## 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
Full Text:

### References:

 [1] ANTOSIEWICZ, P., MIKUSINSKI, J. and SIKORSKI, R. 1973. Theory of Distributions The Sequential \'. Approach. North-Holland, Amsterdam. Z. [2] BURDZY, K. 1993. Some path properties of iterated Brownian motion. In Seminar on Stochastic Z. Processes 1992 E. C \?inlar, K. L. Chung and M. Sharpe, eds. 67 87. Birkhauser, \" Boston. · Zbl 0789.60060 [3] BURDZY, K. 1994. Variation of iterated Brownian motion. In Measure-Valued Processes, Z. Stochastic Partial Differential Equations and Interacting Sy stems D. A. Dawson, ed. 35 53. Amer. Math. Soc., Providence, RI. Z. · Zbl 0803.60077 [4] BURDZY, K. and MA \?DRECKI, A. 1995. An asy mptotically 4-stable process. J. Fourier Anal. Appl. Special Issue. Proceedings of the Conference in Honor of J.-P. Kahane 97 117. Z. · Zbl 0889.60044 [5] FUNAKI, T. 1979. Probabilistic construction of the solution of some higher order parabolic differential equations. Proc. Japan Acad. Ser. A Math. Sci. 55 176 179. Z. · Zbl 0433.35039 [6] HELMS, L. L. 1967. Biharmonic functions and Brownian motion. J. Appl. Probab. 4 130 136. Z. JSTOR: · Zbl 0314.60059 [7] HOCHBERG, K. J. 1978. A signed measure on path space related to Wiener measure. Ann. Probab. 6 433 458. Z. · Zbl 0378.60030 [8] HOCHBERG, K. J. and ORSINGHER, E. 1994. The arc-sine law and its analogues for processes governed by signed and complex measures. Stochastic Process. Appl. To appear. Z. · Zbl 0811.60028 [9] KRy LOV, V. YU. 1960. Some properties of the distribution corresponding to the equation Z.q 1 2 q 2 q u t 1 u x. Soviet Math. Dokl. 1 760 763. Z. MA \?DRECKI, A. and Ry BACZUK, M. 1989. On a Fey nman Kac formula. In Stochastic Methods in Z Experimental Sciences. Proceedings of COSMEX Conference 1989 W. Kasprzak and. A. Weron, eds. 312 321. World Scientific, Singapore. Z. MA \?DRECKI, A. and Ry BACZUK, M. 1993. New Fey nman Kac ty pe formula. Rep. Math. Phy s. 32 301 327. Z. Z. [10] NISHIOKA, K. 1985. Stochastic calculus for a class of evolution equations. Japan J. Math. N.S. 11 59 102. Z. · Zbl 0585.35048 [11] NISHIOKA, K. 1987. A stochastic solution of a high order parabolic equation. J. Math. Soc. Japan 39 209 231. Z. · Zbl 0622.60081 [12] NISHIOKA, K. 1994. The first hitting time and place of a biharmonic pseudo process. Preprint. Z. [13] SAINTY, PH. 1992. Construction of a complex-valued fractional Brownian motion of order N. J. Math. Phy s. 33 3128 3149. Z. · Zbl 0762.60073 [14] VAINSTEIN, A. I., ZAKHAROV, V. I., NOVIKOV, V. A. and SHIFMAN, M. A. 1982. ABC of instantons. Uspekhi Fiz. Nauk 136 553 591. [15] SEATTLE, WASHINGTON 98195 50-370 WROCLAW POLAND · Zbl 0082.00513
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.