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White noise analysis and Tanaka formula for intersections of planar Brownian motion. (English) Zbl 0759.60041

Let \(\{B_ t;t\in R_ +\}\) be a planar Brownian motion. For any bounded measurable function \(g({\mathbf x}), {\mathbf x}\in R^{2(r-1)},\) it holds \[ \int_ D g(B(t_ 2 )-B(t_ 1 ),\cdots,B(t_ r )-B(t_{r-1})) dt_ 1 dt_ 2 \cdots dt_{r-1} = \int_{R^{2(r-1)}} g({\mathbf x})\alpha_ r ({\mathbf x},D)d{\mathbf x}, \] where \(D\subset \{(t_ 1 ,t_ 2 ,\cdots ,t_ r)\), \(0\leq t_ 1 \leq t_ 2 \cdots \leq t_ r < \infty \}\) and \(\alpha_ r ({\mathbf x},D)\) is a measurable function called \(r\)-fold intersection local time of Rosen.
For off diagonal domain \(D=\prod [a_ i ,b_ i ]\), \(0<a_ 1<b_ 1<a_ 2<\cdots \), the author obtains the following iteration formula for \(\alpha_ r\)’s: \[ \begin{aligned} &\int_{a_ r}^{b_ r} \log B(b_{r+1})- B(t_ r ) - x_ r \alpha_ r ({\mathbf x},a_ 1 ,\cdots ,a_ r ,dt_ r)\\ - &\int_{a_ r}^{b_ r} \log B(a_{r+1} ) - B(t_ r) -x_ r \alpha_ r ({\mathbf x},a_ 1,\cdots ,a_ r,dt_ r) \\ =&\int_{a_{r+1}}^{b_{r+1}}\left( \int_{a_ r}^{b_ r} \frac{B(t_{r+1}) - B(t_ r) -x_ r}{B(t_{r+1}) - B(t_ r) -x_ r ^ 2} \alpha_ r({\mathbf x},a_ 1 ,\cdots ,a_ r ,dt_ r)\right)\cdot dB(t_{r+1})\\ & \qquad + \pi \alpha_{r+1}(({\mathbf x},x_ r ),a_ 1,\cdots,a_{r+1},b_{r+1}). \end{aligned} \]
To prove this the author uses the framework of white noise analysis by T. Hida and the technique of approximations of generalized functional by I. Kubo.

MSC:

60G20 Generalized stochastic processes
60G17 Sample path properties
60H99 Stochastic analysis
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