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Zbl 0612.60046
Julia, O.
Temps local pour les martingales a deux indices par rapport a sa variation quadratique. (Local times for two-parameter martingales with respect to their quadratic variation). (French)
[J] Ann. Sci. Univ. Clermont-Ferrand II 88, Probab. Appl. 5, 1-17 (1986). ISSN 0246-1501

The aim of this paper is to find a continuous local time for two- parameter martingales, $\{M\sb z$, $z\in {\bbfR}\sp 2\sb+\}$, continuous and square-integrable, with respect to their quadratic variation. In order to obtain this result we state that the three quadratic variations $<M>\sb z$, $<M\sb{.t}>\sb s$ and $<M\sb{s.}>\sb t$ must have a density, and this density, in the first two items, must be continuous and differentiable with respect to the first coordinate. \par The last part of this paper is destined to get the local time of the process: $$ \{J\sb{st}=\int\sb{R\sb{st}}\int\sb{R\sb{st}}I\sb D(z,z')dW\sb zdW\sb{z'}\quad \forall (s,t)\in {\bbfR}\sp 2\sb+\} $$ where $\{W\sb z$, $z\in {\bbfR}\sp 2\sb+\}$ is a Brownian sheet, $D=\{(z,z')\in {\bbfR}\sp 2\sb+\times {\bbfR}\sp 2\sb+\vert$ $z=(s,t)$, $z'=(s',t')$, $s\le s'$, $t\ge t'\}$ and $R\sb{st}=\{z=(x,y)\in {\bbfR}\sp 2\sb+\vert$ $x\le s$, $y\le t\}$, with respect to this quadratic variation.

MSC 2000:: *60G60 Random fields
60G44 Martingales with continuous parameter

Keywords: continuous local time for two-parameter martingales; quadratic variation

Cited in: Zbl 0647.60057

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