Compound Poisson Jumps

• Compound Poisson jumps are stochastic processes that sum independent and identically distributed jump sizes occurring at Poisson event times, forming a basic pure-jump Lévy process.
• They are widely applied in financial modeling, risk theory, and ergodic control, characterized by stationary, independent increments and explicit moment formulations.
• Recent research employs Bayesian, wavelet, and self-normalization techniques to effectively estimate jump intensities and decompound observed increments.

A compound Poisson jump process is a fundamental stochastic model in which random instantaneous changes (“jumps”) occur at Poisson event times, and each jump has a random size governed by an independent, identically distributed sequence. This framework underlies a large portion of the applied probability literature, including stochastic differential equations, rare-event large deviations, ergodic control, financial modeling, and fractional generalizations. The compound Poisson process serves both as a prototypical example of a pure-jump Lévy process and as a building block in more complex jump-diffusion or time-changed models.

1. Definition and Basic Properties

Let $N_t$ be a Poisson process of rate $\lambda>0$ and $\{Y_i\}_{i\ge1}$ an i.i.d. sequence of random variables, independent of $N$. The classic compound Poisson process is defined as

$X_t = \sum_{i=1}^{N_t} Y_i .$

The corresponding infinitesimal generator, moment generating function, and characteristic function are

$$\mathbb{E}[e^{u X_t}] = \exp\bigl\{ \lambda t [\mathbb{E}[e^{uY_1}]-1] \bigr\},\quad \mathbb{E}[e^{i\theta X_t}] = \exp\bigl\{ \lambda t [\mathbb{E}[e^{i\theta Y_1}]-1] \bigr\}.$$

If $Y_i$ are nonnegative and Lebesgue absolutely continuous with density $f$, the process is used to model aggregated random shocks or claims in risk processes, queueing, and insurance applications (Crescenzo et al., 2015).

Key properties:

• Stationarity and independence of increments: $X_{t+s}-X_t$ is independent of the filtration up to $t$ and has the same law as $X_s$.
• Lévy process: $X_t$ is a finite-activity pure-jump Lévy process.
• Moments: $\mathbb{E}[X_t] = \lambda t\,\mathbb{E}[Y_1]$, $\mathrm{Var}[X_t] = \lambda t\,\mathbb{E}[Y_1^2]$.

If $\lambda$ or the law of $Y_i$ is state-dependent, a "compound Poisson process with state-dependent rate" arises (Hongler et al., 2016).

2. Master Equations and Fractional Extensions

Consider an SDE with a compound Poisson jump term: $dX_t = b(X_{t-})\,dt + \sigma(X_{t-})\,dW_t + dJ_t$ where $J_t=\sum_{i=1}^{N_t} Z_i$ and $W_t$ is Brownian motion. The forward Kolmogorov (Master) equation for the density $P(x,t)$ involves gain–loss terms reflecting the jump structure: $$\frac{\partial P}{\partial t} = -\frac{\partial}{\partial x}(bP) + \frac{1}{2}\frac{\partial^2}{\partial x^2}(\sigma^2 P) - \lambda P(x,t) + \int \varphi(y) \lambda P(x-y,t) dy .$$ For special jump distributions, the integro-differential equation can reduce to a higher-order PDE (e.g., Erlang-distributed jumps yield a spatial $m$-th order PDE) (Hongler et al., 2016).

Fractional generalizations involve time-changing with inverse subordinators $E_f(t)$, yielding the Generalized Fractional Compound Poisson Process (GFCPP): $Y_f(t) = \sum_{i=1}^{N(E_f(t))} X_i$ governed by a fractional Kolmogorov–Feller equation with a generalized Caputo–Džrbašjan derivative: $$\mathcal{D}^f_t p_n(t) = -\lambda p_n(t) + \lambda \sum_{k\ge0} p_{n-k}(t) F_X^{*}(\{k\}) .$$ Special cases cover Mittag–Leffler, discrete uniform, geometric, and negative-binomial jump distributions (Gupta et al., 2023).

3. Estimation and Decompounding

The decompounding problem asks for the recovery of intensity $\lambda$ and jump law $f$ from discretely observed increments.

• Bayesian Approach: Latent count augmentation and Markov Chain Monte Carlo are used for nonparametric $\nu_k = \lambda p_k$ estimation (Gugushvili et al., 2019).
• Wavelet Methods: Adaptive wavelet thresholding estimates convolution powers of the observed mixture density, and inversion formulas reconstruct $f$, achieving minimax rates under certain sampling regimes (Duval, 2012).
• Self-normalization for SDEs: For ergodic diffusions with Poisson jumps, iterative jump-removal based on self-normalized Euler residuals and the Jarque–Bera statistic provides tuning-free, efficient estimators for the diffusion parameters, adapting to unknown volatility (Masuda et al., 2018).

