simulation and inference are modest compared to non-affine jump diffusions, given the closed-form Laplace transform and explicit formulas available for many limit theorems. The limiting behaviors (LLN, CLT, LDP) enable robust long-timescale predictions (ergodicity, Gaussian fluctuations, rare event rates), underpinning both parametric and nonparametric statistical estimation.

6. Summary Table: Model Ingredients and Asymptotic Results

Feature	Mathematical Structure	Practical Significance
Diffusion Term	$b(c - r_t)dt + \sigma \sqrt{r_t} dW_t$	Mean-reversion, stochastic volatility
Jump Term	$a dN_t$, $N_t$ Hawkes with $I_t = \alpha + Br_t$	Self-exciting, state-dependent jump clustering
Laplace Transform	Exponential–affine via ODEs ${(A,B)}$	Closed-form computation for pricing, moments
Limit Theorems	LLN, CLT, LDP explicit in parameters	Quantification of long-term risks
Parameter Constraints	$b > aB$, $2bc > \sigma^2$	Stationarity, positivity of state variable

7. Outlook and Generalizations

Possible extensions include random or state-dependent jump sizes, nonlinear Hawkes intensities, or embedding the Hawkes jump-diffusion into more complex Markovian or even non-Markovian frameworks. Such generalizations facilitate modeling of phenomena with more nuanced excitation and feedback, relevant for credit contagion, systemic risk, or even interacting biological networks.

The Hawkes jump-diffusion process thus provides a powerful theoretical and practical framework capturing mean reversion, heavy tails, clustering, and feedback, with a body of limit laws and closed-form solutions that are highly valuable for advanced statistical analysis and quantitative modeling in stochastic processes.