Equations for probability distributions of local occupation time on a surface for diffusion processes and control problems (Q5915400)

The article is devoted to the multidimensional diffusion process, which is the weak solution of the SDE \[ dy(t)=f(y(t),t)dt+\beta (y(t),t)dw(t),\quad y(0)=a, \quad t\in [0;T], \] with the random initial condition \(a.\) Coefficient \(f\) is bounded and measurable, \(\beta\) is continuous bounded and \(\beta \beta ^T\) is uniformly positively defined. The aim of the paper is the investigation of the local time \(\widehat{t}(T)\) for \(y\) on some surface \(\Gamma.\) To do this the parabolic equation with the generalized potential is treated. For expectation of the functionals from \(\widehat{t}(T)\) the analog of Feynman-Kac formula is obtained. Stochastic optimal control problem, which includes the minimization of \(E\widehat{t}(T)\), is solved.