4. Functional Analysis and Limit Theorems

Key results on distribution tails, functionals, and performance in heavy-tailed or high-intensity regimes include:

• Uniform Large Deviations: For compound Poisson processes with regularly varying jumps ($\mathbb{P}[J>x] \sim L(x) x^{-\alpha}$), the tail of the supremum and functionals such as time to ruin exhibit asymptotics driven by the jump tail, robust uniformly in near-critical vanishing drift (Kamphorst et al., 2015).
• Ornstein–Uhlenbeck with Compound Poisson Jumps: Passage times, ruin probabilities, and undershoots are characterized using partial eigenfunctions and contour integration, with explicit Laplace transform formulas and sharp asymptotics (Rønn-Nielsen, 2016).
• Central Limit and Law of Large Numbers: For finite mean and variance, $X(t)/t \to \mu = \lambda \mathbb{E} Y$ a.s., and the fluctuations are asymptotically Gaussian; in high-intensity limits, scaling yields convergence to Brownian motion (Cinque et al., 10 Apr 2025).

5. Applied and Generalized Settings

Compound Poisson jumps are core in models for finance, control, and beyond:

• Time-Changed Lévy Models in Finance: Option pricing under time-changed Brownian motion with compound Poisson jumps (e.g., variance gamma, normal inverse Gaussian) leads to explicit and quasi-explicit formulas, tractable via conditioning and static hedging arguments (Ivanov et al., 2020).
• Ergodic Control: Stochastic control of jump–diffusion systems with compound Poisson noise is fully characterized by the ergodic HJB equation, with pathwise optimality and fine value function regularity under Lyapunov-type structural assumptions (Arapostathis et al., 2019).
• Prediction Problems: For Gaussian Volterra processes perturbed by compound Poisson jumps, full conditional distributions and prediction laws follow by leveraging independence and additive decomposition, yielding explicit future-forecast statistics (Almani et al., 2023).
• Co-integrated Multivariate Models: Cointegration of dependent Poisson processes induces spread pricing frameworks in energy and commodities, with explicit dependence captured by self-decomposable random variables (Petroni et al., 2015).

6. Rare-Event Simulation and Extreme-Value Regimes

In heavy-tailed settings, rare events frequently result from one or several large jumps:

• Sample-Path Large Deviations: Precise asymptotic rates for rare events are available via the sample-path large deviations principle, identifying the minimal “big-jump” structure required for a given rare event (Chen et al., 2017).
• Strongly Efficient Estimators: Importance sampling strategies that specifically bias toward rare multiple-jump realizations admit provably bounded relative error and are effective across rare-event regimes, including insurance ruin, option barriers, and queue overflows (Chen et al., 2017).

7. Special Constructions and Bell Polynomial Expansions

Time-randomization and iteration yield further classes:

• CPP with Poisson Subordinator: If a CPP is evaluated at Poisson times, closed-form distribution representations result in mixtures characterized by Bell polynomials. Exponential and normal jump cases furnish explicit formulas for densities, moments, and characteristic exponents. The so-called “iterated Poisson process” exhibits convergence to a Poisson process under appropriate scaling (Crescenzo et al., 2015).
• Skellam-Type and Multivariate Compound Poisson: The Skellam process and its generalizations—sums of signed jumps at Poisson times—admit exact CPP representations and clean characterizations of limiting/Gaussian behavior and discrete-time approximations (Cinque et al., 10 Apr 2025).

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References:

• (Crescenzo et al., 2015) Compound Poisson process with a Poisson subordinator
• (Gupta et al., 2023) Fractional Generalizations of the Compound Poisson Process
• (Masuda et al., 2018) Estimating Diffusion With Compound Poisson Jumps Based On Self-normalized Residuals
• (Chen et al., 2017) Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes
• (Cinque et al., 10 Apr 2025) Point processes of the Poisson-Skellam family
• (Hongler et al., 2016) On Jump-Diffusive Driving Noise Sources: Some Explicit Results and Applications
• (Duval, 2012) Adaptive wavelet estimation of a compound Poisson process
• (Ivanov et al., 2020) Option pricing in time-changed Lévy models with compound Poisson jumps
• (Kamphorst et al., 2015) Uniform Asymptotics for Compound Poisson Processes with Regularly Varying Jumps and Vanishing Drift
• (Rønn-Nielsen, 2016) Asymptotics for the ruin time of a piecewise exponential Markov process with jumps
• (Gugushvili et al., 2019) Decompounding discrete distributions: A non-parametric Bayesian approach
• (Almani et al., 2023) Prediction of Gaussian Volterra Processes with Compound Poisson Jumps
• (Petroni et al., 2015) Cointegrating Jumps: an Application to Energy Facilities
• (Arapostathis et al., 2019) Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis