Unified Lévy-Process Framework

• The unified Lévy-process framework is a comprehensive structure that integrates jump components, diffusions, and model uncertainty under one probabilistic model.
• It unifies Lévy–Khintchine triplets with sublinear expectations and viscosity solutions to solve nonlinear partial integro-differential equations.
• The framework supports high-accuracy simulation and statistical inference for applications in finance, econometrics, and physics.

The unified Lévy-process framework integrates stochastic processes with jumps, model uncertainty, and statistical inference under a single principled structure, enabling robust modeling, simulation, and statistical analysis in fields such as finance, econometrics, mathematical physics, and insurance mathematics. This overarching architecture unifies models for purely discontinuous (jump) parts, diffusions, mixed jump-diffusions, and nonlinear or ambiguous local characteristics, with direct connections to sublinear expectations, nonlinear generators, and fully nonlinear partial integro-differential equations (PIDEs).

1. Canonical Lévy Process Structure and Unified Decomposition

A Lévy process $X_t$ is fully specified by its Lévy–Khintchine triplet: $(b, Q, \nu)$, where $b \in \mathbb{R}^d$ is the drift, $Q \in S_+^d$ the covariance matrix, and $\nu$ a Lévy measure on $\mathbb{R}^d \setminus \{0\}$. The canonical Lévy–Itô decomposition writes

$$X_t = b t + Q^{1/2} W_t + \int_0^t\int_{|z|\geq 1} z N(ds, dz) + \lim_{\varepsilon\to0}\int_0^t\int_{\varepsilon<|z|<1} z \left[N(ds, dz) - \nu(dz) ds\right],$$

where $W_t$ is a standard Brownian motion, $N(ds, dz)$ is a Poisson random measure, and $\nu$ satisfies $\int (1 \wedge |z|^2)\nu(dz)<\infty$ (Bouzianis et al., 2019).

This structure extends to multidimensional, nonlinear, and even sublinear-expectation settings, by considering families of triplets $\Theta$ and working with suprema over all admissible characteristics $(b, Q, \nu) \in \Theta$ (Neufeld et al., 2014, Ren, 2011, Hu et al., 2022).

2. Unified Functionals, Weak Convergence, and Model Uncertainty

Functional central limit theorems (CLTs) for subordinated and nonlinear Lévy processes reveal a general principle: scaling limits of continuous-time random walks (CTRWs) and other summation schemes yield time-changed Lévy processes and, in the presence of uncertainty, nonlinear Lévy processes constructed via supremum expectations.

Given i.i.d. arrays $(J_i, \zeta_i)$ in the domain of attraction of stable laws, the limit is $X(t)=Z(D^{-1}(t))$ where $D$ is a stable subordinator and $Z$ a stable process (Søjmark et al., 2023). For nonlinear processes,

$$\tilde L_t = (\tilde{\xi}_t, \tilde{\eta}_t, \tilde{\zeta}_t),\quad t\in [0,1]$$

are constructed as weak limits of normalized sums in a sublinear expectation space, with increments reflecting the supremum over all $(F_\mu, q, Q) \in \Theta$; the associated PDE is

$$\partial_t u - \sup_{(F_\mu, q, Q)\in\Theta} \bigg\{ \int_{\mathbb{R}^d} \delta_\lambda u F_\mu(d\lambda) + \langle D_y u, q\rangle + \frac{1}{2} \operatorname{tr}[D_x^2 u\, Q] \bigg\} = 0$$

with the jump-difference operator $$\delta_\lambda u(t,x,y,z) := u(t,x,y,z+\lambda) - u(t,x,y,z) - \langle D_z u(t,x,y,z), \lambda\rangle$$ (Hu et al., 2022).

Uncertainty sets $\Theta$ encode Knightian ambiguity about drift, diffusion, and jump structure. The sublinear expectation $\mathcal{E}$ is

$$\mathcal{E}[\xi] = \sup_{P\in\mathcal{P}_\Theta} E^P[\xi]$$

with $\mathcal{P}_\Theta$ the set of semimartingale laws with $\Theta$-valued differential characteristics (Neufeld et al., 2014).

3. Nonlinear Generators, PIDEs, and Viscosity Solutions

The infinitesimal generator for the unified framework is

$$\mathcal{G} f(x) = \sup_{(b,Q,\nu)\in\Theta} \left[ b\cdot\nabla f(x) + \frac{1}{2}\operatorname{tr}\big(Q D^2 f(x)\big) + \int \left(f(x+y)-f(x)-\nabla f(x)\cdot h(y)\right)\nu(dy) \right],$$

where $h(y)$ is a truncation function (Neufeld et al., 2014, Ren, 2011).

The value function $u(t,x) = \mathcal{E}[f(x+X_t)]$ solves the nonlinear PIDE

$$\partial_t u(t,x) = \mathcal{G}u(t,x),\qquad u(0,x) = f(x),$$

with existence and uniqueness of bounded viscosity solutions under mild uniform integrability and continuity conditions on $\Theta$. For nonlinear robust limits, the fully degenerate PIDE includes the additional jump-difference structure and allows for degenerate $Q$ (Hu et al., 2022).

