Nonstationary Independent-Increment Processes in Default-Risk Models

Summary

The document distinguishes processes with independent increments from Lévy processes. In a process with independent increments, future increments are independent of the information available at the start of the interval; a Lévy process adds the condition that increment distributions depend only on interval length. This makes stationarity the key difference between the two classes.

The question asks whether nonstationary independent-increment models have financial applications, including possible seasonal parameter changes. The response points to default-risk modeling as an example of their use and mentions Sato processes, described there as self-similar processes with independent increments. The evidence is a reference to a research paper rather than a worked model, calibration procedure, or empirical comparison. The note therefore establishes a relevant application and conceptual distinction, but does not explain how the processes are specified, estimated, or validated in practice. Readers seeking implementation details or evidence about predictive performance would need to consult the cited research.

Key ideas

• A process with independent increments has future changes independent of past information.
• A Lévy process additionally has stationary increment distributions.
• Nonstationary independent-increment processes are reported as being used in default-risk modeling.
• The response also mentions self-similar Sato processes as a related model class.
• The cited discussion does not provide model specifications or empirical performance evidence.

Tags

# Are processes with independent increments (which are not Lévy) used in finance?


# Are processes with independent increments (which are not Lévy) used in finance?












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 !

## Answer by kantadou (score 1)

https://quant.stackexchange.com/a/36758

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.