# Set-valued stochastic integrals and stochastic inclusions in a plane

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2001)

- Volume: 21, Issue: 2, page 249-259
- ISSN: 1509-9407

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topWładysław Sosulski. "Set-valued stochastic integrals and stochastic inclusions in a plane." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 21.2 (2001): 249-259. <http://eudml.org/doc/271484>.

@article{WładysławSosulski2001,

abstract = {We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form
$z_\{s,t\} ∈ φ_\{s,t\} + ∫_\{0\}^\{s\} ∫_\{0\}^\{t\} F_\{u,v\}(z_\{u,v\})dudv + ∫_\{0\}^\{s\} ∫_\{0\}^\{t\}G_\{u,v\}(z_\{u,v\})dw_\{u,v\}$},

author = {Władysław Sosulski},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {stochastic inclusions in the plane; set-valued random field; two-parameter stochastic process; weak compactness; set-valued mapping; stochastic integral; stochastic inclusion; existence theorem; fixed point theorem; semi-continuity of the integrals; Michael's selection theorem},

language = {eng},

number = {2},

pages = {249-259},

title = {Set-valued stochastic integrals and stochastic inclusions in a plane},

url = {http://eudml.org/doc/271484},

volume = {21},

year = {2001},

}

TY - JOUR

AU - Władysław Sosulski

TI - Set-valued stochastic integrals and stochastic inclusions in a plane

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2001

VL - 21

IS - 2

SP - 249

EP - 259

AB - We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form
$z_{s,t} ∈ φ_{s,t} + ∫_{0}^{s} ∫_{0}^{t} F_{u,v}(z_{u,v})dudv + ∫_{0}^{s} ∫_{0}^{t}G_{u,v}(z_{u,v})dw_{u,v}$

LA - eng

KW - stochastic inclusions in the plane; set-valued random field; two-parameter stochastic process; weak compactness; set-valued mapping; stochastic integral; stochastic inclusion; existence theorem; fixed point theorem; semi-continuity of the integrals; Michael's selection theorem

UR - http://eudml.org/doc/271484

ER -

## References

top- [1] R. Cairoli, Sur une équation differentielle stochastique, Compte Rendus Acad. Sc. Paris 274 (A) (1972), 1739-1742. Zbl0244.60045
- [2] R. Cairoli and J.B. Walsh, Stochastic integrals in the plane, Acta Mathematica 134 (1975), 112-183. Zbl0334.60026
- [3] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivar. Anal. 7 (1977), 149-162. Zbl0368.60006
- [4] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ. - PWN, Dordrecht, Boston - London, Warszawa 1991. Zbl0731.49001
- [5] M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. Stoch. Anal. 6 (3) (1993), 217-236. Zbl0796.93106
- [6] L. Ponomarenco, Stochastic integral with respect to the multiparameter Brownian motion and attached stochastic equations (in Russian), Teor. Veroiatn. i Mat. Stat. Kiev 7 (1972) 100-109.
- [7] W. Sosulski, Subtrajectory integrals of set-valued functions depending on parameters, Discuss. Math. 10 (1990), 99-121. Zbl0736.49028
- [8] T.J. Tsarenco, On some scheme of the construction of stochastic integral for the radom field (in Russian), Kibernetika 1 (1972), 113-118.
- [9] C. Tudor, Stochastic integral equations in the plane, Preprint Series in Mathematics, INCREST 29 (1979) 507-538. Zbl0464.60065