A martingale characterization of quantum Poisson processes
Let (\(\Omega\),\({\mathcal F},P)\) be a probability space with an increasing filteration \(\{\) \({\mathcal F}_ t\), \(t\geq 0\}\) of sub-\(\sigma\)-algebras of \({\mathcal F}\). Suppose \(\lambda >0\) and \(\{\) X(t), \(t\geq 0\}\) is a real- valued stochastic process such that (i) \(\{\) X(t), \({\mathcal F}_ t\), \(t\geq 0\}\) is a martingale and (ii) the process \(\{X(t)+\lambda t\), \(t\geq 0\}\) has purely discontinuous trajectories with unit jumps. It is shown that if X(t), \(t>0\), is continuous in probability, then the condition (ii) is equivalent to the fact that \(([X,X])(t)=X(t)\) a.s. where [X,X] denotes the quadratic variation of X. Here the author extends the above characterization result to the quantum Poisson process in analogy with a result for the quantum Brownian motion obtained by \textit{L. Accardi} and \textit{K. R. Parthasarathy} [J. Funct. Anal. 77, No.1, 211-231 (1988; Zbl 0642.60032)].
scientific article; zbMATH DE number 4137117
scientific article