A note on weak solutions to stochastic differential equations

Martin Ondreját; Jan Seidler

Kybernetika (2018)

  • Volume: 54, Issue: 5, page 888-907
  • ISSN: 0023-5954

Abstract

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We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.

How to cite

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Ondreját, Martin, and Seidler, Jan. "A note on weak solutions to stochastic differential equations." Kybernetika 54.5 (2018): 888-907. <http://eudml.org/doc/294488>.

@article{Ondreját2018,
abstract = {We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.},
author = {Ondreját, Martin, Seidler, Jan},
journal = {Kybernetika},
keywords = {stochastic differential equations; continuous coefficients; weak solutions},
language = {eng},
number = {5},
pages = {888-907},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on weak solutions to stochastic differential equations},
url = {http://eudml.org/doc/294488},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Ondreját, Martin
AU - Seidler, Jan
TI - A note on weak solutions to stochastic differential equations
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 5
SP - 888
EP - 907
AB - We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.
LA - eng
KW - stochastic differential equations; continuous coefficients; weak solutions
UR - http://eudml.org/doc/294488
ER -

References

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  1. Bensoussan, A., 10.1007/bf00996149, Acta Appl. Math. 38 (1995), 267-304. MR1326637DOI10.1007/bf00996149
  2. Billingsley, P., 10.1002/9780470316962, Wiley, New York 1999. MR1700749DOI10.1002/9780470316962
  3. Bogachev, V. I., 10.1007/978-3-540-34514-5, Springer, Berlin 2007. MR2267655DOI10.1007/978-3-540-34514-5
  4. Brzeźniak, Z., Ondreját, M., Seidler, J., 10.1016/j.jde.2015.11.007, J. Differential Equations 260 (2016), 4157-4179. MR3437583DOI10.1016/j.jde.2015.11.007
  5. Cherny, A., 10.1007/978-3-540-30788-4_6, In: From Stochastic Calculus to Mathematical Finance, Springer, Berlin 2006, pp. 109-124. MR2233537DOI10.1007/978-3-540-30788-4_6
  6. Debussche, A., Glatt-Holtz, N., Temam, R., 10.1016/j.physd.2011.03.009, Phys. D 240 (2011), 1123-1144. MR2812364DOI10.1016/j.physd.2011.03.009
  7. Dudley, R. M., 10.1017/cbo9780511755347, Cambridge University Press, Cambridge 2002. Zbl1023.60001MR1932358DOI10.1017/cbo9780511755347
  8. Hofmanová, M., Seidler, J., 10.1080/07362994.2012.628916, Stoch. Anal. Appl. 30 (2012), 100-121. MR2870529DOI10.1080/07362994.2012.628916
  9. Hofmanová, M., Seidler, J., 10.1080/07362994.2013.799025, Stoch. Anal. Appl. 31 (2013), 663-670. MR3175790DOI10.1080/07362994.2013.799025
  10. Gyöngy, I., Krylov, N., 10.1007/bf01203833, Probab. Theory Related Fields 105 (1996), 143-158. MR1392450DOI10.1007/bf01203833
  11. Ikeda, N., Watanabe, S., 10.1016/s0924-6509(08)70226-5, North-Holland, Amsterdam 1981. MR0637061DOI10.1016/s0924-6509(08)70226-5
  12. Karatzas, I., Shreve, S. E., 10.1007/978-1-4684-0302-2, Springer, New York 1988. MR0917065DOI10.1007/978-1-4684-0302-2
  13. Kurtz, T. G., Protter, P. E., 10.1007/bfb0093176, In: Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math. 1627, Springer, Berlin 1996, pp. 1-41. MR1431298DOI10.1007/bfb0093176
  14. Skorokhod, A. V., On existence and uniqueness of solutions to stochastic differential equations (in Russian)., Sibirsk. Mat. Ž. 2 (1961), 129-137. MR0132595
  15. Strock, D. W., Varadhan, S. R. S., Multidimensional Diffusion Processes., Springer, Berlin 1979. MR0532498
  16. Taheri, A., 10.1093/acprof:oso/9780198733133.001.0001, Oxford University Press, Oxford 2015. MR3410096DOI10.1093/acprof:oso/9780198733133.001.0001
  17. Triebel, H., 10.1016/s0924-6509(09)x7004-2, North-Holland, Amsterdam 1978. MR0503903DOI10.1016/s0924-6509(09)x7004-2

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