Decomposition of order statistics of semimartingales using local times

arXiv:0807.5001v1 [math.PR] 31 Jul 2008 DECOMPOSITION OF ORDER STATISTICS OF SEMIMARTINGALES USING LOCAL TIMES RAOUF GHOMRASNI AND OLIVIER MENOUKEU PAMEN Abstract. In a recent work [1], given a collection of continuous semimartingales, authors derive a semimartingale decomposition from the corresponding ranked processes in the case that the ranked processes can meet more than two original processes at the same time. This has led to a more general decomposition of ranked processes. In this paper, we derive a more general result for semimartingales (not necessarily continuous) using a simpler approach. Furthermore, we also give a generalization of Ouknine [7, 8] and Yan’s [10] formula for local times of ranked processes. 1. Introduction Some recent developments in mathematical finance and particularly the distribution of capital in stochastic portfolio theory have led to the necessity of understanding dynamics of the kth-ranked amongst n given stocks, at all levels k = 1, · · · , n. For example, k = 1 and k = n correspond to the maximum and minimum process of the collection, respectively. The problem of decomposition for the maximum of n semimartingales was introduced by Chitashvili and Mania in [2]. The authors showed that the maximum process can be expressed in terms of the original processes, adjusted by local times. In [4], Fernholz, defined the more general notion of ranked processes (i.e. order statistic) of n continuous Itˆo processes and gave the decomposition of such processes. However, the main drawback of the latter result is that, triple points do not exist, i.e., not more than two processes coincide at the same time, almost surely. Motivated by the question of extending this decomposition to triple points (and higher orders of incidence) posed by Fernholz in Problem 4.1.13 of [5], Banner and Ghomrasni recently in [1] developed some general formulas for ranked processes of continuous semimartingales. In the setting of problem 4.1.13 in [5], they showed that the ranked processes can be expressed in terms of original processes adjusted by the local times of ranked processes. The proof of those results are based on the generalization of Ouknine’s formula [7, 8, 10]. In the present paper, we give a new decomposition of order statistics of semimartingales (i.e., not necessarily continuous) in the same setting as in [1]. The obtained results are slightly different to the one in [1] in the sense that we express the order statistics of semimartingales firstly in terms of order statistic processes adjusted by their local times and secondly in terms of original processes adjusted by their local times. The proof of this result is a modified and shortened version of the proof given in [1] is based on the homogeneity property. Furthermore, we use the theory of predictable random open sets, introduced by Zheng in [11] and the idea of the proof of Theorem 2.2 in [1] to show that n X i=1 1{X(i)(t−)=0} dX(i)+(t) = n X i=1 1{Xi(t−)=0} dX+ i (t) , where Xi, i = 1, · · · , n represent the original processes and X(i) represent the ranked processes. As a consequence of this result, we are independently able to derive an extension of Ouknine’s formula in the case of general semimartingales. The desired generalization which is essential in the demonstration of Theorem 2.3 in [1] is not used here to prove our decomposition. Key words and phrases. Order-statistics, semimartingales, local times. 2000 Mathematical Subjects Classifications. Primary 60J65, Secondary 60H05. 1 2 RAOUF GHOMRASNI AND OLIVIER MENOUKEU PAMEN The paper is organized as follows. In section 2, we prove the two different decompositions of ranked processes for general semimartingales. In section 3, after showing the above equality, we derive a generalization of Ouknine and Yan’s formula. 2. Decomposition of Ranked Semimartingales We begin by giving the definition of the k-th rank process of a family of n semimartingales. Definition 2.1. Let X1, · · · , Xn be semimartingales. For 1 ≤k ≤n, the k-th rank process of X1, · · · , Xn is defined by X(k) = max 1≤i1<···…(Full text truncated)…

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