Self-Exciting Flexible Residual Point Process

• The paper introduces the SE-FRPP framework, replacing standard exponential residuals with flexible distributions while preserving Hawkes-style self-excitation.
• The methodology captures both ultra-short interarrival spikes and heavy-tailed durations using a discrete Markov recursion and flexible residual embedding.
• Likelihood estimation and simulation are facilitated by a tractable recursion, enabling rigorous stability analysis and improved forecasting in high-frequency settings.

Searching arXiv for relevant papers on the Self-Exciting Flexible Residual Point Process and closely related formulations. The Self-Exciting Flexible Residual Point Process (SE-FRPP) is a point-process framework that retains Hawkes-style self-excitation while replacing the standard exponential residual structure with a flexible residual distribution. In the one-dimensional formulation introduced as a self-exciting point process embedding flexible residuals and discretely Markovian intensity dynamics, the model is driven by positive i.i.d. residuals with mean one and a latent event-time state variable that evolves through a first-order recursion (Lee, 2024). A later forecasting-oriented treatment presents the SE-FRPP as a generative model for irregularly spaced high-frequency events in which each event instantaneously excites future arrivals, the excitation decays exponentially in calendar time, and the interarrival law is made arbitrarily flexible by choosing a heavy-tailed residual distribution (Lee, 1 Apr 2026). In this sense, the SE-FRPP extends the exponential-kernel Hawkes process by embedding the hazard of a non-exponential residual law directly into the conditional intensity.

1. Formal definition and relationship to Hawkes processes

In the one-dimensional construction, let $\{\varepsilon_n\}_{n\ge 1}$ be an i.i.d. sequence of positive random variables with $\mathbb E[\varepsilon_n]=1$, density $f_{\varepsilon}(u)$, and distribution function $F_{\varepsilon}(u)$. For positive parameters $\mu,\alpha,\beta$ with $\alpha<\beta$, one defines a discrete-time Markov chain $\{(\tau_n,\lambda_n)\}_{n\ge 0}$ through

$$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$

starting from $\lambda_0>\mu$, where

$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$

and

$\mathbb E[\varepsilon_n]=1$0

The inverse $\mathbb E[\varepsilon_n]=1$1 is taken with respect to the first argument (Lee, 2024).

Between successive events $\mathbb E[\varepsilon_n]=1$2 and $\mathbb E[\varepsilon_n]=1$3, the conditional intensity for $\mathbb E[\varepsilon_n]=1$4 is

$\mathbb E[\varepsilon_n]=1$5

The multiplicative factor is the hazard ratio of the residual distribution, so the continuous-time intensity is the usual exponentially decaying excitation term modulated by residual hazard (Lee, 2024).

A later presentation rewrites the same structure in terms of an event-time state process $\mathbb E[\varepsilon_n]=1$6 and elapsed-time intensity

$\mathbb E[\varepsilon_n]=1$7

where

$\mathbb E[\varepsilon_n]=1$8

and

$\mathbb E[\varepsilon_n]=1$9

This makes explicit that the Hawkes-type decay is preserved in $f_{\varepsilon}(u)$0, while the residual law enters entirely through $f_{\varepsilon}(u)$1 (Lee, 1 Apr 2026).

If $f_{\varepsilon}(u)$2, then $f_{\varepsilon}(u)$3, and the model reduces to the classical exponential-kernel Hawkes intensity (Lee, 2024). The same nesting result is emphasized in the forecasting formulation, which states that the SE-FRPP nests the exponential Hawkes process when $f_{\varepsilon}(u)$4 is exponential (Lee, 1 Apr 2026).

2. Residual embedding and distributional flexibility

The defining feature of the SE-FRPP is the replacement of the standard exponential residual by a flexible positive law with mean one. In the earlier formulation, the residual density may be any parametric family on $f_{\varepsilon}(u)$5 with mean one. Explicit examples include a Gamma$f_{\varepsilon}(u)$6 family with $f_{\varepsilon}(u)$7 and a two-piece trapezoid-plus-exponential density with constants chosen so that the density integrates to one and has unit mean (Lee, 2024).

Under this embedding, the conditional density of the interarrival time given the previous state is obtained by change of variables: $f_{\varepsilon}(u)$8 Thus the interarrival law inherits both the Hawkes-style state dependence through $f_{\varepsilon}(u)$9 and the chosen residual shape through $F_{\varepsilon}(u)$0 (Lee, 2024).

