Stochastinis

Stochastinis is a term used to describe a hypothetical class of stochastic processes that combine continuous dynamics with discrete, random events. In this framework, a process X_t evolves according to a drift term a(X_t, t) that acts continuously, a diffusion term b(X_t, t) dW_t representing random fluctuations, and a jump component that captures abrupt changes. A common mathematical representation uses a jump-diffusion model: dX_t = a(X_t,t) dt + b(X_t,t) dW_t + ∫ κ(X_{t-}, z) N(dt, dz), where W_t is a standard Brownian motion, N is a Poisson random measure, and κ describes the impact of each jump.

Key properties: stochastinis are typically semimartingales with càdlàg paths, possess the strong Markov property under suitable conditions, and have moments that satisfy certain integrability constraints. Depending on the choice of a, b, and κ, they can exhibit heavy tails, skewness, and jump clustering.

Common constructions: combining a diffusion process with a Poisson or Lévy jump component, or using stochastic volatility with jumps. Tractable subclasses include linear drift with constant diffusion and Poisson-driven jumps.

Applications: used in teaching to illustrate how diffusion and jumps interact; in finance to model asset prices with sudden moves, in physics to model systems subject to random shocks, and in ecology to reflect abrupt environmental disturbances.

History and status: stochastinis are primarily a pedagogical term or a fictional construct in some texts, and are not a standard category in established stochastic process literature. Variants and related concepts include jump-diffusion processes and Lévy processes.

See also: stochastic process, diffusion process, jump diffusion, Lévy process, semimartingale.