Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions
Discussiones Mathematicae. Probability and Statistics, Tome 35 (2015) no. 1-2, pp. 7-27.

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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.
Keywords: forward, backward and symmetric integral, time-reversible process, semimartingale measure, set-valued stochastic integral, Stratonovich inclusion
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Syga, Joachim. Semimartingale measure in the investigation of Stratonovich-type stochastic integrals and inclusions. Discussiones Mathematicae. Probability and Statistics, Tome 35 (2015) no. 1-2, pp. 7-27. http://geodesic.mathdoc.fr/item/DMPS_2015_35_1-2_a0/

[1] K.K. Aase and P.Guttrup, Estimation in models for security prices, Scand. Actuarial J. 3/4 (1987), 211-225.

[2] N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach spaces, Stoch. Anal. Appl. 12 (1994), 1-10.

[3] N.U. Ahmed, Impulsive perturbation of C₀ semigroups and stochastic evolution inclusions, Discuss. Math. DICO 22 (1) (2002), 125-149.

[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] J.P. Aubin and A. Cellina, Differential Inclusions (Springer-Verlag, Berlin, Heidelberg, New York, 1984).

[6] J.P. Aubin and H. Frankowska, Set-Valued Analysis (Birkhäuser, Boston - Basel - Berlin, 1990).

[7] J.P. Aubin and G. Da Prato, The viability theorem for stochastic differential inclusions, Stoch. Anal. Appl. 16 (1998), 1-15.

[8] E.P. Avgerinos and N.S. Papageorgiou, Random nonlinear evolution inclusions in reflexive Banach spaces, Proc. Amer. Math. Soc. 104 (1988), 293-299.

[9] K.L. Chung and R.J. Williams, Introduction to stochastic integration (Birkhäuser, Boston - Basel - Berlin, 1990).

[10] G. Da Prato and H. Frankowska, A stochastic Filippov theorem, Stoch. Anal. Appl. 12 (4) (1994), 409-426.

[11] D. Duffie, Dynamic Asset Pricing Theory (Princeton Univ. Press Princeton, New Jersey, 1996).

[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.

[13] A. Fryszkowski, Continuous selections of Aumann integrals, J. Math. Anal. Appl. 145 (2) (1990), 431-446.

[14] Góralczyk and J. Motyl, Stratonovich stochastic inclusion, Dynam. Systems Appl. 18 (2) (2009), 191-204.

[15] F. Hiai and H. Umegaki, Integrals, conditional expectations, J. Multivar. Anal. 7 (1977), 149-182.

[16] M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. Stoch. Anal. 6 (3) (1993), 217-236.

[17] M. Kisielewicz, Existence theorem for nonconvex stochastic inclusions, J. Appl. Math. Stoch. Anal. 2 (1994), 151-159.

[18] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997), 783-800.

[19] M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer Acad. Publ. and Polish Sci. Publ. (Warszawa - Dordrecht - Boston - London, 1991).

[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.

[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.

[22] M. Michta, On set-valued stochastic integrals and fuzzy stochastic equations, Fuzzy Sets and Systems 177 (2011), 1-19.

[23] M. Michta and J. Motyl, Differentiable selections of multifunctions and their applications, Nonlinear Anal. 66 (2) (2007), 536-545.

[24] M. Michta and J. Motyl, Martingale problem to Stratonovich stochastic inclusion, Nonlinear Anal. 71 (12) (2009), e1307-e1311.

[25] J. Motyl, On the solution of a stochastic differential inclusion, J. Math. Anal. Appl. 192 (1) (1995), 117-132.

[26] J. Motyl, Note on strong solutions of a stochastic inclusion, J. Appl. Math. Stoch. Anal. 8 (3) (1995), 291-297.

[27] J. Motyl, Existence of solutions of set-valued Itô equation, Bull. Acad. Pol. Sci. 46 (1998), 419-430.

[28] J. Motyl and J. Syga, Properties of set-valued stochastic integrals, Discuss. Math. Probab. Stat. 26 (2006), 83-103.

[29] J. Motyl and J. Syga, Selection property of Stratonovich set-valued integral, Dynamics of Continuous, Discrete and Impulsive Systems 17 (2010), 431-443.

[30] P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, 2nd Edition, Version 2.1 (Berlin - Heideberg - New York, 2005).

[31] D. Repovš and P.V. Semenov, Continuous selections of multivalued mappings (Kluwer Academic Publishers (Dordrecht, Boston, London, 1998)).

[32] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Relat. Fields 97 (1993), 403-421.

[33] F. Russo and P. Vallois, The generalized covariation process and Ito formula, Stochastic Process. Appl. 59 (1995), 81-104.

[34] J. Syga, Application of semimartingale measure to the investigation of stochastic inclusion, Dynam. Systems Appl. 21 (2012), 393-406.