The Tanaka Formula for Symmetric Stable Processes with Index $\alpha$, $0&lt;\alpha&lt;2$

Engelbert, H. J.; Kurenok, V. P.

doi:10.1137/s0040585x97t989489

Cited by 3 publications

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“…In general, we cannot handle the case when μ is not compactly supported, due to the integrability restrictions of strictly stable processes. Nevertheless, in the following particular case, we obtain a generalization of the works of Salminen and Yor in [23] and Engelbert and Kurenok [2] from the symmetric to the general case. Formally, the result would follow from applying Theorem 2 to the infinite measure μ(dy) = |y| γ −α [k − 1l y>0 +k + 1l y<0 ] dy.…”

Section: Theorem 2 (Occupational Meyermentioning

confidence: 69%

“…Protter [9,IV.7]) which features a non-zero local time term. This allows for a concrete semimartingale decompositions for power functions applied to stable processes which were recently obtained for symmetric stable processes in Engelbert and Kurenok [2].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 90%

“…The main result of Engelbert and Kurenok [2], for symmetric stable processes, is that the finite variation part in the semimartingale decomposition of |X t − x| γ is increasing so that this process is a submartingale. One thus obtains a Doob-Meyer decomposition.…”

Section: Theorem 2 (Occupational Meyermentioning

confidence: 99%

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The infinitesimal generator of a one-dimensional strictly $$\alpha $$ α -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $$\alpha $$ α and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and the recurrent potential if $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $$\alpha $$ α . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada [1]. This result is used to find a Doob-Meyer (or semimartingale) decomposition for $$|X_t - x|^{\gamma }$$ | X t - x | γ with X a recurrent strictly stable process of index $$\alpha $$ α and $$\gamma \in (\alpha -1,\alpha )$$ γ ∈ ( α - 1 , α ) , generalizing the work of Engelbert and Kurenok [2] to the asymmetric case.

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Section: Theorem 2 (Occupational Meyermentioning

confidence: 69%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 90%