• Rajeeva L Karandikar

Articles written in Proceedings – Mathematical Sciences

• On Quadratic Variation of Martingales

We give a construction of an explicit mapping

$$\Psi: D([0,\infty),\mathbb{R})\to D([0,\infty),\mathbb{R}),$$

where $D([0,\infty), \mathbb{R})$ 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 $M_t^2-B_t$ is a local martingale, $B_0=0$ and

$$\mathbb{P}((\Delta B)_t=[(\Delta M)_t]^2, 0 &lt; \infty)=1.$$

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.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019