On quadratic variation of martingales

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.