×

Viability property of jump diffusion processes on manifolds. (English) Zbl 1336.60163

Summary: In this note, we give a necessary and sufficient condition for viability property of diffusion processes with jumps on closed submanifolds of \(\mathbb{R}^m\). Our result is the system is viable in a closed submanifold \(K\) iff the coefficients are tangent to \(K\) along \(K\) if the equation is in the sense of Stratonovich integral and the solution jumps from \(K\) to \(K\).

MSC:

60J75 Jump processes (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubin, J.P., Da prato, G. Stochastic Viability and Invariance. Ann. Scu. Norm. di Pisa, 27: 595-694 (1990) · Zbl 0741.60046
[2] Buckdahn, R., Peng, S., Quincampoix, M., Rainer, C. Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris, t.327, série I: 17-22 (1998) · Zbl 1036.49026 · doi:10.1016/S0764-4442(98)80096-7
[3] Gautier, S., Thibault, L. Viability for constrained stochastic differential equations. Differ. Integ., 6: 1394-1414 (1993) · Zbl 0785.76050
[4] Hsu, P., Elton P. Stochastic Analysis on Manifolds. American Mathematical Society, 2002 · Zbl 0994.58019 · doi:10.1090/gsm/038
[5] Mazliak, L. A note on weak viability for controlled diffusions. Statist. Probab. Lett., 49: 331-336 (2000) · Zbl 1145.60325 · doi:10.1016/S0167-7152(00)00065-1
[6] Mazliak, L., Rainer, C. Exact and posible viability for controlled diffusions. Statist. Probab. Lett., 62: 155-161 (2003) · Zbl 1116.60346 · doi:10.1016/S0167-7152(03)00006-3
[7] Michta, M. A note on viability under distribution constraints. Discuss. Math. Probab. Statist., 20: 249-260 (1998) · Zbl 0984.60048 · doi:10.7151/dmps.1015
[8] Peng, S., Zhu, X. The viability property of controlled jump diffusion processes. Acta. Math. Sinica (English Series), 24(8): 1351-1368 (2008) · Zbl 1156.60317 · doi:10.1007/s10114-008-4528-x
[9] Peng, S., Zhu, X. Viability property on Riemannian manifolds. C.R. Acad. Sci. Paris, Ser. I, 347: 1423-1428 (2009) · Zbl 1211.58022 · doi:10.1016/j.crma.2009.10.007
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.