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Register a new userThe n-term Approximation of Periodic Generalized Levy Processes
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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.
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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.
We extend the concept of packing dimension profiles, due to Falconer and Howroyd (1997) and Howroyd (2001), and use our extension in order to determine the packing dimension of an arbitrary image of a general Levy process.
We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed ones and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Levy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Levy processes form a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Levy processes.
We give equivalent conditions for the existence of generalized moments of a Levy process $(X_t)_{tgeq 0}$. We show, in particular, that the existence of a generalized $g$-moment is equivalent to uniform integrability of $(g(X_t))_{tin [0,1]}$. As an application, it turns out that certain functions of a Levy process which are integrable and local martingales are already true martingales.
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.
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