×

zbMATH — the first resource for mathematics

Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. (English) Zbl 1111.60038
The paper considers stochastic differential equations (SDE) \(dX_t=\varphi (X_{t-})dZ_t\), where \(Z_t\) is a one-dimensional symmetric stable process of order \(\alpha\) with \(0<\alpha <2\) and proves that pathwise uniqueness can fail even when \(\varphi\) is a Hölder continuous function of order \(\beta<\min(1/\alpha,1)\) bounded above and below by strictly positive finite constants. The result is sharp in what concerns \(\beta\). This is an extension of a similar result by M. T. Barlow [J. Lond. Math. Soc., II. Ser. 26, 335–347 (1982; Zbl 0456.60062)] for SDE \(dX_t=\varphi(X_t)dB_t\) driven by a one-dimensional Brownian motion \(B_t\).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G52 Stable stochastic processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barlow, M.T., One-dimensional stochastic differential equations with no strong solution, J. London math. soc., 26, 335-347, (1982) · Zbl 0456.60062
[2] Bass, R.F., Uniqueness in law for pure jump Markov processes, Probab. theory related fields, 79, 271-287, (1988) · Zbl 0664.60080
[3] Bass, R.F., 2003. Stochastic differential equations driven by symmetric stable processes. Séminaire de Probabilités, Vol. XXXVI. Springer, New York, pp. 302-313. · Zbl 1039.60056
[4] Bertoin, J., Lévy processes, (1996), Cambridge University Press Cambridge
[5] Blumenthal, R.M.; Getoor, R.K.; Ray, D.B., On the distribution of first hits for the symmetric stable processes, Trans. amer. math. soc., 99, 540-554, (1961) · Zbl 0118.13005
[6] Kolokoltsov, V., Symmetric stable laws and stable-like jump-diffusions, Proc. London math. soc., 80, 725-768, (2000) · Zbl 1021.60011
[7] Komatsu, T., On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type, Proc. Japan acad. ser. A math. sci., 58, 353-356, (1982) · Zbl 0511.60057
[8] Meyer, P.-A., 1976. Un cours sur les intégrales stochastiques. Séminaire de Probabilités X. Springer, Berlin, pp. 245-400.
[9] Nakao, S., On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. math., 9, 513-518, (1972) · Zbl 0255.60039
[10] Rosiński, J.; Woyczyński, W.A., On ito stochastic integration with respect to p-stable motioninner clock, integrability of sample paths, double and multiple integrals, Ann. probab., 14, 271-286, (1986) · Zbl 0594.60056
[11] Williams, D.R., Path-wise solutions of stochastic differential equations driven by Lévy processes, Rev. mat. iberoamericana, 17, 295-329, (2001) · Zbl 1002.60060
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.