Local times of continuous N-parameter strong martingales
This paper studies the properties of local times of continuous N- parameter strong martingales. Suppose that \(M=\{M(z)\), \(z\in [0,1]^ N\}\) is a 4N-integrable, real-valued continuous N-parameter strong martingale with respect to a family of \(\sigma\)-fields \(\{\) \({\mathcal F}_ z\), \(z\in [0,1]^ N\}\) verifying the usual conditional independence property of \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta Math. 134, 111- 183 (1975; Zbl 0334.60026)]. In a previous paper [A stochastic calculus for continuous N-parameter strong martingales. Stochastic Processes Appl. 20, 1-40 (1985)] the author has obtained two kinds of Itô formulas for M. By applying these formulas to a function whose 2N th derivative is an approximation of a Dirac's \(\delta\)-distribution, Tanaka-type formulas for M can be derived. They provide expressions for the local time of M (and its k th derivatives in the space variable, for \(k\leq N-1)\) with respect to occupation time defined using a random measure which involves the quadratic variation of M. From these explicit representations of the local time and its derivatives, and using Kolmogorov's criterion, continuity properties can be deduced: joint continuity in space and time for the local time, and space continuity for the derivatives up to order N-2. For derivatives of order N-1, space continuity holds in general if N is odd and requires an additional assumption if N is even. The case \(N=2\) has been investigated by the reviewer [Ann. Inst. Henri Poincaré, Probab. Stat. 20, 251-275 (1984; Zbl 0543.60062)] in the more general context of non-strong continuous martingales bounded in \(L^ 4\), obtaining similar results.
local times of continuous N-parameter strong martingales
scientific article; zbMATH DE number 3969781
scientific article