Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Besov Regularity of Periodic Levy Noises
101
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study the Besov regularity of Levy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Levy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$alpha$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $sigma$-field of the space of generalized functions. These results pave the way to the characterization of the $n$-term wavelet approximation properties of stochastic processes.
rate research
Read More
In the paper, we consider nonlinear filtering problems of multiscale systems in two cases-correlated sensor Levy noises and correlated Levy noises. First of all, we prove that the slow part of the origin system converges to the homogenized system in the uniform mean square sense. And then based on the convergence result, in the case of correlated sensor Levy noises, the nonlinear filtering of the slow part is shown to approximate that of the homogenized system in $L^1$ sense. However, in the case of correlated Levy noises, we prove that the nonlinear filtering of the slow part converges weakly to that of the homogenized system.
In this paper, we study the compressibility of random processes and fields, called generalized Levy processes, that are solutions of stochastic differential equations driven by $d$-dimensional periodic Levy white noises. Our results are based on the estimation of the Besov regularity of Levy white noises and generalized Levy processes. We show in particular that non-Gaussian generalized Levy processes are more compressible in a wavelet basis than the corresponding Gaussian processes, in the sense that their $n$-term approximation error decays faster. We quantify this compressibility in terms of the Blumenthal-Getoor index of the underlying Levy white noise.
We investigate the space-time regularity of the local time associated to Volterra-Levy processes, including Volterra processes driven by $alpha$-stable processes for $alphain(0,2]$. We show that the spatial regularity of the local time for Volterra-Levy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbaPtion of ODEs by a Volterra-Levy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.
Properties of systems driven by white non-Gaussian noises can be very different from these systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by $alpha$-stable Levy type noises, which provide natural extension to the Gaussian noise having however a new property mainly a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic and in generic double well potential models subjected to the action of $alpha$-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.
We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or Levy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of Levy processes. Numerical results are also given.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
