Viability of moving sets for stochastic differential equation.
The paper provides a necessary and sufficient characterization that guarantees the solution \( X_s^{t,x} , s \in [0,T]\), of a given forward stochastic differential equation stays (a.s. and \( \forall s \in [t,T] \)) in a prescribed set of closed, time dependent constraint \( K_0 (t), t \in [0,T]\) for all \( t \in [0,T] \) and \( x \in K_0 (t)\). This characterization is given in terms of viscosity super- and subsolution of some suitable partial differentiable equations (PDE). The above property, called viability, is stated for both forward and backward stochastic differential equations (BSDE) and is equivalent to the fact that the square of the distance function is a viscosity supersolution (subsolution, respectively) of the PDE for the forward SDE (BSDE, respectively) [see, e.g., \textit{R. Buckdahn}, \textit{M. Quincampoix} and \textit{A. Răşcanu}, Probab. Theory Relat. Fields 116, 485--504 (2000; Zbl 0969.60061)].
forward stochastic differential equations
semilinear parabolic partial differential equations
backward stochastic differential equations
scientific article; zbMATH DE number 1977261