Weak compactness of weak solutions sets of forward-backward stochastic differential inclusions. (English) Zbl 1535.60100
Summary: The main result of the article deals with weak compactness of weak solutions sets of forward-backward stochastic differential inclusions. The main result is preceded by existence theorem for considered forward-backward stochastic differential inclusions. More precisely, we shall prove that by given set-valued mappings there exists a system consisting of a complete probability space \((\Omega,\mathcal{F},P)\), an \(d\)-dimensional continuous stochastic process \(X\), an \(\ell\)-dimensional càdlàg process \(Y\), and an \(m\)-dimensional Brownian motion \(B\) defined on the space \((\Omega,\mathcal{F},P)\) satisfying considered forward-backward stochastic differential inclusion and that each \(\mathbb{F}^X\)-martingale is an \(\mathbb{F}^{X Y}\)-martingale.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 47H04 | Set-valued operators |
Keywords:
Brownian motion; backward stochastic differential equations; forward-backward stochastic differential inclusions; continuous and càdlàg stochastic processes; weak compactness of sequences of stochastic processesReferences:
| [1] | Duffie, D., Epstein, L. G. (1992). Stochastic differential utility. Econometrica. 60(2): 353. DOI: . · Zbl 0763.90005 |
| [2] | Pardoux, E., Peng, S. (1990). Adapted solutions of a backward stochastic differential equation. System Control Lett. 14: 55-61. DOI: . · Zbl 0692.93064 |
| [3] | Buckdahn, R., Engelbert, H. J., Ruascanu, A. (2005). On weak solutions of backward stochastic differential equations. Theory Probab. Appl.49(1): 16-50. DOI: . · Zbl 1095.60019 |
| [4] | Billingsley, P. (1999). Convergence of Probability Measures. New York, London, Toronto: Wiley and Sons. · Zbl 0944.60003 |
| [5] | Ma, J., Protter, P., Yong, J. (1994). Solving forward-backward stochastic differential equations explicitly - a four step scheme. Probab. Theory Rel. Fields. 98(3): 339-359. DOI: . · Zbl 0794.60056 |
| [6] | Kisielewicz, M. (2020). Set-Valued Stochastic Integrals and Applications. Switzerland: Springer Nature Switzerland. · Zbl 1440.60002 |
| [7] | Hu, S., Papageourgiou, N. S. (1997). Handbook of Multivalued Analysis I. Dordrecht: Kluwer Academic Publishers. · Zbl 0887.47001 |
| [8] | Meyer, P. A., Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Probab. Stat. 20(4): 353-372. · Zbl 0551.60046 |
| [9] | Whitt, W. (2007). Proofs of the martingale FCTT. Probab. Surv.4. DOI: . |
| [10] | Kisielewicz, M., Michta, M. (2019). Weak solutions of set-valued stochastic differential equations. J. Math. Anal. Appl. 473(2): 1026-1052. DOI: . · Zbl 1426.60073 |
| [11] | Øksendal, B. (1998). Equations Stochastic Differential. Berlin: Springer. · Zbl 0897.60056 |
| [12] | Kisielewicz, M. (2020). Weak compactness of weak solutions sets to stochastic differential inclusions. Stoch. Anal. Appl. 38(3): 506-526. DOI: . · Zbl 1443.60053 |
| [13] | Hiai, F., Umegaki, H. (1977). Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7(1): 149-182. DOI: . · Zbl 0368.60006 |
| [14] | Kisielewicz, M. (2013). Stochastic Differential Inclusions and Applications. New York: Springer. · Zbl 1277.93002 |
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