##  [Jump Diffusion Process](/jump-diffusion-process-0) 

 Definition

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

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

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

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

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

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

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

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

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.