Stochastic Processes and Their Applications

Itô, Kiyosi; Hida, Takeyuki

doi:10.1007/bfb0076868

Lecture Notes in Mathematics

1986

DOI: 10.1007/bfb0076868

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Stochastic Processes and Their Applications

Kiyosi Itô¹,

Takeyuki Hida²

Abstract: In this paper, we study the problem of the nonparametric estimation of the marginal density f of a class of continuous time processes. To this aim, we use a projection estimator and deal with the integrated mean square risk. Under Castellana and Leadbetter's condition (Stoch. Proc. Appl. 21 (1986) 179), we show that our estimator reaches a parametric rate of convergence and coincides with the projection of the local time estimator. Discussions about the optimality of this condition are provided. We also deal w… Show more

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Cited by 4 publications

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Fractional noise destroys or induces a stochastic bifurcation

Yang

Zeng

2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Little seems to be known about the stochastic bifurcation phenomena of non-Markovian systems. Our intention in this paper is to understand such complex dynamics by a simple system, namely, the Black-Scholes model driven by a mixed fractional Brownian motion. The most interesting finding is that the multiplicative fractional noise not only destroys but also induces a stochastic bifurcation under some suitable conditions. So it opens a possible way to explore the theory of stochastic bifurcation in the non-Markovian framework.

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Fractional noise destroys or induces a stochastic bifurcation

Yang

Zeng

2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Bifurcation dynamics of the tempered fractional Langevin equation

Zeng

Yang

Chen

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.

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Evolution systems and continuity equations for white noise functionals

2022

Journal of Mathematical Physics

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In this paper, by applying the quantum decomposition of white noise functionals, the continuity equations for white noise functionals are reformulated as time inhomogeneous white noise differential equations. We establish a systematic study of the evolution systems whose infinitesimal generators are quadratic quantum white noises and then, as an application, we study the continuity equations for white noise functionals of which the coefficient has up to the first chaos (Gaussian process). The restriction for the coefficient is comparable (not necessarily equivalent) to the integrability condition of the exponential of the coefficient by Ambrosio and Figalli [J. Funct. Anal. 256, 179–214 (2009)] for the well-posedness of the continuity equation in the abstract Wiener space setting.

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Stochastic Processes and Their Applications

Cited by 4 publications

References 8 publications

Fractional noise destroys or induces a stochastic bifurcation

Fractional noise destroys or induces a stochastic bifurcation

Bifurcation dynamics of the tempered fractional Langevin equation

Evolution systems and continuity equations for white noise functionals

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