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Proceedings - Mathematical Sciences

, Volume 124, Issue 3, pp 457–469 | Cite as

On quadratic variation of martingales

  • RAJEEVA L KARANDIKAR
  • B V RAO
Article

Abstract

We give a construction of an explicit mapping

\(\Psi :\mathsf {D}([0,\infty ), \mathbb {R} )\rightarrow \mathsf {D}([0,\infty ), \mathbb {R} ),\)

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,

Ψ(M.(ω)) = A.(ω)

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 the 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.

Keywords

Doob–Meyer decomposition martingales quadratic variation. 

Mathematics Subject Classification.

60G44, 60G17, 60H05. 

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Copyright information

© Indian Academy of Sciences 2014

Authors and Affiliations

  1. 1.Chennai Mathematical InstituteSiruseriIndia

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