1,568 research outputs found
Fractional L\'{e}vy-driven Ornstein--Uhlenbeck processes and stochastic differential equations
Using Riemann-Stieltjes methods for integrators of bounded -variation we
define a pathwise integral driven by a fractional L\'{e}vy process (FLP). To
explicitly solve general fractional stochastic differential equations (SDEs) we
introduce an Ornstein-Uhlenbeck model by a stochastic integral representation,
where the driving stochastic process is an FLP. To achieve the convergence of
improper integrals, the long-time behavior of FLPs is derived. This is
sufficient to define the fractional L\'{e}vy-Ornstein-Uhlenbeck process (FLOUP)
pathwise as an improper Riemann-Stieltjes integral. We show further that the
FLOUP is the unique stationary solution of the corresponding Langevin equation.
Furthermore, we calculate the autocovariance function and prove that its
increments exhibit long-range dependence. Exploiting the Langevin equation, we
consider SDEs driven by FLPs of bounded -variation for and construct
solutions using the corresponding FLOUP. Finally, we consider examples of such
SDEs, including various state space transforms of the FLOUP and also fractional
L\'{e}vy-driven Cox-Ingersoll-Ross (CIR) models.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ281 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion
In this note we prove an existence and uniqueness result for the solution of
multidimensional stochastic delay differential equations with normal
reflection. The equations are driven by a fractional Brownian motion with Hurst
parameter . The stochastic integral with respect to the fractional
Brownian motion is a pathwise Riemann--Stieltjes integral.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ327 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach
We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions
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