Control of diffusion processes in RN

✍ Scribed by P. L. Lions

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 799 KB
• Volume: 34
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160340106

✦ Synopsis

Notations rind Assumptions

1.1. Notations. Let V be a closed, non-empty, convex set in R" ( V is the set of values of the control). We assume that the following functions are given:

and we denote a ( x , u ) = + ( x , u ) * d ( X , O ) . ' We assume in the following (except when mentioned explicitly) that: for all u in V and cp = uii, bi, c, where (3) c(x,u) > a > 0 for all x E R N and all u E V, and we denote h = infxERNc(x,u), there is a functionp E C ( R + , R + ) such that p ( 0 ) = 0 and V E v Jcp(x,u)cp(x,u')( < p ( ( uu'l) for all cp = uii, bi, c, and all x E R N . (4) 1.2. The stochastic control problem and the HJB equations. By definition, an 1. a probability space (Q,F,F,,P,W,) with a Brownian motion W,, 2. a non-anticipative process (sometimes improperly called the control) u ( t , w ) taking values in a compact subset of V (depending on u ( -; ) ) , 3. a family of solutions ( y x ( f , w ) ) , ~~~ satisfying the stochastic differential admissible system &! will consist of ' o r is the adjoint of o.