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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 !

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  • $\begingroup$ A process to be used in finance needs to make economic sense such as the geometric Brownian motion, where the drift and volatility both have good economic interpretation. In addition, the model needs to be simple enough so that it can be calibrated to market. $\endgroup$ – Gordon Sep 29 '17 at 17:14
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After some research, I found that PIIs are used in the modelling of Default Risk. See for example : http://www.tandfonline.com/doi/abs/10.1080/13504860903357292

In this paper, the authors also use "Sato processes" which the authors define as a PII with self-similarity.

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