Linearization of a dynamic programming equation

Let x(t) = (x(1)(t),x(2)(t)) be defined by the stochastic differential equations dx(i)(t) = a(i)[x(t)]dt + Sigma (2)(j=1) b(ij)[x(t)]u(j)(t)dt + c(i)(1/ 2)[x(t)]dW(i)(t), where W-i is a standard Brownian motion, for i = 1, 2. There are two optimizers. The first one, using u(1) (t), tries to minimize the expected value of a quadratic cost criterion J, while the second one, using u(2)(t), wants to maximize this expected value. The game ends the first time x(t) reaches a subset of IR2. It is shown that it is sometimes possible to linearize the dynamic programming equation that must be solved to obtain the optimal value of u(i)(t). Examples are solved explicitly.