Thus, Markovian expectations of functionals reduce to solving a single PIDE under model uncertainty, yielding tractable computation and analysis.

4. Statistical Inference and Empirical Characteristic Function Methods

A unified approach to infer the volatility, the jump measure, and the Blumenthal–Getoor index of a Lévy process employs the empirical characteristic function (ECF)

$$\varphi_n(u) = \frac{1}{n}\sum_{k=1}^n e^{iu\,\Delta_k^n X}$$

and its logarithm $\psi_n(u)=\log\varphi_n(u)$ (Reiß, 2013).

Hypothesis testing for triplet components is formulated for all sampling regimes (high, low, and intermediate frequency), using the signal–noise tradeoff in characteristic exponents. For example, testing for volatility exploits

$$\left| \varphi_n(U_n) - e^{-\frac{1}{2}\Delta_n\sigma_0^2 U_n^2} \right| > c_n$$

at frequency $U_n$ chosen to balance bias and variance, with minimax rates depending on the Blumenthal–Getoor index (Reiß, 2013).

Sufficiently general inference encompasses both parametric and nonparametric settings and allows for optimal separation rates in each regime.

5. Simulation and Fluctuation Functionals: Wiener–Hopf and Unified Inversion

Unified simulation methodologies leverage Wiener–Hopf factorization and contour integration (sinh–acceleration) for exact or high-accuracy sampling of path functionals such as $(X_T,\bar X_T,\tau_T)$, where $\bar X_T = \sup_{t\leq T} X_t$ and $\tau_T$ the time at which the supremum is first achieved (Boyarchenko et al., 2023).

All marginal, joint, and conditional functionals can be obtained by Laplace–Fourier inversion involving the characteristic exponent and Wiener–Hopf factors, with the transformations performed by conformal changes of variable and infinite-trapezoid rules. Complexity is logarithmic in the desired precision, and the method directly accommodates arbitrary Lévy exponents $\psi(\xi)$—not requiring dedicated small/large jump decompositions.

This covers simulation tasks for log-prices, barrier options, first-passage/ruin problems, and joint functionals of drawdown and running maxima across broad classes of Lévy models.

6. Applications, Extensions, and Model Classes

Finance

The unified Lévy-Itô model in finance specifies asset price processes as exponentials of general Lévy processes, supporting flexible jump structures (Merton, Variance-Gamma, CGMY, etc.) and a pricing kernel approach encoding the risk-premium for both Brownian and jump risk. Closed-form characteristic functions permit fast Fourier-based option pricing (Bouzianis et al., 2019).

Non-Gaussian and Stable State-Space Models

State-space models driven by non-Gaussian Lévy processes integrate heavy tail and self-similarity properties, with tractable inference via conditionally Gaussian “shot noise” expansions, Kalman filtering for conjugate priors, and Rao–Blackwellised sequential Monte Carlo for continuous-time filtering with irregular data (Godsill et al., 2019).

Nonlinear and Robust Lévy Processes

Unifying frameworks for “nonlinear Lévy” or “G-Lévy” processes provide robust modeling and inference under model uncertainty, via sublinear expectations, viscosity PDEs, and supremum generators. These permit precise control of volatility and jump law ambiguity, with Markovian functionals equated to solutions (viscosity/weak) of nonlinear PIDEs (Ren, 2011, Neufeld et al., 2014, Hu et al., 2022).

Functional Convergence and Subordination

Functional CLT limits unify CTRW scaling and time-changed processes under both $J_1$ and $M_1$ Skorokhod topologies, capturing the continuum of applications from statistical physics to econometrics. All limits are schemes of time-changed stable Lévy processes, with only the topology dictated by the jump-correlation structure (Søjmark et al., 2023).

7. Key Synthesis and Theoretical Insights

The unified Lévy-process framework merges model construction, robust limit theorems, statistical testing, path simulation, and application-motivated modeling under the lens of suprema over sets of coefficients, sublinear expectations, and unified analytic and probabilistic representations. Central themes include:

• All model ambiguity is parametrized by uncertainty sets $\Theta$ over triplets; nonlinear expectations correspond to a supremum over associated semimartingale or Lévy laws.
• Markovian functionals reduce to solving fully nonlinear PIDEs with explicit supremum-form generators; viscosity theory yields existence and uniqueness.
• Unified high/low/intermediate frequency inferential theory via the empirical characteristic function.
• Simulation and option pricing for path-dependent functionals are achieved via unified Wiener–Hopf/Laplace-Fourier integral methods, applicable to arbitrary Lévy exponents.
• Theoretical generality extends robust limit results, weak convergence, and probabilistic constructions to multidimensional, nonlinear, and degenerate cases.

This systematic approach enables robust, unified modeling and analysis for stochastic processes with jumps under both classical and model-uncertainty paradigms, furnishing explicit computational tools and theoretical guarantees across mathematical finance, econometrics, and applied probability (Neufeld et al., 2014, Søjmark et al., 2023, Hu et al., 2022, Reiß, 2013, Ren, 2011, Boyarchenko et al., 2023, Gusak et al., 2013, Bouzianis et al., 2019, Godsill et al., 2019).