Existence of strong viable solutions of backward stochastic differential inclusions (Q2925769)

The paper deals with strong solutions to the viability problem of the form NEWLINENEWLINE\[NEWLINE \begin{cases} x_s \in \mathbb{E} \left[x_t+ \int_s^t F(\tau,x_\tau)d\tau|\mathcal{F}_s \right] &\text{ a.s. for } 0\leq t\leq T\\ x_t \in K(t) &\text{ a.s. for } 0\leq t\leq T. \end{cases}\tag{1}NEWLINE\]NEWLINE NEWLINEIn (1), \(F:[0,T]\times \mathbb{R}^m \to Cl(\mathbb{R})\) and \(K:[0,T]\times \mathbb{R}^m \to Cl(\mathbb{R})\) are assumed to be given a set-valued mapping and a set-valued process, respectively, and \(\mathcal{P}_\mathbb{F}=(\Omega,\mathcal{F},P,\mathbb{F})\) denotes a given complete filtered probability space.NEWLINENEWLINEBy a backward stochastic differential inclusion, for short \(\mathrm{BSDI}(F,K)\), we mean relations of the form (1) that are satisfied by a measurable \(m\)-dimensional càdlág process \(x=(x_t)_{0\leq t\leq T}\) on \(\mathcal{P}_\mathbb{F}\).NEWLINENEWLINEThe paper consists of five sections. The first two sections have a introductory character. They present the problem and some useful properties of the set-valued conditional expectation of Aumann's integral.NEWLINENEWLINESection 3 provides a measurable selection theorem. In Sections 4 and 5, the main results of the paper are formulated. Theorem 4.2 supplies a viable approximation result which will imply the existence of a solution to the viability problem (1). Theorem 5.2 gives sufficient conditions on the mappings \(F\) and \(K\) in (1) for the existence of a strong viable solution to the \(\mathrm{BSDI}(F,K)\) given by (1).NEWLINENEWLINEThe paper seems to be an interesting and valuable contribution to the theory of stochastic differential inclusions.