Advances and Applications in Statistics
Volume 44, Issue 3, Pages 219 - 239
(March 2015) http://dx.doi.org/10.17654/ADASMar2015_219_239 |
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INDEPENDENCE AMONG RANDOM VARIABLES BETWEEN SEMI-MARTINGALES
Shengxi Wang and Junfeng Shang
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Abstract: Independence is an optimal property among random variables within a semi-martingale and between semi-martingales, and the verification of independence can effectively improve and simplify the applications of semi-martingale processes. This paper investigates the conditions for independence among random variables in a semi-martingale and between semi-martingales. Specifically, we establish the conditions for independence among random variables in a continuous semi-martingale, for independence among random variables in a jump process, and for independence among random variables between a jump process and a continuous semi-martingale. Corresponding to each case of the above independence, the theorems and corollaries are stated and proved to demonstrate the conditions for the purpose of identifying independence among random variables in a semi-martingale and between martingales. In the proofs of theorems, Itô’s formula is employed. To further detect the conditions for independence, we propose the definitions of local independence and progressive independence. To explore the conditions with respect to independence among random variables in a jump process, the linear decompositions of a jump process are initially utilized to prove the theorems where a jump process is involved. The results regarding independence from the theorems and corollaries are applied to show the independence conditions between Poisson processes in Bertoin [1] and Poisson process and Brownian motion in Shreve [6]. |
Keywords and phrases: semi-martingale, local martingale, pure jump process, local independence, Itô’s formula. |
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