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Itô formula for an asymptotically 4-stable process. (English) Zbl 0856.60042

The authors consider a generalization of the notion of random variable; a random variable is a sequence \(Z= (Z_n)\) of complex variables for which \(E\exp(itZ_n)\) converges for \(t\) real (the limit is called the characteristic function of \(Z\)). With the framework, the authors introduce the notion of 4-stable variables, of 4-stable processes, and list some of their properties (for the proofs of these results, they refer to their previous paper published in 1995 in a special issue of J. Fourier Anal. Appl.). Then they construct the stochastic integral with respect to a 4-stable process \(Z_t\) as a limit of Riemann sums, and prove an Itô formula which gives an expression for \(f(Z_t)\) when \(f\) is a polynomial; this formula involves the 4th derivative of \(f\), so this study can be viewed as a probabilistic approach to the 4th order partial differential equation \(\partial_t u= -\partial^4_x u\).

MSC:

60G18 Self-similar stochastic processes
60H05 Stochastic integrals
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