Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions

Joachim Syga

Discussiones Mathematicae Probability and Statistics (2015)

  • Volume: 35, Issue: 1-2, page 7-27
  • ISSN: 1509-9423

Abstract

top
A random measure associated to a semimartingale is introduced. We use it to investigate properties of several types of stochastic integrals and properties of the solution set of Stratonovich-type stochastic inclusion.

How to cite

top

Joachim Syga. "Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions." Discussiones Mathematicae Probability and Statistics 35.1-2 (2015): 7-27. <http://eudml.org/doc/276511>.

@article{JoachimSyga2015,
abstract = {A random measure associated to a semimartingale is introduced. We use it to investigate properties of several types of stochastic integrals and properties of the solution set of Stratonovich-type stochastic inclusion.},
author = {Joachim Syga},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {forward; backward and symmetric integral; time-reversible process; semimartingale measure; set-valued stochastic integral; Stratonovich inclusion},
language = {eng},
number = {1-2},
pages = {7-27},
title = {Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions},
url = {http://eudml.org/doc/276511},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Joachim Syga
TI - Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions
JO - Discussiones Mathematicae Probability and Statistics
PY - 2015
VL - 35
IS - 1-2
SP - 7
EP - 27
AB - A random measure associated to a semimartingale is introduced. We use it to investigate properties of several types of stochastic integrals and properties of the solution set of Stratonovich-type stochastic inclusion.
LA - eng
KW - forward; backward and symmetric integral; time-reversible process; semimartingale measure; set-valued stochastic integral; Stratonovich inclusion
UR - http://eudml.org/doc/276511
ER -

References

top
  1. [1] K.K. Aase and P.Guttrup, Estimation in models for security prices, Scand. Actuarial J. 3/4 (1987), 211-225. 
  2. [2] N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach spaces, Stoch. Anal. Appl. 12 (1994), 1-10. 
  3. [3] N.U. Ahmed, Impulsive perturbation of C₀ semigroups and stochastic evolution inclusions, Discuss. Math. DICO 22 (1) (2002), 125-149. Zbl1039.34055
  4. [4] N.U. Ahmed, A brief summary of optimal control of uncertain systems governed by parabolic inclusions, Arab. J. Sci. Eng. ASJE Math. 4 (1D) (2009), 13-23. 
  5. [5] J.P. Aubin and A. Cellina, Differential Inclusions (Springer-Verlag, Berlin, Heidelberg, New York, 1984). 
  6. [6] J.P. Aubin and H. Frankowska, Set-Valued Analysis (Birkhäuser, Boston - Basel - Berlin, 1990). Zbl0713.49021
  7. [7] J.P. Aubin and G. Da Prato, The viability theorem for stochastic differential inclusions, Stoch. Anal. Appl. 16 (1998), 1-15. Zbl0931.60059
  8. [8] E.P. Avgerinos and N.S. Papageorgiou, Random nonlinear evolution inclusions in reflexive Banach spaces, Proc. Amer. Math. Soc. 104 (1988), 293-299. Zbl0663.60050
  9. [9] K.L. Chung and R.J. Williams, Introduction to stochastic integration (Birkhäuser, Boston - Basel - Berlin, 1990). Zbl0725.60050
  10. [10] G. Da Prato and H. Frankowska, A stochastic Filippov theorem, Stoch. Anal. Appl. 12 (4) (1994), 409-426. Zbl0810.60059
  11. [11] D. Duffie, Dynamic Asset Pricing Theory (Princeton Univ. Press Princeton, New Jersey, 1996). Zbl1140.91041
  12. [12] M. Errami, F. Russo and P. Vallois, Itôs formula for C 1 , λ -functions of a càdlàg process and related calculus, Probab. Theory Relat. Fields 122 (2002), 191-221. Zbl0999.60048
  13. [13] A. Fryszkowski, Continuous selections of Aumann integrals, J. Math. Anal. Appl. 145 (2) (1990), 431-446. Zbl0704.28006
  14. [14] Góralczyk and J. Motyl, Stratonovich stochastic inclusion, Dynam. Systems Appl. 18 (2) (2009), 191-204. Zbl1177.49030
  15. [15] F. Hiai and H. Umegaki, Integrals, conditional expectations, J. Multivar. Anal. 7 (1977), 149-182. Zbl0368.60006
  16. [16] M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. Stoch. Anal. 6 (3) (1993), 217-236. Zbl0796.93106
  17. [17] M. Kisielewicz, Existence theorem for nonconvex stochastic inclusions, J. Appl. Math. Stoch. Anal. 2 (1994), 151-159. Zbl0817.93065
  18. [18] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997), 783-800. Zbl0891.93070
  19. [19] M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer Acad. Publ. and Polish Sci. Publ. (Warszawa - Dordrecht - Boston - London, 1991). Zbl0731.49001
  20. [20] M. Kisielewicz, M. Michta and J. Motyl, Set valued approach to stochastic control. Part I (Existence and Regularity Properties), Dynam. Systems Appl. 12 (2003), 405-432. Zbl1063.93047
  21. [21] M. Kisielewicz, M. Michta and J. Motyl, Set valued approach to stochastic control. Part II (Viability and Semimartingale Issues), Dynam. Systems Appl. 12 (2003), 433-466. Zbl1064.93042
  22. [22] M. Michta, On set-valued stochastic integrals and fuzzy stochastic equations, Fuzzy Sets and Systems 177 (2011), 1-19. 
  23. [23] M. Michta and J. Motyl, Differentiable selections of multifunctions and their applications, Nonlinear Anal. 66 (2) (2007), 536-545. Zbl1104.49017
  24. [24] M. Michta and J. Motyl, Martingale problem to Stratonovich stochastic inclusion, Nonlinear Anal. 71 (12) (2009), e1307-e1311. Zbl1238.60078
  25. [25] J. Motyl, On the solution of a stochastic differential inclusion, J. Math. Anal. Appl. 192 (1) (1995), 117-132. Zbl0826.60053
  26. [26] J. Motyl, Note on strong solutions of a stochastic inclusion, J. Appl. Math. Stoch. Anal. 8 (3) (1995), 291-297. Zbl0831.93061
  27. [27] J. Motyl, Existence of solutions of set-valued Itô equation, Bull. Acad. Pol. Sci. 46 (1998), 419-430. Zbl0916.93069
  28. [28] J. Motyl and J. Syga, Properties of set-valued stochastic integrals, Discuss. Math. Probab. Stat. 26 (2006), 83-103. Zbl1129.93046
  29. [29] J. Motyl and J. Syga, Selection property of Stratonovich set-valued integral, Dynamics of Continuous, Discrete and Impulsive Systems 17 (2010), 431-443. Zbl1189.93071
  30. [30] P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, 2nd Edition, Version 2.1 (Berlin - Heideberg - New York, 2005). 
  31. [31] D. Repovš and P.V. Semenov, Continuous selections of multivalued mappings (Kluwer Academic Publishers (Dordrecht, Boston, London, 1998)). Zbl0915.54001
  32. [32] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Relat. Fields 97 (1993), 403-421. Zbl0792.60046
  33. [33] F. Russo and P. Vallois, The generalized covariation process and Ito formula, Stochastic Process. Appl. 59 (1995), 81-104. Zbl0840.60052
  34. [34] J. Syga, Application of semimartingale measure to the investigation of stochastic inclusion, Dynam. Systems Appl. 21 (2012), 393-406. Zbl1259.93108

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.