Properties of bounded stochastic processes employed in biophysics. (English) Zbl 1457.60107
Summary: Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated within a solid mathematical ground. Four families are object of investigation: the Sine-Wiener (SW), the Doering-Cai-Lin (DCL), the Tsallis-Stariolo-Borland (TSB), and the Kessler-Sørensen (KS) families. We address mathematical questions on existence and uniqueness of the processes defined through Stochastic Differential Equations, which often conceal non-obvious behavior, and we explore the behavior of the solutions near the boundaries of the state space. The expression of the time-dependent probability density of the Sine-Wiener noise is provided in closed form, and a close connection with the Doering-Cai-Lin noise is shown. Further relationships among the different families are explored, pathwise and in distribution. Finally, we illustrate an analogy between the Kessler-Sørensen family and Bessel processes, which allows to relate the respective local times at the boundaries.
Keywords:
bounded noises; stochastic differential equations; strong uniqueness; local times; transformationsReferences:
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