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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/124/03/0457-0469

    • Keywords

       

      Doob–Meyer decomposition; martingales; quadratic variation.

    • Abstract

       

      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

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

      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.

    • Author Affiliations

       

      Rajeeva L Karandikar1 B V Rao1

      1. Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri 603 103, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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