Proceedings - Mathematical Sciences

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

On quadratic variation of martingales

  • B V RAO


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.


Doob–Meyer decomposition martingales quadratic variation. 

Mathematics Subject Classification.

60G44, 60G17, 60H05. 


  1. [1]
    Bass R F, The Doob–Meyer decomposition revisited, Canad. Math. Bull. 39 (1996) 138–150.Google Scholar
  2. [2]
    Beiglbock M, Schachermayer W and Veliyev B, A short proof of the Doob–Meyer theorem, Stoch. Process. Appl. 122 (2012) 1204–1209Google Scholar
  3. [3]
    Karandikar R L, Pathwise solution of stochastic differential equations, Sankhya A 43 (1981) 121–132Google Scholar
  4. [4]
    Karandikar R L, On quadratic variation process of a continuous martingales, Ill. J. Math. 27 (1983) 178–181Google Scholar
  5. [5]
    Karandikar R L, Stochastic integration w.r.t. continuous local martingales, Stoch. Process. Appl. 15 (1983) 203–209Google Scholar
  6. [6]
    Karandikar R L, On pathwise stochastic integration, Stoch. Process. Appl. 57 (1995) 11–18Google Scholar
  7. [7]
    Meyer P A, A decomposition theorem for supermartingales, Ill. J. Math. 6 (1962) 193–205Google Scholar
  8. [8]
    Meyer P A, Decomposition of supermartingales: the uniqueness theorem, Ill. J. Math. 7 (1963) 1–17Google Scholar
  9. [9]
    Meyer P A, Integrales stochastiques, I–IV. Seminaire de Probabilites I. Lecture Notes in Math. 39 (1967) (Springer: Berlin) pp. 72-162Google Scholar
  10. [10]
    Meyer P A, Un cours sur les integrales stochastiques, Seminaire Probab. X, Lecture Notes in Math. 511 (Springer Berlin) pp. 245–400Google Scholar
  11. [11]
    Rao K M, On decomposition theorems of Meyer. Math. Scand. 24 (1969) 66–78Google Scholar

Copyright information

© Indian Academy of Sciences 2014

Authors and Affiliations

  1. 1.Chennai Mathematical InstituteSiruseriIndia

Personalised recommendations