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In this paper, we consider an extension of the Poisson random measure for theformulation of continuous-time reinforcement learning, such that both thefrequency and the width of the jumps depend on the path. Starting from ageneral point process, we define a new Poisson random measure as limit of thelinear sum of these counting processes, and name it the Mesgaki random measure.We also construct its Stochastic integral and It\^o's formula.

Path-dependent Poisson random measures and stochastic integrals constructed from general point processes