Show that the existence of a weak solution to XXXXXXXXXXis equivalent to the existence of a solution to the martingale problem for L. Suppose the ai j are Lipschitz functions in x and the matrices...

Show that the existence of a weak solution to (39.1) is equivalent to the existence of a solution to the martingale problem for L.

Suppose the ai j are Lipschitz functions in x and the matrices a(x) are positive definite, uniformly in x; Show that we can find matrices σ(x)so that each σi j is a Lipschitz function of x and a(x) = σ(x)σ^T
(x) for each x.

If X is a solution to (39.1), give formulas for At and Mt in terms of σ and b, where Mt is a local martingale, At is a process whose paths are locally of bounded variation, and