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Free CIR processes

Faculty Business Administration and International Finance, Nuertingen-Geislingen University, Sigmaringer Street 25, 72622 Nürtingen, Germany

Center for Quantitative Risk Analysis CEQURA, Ludwig-Maximilians-Universität München, Akademiestraße 1/I, 80799 Munich, Germany

E-mail Address: holger.graf@hfwu.de

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Henry Port

Center for Quantitative Risk Analysis CEQURA, Ludwig-Maximilians-Universität München, Akademiestraße 1/I, 80799 Munich, Germany

Chair of Financial Econometrics, Institute of Statistics, Ludwig-Maximilians-Universität München, Akademiestraße 1/I, 80799 Munich, Germany

E-mail Address: henry.port@stat.uni-muenchen.de

Corresponding author.

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Georg Schlüchtermann

Department of Mechanical, Automotive and Aeronautical Engineering, Munich University of Applied Sciences, Munich, Germany

Faculty of Mathematics, Computer Science and Statistics, Ludwig-Maximilians-Universität München, Theresienstrasse 39/I, 80333 Munich, Germany

E-mail Address: georg.schluechtermann@hm.edu

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https://doi.org/10.1142/S0219025722500126Cited by:1 (Source: Crossref)

Abstract

For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the global existence for a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion.

Communicated by B. V. Rajarama Bhat

Keywords:

AMSC: 60H10, 65C30, 46L54, 97M30

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