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On quadratic variation of martingales. (English) Zbl 1308.60050
Summary: We give a construction of an explicit mapping \[ \Psi:\mathsf D([0,\infty),\mathbb R)\to\mathsf D([0,\infty),\mathbb R), \] where \(\mathsf D([0,\infty),\mathbb R)\) denotes the class of real valued r.c.l.l. functions on \([0,\infty)\) such that for a locally square integrable martingale \((M_t)\) with r.c.l.l. paths, \[ \Psi(M.(\omega))=A.(\omega) \] gives the quadratic variation process (written usually as \([M,M]_t\)) of \((M_t)\). We also show that this process \((A_t)\) is the unique increasing process \((B_t)\) such that \(M^2_t-B_t\) is a local martingale, \(B_0=0\) and \[ \mathbb P((\Delta B)_t=[(\Delta M)_t]^2,\, 0<t<\infty)=1. \] Apart from elementary properties of martingales, the only result used is Doob’s maximal inequality. This result can be the starting point of the development of the stochastic integral with respect to r.c.l.l. martingales.

60G44 Martingales with continuous parameter
60G17 Sample path properties
60H05 Stochastic integrals
Full Text: DOI
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