Nonlinear Lévy Processes

• Nonlinear Lévy processes are stochastic processes defined via convex or sublinear expectations that incorporate uncertainty in drift, volatility, and jump phenomena.
• They are constructed from measurable sets of Lévy triplets and governed by fully nonlinear partial integro-differential equations that extend classical Fokker-Planck models.
• Applications span robust control, mean-field interactions, and financial modeling, leveraging nonlinear Markov semigroups and kinetic equations.

Nonlinear Lévy processes generalize classical Lévy processes by replacing the linearity of expectation in the governing laws with convex or sublinear expectation structures, introducing uncertainty in drift, volatility, and jump characteristics. This framework allows modeling phenomena such as Knightian uncertainty, robust control, and mean-field interactions, producing processes whose statistical properties and evolution equations are governed by worst-case or law-dependent integro-differential operators. The theory encompasses nonlinear Markov semigroups, kinetic equations, nonlinear Fokker-Planck PDEs, and generalizations to free probability and multivariate dependence configurations.

1. Convex and Sublinear Expectation Spaces

Let $(\Omega, \mathcal{F})$ be a measurable space. A convex expectation $$\mathcal{E}: L^\infty(\Omega, \mathcal{F}) \to \mathbb{R}$$ satisfies monotonicity, constant-preservation, convexity, and continuity from below. If $\mathcal{E}$ is also positively homogeneous, it is called sublinear. Crucially, nonlinear expectations allow modeling "worst-case" or robust evaluations over an admissible set of probability measures or Lévy triplets, as in

$$\mathcal{E}(\xi) = \sup_{P \in \mathcal{P}} E^P[\xi].$$

This sublinear framework underpins the construction and analysis of nonlinear Lévy processes (Denk et al., 2017, Neufeld et al., 2014, Hu et al., 2022).

2. Definition and Construction

A process $X = (X_t)_{t\ge0}$ valued in a separable topological group $G$ is an $\mathcal{E}$-Lévy process if:

• $X_0 = 0$ in distribution (under $\mathcal{E}$).
• $X_{s+t} - X_s$ is stationary and independent of $\mathcal{F}_s$, for all $s, t \ge 0$.
• The increments are independent in the convex-expectation sense.

Given a measurable set of Lévy triplets $\Theta \subset \mathbb{R}^d \times S^d_+ \times L$ (with $L$ the set of admissible Lévy measures), the canonical process $X$ under the sublinear expectation

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

inherits stationary independent increments and law-governing uncertainty encoded by $\Theta$ (Neufeld et al., 2014, Hu et al., 2022). The generator of such a process is explicitly given by

$$G\varphi(x) = \sup_{(b, c, F) \in \Theta} \left\{ b \cdot \nabla \varphi(x) + \frac{1}{2} \mathrm{tr}[c D^2 \varphi(x)] + \int_{\mathbb{R}^d}[\varphi(x+z) - \varphi(x) - \nabla \varphi(x) \cdot h(z)] F(dz) \right\}$$

for test function $\varphi \in C_b^2(\mathbb{R}^d)$ and a truncation function $h$ (Neufeld et al., 2014, Denk et al., 2017).

3. Fully Nonlinear PDE Characterization

The law of functionals of nonlinear Lévy processes is governed by fully nonlinear partial integro-differential equations (PIDEs). For a value function $u(t,x) = \mathcal{E}[\psi(x+X_t)]$ with Lipschitz $\psi$, the unique viscosity solution satisfies

$$\partial_t u(t,x) - \sup_{(b, c, F) \in \Theta} \left\{ b \cdot \nabla_x u + \frac{1}{2} \mathrm{tr}[c D_{xx}^2 u] + \int_{\mathbb{R}^d}[u(t, x+z) - u(t, x) - \nabla_x u(t, x) \cdot h(z)] F(dz) \right\} = 0$$

with $u(0,x) = \psi(x)$ (Neufeld et al., 2014, Hu et al., 2022, Denk et al., 2017). This PIDE structure generalizes the classical Kolmogorov and Fokker-Planck equations, incorporating nonlinearity via the supremum over the uncertainty set $\Theta$.

4. Kinetic Equations and Law-Dependence

Nonlinear Lévy processes are closely related to kinetic equations and nonlinear Markov propagators. For a probability law $\mu_t$ on $\mathbb{R}^d$, the kinetic equation governing its evolution is

$$\partial_t \langle g, \mu_t \rangle = \langle A[\mu_t] g, \mu_t \rangle, \qquad g \in C^2_b(\mathbb{R}^d)$$

where $A[\mu]$ is a nonlinear (possibly measure-dependent) Lévy-Khintchine generator: $$A[\mu] f(x) = \frac{1}{2} \operatorname{tr}(G(\mu) D^2 f(x)) + b(\mu) \cdot \nabla f(x) + \int_{\mathbb{R}^d}[f(x+y) - f(x) - \nabla f(x) \cdot y 1_{|y|<1}] \nu(\mu, dy)$$ (Kolokoltsov, 2011). Existence and uniqueness of solution flows $\mu_t$ and regularity properties are established under Lipschitz continuity of $A[\cdot]$.