The forecasting-oriented treatment focuses particularly on heavy-tailed interarrival behavior in high-frequency financial data. It allows $F_{\varepsilon}(u)$1 to follow any absolutely continuous law on $F_{\varepsilon}(u)$2 with finite mean $F_{\varepsilon}(u)$3, and lists three common choices: Gamma$F_{\varepsilon}(u)$4 with $F_{\varepsilon}(u)$5, generalized Gamma$F_{\varepsilon}(u)$6 with scale fixed so $F_{\varepsilon}(u)$7, and Burr Type XII$F_{\varepsilon}(u)$8 with scale $F_{\varepsilon}(u)$9 chosen so $\mu,\alpha,\beta$0 (Lee, 1 Apr 2026).

A central property is that the transformed-time residual

$\mu,\alpha,\beta$1

is exactly standard exponential for any $\mu,\alpha,\beta$2 (Lee, 1 Apr 2026). This is the analytical basis for residual diagnostics and also clarifies the role of the residual law: the model alters the calendar-time duration distribution without breaking the time-change structure familiar from point-process theory.

This suggests that the SE-FRPP should not be viewed merely as a heavy-tailed duration model. Its construction preserves the continuous-time excitation-decay mechanism while relocating distributional flexibility into the residual hazard, thereby separating excitation dynamics from the choice of interarrival family.

3. Discrete Markovian state dynamics and self-excitation

The SE-FRPP is built around a discrete-time Markov recursion at event times. In the earlier notation, the transition from $\mu,\alpha,\beta$3 to $\mu,\alpha,\beta$4 proceeds in three steps: draw $\mu,\alpha,\beta$5, solve $\mu,\alpha,\beta$6, and set

$\mu,\alpha,\beta$7

Because $\mu,\alpha,\beta$8 and $\mu,\alpha,\beta$9 depend only on $\alpha<\beta$0, the state process is first-order discrete Markov (Lee, 2024).

The same recursion is written in the later formulation as

$\alpha<\beta$1

with state space $\alpha<\beta$2 (Lee, 1 Apr 2026). Each event increases the effective intensity level through the jump parameter $\alpha<\beta$3, after which the effect decays exponentially at rate $\alpha<\beta$4 until the next event.

The self-exciting character therefore has the standard Hawkes interpretation: each arrival lifts future intensity, and that lift decays over calendar time. What distinguishes the SE-FRPP is that the realized waiting time is generated via the inverse integrated intensity map $\alpha<\beta$5 acting on a non-exponential residual. In the later empirical treatment, autocorrelation-function simulations are reported to confirm that the ratio $\alpha<\beta$6 controls memory, and that as $\alpha<\beta$7 the ACF decays more slowly, generating stronger clustering (Lee, 1 Apr 2026).

A possible misconception is that flexible residuals eliminate the Hawkes mechanism. The formulations explicitly show the opposite. The term $\alpha<\beta$8 preserves a Hawkes-style excitation-decay structure, while $\alpha<\beta$9 modifies the elapsed-time hazard (Lee, 1 Apr 2026). The earlier paper states correspondingly that if the residual is unit exponential, one recovers the classical exponential-kernel Hawkes process (Lee, 2024).

4. Likelihood-based estimation and simulation

For observed arrival times $\{(\tau_n,\lambda_n)\}_{n\ge 0}$0 or interarrivals $\{(\tau_n,\lambda_n)\}_{n\ge 0}$1, the log-likelihood in the one-dimensional model is

$\{(\tau_n,\lambda_n)\}_{n\ge 0}$2

Maximization over $\{(\tau_n,\lambda_n)\}_{n\ge 0}$3 yields the maximum-likelihood estimator (Lee, 2024). The same form is presented in the forecasting paper as

$\{(\tau_n,\lambda_n)\}_{n\ge 0}$4

with recursive reconstruction of the latent state (Lee, 1 Apr 2026).

The earlier paper states that practical optimization can be carried out using standard routines such as Newton-Raphson, quasi-Newton, and Nelder-Mead, relying on the recursion $\{(\tau_n,\lambda_n)\}_{n\ge 0}$5 (Lee, 2024). The later paper specifically notes numerical maximization by BFGS over $\{(\tau_n,\lambda_n)\}_{n\ge 0}$6, simultaneously updating $\{(\tau_n,\lambda_n)\}_{n\ge 0}$7 recursively (Lee, 1 Apr 2026).

