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Backward stochastic generalized variational inequality. (English) Zbl 1168.60023

Cârjă, Ovidiu (ed.) et al., Applied analysis and differential equations. Selected papers from the international conference, “Al. I. Cuza” University of Iaşi, Iaşi, Romania, September 4–9, 2006. Hackensack, NJ: World Scientific (ISBN 978-981-270-594-5/hbk). 217-226 (2007).
Given a complete probability space endowed with a \(d\)-dimensional Brownian motion \(W\) and the filtration generated by \(W\), the authors of the present paper study the existence and the uniqueness of the backward stochastic variational inequality \(dY_t+F(t,Y_t,Z_t)dt+G(t,Y_t)dA_t\in \partial\varphi(Y_t)dt+ \partial\psi(Y_t)dA_t+Z_tdW_t,\, t \in[0,\tau],\, Y_\tau=\xi,\) where \(\tau\) is a finite-valued stopping time, \(\phi\) and \(\psi\) are proper convex lower semicontinuous functions and \(\partial\varphi(x),\partial\psi(x)\) their subdifferentials at \(x\in \mathbb{R}^d.\)
The present paper generalises former results on backward stochastic variational inequalities, in particular those by E. Pardoux and A. Rascanu in [Stochastic Processes Appl. 76, No. 2, 191–215 (1998; Zbl 0932.60070)]. The generalisation concerns at one side the introduction of a one-dimensional continuous, increasing process \(A\) in the equation and at the other hand the study of a random terminal time \(\tau\). Such generalisations have been studied recently also for backward stochastic differential equations and, in particular, they have allowed to give a stochastic interpretation to elliptic PDEs with nonlinear Neumann boundary [see E. Pardoux and S. Zhang, Probab. Theory Relat. Field 110, No. 4, 535–558 (1998; Zbl 0909.60046)].
For the entire collection see [Zbl 1154.34003].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
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