A maximal inequality for stochastic convolution integrals on hilbert spaces and space-time regularity of linear stochastic partial differential equations

We consider optimal control problems for one-dimensional diffusion processes ILM0001] where the control processes υ_t are increasing, positive, and adapted. Several types of expected cost structures associated with each policy υ(.) are adopted, e.g. discounted cost, long term average cost and time average cost. Our work is related to 2,6,12,14,16 and 21], where diffusions are allowed to evolve in the whole space, and to 13] and 20], where diffusions evolve only in bounded regions. We shall present some analytic results about value functions, mainly their characterizations, by simple dynamic programming arguments. Several simple examples are explicitly solved to illustrate the singular behaviour of our problems.