Simulation is correspondingly direct. In the one-dimensional construction, exact simulation proceeds by initializing $\{(\tau_n,\lambda_n)\}_{n\ge 0}$8, drawing $\{(\tau_n,\lambda_n)\}_{n\ge 0}$9, computing $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$0, updating event times, and setting $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$1 until the time horizon is exceeded (Lee, 2024). The paper states that in the special case $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$2, this recovers the exact exponential-kernel Hawkes simulation of Dassios and Zhao (Lee, 2024).

The later forecasting study applies a rolling-window estimation scheme in empirical work: a window of the last 5 000 events is used to re-estimate parameters every 100 new events, reconstruct $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$3 on the window, and compute forecasts for the next 100 durations (Lee, 1 Apr 2026). This emphasizes that the model is intended not only for static fitting but also for operational sequential forecasting.

5. Stability, stationarity, and ergodic properties

The two principal papers present related but not identical stability conditions. In the earlier one-dimensional formulation, stationarity and no-explosion require $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$4 in the one-dimensional case, so that

$$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$5

and in the multivariate case the spectral radius of the branching matrix $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$6 must be less than $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$7 (Lee, 2024). The paper also notes that because $$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$8, the long-run average intensity satisfies

$$\tau_n=\phi^{-1}(\varepsilon_n;\lambda_{n-1},\mu,\alpha,\beta),\qquad \lambda_n=\psi(\tau_n;\lambda_{n-1},\mu,\alpha,\beta),$$9

It further states that asymptotic normality and consistency of the MLE follow from martingale-based theory for ergodic point processes once stationarity and an ergodicity argument for the underlying Markov chain are in force (Lee, 2024).

The later paper develops a more explicit stochastic-stability theory by treating the state recursion as a general state-space Markov chain. Under the condition

$\lambda_0>\mu$0

Theorem 3.1 is stated to show that $\lambda_0>\mu$1 on $\lambda_0>\mu$2 is Lebesgue-irreducible, aperiodic, a $\lambda_0>\mu$3-chain with every compact set petite, and admits a Foster-Lyapunov drift with $\lambda_0>\mu$4 toward a compact petite set (Lee, 1 Apr 2026). From this, the chain is positive Harris recurrent and ergodic, admits a unique stationary law $\lambda_0>\mu$5, and arbitrary initial distributions converge in total variation to $\lambda_0>\mu$6 (Lee, 1 Apr 2026).

The same result yields the stationary mean duration

$\lambda_0>\mu$7

The proof sketch reported in the paper attributes irreducibility to the continuous positive density of $\lambda_0>\mu$8 on $\lambda_0>\mu$9, aperiodicity to overlap of these random-decay ranges, and recurrence to a drift inequality of the form $$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$0 outside a compact set (Lee, 1 Apr 2026).

These results are important because the SE-FRPP embeds flexible residual laws without abandoning a tractable Markov structure. A plausible implication is that the model is unusually amenable to both rigorous stability analysis and routine likelihood estimation, which is not always the case for non-Markovian flexible point-process constructions.

6. Multivariate extension, empirical use, and diagnostics

The earlier paper extends the model to $$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$1 interconnected event types. For each component $$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$2, one defines candidate interarrival times

$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$3

where $$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$4 measures excitation on type $$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$5 triggered by a type-$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$6 event. The next event time and type are then

$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$7

and the post-event state updates according to

$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$8

The stepwise log-likelihood contribution becomes

$$\psi(t;\lambda,\mu,\alpha,\beta)=\mu+(\lambda-\mu+\alpha)e^{-\beta t},$$9

and simulation proceeds by drawing all residuals, computing all candidate durations, selecting the minimum, and updating all intensities (Lee, 2024). This preserves manageable estimation and simulation implementations in the mutually exciting setting.

Empirically, the earlier paper fits both the standard Hawkes model and the flexible-residual version to mid-price change times of AAPL on November 7, 2019. It reports that under the Hawkes fit the Q-Q plot of inferred exponential residuals shows marked departures in both tails, whereas under the flexible-residual model the fitted residual histogram becomes accurate across the entire range (Lee, 2024). In filtered historical simulation, the flexible-residual model is reported to reproduce both ultra-short and long interarrival times more faithfully than standard Hawkes (Lee, 2024).

