From Jacod and Shiryaev's Limit Theorems for Stochastic Processes, we get the following definitions.
Definitions:
- A process with independent increments (abbreviated PII) $X = (X_t)_{t \geq 0}$ on a stochastic basis $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ is a càdlàg adapted real-valued process with $X_0 = 0$ and for all $0 \leq s \leq t < +\infty$, $X_t - X_s$ is independent of $\mathcal{F}_s$.
- A Lévy process (also called process with independent and stationary increments) on a stochastic basis $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ is a PII $X$ such that the distribution of the increment $X_t - X_s$ depends only on $t-s$, for all $0 \leq s \leq t$.
Lévy processes are ubiquitous in mathematical finance. For example, most models for the return of financial assets (Brownian motion, Kou, Merton, CGMY, etc...) are Lévy processes.
I would like to know if processes with independent increments which are not Lévy (i.e. not stationary) are used in finance. One possible application could be models with seasonal changes in parameters of the distribution. Thanks a lot !
