×

Viability of moving sets for stochastic differential equation. (English) Zbl 1037.60055

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)].

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.)

Citations:

Zbl 0969.60061
PDFBibTeX XMLCite