The later study provides a forecasting evaluation on October-December 2022 AAPL mid-price one-tick durations, with 1 023 480 observations in December. It reports that the SE-FRPP with Burr residual, denoted “SE-Burr,” outperforms exponential Hawkes (“SE-Exp”), ACD, log-ACD, and fractionally integrated log-ACD in several reported metrics: root-MSE ratio $\mathbb E[\varepsilon_n]=1$00 versus $\mathbb E[\varepsilon_n]=1$01 for SE-Exp, $\mathbb E[\varepsilon_n]=1$02 equal to $\mathbb E[\varepsilon_n]=1$03 versus $\mathbb E[\varepsilon_n]=1$04, KS equal to $\mathbb E[\varepsilon_n]=1$05 versus $\mathbb E[\varepsilon_n]=1$06, and Wasserstein $\mathbb E[\varepsilon_n]=1$07 equal to $\mathbb E[\varepsilon_n]=1$08 versus $\mathbb E[\varepsilon_n]=1$09 (Lee, 1 Apr 2026). Representative SE-Burr estimates for December 16, 2022 are

$\mathbb E[\varepsilon_n]=1$10

The same paper states that P-P and exponential-residual diagnostic plots show SE-Burr residuals lying very close to the $\mathbb E[\varepsilon_n]=1$11 reference line and to the unit-exponential histogram, outperforming ACD-Burr and SE-Exp (Lee, 1 Apr 2026).

Together, these studies position the SE-FRPP as a framework for high-frequency duration modeling in which the principal empirical gain comes from matching both short-duration spikes and long waiting times without discarding the self-exciting structure of Hawkes-type intensity dynamics (Lee, 1 Apr 2026).

7. Relation to adjacent flexible and stochastic-excitation point-process models

The SE-FRPP belongs to a broader line of work seeking to relax the standard Hawkes specification without sacrificing tractability. One nearby direction models excitation magnitudes themselves as stochastic processes. In “Hawkes Processes with Stochastic Excitations,” the conditional intensity is

$\mathbb E[\varepsilon_n]=1$12

where the nonnegative contagion levels $\mathbb E[\varepsilon_n]=1$13 are realized from an SDE-driven process rather than fixed marks (Lee et al., 2016). Canonical choices include geometric Brownian motion and exponentiated Ornstein-Uhlenbeck dynamics for $\mathbb E[\varepsilon_n]=1$14, and the paper develops complete-data likelihoods, Gibbs updates for branching indicators, Metropolis-Hastings updates for latent excitation levels, and an $\mathbb E[\varepsilon_n]=1$15 simulator (Lee et al., 2016).

The contrast is instructive. In stochastic-excitation Hawkes models, flexibility is introduced through random contagion levels $\mathbb E[\varepsilon_n]=1$16 that generate serially correlated excitation strengths (Lee et al., 2016). In the SE-FRPP, flexibility is instead introduced by embedding a non-exponential residual law into the hazard factor multiplying an exponentially decaying excitation term (Lee, 2024, Lee, 1 Apr 2026). This suggests two distinct axes of generalization: one modifies the mark or excitation process, while the other modifies the residual-time law.

Another related development is mixture modeling for temporal point processes with memory. The MTD-mixture duration model specifies the conditional duration density as a weighted mixture of first-order lag-conditioned kernels and shows that the implied conditional intensity is a local mixture of first-order hazard functions (Zheng et al., 2024). By choosing decreasing hazard components such as Lomax or Weibull with shape parameter in $\mathbb E[\varepsilon_n]=1$17, the construction can induce self-exciting point-process behavior, and the paper develops Bayesian inference, residual diagnostics, and a renewal-equation approach to stationary marginals (Zheng et al., 2024).

Relative to that approach, the SE-FRPP is more tightly tied to a Hawkes-style exponential decay and event-time Markov recursion (Lee, 2024, Lee, 1 Apr 2026), whereas the mixture-duration framework emphasizes high-order memory through mixtures over lagged conditional densities (Zheng et al., 2024). A plausible implication is that the SE-FRPP occupies an intermediate position between classical Hawkes models and more general duration-based point-process constructions: it preserves a parsimonious self-exciting backbone while allowing substantial freedom in the interarrival distribution.

A recurring misconception in this area is that flexible duration models and self-exciting intensity models are disjoint categories. The SE-FRPP is explicitly designed to bridge them: it is a self-exciting process with exponential decay in intensity, but its interarrivals are governed by a residual distribution that need not be exponential and may be heavy-tailed (Lee, 1 Apr 2026). That synthesis explains why the framework has been used both for structural generalization of Hawkes processes and for forecasting in ultra-high-frequency financial settings (Lee, 2024, Lee, 1 Apr 2026).