On Quadratic Variation of Martingales

Proceedings Mathematical Sciences

Rajeeva L Karandikar²  B V Rao¹ 

[1] $$;Chennai Mathematical Institute, H, SIPCOT IT Park, Siruseri 0 0, India$$

关键词: Doob–Meyer decomposition;    martingales;    quadratic variation.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We give a construction of an explicit mapping$$Psi: D([0,∞),mathbb{R})→ D([0,∞),mathbb{R}),$$where $D([0,∞), mathbb{R})$ denotes the class of real valued r.c.l.l. functions on $[0,∞)$ such that for a locally square integrable martingale $(M_t)$ with r.c.l.l. paths,$$Psi(M.(𝜔))=A.(𝜔)$$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 $M_t^2-B_t$ is a local martingale, $B_0=0$ and$$mathbb{P}((𝛥 B)_t=[(𝛥 M)_t]^2, 0 < ∞)=1.$$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.

【 授权许可】

Unknown