Stochastic Analysis of Lévy Processes

The stochastic analysis of Lévy processes constitutes a robust framework for modelling random phenomena exhibiting both continuous diffusion and discontinuous jumps. Lévy processes are characterised by stationary independent increments and are governed by an infinitesimal generator that takes the form of a pseudo-differential operator. Central topics include the Lévy–Khintchine representation of the characteristic exponent, path decomposition into drift, diffusion and jump components, and the study of stochastic differential equations driven by Lévy noise. Analytical tools range from symbol calculus and martingale problem formulations to potential theory and heat kernel estimates. Interdisciplinary methods combining spectral theory, functional inequalities and Malliavin calculus have enhanced precise estimates of exit times and survival probabilities in bounded domains. Recent advances have deepened our understanding of sample path regularity, long-time behaviour and ergodicity, illuminating the interplay between non-local operators and probabilistic techniques. Applications span quantitative finance, where heavy-tailed distributions capture market shocks, physicochemical systems displaying anomalous transport, and biological models of gene expression bursts. The global significance of this field lies in its capacity to bridge stochastic processes with non-local partial differential equations, facilitating tractable models for complex systems across the natural and social sciences.

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Technical terms

Lévy process: A stochastic process with stationary independent increments and càdlàg paths, extending Brownian motion to include jumps.

Stochastic differential equation (SDE): An equation describing the evolution of a random variable driven by noise, accommodating both continuous diffusion and jumps.

Infinitesimal generator: The limiting operator governing the time evolution of expectation semigroups, encoding drift and jump dynamics.

Pseudo-differential operator: An operator defined via its symbol in Fourier space, capturing non-local and fractional behaviour.

Symbol: The function in the Fourier representation of an operator or generator, encapsulating the intensity and distribution of jumps.

Heat kernel: The fundamental solution of the time-dependent equation associated with a generator, representing transition probability densities.

References

1. Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials. Communications in Mathematical Physics (2023).
2. The Symbol Associated with the Solution of a Stochastic Differential Equation. Electronic Journal of Probability (2010).
3. Barriers, exit time and survival probability for unimodal Lévy processes. Probability Theory and Related Fields (2014).
4. On martingale problems and Feller processes. Electronic Journal of Probability (2018).
5. Strong Feller Property for SDEs Driven by Multiplicative Cylindrical Stable Noise. Potential Analysis (2020).