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Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

D. Loukianova, O. Loukianov (2008)

Annales de l'I.H.P. Probabilités et statistiques

Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach...

Uniform exponential ergodicity of stochastic dissipative systems

Beniamin Goldys, Bohdan Maslowski (2001)

Czechoslovak Mathematical Journal

We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in ℝ d with d ≤ 3 .

Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Fabrice Baudoin, Cheng Ouyang, Samy Tindel (2014)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H g t ; 1 / 3 . We show that under some geometric conditions, in the regular case H g t ; 1 / 2 , the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H g t ; 1 / 3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper...

Valuation and optimal design to defaultable security

Jianhui Huang, Na Li (2006)

Applicationes Mathematicae

Herein, we develop a backward stochastic differential equation (BSDE) valuation of securities with default risk. Consequently, the optimal recovery problem with quasi-linear utility functions is discussed with the help of the stochastic maximum principle. Finally, two important examples: the exponential and power utility cases are studied and their business implications are considered.

Weak solutions to stochastic differential equations driven by fractional Brownian motion

J. Šnupárková (2009)

Czechoslovak Mathematical Journal

Existence of a weak solution to the n -dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter H ∈ ( 0 , 1 ) ∖ { 1 2 } is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

If the space 𝒬 of quadratic forms in ℝ n is splitted in a direct sum 𝒬 1 ⊕ ... ⊕ 𝒬 k and if X and Y are independent random variables of ℝ n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when ℝ n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

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