It follows from Michael's selection theory that a closed convex nonempty-valued mapping from the Sorgenfrey line to a euclidean space is inner semicontinuous if and only if the mapping can be represented as the im...
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It follows from Michael's selection theory that a closed convex nonempty-valued mapping from the Sorgenfrey line to a euclidean space is inner semicontinuous if and only if the mapping can be represented as the image closure of right-continuous selections of the mapping. This article gives necessary and sufficient conditions for the representation to hold for cadlag selections, i.e., for selections that are right-continuous and have left limits. The characterization is motivated by continuous time stochastic optimization problems over cadlag processes. Here, an application to integral functionals of cadlag functions is given.
We revisit Follmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of...
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We revisit Follmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.
Quicksort on the fly returns the input of n reals in increasing natural order during the sorting process. Correctly normalized the running time up to returning the l-th smallest out of n seen as a process in 1 converg...
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Quicksort on the fly returns the input of n reals in increasing natural order during the sorting process. Correctly normalized the running time up to returning the l-th smallest out of n seen as a process in 1 converges weakly to a limiting process with path in the space of cadlag functions. (C) 2013 Elsevier B.V. All rights reserved.
In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form X D = Sigma(i) AiXi o Bi + C on the space D of cadlag fu...
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In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form X D = Sigma(i) AiXi o Bi + C on the space D of cadlag functions. The distribution of the D-valued process X is invariant by some random linear affine transformation of space and random time change. We show the existence of solutions in some generality via the Weighted Branching Process. Finite exponential moments are connected to stochastic fixed point of supremum type X D = sup(i) (A(i)X(i) + C-i) on the positive reals. Specifically we present a running time analysis of m-median and adapted versions of Find. The finite dimensional distributions converge in L-1 and are continuous in the cylinder coordinates. We present the optimal adapted version in the sense of low asymptotic average number of comparisons. The limit distribution of the optimal adapted version of Find is a point measure on the function [0, 1] there exists t -> 1 + min{t, 1 - t}.
Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This paper provides some tools for working with such processes. We develop ...
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Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This paper provides some tools for working with such processes. We develop a sample path formula for the cumulative jump height over a given time interval. From this formula an expression for the expected value of the cumulative jump random variable is developed under reasonable conditions. The results are applied to finding the expected number of failures in the separate maintenance model over a stated time interval and to the expected number of occurrences of a regenerative event over a stated time interval.
We derive a change of variable formula for non-anticipative functionals defined on the space of R(d)-valued right-continuous paths with left limits. The functionals are only required to possess certain directional der...
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We derive a change of variable formula for non-anticipative functionals defined on the space of R(d)-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Ito formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations. (C) 2010 Elsevier Inc. All rights reserved.
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