On local times of a symmetric stable process as a doubly indexed process. (English) Zbl 0966.60046

The paper is devoted to the study of a symmetric stable process \((X_t, t \geq 0)\) with index \(\beta \in (1, 2]\). The author considers the local time \(L_x^t, x \in {\mathbb R}, t \geq 0\), as a doubly indexed stochastic process. The main attention is paid to the computation in \(L^1\) of the limit of the following sum as \(k\) tends to \(\infty\): \(\sum_{(x_i, t_j)\in \Delta_k} \left|L_{t_{j+1}}^{x_{i+1}} - L_{t_{j}}^{x_{i+1}} - L_{t_{j+1}}^{x_{i}} + L_{t_{j}}^{x_{i}} \right|^{2/(\beta - 1)},\) where \((\Delta_k)_{k\in {\mathbb N}}\) is a sequence of grids of \( [a, b] \times [s, t] \subset {\mathbb R} \times [0, +\infty) \), \( L_{t_{j}}^{x_{i}} \) are values of the local time at corresponding points. The case \(\beta = 2\) (Brownian motion) is of special interest because of the connection of the results of the paper with the Tanaka’s formula. The author calculates also the quadratic variation of the local time in this case. On the base of the considered approach a new representation of Itô formula is obtained, namely \[ F(X_t, t) = F(X_0, 0) + \int_0^t \frac{\partial F}{\partial s} (X_s, s)ds + \int_0^t \frac{\partial F}{\partial x} (X_s, s) d X_s -\frac{1}{2} \int_0^t \int_{\mathbb R} \frac{\partial F}{\partial x} (x, s) d L_x^s . \]


60G52 Stable stochastic processes
60H05 Stochastic integrals
60J55 Local time and additive functionals
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