Unlike Brownian motion, which propagates diffusively and whose sample-path trajectories are continuous, non-Brownian Lévy motions propagate via jumps (flights) and their sample-path trajectories are purely discontinuous. When analyzing systems involving non-Brownian Lévy motions, the common practice is to use either spectral or fractional-calculus methods. In this manuscript we suggest an alternative analytical approach: using the Poisson-superposition jump structure of non-Brownian Lévy motions. We demonstrate this approach in two exemplary topics: (i) systems governed by L évy-driven Ornstein-Uhlenbeck dynamics; and, (ii) systems subject to temporal Lévy subordination. We show that this approach yields answers and insights that are not attainable using spectral methods alone.
- Lévy-driven Ornstein-Uhlenbeck dynamics
- Non-Brownian Lévy motions
- Poisson superposition
- Selfsimilar Lévy motions
- Temporal Lévy subordination