Buckdahn, Rainer; Quincampoix, Marc; Rainer, Catherine; Răşcanu, Aurel Viability of moving sets for stochastic differential equation. (English) Zbl 1037.60055 Adv. Differ. Equ. 7, No. 9, 1045-1072 (2002). 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., R. Buckdahn, M. Quincampoix and A. Răşcanu, Probab. Theory Relat. Fields 116, 485–504 (2000; Zbl 0969.60061)]. Reviewer: Tran Nhu Pham (Hanoi) Cited in 11 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 35K65 Degenerate parabolic equations 60H30 Applications of stochastic analysis (to PDEs, etc.) Keywords:forward stochastic differential equations; semilinear parabolic partial differential equations; viability; viscosity subsolution; closed and time depending constraint; backward stochastic differential equations; viscosity supersolution Citations:Zbl 0969.60061 PDFBibTeX XMLCite \textit{R. Buckdahn} et al., Adv. Differ. Equ. 7, No. 9, 1045--1072 (2002; Zbl 1037.60055)