×

zbMATH — the first resource for mathematics

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.

MSC:
60G44 Martingales with continuous parameter
60G17 Sample path properties
60H05 Stochastic integrals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bass R F, The Doob-Meyer decomposition revisited, Canad. Math. Bull.39 (1996) 138-150. · Zbl 0880.60046
[2] Beiglbock M, Schachermayer W and Veliyev B, A short proof of the Doob-Meyer theorem, Stoch. Process. Appl.122 (2012) 1204-1209
[3] Karandikar R L, Pathwise solution of stochastic differential equations, Sankhya A43 (1981) 121-132 · Zbl 0507.60047
[4] Karandikar R L, On quadratic variation process of a continuous martingales, Ill. J. Math.27 (1983) 178-181 · Zbl 0532.60039
[5] Karandikar R L, Stochastic integration w.r.t. continuous local martingales, Stoch. Process. Appl.15 (1983) 203-209 · Zbl 0511.60048
[6] Karandikar R L, On pathwise stochastic integration, Stoch. Process. Appl.57 (1995) 11-18 · Zbl 0816.60047
[7] Meyer P A, A decomposition theorem for supermartingales, Ill. J. Math.6 (1962) 193-205 · Zbl 0133.40304
[8] Meyer P A, Decomposition of supermartingales: the uniqueness theorem, Ill. J. Math.7 (1963) 1-17 · Zbl 0133.40401
[9] Meyer P A, Integrales stochastiques, I-IV. Seminaire de Probabilites I. Lecture Notes in Math. 39 (1967) (Springer: Berlin) pp. 72-162
[10] Meyer P A, Un cours sur les integrales stochastiques, Seminaire Probab. X, Lecture Notes in Math. 511 (Springer Berlin) pp. 245-400 · Zbl 0374.60070
[11] Rao K M, On decomposition theorems of Meyer. Math. Scand.24 (1969) 66-78 · Zbl 0193.45501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.