5. Nonlinear Stochastic Differential Equations and Stationary Laws

Nonlinear Lévy noise enters as multiplicative or law-dependent terms in stochastic differential equations. For the nonlinear Langevin equation

$\dot X(t) = F(X(t)) + G(X(t)) \eta(t)$

with $\eta(t)$ an $\alpha$-stable Lévy white noise and power law form for $F$ and $G$ (Srokowski, 2010), the stationary density (Stratonovich interpretation) admits a tail exponent

$$p_s(x) \sim |x|^{-(\alpha + \theta + \gamma)} \quad (|x| \to \infty)$$

with finite second moment iff $\alpha + \theta + \gamma > 3$. The behavior under Stratonovich and Itô interpretations diverges in tail dependence and escape kinetics, and stationary laws are tractable in terms of Fox $H$-functions and fractional Fokker-Planck equations.

6. Universal Robust Limit Theorems and Nonlinear Lévy–Khintchine Formula

Under sublinear expectation, a universal robust limit theorem establishes that appropriately normalized i.i.d. sums converge in distribution to a nonlinear multidimensional Lévy process $(\tilde{L}_t)$, characterized by an uncertainty set $\Theta$ of Lévy triplets (Hu et al., 2022). The infinitesimal generator is encoded via a nonlinear Lévy–Khintchine formula: $$\lim_{t \downarrow 0} \frac{1}{t} \mathcal{E}\left[ \varphi(\tilde{\zeta}_t) + \langle p, \tilde{\eta}_t \rangle + \frac{1}{2} \langle A \tilde{\xi}_t, \tilde{\xi}_t \rangle \right] = \sup_{(F_\mu, q, Q) \in \Theta} \left\{ \int [\varphi(z) - \nabla \varphi(0) \cdot z] F_\mu(dz) + \langle p, q \rangle + \frac{1}{2} \operatorname{tr}(A Q) \right\}$$ and the law of $(\tilde{L}_t)$ is recovered as the unique viscosity solution to the associated PIDE.

7. Extensions: Free Probability and Multivariate Nonlinear Dependence

Nonlinear Lévy processes extend to the free probability framework (Biane, 2019), where semigroups are constructed via nonlinear free convolution and Nevanlinna function generators $\varphi(z) = \psi(\Phi(z))$ with explicit characterization. In the context of multivariate asset returns, non-linear dependence is quantified using joint cumulants computed via multi-index Bell polynomials, encoding higher-order comovements, tail dependence, and providing closed-form sensitivity to model parameters (Nardo et al., 2020).

8. Fokker-Planck Equations for Nonlinear Lévy Dynamical Systems

In nonlinear dynamical systems driven by non-Gaussian Lévy processes, the Fokker-Planck (forward Kolmogorov) equation for the probability density $p(x,t)$ of the state variable is

$$\partial_t p(x,t) = -\partial_x[(f(x,t)+b \sigma(x,t)) p(x,t)] + \frac{1}{2} \partial_x^2 [\sigma^2(x,t) p(x,t)] + \int_{\mathbb{R}\setminus\{0\}} \Bigl\{ \sum_{k=1}^\infty \frac{(-y)^k}{k!} \partial_x^k [\sigma^k(x,t) p(x,t)] + I_{(-1,1)}(y)y \partial_x[\sigma(x,t)p(x,t)] \Bigr\} \nu(dy)$$

This infinite-series expansion captures the nonlocality induced by the jump structure (Sun et al., 2012).

• Srokowski, T., "Nonlinear stochastic equations with multiplicative Lévy noise" (Srokowski, 2010).
• Neufeld, A., Nutz, M., "Nonlinear Lévy Processes and their Characteristics" (Neufeld et al., 2014).
• Denk, R., Kupper, M., Nendel, R., "A semigroup approach to nonlinear Lévy processes" (Denk et al., 2017).
• Kolokoltsov, V. N., "Nonlinear Lévy and nonlinear Feller processes: an analytic introduction" (Kolokoltsov, 2011).
• Peng, S., Pan, X., "A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation" (Hu et al., 2022).
• Sun, L., Duan, J., "Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes" (Sun et al., 2012).
• Di Nardo, E., Marena, M., Semeraro, P., "On non-linear dependence of multivariate subordinated Lévy processes" (Nardo et al., 2020).
• Barndorff-Nielsen, Sato, Pedersen, foundational results for multivariate and subordinated Lévy processes.
• Biane, P., "Nonlinear free Lévy-Khinchine formula and conformal mapping" (Biane, 2019).