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Register a new userLevy Processes, Martingales and Uniform Integrability
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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.
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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 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.
We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.
Scale functions play a central role in the fluctuation theory of spectrally negative Levy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of It^o calculus. The reason for the latter is that standard It^o calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Levy measure. We place particular emphasis on spectrally negative Levy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
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