dorsal/arxiv
View SchemaProperties of a simple bilinear stochastic model: estimation and predictability
| Authors | D. Sornette, V. F. Pisarenko |
|---|---|
| Categories | |
| ArXiv ID | physics/0703217 |
| URL | https://arxiv.org/abs/physics/0703217 |
| DOI | 10.1016/j.physd.2007.08.020 |
Abstract
We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictbility in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distribution from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impluse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describes a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial times series and of other complex systems.
{
"annotation_id": "15262252-ac9b-4dbc-9ba4-06684a9dec00",
"date_created": "2026-03-02T18:01:18.376000Z",
"date_modified": "2026-03-02T18:01:18.376000Z",
"file_hash": "51306e8b2c004bb8edbc3257709cd42cee52e6ceae815542c1aef41072b1d279",
"private": false,
"record": {
"abstract": "We analyze the properties of arguably the simplest bilinear stochastic\nmultiplicative process, proposed as a model of financial returns and of other\ncomplex systems combining both nonlinearity and multiplicative noise. By\nconstruction, it has no linear predictability (zero two-point correlation) but\na certain nonlinear predictability (non-zero three-point correlation). It can\nthus be considered as a paradigm for testing the existence of a possible\nnonlinear predictbility in a given time series. We present a rather exhaustive\nstudy of the process, including its ability to produce fat-tailed distribution\nfrom Gaussian innovations, the unstable characteristics of the inversion of the\nkey nonlinear parameters and of the two initial conditions necessary for the\nimplementation of a prediction scheme and an analysis of the associated\nsuper-exponential sensitivity of the inversion of the innovations in the\npresence of a large impluse. Our study emphasizes the conditions under which a\ndegree of predictability can be achieved and describes a number of different\nattempts, which overall illuminates the properties of the process. In\nconclusion, notwithstanding its remarkable simplicity, the bilinear stochastic\nprocess exhibits remarkably rich and complex behavior, which makes it a serious\ncandidate for the modeling of financial times series and of other complex\nsystems.",
"arxiv_id": "physics/0703217",
"authors": [
"D. Sornette",
"V. F. Pisarenko"
],
"categories": [
"physics.data-an",
"physics.soc-ph",
"q-fin.ST"
],
"doi": "10.1016/j.physd.2007.08.020",
"title": "Properties of a simple bilinear stochastic model: estimation and predictability",
"url": "https://arxiv.org/abs/physics/0703217"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "1dd8b17d-90a1-4991-baea-c984c51824f7",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}