A stochastic process that combines continuous diffusion-like evolution (typically a Brownian component) with discrete jump events governed by a jump measure or point process, producing sample paths that are continuous between jump times and discontinuous at jumps.

Principle

The dynamics are governed by a diffusion term capturing small, frequent fluctuations and a jump term specifying arrival times and sizes of sudden moves; mathematically represented by an SDE with both a Brownian-driven part and a compensated Poisson random measure or jump process.

Demonstration

An asset‑price model where the logarithm of price follows geometric Brownian motion most of the time but occasionally experiences sudden percentage changes at random times drawn from a Poisson process with a specified jump-size distribution; this induces fat tails and skewness in return distributions.

Misapplication

Ignoring the jump component when calibrating models to data with pronounced discontinuities, or using a jump specification with unrealistic jump intensity or size distribution that misrepresents empirical tail behavior.

Consequence

Correctly incorporating jumps yields more accurate risk assessments for extreme events, changes option pricing and hedging strategies, and affects first‑passage times and ruin probabilities in applied settings.

Reversal

Removing the jump term yields a pure diffusion process (continuous paths, e.g., geometric Brownian motion) that cannot represent sudden discontinuities; conversely, setting diffusion to zero yields a pure jump process with piecewise constant trajectories between jumps.

Boundary

Applies when both small-scale fluctuations and occasional large shocks are relevant and when jump arrivals can be modeled by point processes; excludes models where jumps are deterministic, fully endogenous without separable intensity, or when microscopic microstructure invalidates continuous-time approximations.

Semantic Tension

Tension appears between modeling with continuous diffusions (convenient analytically) and discrete jumps (necessary for tail risk), and between calibrating parsimony and capturing empirically observed extreme moves.

Synthesis

A jump diffusion process extends diffusion models by superimposing a stochastic jump mechanism so that typical evolution is diffusive while rare sudden moves are explicitly represented, enabling joint modeling of everyday variability and abrupt shocks.