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To Professor Kiyoshi Noshiro on the occasion of his 60th birthday. § 0. Introduction. We shall show in § 2 that when X t is a continuous martingale, the above formula is still valid if ds is replaced by dζ xy s .When X t is not continuous, the formula becomes a more complicated form (See § 5). There, Levy system introduced by one of the authors [11] plays an important role.The formula on stochastic integral will be applied to two problems. In § 6, we shall discuss the structure of multiplicative functionals of a Markov process. Roughly speaking, every multiplicative functional with mean 1 is factorized into two mutually orthogonal martingales; one is continuous multiplicative functional and the other is a jump type one. § 7 is devoted to giving another proof of Levy-Itό's decomposition of additive process (process with independent 209 https://www.cambridge.org/core/terms. https://doi

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