Stochastic Control of One-Dimensional Diffusions Whose Generators Have Discontinuous Coefficients
Hideo Nagai · SIAM Journal on Control and Optimization · 1985
Stochastic control of one-dimensional singular diffusion processes on an open interval $(\alpha ,\beta )$ is studied. Coefficients $m_i (x)$, $s_i (x)$ of those generators $({1 / {m_i (x)}})({d / {dx}})({1 /{s_i (x)}})({d / {dx}})$ are so singular that controlled processes cannot be defined by solutions of stochastic differential equations. We take up Markovian switching controlled processes and characterize the pay-off function, which is defined by additive functionals of the diffusion processes corresponding to given measures $\mu ^i $, by the unique solution of \[\begin{gathered} \mathcal{E}^i (u,v - u) \geqq \left\langle {\mu ^i ,v - u} \right\rangle \quad \forall v \in \mathcal{S},\quad i = 1,2,3, \cdots , \hfill \\ u \in \mathcal{S}. \hfill \\ \end{gathered} \] Here \[ \begin{gathered} \mathcal{E}^i (u,v) = \int_\alpha ^\beta {\frac{{du}} {{dx}}\frac{{dv}} {{dx}}\frac{1} {{s_i (x)}}dx,} \hfill \\ \mathcal{S} = \left\{ {v \in H_0^1 (\alpha ,\beta );\mathcal{E}^i (u,\phi ) \leqq \left\langle {\mu ^i ,\phi } \right\rangle \forall \phi \geqq 0, \in H_0^1 (\alpha ,\beta )} \right\}. \hfill \\ \end{gathered} \]