Diffusion Processes

Here a is a positive semidefinite symmetric matrix for each t and x, and b is a d vector for each t and x. There are various ways of describing exactly what we mean by a diffusion process corresponding to the specified set of coefficients. We shall adopt the following approach. Let Q be the space of Rd valued continuous functions on [0, cc). The value of a function ow = x( ) in Q at time t will be denoted by x(t). The a-field generated by x(s) for t1 . s < t2 will be denoted by M'8. If t1 = 0, we will denote this by M,2 and by M" in case t2 = oc, where M is the a-field generated by x(s) for 0 . s < cc. The space Q can be viewed as a complete separable metric space, with uniform convergence on bounded intervals defining the topology. Then M is the Borel a-field in Q. A stochastic process with values in Rd, defined for t _ to, is a probability measure on (Q, Mio). Given the coefficients {aij(t, x)} and {bj(t, x)}, we define an operator L, acting on functions f(x) e Ct (Rd) by