Causal predictability between stochastic processes and filtrations

Abstract

In this paper we further develop a notion of causal predictability defined in [A. Merkle, Predictability and Uniqueness of Weak Solutions of Stochastic Differential Equations, Analele Stiintifice ale Universitatii Ovidius Constanta, 2022] as a concept of dependence which is based on Granger's definition of causality. More precisely, in [A. Merkle, Predictability and Uniqueness of Weak Solutions of Stochastic Differential Equations, Analele Stiintifice ale Universitatii Ovidius Constanta, 2022] causal predictability is defined between filtrations, but now we introduce causal predictability between stochastic processes and filtrations. Also, we provide some properties of this new concept. Then we apply the given causality concept to the uniqueness of weak solutions of the stochastic differential equations and in financial mathematics. Granger [Investigating causal relations by econometric models and cross spectral methods, Econometrica. 37 (1969), pp. 424–438] has considered causality concept between time series. In this paper we consider continuous time processes, since continuous time models represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc.

Keywords:

• Filtration
• causal predictability
• stochastic differential equations
• weak uniqueness of weak solution
• representation theorem
• default risk

Acknowledgments

The author would like to thank the editor and the referees for their comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Ministarstvo Prosvete, Nauke i Tehnolokog Razvoja [174012].