Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski

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

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

Abstract

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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 ) d u d v + ∫ 0 s ∫ 0 t G u , v ( z u , v ) d w u , v

How to cite

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Władysław 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

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

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